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DYNARE User Guide

1. oo Di Pi 1 1 80 80 J Ei fries i k 0 gt Il 0 46 CHAPTER 4 SOLVING DSGE MODELS ADVANCED TOPICS where Mik MC4k H is the deviation of the marginal cost from its natural rate defined as the marginal cost when prices are perfectly flexible The trick now is to note that the above can be written recursively by writing the right hand side as the first term of the sum with k 0 plus the remainder of the sum which can be written as the left hand side term scrolled forward one period and appropriately discounted Mathematically this yields pi pia 1 80 me m 8 0E pr pe which has gotten rid of our infinite sum That would be enough for Dynare but for convenience we can go one step further and write the above as T BTE i a Ame where eee which is the recognizable inflation equation in the new Keynesian or new Neoclassical monetary literature 4 3 4 Infinite sums with changing timing of expectations When you are not able to write an infinite sum recursively as the index of the expectations changes with each element of the sum as in the following example a different approach than the one mentioned above is necessary Suppose your model included the following sum oo Us gt E jtt j 0 where y and x are endogenous variables In Dynare the best way to handle this is to write out the first k terms explicitly and enter
2. NORMAL PDF N 11 0 R GAMMA PDF G u 0 pa p3 00 BETA PDF B 4 0 pa pa ps pal INV GAMMA PDF IGi p 0 R UNIFORM PDF U p3 pa p3 pa where u is the PRIOR MEAN c is the PRIOR_STANDARD_ERROR pa is the PRIOR 3 d PARAMETER whose default is 0 and p4 is the PRIOR 4 PARAMETER whose default is 1 TIP When specifying a uniform distribution between 0 and 1 as a prior for a parameter say o you therefore have to put two empty spaces for parameters u and c and then specify parameters p3 and p4 since the uniform distribution only takes p3 and p4 as arguments For instance you would write alpha uniform pdf 0 1 For a more complete review of all possible options for declaring priors as well as the syntax to declare priors for maximum likelihood estimation not Bayesian see the Reference Manual Note also that if some parameters in a model are calibrated and are not to be estimated you should declare them as such by using the parameters command and its related syntax as explained in chapter 3 TIP Choosing the appropriate prior for your parameters is a tricky yet very important endeavor It is worth spending time on your choice of priors and to test the robustness of your results to your priors Some considerations may prove helpful First think about the domain of your prior over each pa rameter Should it be bounded Should it be opened on either or both sides Remember also that if you specify a probabilit
3. 3 10 2 Results deterministic models Solving DSGE models advanced topics 4 1 Dynare features and functionality 4 4 1 QOtherexamples o sa s a 65 4b o RR 4 1 2 Alternative complete example 4 1 3 Finding saving and viewing your output 4 1 4 Referring to external files 4 1 5 Infinite eigenvalues 2l 4 2 Files created by Dynare o so erase cuasan citanas 23 JIModelnpdipB so n fusa koe a e a 4 3 1 Stationarizing your model 6464 64 see kes 4 3 2 Expectations taken in the past 43 3 Infinitesums sa s su 05 4 ee c 39 ho 4 XR a 4 3 4 Infinite sums with changing timing of expectations Estimating DSGE models basics 5 1 Introducing an example lt ao ca petsa boa sa s o 5 2 Declaring variables and parameters ss So Declarine the mod l 22 2 4 scana eg mo tee d Rs 5 4 Declaring observable variables a 5 5 Specifying the steady state Dio JDeelanng Driers soeces cha i Di e a Qe EE RUE E 5 7 Launching the estimation c 5 8 Thecomplete modfile o s sa ss csa cac saaa les 0 9 Inberpreting output lt soe pa aaee ooo m RE dee RE S91 Tabular T SuliS ook sos eu yo oo ea 37 37 37 38 41 42 43 43 44 44 44 44 46 0 9 2 Graphical Toig oce vol oe Row or odo ERR etx a 57 6 Estimating DSGE models advanced topics 61 6 1 Alternative and non stationary examp
4. 1 c beta 1 c 1 1 r 1 delta psi c 1 1 vw cti yj y k 1 Yalpha exp z 1 1 alpha w y Cepsilon 1 epsilon 1 alpha 1 r y epsilon 1 epsilon alpha k 1 i k 1 delta k 1 y l y l z rho z 1 e end Just in case you need a hint or two to recognize these equations here s a brief description the first equation is the Euler equation in consumption The second the labor supply function The third the accounting identity The fourth is the production function The fifth and sixth are the marginal cost equal to markup equations The seventh is the investment equality The eighth an identity that may be useful and the last the equation of motion of technology NOTE that the above model specification corresponds to the stochastic case indeed notice that the law of motion for technology is included as per our discussion of the preamble The corresponding model for the determin istic casce would simply loose the last equation 3 5 SPECIFYING THE MODEL 19 3 5 2 General conventions The above example illustrates the use of a few important commands and conventions to translate a model into a Dynare readable mod file e The first thing to notice is that the model block of the mod file begins with the command model and ends with the command end e Second in between there need to be as many equations as you declared endogenous variables this is actually one of the first things that
5. Pair e BY Paes Sa j l j 0 This formulation turns out to be useful in problems of the following form oo oo 25 Pigs Pt 5 Fii j 0 j 0 which can be written as a recursive system of the form Sl zt PS 52 Yt y52144 This is particularly helpful for instance in a Calvo type setting as illustrated in the following brief example The RBC model with monopolistic competition introduced in chapter 3 involved flexible prices The extension with sticky prices la Calvo for instance is instead typical of the new Key nesian monetary literature exemplified by papers such as Clarida Gali and Gertler 1999 The optimal price for a firm resetting its price in period t given that it will be able to reset its price only with probability 1 0 each period is co pi i u 1 80 S 80 Elmeti k 0 where u is the markup is a discount factor i represents a firm of the contin uum between 0 and 1 and mc is marginal cost as described in the example in chapter 3 The trouble of course is how to input this infinite sum into Dynare It turns out that the Calvo price setting implies that the aggregate price follows the equation of motion p 0p 1 1 0 pz thus implying the following inflation relationship 7 1 0 pf pi 1 Finally we can also rewrite the optimal price setting equation after some algebraic manipulations as Me 30 E eu
6. exp ww end where this time repeating a letter for a variable means log of that variable so that the level of a variable is given by exp repeatedvariable 3 6 Specifying steady states and or initial values Material in this section has created much confusion in the past But with some attention to the explanations below you should get through unscathed Let s start by emphasizing the uses of this section of the mod file First recall that stochastic models need to be linearized Thus they need to have a steady state One of the functions of this section is indeed to provide these steady state values or approximations of values Second irrespective of whether you re working with a stochastic or deterministic model you may be inter ested to start your simulations or impulse response functions from either a steady state or another given point This section is also useful to specify this starting value Let s see in more details how all this works In passing though note that the relevant commands in this section are initval endval or more rarely histval which is covered only in the Ref erence Manual The first two are instead covered in what follows 3 6 1 Stochastic models and steady states In a stochastic setting your model will need to be linearized before it is solved To do so Dynare needs to know your model s steady state more details on finding a steady state as well as tips to do so more efficiently are provi
7. model dA exp gamte a log m 1 rho log mst rho log m 1 em P c 1 P 1 m bet P 1 alp exp alp gamtlog e 1 K alp 1 n 1 1 alp 1 del exp gamtlog e 1 c 2 P 2 m 1 0 W 1 n psi 1 psi c P 1 n 1 n 0 R P 1 alp exp alp gamt e_a k 1 Yalp n alp W 1 c P bet P 1 alp exp alp gamte a k 1 alp n 1 alp m 1 c 1 P 1 0 c k exp alp gamte a k 1 alp n 1 alp 1 del exp gamte a k 1 P c m m 1 d 1 e exp e a y k 1 alp n 1 alp exp alp gamte a Y_obs Y_obs 1 dA y y 1 P_obs P_obs 1 p p 1 m 1 dA end varobs P_obs Y_obs observation_trends 6 1 ALTERNATIVE AND NON STATIONARY EXAMPLE 73 P_obs log mst gam Y_obs gam end unit root vars P obs Y obs initval 6 mst 2 25 0 45 1 4 1 02 0 85 0 19 0 86 0 6 A exp gam po lt ppa9 Z 0 01 8 Il the above is really only useful if you want to do a stoch simul of your model since the estimation will use the Matlab steady state file also provided and discussed above steady estimated params alp beta pdf 0 356 0 02 bet beta pdf 0 993 0 002 gam normal pdf 0 0085 0 003 mst normal pdf 1 0002 0 007 rho beta pdf 0 129 0 223 psi beta pdf 0 65 0 05 del beta pdf 0 01 0 005 stderr e a inv gamma pdf 0 035449 inf stderr em inv gamma pdf 0 008862 inf
8. Ho s t PC lt Mi Di WH 0 lt Di Misi Mi Di WH PCs Rgi Di Fi Bi where the second equation spells out the cash in advance constraint including wage revenues the third the inability to borrow from the bank and the fourth the intertemporal budget constraint emphasizing that households accumulate the money that remains after bank deposits and purchases on goods are de ducted from total inflows made up of the money they receive from last period s cash balances wages interests as well as dividends from firms F and from banks B which in both cases are made up of net cash inflows defined below Banks on their end receive cash deposits from households and a cash injection X from the central bank which equals the net change in nominal money balances M Mi It uses these funds to disburse loans to firms Li on which they make a net return of Rr 1 Of course banks are con strained in their loans by a credit market equilibrium condition Ly X Di Finally bank dividends B are simply equal to D 4 Rg L RaiDi Lit Xi Finally firms maximize the net present value of future dividends dis counted by the marginal utility of consumption since they are owned by households by choosing dividends next period s capital stock K 1 labor demand N and loans Its problem is summarized by Fo Keri Nola Eo Pe pr ni st FEE Lib B KS GUN Kia 0 9 Ki WiN LRri Wi
9. Note that observations do not need to show up in any order but vectors of observations need to be named with the same names as those in var_obs In Excel files for instance observations could be ordered in columns and variable names would show up in the first cell of each column 2 nobs INTEGER the number of observations to be used default all observations in the file 3 first obs INTEGER the number of the first observation to be used default 1 This is useful when running loops or instance to divide the observations into sub periods 4 prefilter 1 the estimation procedure demeans the data default 0 no prefiltering This is useful if model variables are in deviations from steady state for instance and therefore have zero mean Demeaning the observations would also impose a zero mean on the observed variables 5 nograph no graphs should be plotted 6 conf sig INTEGER DOUBLE the level for the confidence in tervals reported in the results default 0 90 T mh replic INTEGER number of replication for Metropolis Hasting algorithm For the time being mh replic should be larger than 1200 default 20000 8 mh nblocks 2 INTEGER number of parallel chains for Metropolis Hast ing algorithm default 2 Despite this low default value it is advisable to work with a higher value such as 5 or more This improves the com putation of between group variance of the parameter means one of the
10. values 0 end 3 7 3 Stochastic models Recall from our earlier description of stochastic models that shocks are only allowed to be temporary A permanent shock cannot be accommodated due to the need to stationarize the model around a steady state Furthermore shocks can only hit the system today as the expectation of future shocks must be zero With that in mind we can however make the effect of the shock propa gate slowly throughout the economy by introducing a latent shock variable such as e in our example that affects the model s true exogenous variable z in our example which is itself an AR 1 exactly as in the model we introduced from the outset In that case though we would declare z as an endogenous variable and e as an exogenous variable as we did in the preamble of the mod file in section 3 4 Supposing we wanted to add a shock with variance c where c is determined in the preamble block we would write shocks var e sigma 2 end TIP You can actually mix in deterministic shocks in stochastic models by using the commands varexo det and listing some shocks as lasting more than one period in the shocks block For information on how to do so please see the Reference Manual This can be particularly useful if you re studying the effects of anticipated shocks in a stochastic model For instance you may be interested in what happens to your monetary model if agents began ex pecting higher inflation or a de
11. key criteria to evaluate the efficiency of the Metropolis Hastings to eval uate the posterior distribution More details on this subject appear in Chapter 6 5 7 LAUNCHING THE ESTIMATION 53 10 11 12 13 14 mh drop DOUBLE the fraction of initially generated parameter vec tors to be dropped before using posterior simulations default 0 5 this means that the first half of the draws from the Metropolis Hastings are discarded mh jscale DOUBLE the scale to be used for the jumping distribu tion in MH algorithm The default value is rarely satisfactory This option must be tuned to obtain ideally an acceptance rate of 2596 in the Metropolis Hastings algorithm default 0 2 The idea is not to reject or accept too often a candidate parameter the literature has set tled on a value of between 0 2 and 0 4 If the acceptance rate were too high your Metropolis Hastings iterations would never visit the tails of a distribution while if it were too low the iterations would get stuck in a subspace of the parameter range Note that the acceptance rate drops if you increase the scale used in the jumping distribution and vice a versa mh init scale DOUBLE the scale to be used for drawing the initial value of the Metropolis Hastings chain default 2 mh_jscale The idea here is to draw initial values from a stretched out distribution in order to maximize the chances of these values not being too close together
12. var u stderr 0 009 var e u phi 0 009 0 009 end stoch simul periods 2100 4 1 3 Finding saving and viewing your output Where is output stored Most of the moments of interest are stored in global variable oo You can easily browse this global variable in Matlab by either calling it in the command line or using the workspace interface In global variable oo you will find the following NOTE variables will always appear in the order in which you declared them in the preamble block of your mod file Steady state the steady state of your variables mean the mean of your variables var the variance of your variables autocorr the various autocorrelation matrices of your variables Each row of these matrices will correspond to a variables in time t and columns correspond to the variables lagged 1 for the first matrix then lagged 2 for the second matrix and so on Thus the matrix of auto correlations that is automatically displayed in the results after running stoch simul has running down each column the diagonal elements of each of the various autocorrelation matrices described here gamma y the matrices of autocovariances gamma y 1 represents vari ances while gamma y 2 represents autocovariances where variables on each column are lagged by one period and so on By default Dynare will return autocovariances with a lag of 5 The last matrix gamma y 7 in 42 CHAPTER 4 SOLVING DSGE MODELS ADVANCED TOPICS
13. 8 Dynare modfilestructure ss se ctg p d a ea r 3 4 Filling out the preamble LL 3 4 1 The deterministic case LL 3 2 The stochastic case LL 3 4 3 Comments on your first lines of Dynare code Qo spedlving ie model uon cedo e xw eR eR Rus 3 5 1 Modelin Dynare notation acr 24 939 n RR 3 5 2 General conventions saoo e 3 5 3 Notationalconventions LL 3 9 4 Timing conventions sa aces s woi eee ae ees 3 5 5 Conventions specifying non predetermined variables 3 5 6 Linear and log linearized models il iii vi NNN Oo 10 11 15 15 16 16 17 18 18 19 19 19 20 CONTENTS 3 6 Specifying steady states and or initial values 3 6 1 Stochastic models and steady states 3 6 2 Deterministic models and initial values 3 6 3 Findingasteady state 3 6 4 Checking system stability Ar Addnp ghocas ce deces me a e ed ORES e 3 7 1 Deterministic models temporary shocks 3 7 2 Deterministic models permanent shocks dtd itochusbe qHOdelS v lube ree a 3 8 Selecting a computation gt s s e s aos s s ete lees 3 8 1 For deterministic models 00 3 8 2 For stochastic models osa soaa onanera naga 3 9 Thecomplete modfile ek e a 3 9 1 The stochastic model 3 9 2 The deterministic model case of temporary shock 3 10 File execution and results 3 10 1 Results stochastic models
14. Dynare checks it will immediately let you know if there are any problems e Third as in the preamble and everywhere along the mod file each line of instruction ends with a semicolon except when a line is too long and you want to break it across two lines This is unlike Matlab where if you break a line you need to add e Fourth equations are entered one after the other no matrix representa tion is necessary Note that variable and parameter names used in the model block must be the same as those declared in the preamble TIP remember that variable and parameter names are case sensitive 3 5 3 Notational conventions e Variables entering the system with a time t subscript are written plainly For example x would be written x e Variables entering the system with a time t n subscript are written with n following them For example x 2 would be written z 2 incidentally this would count as two backward looking variables e In the same way variables entering the system with a time t n subscript are written with n following them For example r 2 would be written x 2 Writing x 2 is also allowed but this notation makes it slightly harder to count by hand the number of forward looking variables a useful measure to check more on this below 3 5 4 Timing conventions e In Dynare the timing of each variable reflects when that variable is de cided For instance our capital stock is not decided toda
15. Hastings default diagnostics are computed and displayed Actually seeing if the various blocks of Metropolis Hastings runs converge is a powerful and useful option to build confidence in your model estima tion More details on these diagnostics are given in Chapter 6 17 bayesian irf triggers the computation of the posterior distribution of im pulse response functions IRFs The length of the IRFs are controlled by the irf option as specified in chapter 3 when discussing the options for stoch simul To build the posterior distribution of the IRFs Dynare pulls parameter and shock values from the corresponding estimated dis tributions and for each set of draws generates an IRF Repeating this process often enough generates a distribution of IRFs TIP If you stop the estimation procedure after calculating the posterior mode or carry out maximum likelihood estimation only the corresponding parameter estimates will be used to generate the IRFs If you instead carry out a full Metropolis Hastings estimation on the other hand the IRFs will use the parameters the posterior distributions including the variance of the shocks 18 All options available for stoch simul can simply be added to the above options separated by commas To view a list of these options either see the Reference Manual or section 3 8 of chapter 3 19 moments varendo triggers the computation of the posterior distribution of the theoretical moments of the endogenou
16. In mi 2 In m ey j A where all terms on the right hand side are constant except for In m and y 6 1 ALTERNATIVE AND NON STATIONARY EXAMPLE 71 initial condition in the case of a deterministic trend since the variance is finite 6 1 10 Specifying the steady state Declaring the steady state is just as explained in details and according to the same syntax explained in chapter 3 covering the initval steady and check commands In chapter 5 section 5 5 we also discussed the usefulness of providing an external Matlab file to solve for the steady state In this case you can find the corresponding steady state file in the models folder under UserGuide The file is called fs2000ns_steadystate m There are some things to notice First the output of the function is the endogenous variables at steady state the ys vector The check 0 limits steady state values to real numbers Second notice the declaration of parameters at the beginning intuitive but tedious This functionality may be updated in later versions of Dynare Third note that the file is really only a sequential set of equalities defining each variable in terms of parameters or variables solved in the lines above So far nothing has changed with respect to the equivalent file of chapter 5 The only novelty is the declaration of the non stationary variables P_obs and Y_obs which take the value of 1 This is Dynare convention and must be the case for all your no
17. Note that a variable may occur both as predetermined and non predetermined For instance consumption could appear with a lead in the Euler equa tion but also with a lag in a habit formation equation if you had one In this case the second order difference equation would have two eigen values one needing to be greater and the other smaller than one for stability 3 5 6 Linear and log linearized models There are two other variants of the system s equations which Dynare accom modates First the linear model and second the model in exp logs In the first case all that is necessary is to write the term linear next to the command model Our example with just the equation for y for illustration would look like model linear yy l yy 11 end where repeating a letter for a variable means difference from steady state Otherwise you may be interested to have Dynare take Taylor series ex pansions in logs rather than in levels this turns out to be a very useful option 3 6 SPECIFYING STEADY STATES AND OR INITIAL VALUES 21 when estimating models with unit roots as we will see in chapter 5 If so simply rewrite your equations by taking the exponential and logarithm of each variable The Dynare input convention makes this very easy to do Our ex ample would need to be re written as follows just shown for the first two equations model 1 exp cc beta 1 exp cc 1 1 exp rr 1 delta psi exp cc 1 exp 11
18. We come back to this important topic in details in section 6 1 3 below For now we pause a moment to give some intuition for the above equations In order these equations correspond to 1 The Euler equation in the goods market representing the tradeoff to the economy of moving consumption goods across time 2 The firms borrowing constraint also affecting labor demand as firms use borrowed funds to pay for labor input 3 The intertemporal labor market optimality condition linking labor sup ply labor demand and the marginal rate of substitution between con sumption and leisure 4 The equilibrium interest rate in which the marginal revenue product of labor equals the cost of borrowing to pay for that additional unit of labor 66 CHAPTER 6 ESTIMATING DSGE MODELS ADVANCED TOPICS 5 The Euler equation in the credit market which ensures that giving up one unit of consumption today for additional savings equals the net present value of future consumption 6 The aggregate resource constraint 7 The money market equilibrium condition equating nominal consumption demand to money demand to money supply to current nominal balances plus money injection 8 The credit market equilibrium condition 9 The production function 10 The stochastic process for money growth 11 The stochastic process for technology 12 The relationship between observable variables and stationary variables more details on these last two
19. below will either comfort you in realizing that Dynare does what you expected of it and what you would have also done if you had had to code it all yourself with a little extra time on your hands or will spur your curiosity to have a look at more detailed material If so you may want to go through Michel Juillard s presentation on solving DSGE models to a first and second order available on Michel Juillard s website or read Collard and Juillard 2001a or Schmitt Grohe and Uribe 2004 which gives a good overview of the most recent solution techniques based on perturbation methods Finally note that in this chapter we will focus on stochastic models which is where the major complication lies as explained in section 3 1 1 of chapter 3 For more details on the Newton Raphson algorithm used in Dynare to solve deterministic mod els see Juillard 1996 7 2 What is the advantage of a second order approximation As noted in chapter 3 and as will become clear in the section below lin earizing a system of equations to the first order raises the issue of certainty equivalence This is because only the first moments of the shocks enter the linearized equations and when expectations are taken they disappear Thus 77 CHAPTER 7 SOLVING DSGE MODELS BEHIND THE SCENES OF 78 DYNARE unconditional expectations of the endogenous variables are equal to their non stochastic steady state values This may be an acceptable simplification
20. end estimation datafile fsdat nobs 192 1oglinear mh replic 2000 mode compute 4 mh nblocks 2 mh drop 0 45 mh jscale 0 65 74 CHAPTER 6 ESTIMATING DSGE MODELS ADVANCED TOPICS Figure 6 2 At a high level there are five basic steps to translate a model into Dynare for successful estimation 6 1 14 Summing it up The explanations given above of each step necessary to translate the Schorfheide 2000 example into language that Dynare can understand and process was quite lengthy and involved a slew of new commands and information It may therefore be useful to gain a bird s eyeview on what we have just accom plished and summarize the most important steps at a high level This is done in figure 6 1 14 6 2 Comparing models based on their posterior distributions TBD 6 3 WHERE IS YOUR OUTPUT STORED 75 6 3 Where is your output stored The output from estimation can be extremely varied depending on the in structions you give Dynare The Reference Manual overviews the complete set of potential output files and describes where you can find each one Chapter 7 Solving DSGE models Behind the scenes of Dynare 7 1 Introduction The aim of this chapter is to peer behind the scenes of Dynare or under its hood to get an idea of the methodologies and algorithms used in its com putations Going into details would be beyond the scope of this User Guide which will instead remain at a high level What you will find
21. f p 07 T x p Y7 6z 2 d07 Or Or Note that the expression inside the integral sign is exactly the posterior kernel To remind you of this you may want to glance back at the first subsection above specifying the basic mechanics of Bayesian estimation To obtain the marginal density of the data conditional on the model there are two options The first is to assume a functional form of the posterior kernel that we can integrate The most straightforward and the best approximation especially for large samples is the Gaussian called a Laplace approxima tion In this case we would have the following estimator A k T m m P Yr T 27 Dom 2p 07 Yr I p 07 T where 07 is the posterior mode The advantage of this technique is its com putational efficiency time consuming Metropolis Hastings iterations are not 8 4 COMPARING MODELS USING POSTERIOR DISTRIBUTIONS 91 necessary only the numerically calculated posterior mode is required The second option is instead to use information from the Metropolis Hastings runs and is typically referred to as the Harmonic Mean Esti mator The idea is to simulate the marginal density of interest and to simply take an average of these simulated values To start note that f 6r Ki Tio ii where f is a probability density function since f 07 0 z u Jo 0 d0 p Or Z p Yr r 2 Jo P 0x1Dp Y7 07 2 do7 p Yr T and the numerator integrates
22. focusses on advanced topics and features of Dynare in the area of model estimation The chapter begins by presenting a more complex example than the one used for illustration purposes in chapter 5 The goal is to show how Dynare would be used in the more realistic setting of reproducing a recent academic paper The chapter then follows with sections on comparing models to one another and then to BVARs and ends with a table summariz ing where output series are stored and how these can be retrieved 6 1 Alternative and non stationary example The example provided in chapter 5 is really only useful for illustration pur poses So we thought you would enjoy and continue learning from a more realistic example which reproduces the work in a recent and highly regarded academic paper The example shows how to use Dynare in a more realistic setting while emphasizing techniques to deal with non stationary observations and stochastic trends in dynamics 6 1 1 Introducing the example The example is drawn from Schorfheide 2000 This first section introduces the model its basic intuitions and equations We will then see in subsequent sections how to estimate it using Dynare Note that the original paper by Schorfheide mainly focusses on estimation methodologies difficulties and so lutions with a special interest in model comparison while the mathematics and economic intuitions of the model it evaluates are drawn from Nason and Cogley 19
23. mod file below Fist though note that to introduce shocks into Dynare we have two options this was not discussed in the earlier chapter Either write shocks var e stderr 0 009 var u stderr 0 009 40 CHAPTER 4 SOLVING DSGE MODELS ADVANCED TOPICS var e u phi 0 009 0 009 end where the last line specifies the contemporaneous correlation between our two exogenous variables Alternatively you can also write shocks var e 0 009 2 var u 0 009 2 var e u phi 0 009 0 009 end So that you can gain experience by manipulating the entire model here is the complete mod file corresponding to the above example You can find the corresponding file in the models folder under UserGuide in your installation of Dynare The file is called Alt Ex1 mod var y c k a h b varexo e u parameters beta rho alpha delta theta psi tau alpha 0 36 rho 0 95 tau 0 025 beta 0 99 delta 0 025 psi 0 theta 2 95 phi 0 1 model c theta h 1 psi 1 alpha y k beta exp b c exp b 1 c 1 exp b 1 alpha y 1 1 delta k y exp a k 1 alpha H 1 alpha k exp b y c 1 delta k 1 rho a 1 tau b 1 e tau a 1 rho b 1 u a b end initval y 1 08068253095672 4 1 DYNARE FEATURES AND FUNCTIONALITY 41 c 0 80359242014163 h 0 29175631001732 k 5 a 0 b 0 e 0 u 0 end shocks var e stderr 0 009
24. model but you may also want to go a bit further Typically you may be interested in how this system behaves in response to shocks whether temporary or permanent Likewise you may want to explore how the system comes back to its steady state or moves to a new one This chapter covers all these topics But instead of skipping to the topic closest to your needs we recommend that you read this chapter chronologically to learn basic Dynare commands and the process of writing a proper mod file this will serve as a base to carry out any of the above computations 3 1 A fundamental distinction Before speaking of Dynare it is important to recognize a distinction in model types This distinction will appear throughout the chapter in fact it is so fundamental that we considered writing separate chapters altogether But the amount of common material Dynare commands and syntax is notable and writing two chapters would have been overly repetitive Enough suspense here is the important question is your model stochastic or determinis tic The distinction hinges on whether future shocks are known In de terministic models the occurrence of all future shocks is known exactly at the time of computing the model s solution In stochastic models instead only the distribution of future shocks is known Let s consider a shock to a model s innovation only in period 1 In a deterministic context agents will take their decisions knowing that future v
25. non stationary model Recall from chapter 3 that we are dealing with an RBC model with mo nopolistic competition Suppose we had data on business cycle variations of output Suppose also that we thought our little RBC model did a good job of reproducing reality We could then use Bayesian methods to estimate the pa rameters of the model o the capital share of output 68 the discount factor 47 48 CHAPTER 5 ESTIMATING DSGE MODELS BASICS the depreciation rate v the weight on leisure in the utility function p the degree of persistence in productivity and e the markup parameter Note that in Bayesian estimation the condition for undertaking estimation is that there be at least as many shocks as there are observables a less stringent condition than for maximum likelihood estimation It may be that this does not allow you to identify all your parameters yielding posterior distributions identical to prior distributions but the Bayesian estimation procedure would still run successfully Let s see how to go about doing this 5 2 Declaring variables and parameters To input the above model into Dynare for estimation purposes we must first declare the model s variables in the preamble of the mod file This is done exactly as described in chapter 3 on solving DSGE models We thus begin the mod file with var yckilylwrz varexo e parameters beta psi delta alpha rho epsilon 5 3 Declaring the model Suppose that the e
26. shocks of any dimension The solution to this system is a set of equations relating variables in the current period to the past state of the system and current shocks that satisfy the original system This is what we call the policy function Sticking to the above notation we can write this function as Yt g Yt 1 Ut 7 3 HOW DOES DYNARE SOLVE STOCHASTIC DSGE MODELS 79 Then it is straightforward to re write y41 as Ut 1 g yr Ut41 g g Yt 1 Ut uti We can then define a new function F such that F yt 1 Ut Ut 41 f g g yi 1 Ut uda G Yt 1 Ut Yt 1 Ut which enables us to rewrite our system in terms of past variables and current and future shocks E F yr 1 ut ut41 0 We then venture to linearize this model around a steady state defined as f 9 9 9 0 0 having the property that y g y 0 The first order Taylor expansion around 7 yields g FO gripe j Ei ADD fu Gy Gui gut gu fos yi guu fyG fut 0 A 2 to 0 with Y Yt 1 T Y U Ut u Ut 1 Jus ae fyo a fy gl _ Of 0g _ Og fu Fig 9v By IU Bap Taking expectations we re almost there E EO Ya Ut ust TU y y RE fy gy 949 T guu fs Gull gut fy 9 fuu fy Quy T fyoIy fu y Fu Gy Gi yogu d fa u 0 As you can see since future shocks only enter with their first moments which are ze
27. to know the value of dy at the final equilibrium Note that in a stationary model it is expected that variables will eventually go back to steady state after the initial shock If you expect to see a growing curve for a variable you are thinking about a growth model Because growth models are nonstationary it is easier to work with the stationarized version of such models Again if you know the trend you can always add it back after the simulation of the stationary components of the variables 4 3 2 Expectations taken in the past For instance to enter the term E 1y define s E yz41 and then use s 1 in your mod file Note that because of Jensen s inequality you cannot do this for terms that enter your equation in a non linear fashion If you do have non linear terms on which you want to take expectations in the past you would need to apply the above manipulation to the entire equation as if y were an equation not just a variable 4 3 3 Infinite sums Dealing with infinite sums is tricky in general and needs particular care when working with Dynare The trick is to use a recursive representation of the sum For example suppose your model included CO gt Pag 0 j 0 4 3 MODELING TIPS 45 Note that the above can also be written by using an auxiliary variable Sy defined as M S M Bi j 0 which can also be written in the following recursive manner oo oo oo Se YO Pan
28. to make But depending on the context it may instead be quite misleading For instance when using sec ond order welfare functions to compare policies you also need second order approximations of the policy function Yet more clearly in the case of asset pricing models linearizing to the second order enables you to take risk or the variance of shocks into consideration a highly desirable modeling feature It is therefore very convenient that Dynare allows you to choose between a first or second order linearization of your model in the option of the stoch simul command 7 3 How does dynare solve stochastic DSGE models In this section we shall briefly overview the perturbation methods employed by Dynare to solve DSGE models to a first order approximation The sec ond order follows very much the same approach although at a higher level of complexity The summary below is taken mainly from Michel Juillard s presentation Computing first order approximations of DSGE models with Dynare which you should read if interested in particular details especially regarding second order approximations available on Michel Juillard s web site To summarize a DSGE model is a collection of first order and equilibrium conditions that take the general form E Lf yt Yt Yi 1 w 0 E u 0 E uru Xu and where y vector of endogenous variables of any dimension u vector of exogenous stochastic
29. too small the acceptance rate the fraction of candidate parameters that are accepted in a window of time will be too high and the Markov Chain of candidate parameters will mix slowly meaning that the distribution will take a long time to converge to the posterior distribution since the chain is likely to get stuck around a local maximum On the other hand if the scale factor is too large the acceptance rate will be very low as the candidates are likely to 8 3 DSGE MODELS AND BAYESIAN ESTIMATION 89 Step 1 Pick candidate 9 from jumping distribution Ae Step 3 Accept or Final step Build discard candidate histogram of values m Step 2 Compute acceptance ratio Figure 8 1 The above sketches the Metropolis Hastings algorithm used to build the posterior distribution function Imagine repeating these steps a large number of times and smoothing the histogram such that each bucket has infinitely small width land in regions of low probability density and the chain will spend too much time in the tails of the posterior distribution While these steps are mathematically clear at least to a machine needing to undertake the above calculations several practical questions arise when carrying out Bayesian estimation These include How should we choose the scale factor c variance of the jumping distribution What is a satisfactory acceptance rate How many draws are ideal How is convergence of the
30. usually called closed loop solutions and those to stochastic models are referred to as open loop Because this distinction will resurface again and again throughout the chapter but also because it has been a source of significant confusion in the past the following gives some additional details 3 1 1 NOTE Deterministic vs stochastic models Deterministic models have the following characteristics 1 As the DSGE read stochastic i e not deterministic literature has gained attention in economics deterministic models have become somewhat rare Examples include OLG models without aggregate un certainty 2 These models are usually introduced to study the impact of a change in regime as in the introduction of a new tax for instance 3 Models assume full information perfect foresight and no uncertainty around shocks 4 Shocks can hit the economy today or at any time in the future in which case they would be expected with perfect foresight They can also last one or several periods 5 Most often though models introduce a positive shock today and zero shocks thereafter with certainty 3 2 INTRODUCING AN EXAMPLE 11 6 The solution does not require linearization in fact it doesn t even really need a steady state Instead it involves numerical simulation to find the exact paths of endogenous variables that meet the model s first order conditions and shock structure T l his solution method can therefor
31. 4 Stationarizing variables Let s illustrate this transformation on output and leave the transformations of the remaining equations as an exercise if you wish Nason and Cogley 1994 includes more details on the transformations of each equation We stationarize output by dividing its real variables except for labor by Az We define Y to equal Y A and K as Ky Ax NOTE Recall from section 3 5 in chapter 3 that in Dynare variables take the time subscript of the period in which they are decided in the case of the capital stock today s capital stock is a result of yesterday s decision Thus in the output equation we should actually work with Ay Ky 1 A 1 The resulting equation made up of stationary variables is Y Ky 9 xoc GE Aeneaan 2 n A Xo Y Renee At Rg N exp a y cA4 where we go from the second to the third line by taking the exponential of both sides of the equation of motion of technology The above is the equation we retain for the mod file of Dynare into which we enter y k 1 alp n 1 alp exp alp gam e_a The other equations are entered into the mod file after transforming them in exactly the same way as the one above A final transformation to consider that turns out to be useful since we often deal with the growth rate of tech nology is to define dA exp gamte_a by simply taking the exponential of both sides of the stochastic process of technology defined in th
32. 94 That paper should serve as a helpful reference if anything is 61 62 CHAPTER 6 ESTIMATING DSGE MODELS ADVANCED TOPICS Deposits Money House Dividends Cash Central holds Bank 4 I I um Wages Loans Goods Dividends N Labor N Figure 6 1 Continuous lines show the circulation of nominal funds while dashed lines show the flow of real variables left unclear in the description below In essence the model studied by Schorfheide 2000 is one of cash in ad vance CIA The goal of the paper is to estimate the model using Bayesian techniques while observing only output and inflation In the model there are several markets and actors to keep track of So to clarify things figure 6 1 1 sketches the main dynamics of the model You may want to refer back to the figure as you read through the following sections The economy is made up of three central agents and one secondary agent households firms and banks representing the financial sector and a mon etary authority which plays a minor role Households maximize their utility function which depends on consumption C and hours worked H while deciding how much money to hold next period in cash M 1 and how much to deposit at the bank D in order to earn Ry 1 interest Households 6 1 ALTERNATIVE AND NON STATIONARY EXAMPLE 63 therefore solve the problem Ce He Mica Di Eo 22 5 6 1 6 In C oln 1
33. DYNARE User Guide An introduction to the solution amp estimation of DSGE models Tommaso Mancini Griffoli 2007 2008 Dynare v4 User Guide Public beta version Tommaso Mancini Griffoli tommaso mancini stanfordalumni org This draft March 2007 ii Copyright 2007 2008 Tommaso Mancini Griffoli Permission is granted to copy distribute and or modify this document under the terms of the GNU Free Documentation License Version 1 3 or any later version published by the Free Software Foundation with no Invariant Sections no Front Cover Texts and no Back Cover Texts A copy of the license can be found at http www gnu org licenses fdl txt Contents Contents List of Figures 1 Introduction 1 1 About this Guide approach and structure 12 Whatis Dynare ors cina pa e hb a RU RR A REOR 1 3 Additional sources of help sos o eo ey la Nomenclature oo aun ai a i e e a 1 5 v4 what s new and backward compatibility 2 Installing Dynare ZU Dynas Versione x sss 4k Bao ok Ros S ie e SE FOE mie RE 2 20 System requirements lt o c a ea 2l 23 Installing Dymato i see xke Rowe e a e e a 2 3 1 Installing on Windows ana s ea aia 24 Matlab particularities aos ma 534 04 4 x R Xr be wars 3 Solving DSGE models basics 3 1 A fundamental distinction gt o see sa ssw wad sad aaa 3 1 1 NOTE Deterministic vs stochastic models 3 4 Introducing an example 2 45 4 2 46 Ha xh RR 3
34. Metropolis Hastings iterations assessed These are all important questions that will come up in your usage of Dynare They are addressed as clearly as possible in section 5 7 of Chapter 5 CHAPTER 8 ESTIMATING DSGE MODELS BEHIND THE SCENES 90 OF DYNARE 8 4 Comparing models using posterior distributions As mentioned earlier while touting the advantages of Bayesian estimation the posterior distribution offers a particularly natural method of comparing models Let s look at an illustration Suppose we have a prior distribution over two competing models p A and p B Using Bayes rule we can compute the posterior distribution over models where 7 A B P 1 p Yr Z p I Yr P AB p Z p Yr Z where this formula may easily be generalized to a collection of N models Then the comparison of the two models is done very naturally through the ratio of the posterior model distributions We call this the posterior odds ratio p A Y T p A pY TIA p B Yr p B p Yr B The only complication is finding the magrinal density of the data condi tional on the model p Y7 Z which is also the denominator of the posterior density p 0 Y r discussed earlier This requires some detailed explanations of their own For each model Z A B we can evaluate at least theoretically the marginal density of the data conditional on the model by integrating out the deep parameters 07 from the posterior kernel P Yr T f p 07 YT 07 2 d07
35. N lt Li where the second equation makes use of the production function Y K ALN and the real aggregate accounting constraint goods market equilibrium C I Y where I Ky41 1 6 K and where is the rate of depreciation Note that it is the firms that engage in investment in this model by trading off investment for dividends to consumers The third equation simply specifies that bank loans are used to pay for wage costs To close the model we add the usual labor and money market equilib rium equations H N and P C My Xi as well as the condition that Rut Rp due to the equal risk profiles of the loans 64 CHAPTER 6 ESTIMATING DSGE MODELS ADVANCED TOPICS More importantly we add a stochastic elements to the model The model allows for two sources of perturbations one real affecting technology and one nominal affecting the money stock These important equations are In A y In Aja 44 EAL N 0 0 and Inm 1 p nm pln mi 4 mg EMt N 0 025 where m Mr i M is the growth rate of the money stock Note that theses expressions for trends are not written in the most straightforward manner nor very consistently But we reproduced them never the less to make it easier to compare this example to the original paper The first equation is therefore a unit root with drift in the log of tech nology and the second an autoregressive stationary process in the growth r
36. P Aln B In mii 4 44 Note from the original equation of motion of Inm that in steady state In m In m so that the drift terms in the above equation are In m y In Dynare any trends whether deterministic or stochastic the drift term must be declared up front In the case of our example we therefore write in a somewhat cumbersome manner observation trends P obs log mst gam Y obs gam end In general the command observation trends specifies linear trends as a function of model parameters for the observed variables in the model 6 1 9 Declaring unit roots in observable variables And finally since P_obs and Y _obs inherit the unit root characteristics of their driving variables technology and money we must tell Dynare to use a diffuse prior infinite variance for their initialization in the Kalman filter Note that for stationary variables the unconditional covariance matrix of these variables is used for initialization The algorithm to compute a true diffuse prior is taken from Durbin and Koopman 2001 To give these instructions to Dynare we write in the mod unit root vars P obs Y obs NOTE You don t need to declare unit roots for any non stationary model Unit roots are only related to stochastic trends You don t need to use a diffuse This can also be see from substituting for In mii in the above equation with the equation of motion of In m to yield A ln P A ln P ln m p
37. Y T the posterior density Using the Bayes theorem twice we obtain this density of parameters knowing the data Generally we have 6 Y DIO IT p 0 Yr 07 P Yr We also know that p 9 Yr p 0 By using these identities we can combine the prior density and the likelihood function discussed above to get the posterior density P Yr 0A A p 04 A p 0A Y r A p Y7 A Yr 0 p 6 Yr p Yr 0 x p 0 where p Y 7 A is the marginal density of the data conditional on the model DYTA f pOL Yr Ades OA Finally the posterior kernel or un normalized posterior density given that the marginal density above is a constant or equal for any parameter corresponds to the numerator of the posterior density P 04 Yr A x p Yr 04 A p 04 A K 0A Yr A This is the fundamental equation that will allow us to rebuild all posterior mo ments of interest The trick will be to estimate the likelihood function with the help of the Kalman filter and then simulate the posterior kernel using a sampling like or Monte Carlo method such as the Metropolis Hastings These topics are covered in more details below Before moving on though the subsection below gives a simple example based on the above reasoning of what we mean when we say that Bayesian estimation is somewhere in between calibration and maximum likelihood estimation The example is drawn from Zellner 1971 although other similar examples can b
38. adratic approximation of the decision rules 0 uses a pure perturbation approach as in Schmitt Grohe and Uribe 2004 default and 1 moves the point around which the Taylor expansion is computed toward the means of the distribution as in Collard and Juillard 2001b e drop INTEGER number of points dropped at the beginning of sim ulation before computing the summary statistics default 100 e hp filter INTEGER uses HP filter with lambda INTEGER before computing moments default no filter e hp ngrid INTEGER number of points in the grid for the discreet In verse Fast Fourier Transform used in the HP filter computation It may be necessary to increase it for highly autocorrelated processes default 512 e irf INTEGER number of periods on which to compute the IRFs default 40 Setting IRF 0 suppresses the plotting of IRF s e relative irf requests the computation of normalized IRFs in percentage of the standard error of each shock e nocorr doesn t print the correlation matrix printing is the default e nofunctions doesn t print the coefficients of the approximated solution printing is the default 30 CHAPTER 3 SOLVING DSGE MODELS BASICS nomoments doesn t print moments of the endogenous variables print ing them is the default noprint cancel any printing usefull for loops order 1 or 2 order of Taylor approximation default 2 unless you re working with a linear model
39. alues of the innovations will be zero in all periods to come In a stochastic context agents will take their decisions 9 10 CHAPTER 3 SOLVING DSGE MODELS BASICS knowing that the future value of innovations are random but will have zero mean This isn t the same thing because of Jensen s inequality Of course if you consider only a first order linear approximation of the stochastic model or a linear model the two cases become practically the same due to certainty equivalence A second order approximation will instead lead to very different results as the variance of shocks will matter The solution method for each of these model types differs significantly In deterministic models a highly accurate solution can be found by numerical methods The solution is nothing more than a series of numbers that match a given set of equations Intuitively if an agent has perfect foresight she can specify today at the time of making her decision what each of her precise actions will be in the future In a stochastic environment instead the best the agent can do is specify a decision policy or feedback rule for the future what will her optimal actions be contingent on each possible realization of shocks In this case we therefore search for a function satisfying the model s first order conditions To complicate things this function may be non linear and thus needs to be approximated In control theory solutions to determin istic models are
40. are will display the following results 5 9 1 Tabular results The first results to be displayed and calculated from a chronological stand point are the steady state results Note the dummy values of 1 for the non stationary variables Y obs and P obs These results are followed by the eigen values of the system presented in the order in which the endogenous variables are declared at the beginning of the mod file The table of eigenvalues is completed with a statement about the Blanchard Kahn condition being met hopefully The next set of results are for the numerical iterations necessary to find the posterior mode as explained in more details in Chapter 6 The improve ment from one iteration to the next reaches zero Dynare give the value of the objective function the posterior Kernel at the mode and displays two important table summarizing results from posterior maximization The first table summarizes results for parameter values It includes prior means posterior mode standard deviation and t stat of the mode based on the assumption of a Normal probably erroneous when undertaking Bayesian estimation as opposed to standard maximum likelihood as well as the prior distribution and standard deviation pstdev It is followed by a second table summarizing the same results for the shocks It may be entirely possible that you get an infinite value for a standard deviation this is simply the limit case of the inverse gamma distri
41. articular platform feel free to write directly to Michel Juillard michel juillard AT ens fr or visit the Dynare forums To run Dynare it is recommended to allocate at least 256MB of RAM to the platform running Dynare although 512MB is preferred Depending on the type of computations required like the very processor intensive Metropolis Hastings algorithm you may need up to 1GB of RAM to obtain acceptable T 8 CHAPTER 2 INSTALLING DYNARE computational times 2 3 Installing Dynare 2 3 1 Installing on Windows The following assumes you have Matlab version 6 5 1 or later installed on your Windows system The current way to install Dynare version 4 may not yet be on par with the procedure described below If a discrepancy exists please follow downloading and installation instructions on the Dynare website 1 Download the latest stable version of Dynare for Matlab Windows from the Dynare website 2 You will now have on your computer a zip file which you should un zip This will create a folder called by default Dynare and its version number for example Dynare v4 x where x stands for any subsequent upgrades 3 This directory contains several sub directories among which i matlab ii doc and iii examples 4 Place the Dynare folder Dynare v4 x in our example in the c directory and note that location The easiest is probably to put it in the root of c as in c dynare_v4 x 5 Start Matlab and us
42. ate of money but an AR 2 with a unit root in the log of the level of money This can be seen from the definition of my which can be rewritten as In M 1 In M Inm When the above functions are maximized we obtain the following set of first order and equilibrium conditions We will not dwell on the derivations here to save space but encourage you to browse Nason and Cogley 1994 for additional details We nonetheless give a brief intuitive explanation of each Alternatively we could have written the AR 2 process in state space form and realized that the system has an eigenvalue of one Otherwise said one is a root of the second order autoregressive lag polynomial As usual if the logs of a variable are specified to follow a unit root process the rate of growth of the series is a stationary stochastic process see Hamilton 1994 chapter 15 for details 6 1 ALTERNATIVE AND NON STATIONARY EXAMPLE 65 equation The system comes down to id Bj CP Yi Yia Pf Pia Be 4041 P 1 jake NEP Ld ea B ome s N LN 1 a Pre 0144 Ro NF AV 8 1 a Pie sette KP N a 2 A Ja xE Pim P e erre Ra N12 E 1 dye tA R 1 My Re NI Nt 1 p In m pIn my1 ey exp y 44 ert Yi B B mia ert where importantly hats over variables no longer mean deviations from steady state but instead represent variables that have been made stationary
43. bution 5 9 2 Graphical results corresponding graphs will be reproduced below The first figure comes up soon after launching Dynare as little computa tion is necessary to generate it T he figure returns a graphical representation of the priors for each parameter of interest 58 CHAPTER 5 ESTIMATING DSGE MODELS BASICS The second set of figures displays MCMC univariate diagnostics where MCMC stands for Monte Carlo Markov Chains This is the main source of feedback to gain confidence or spot a problem with results Recall that Dynare completes several runs of Metropolis Hastings simulations as many as determined in the option mh nblocks each time starting from a different ini tial value If the results from one chain are sensible and the optimizer did not get stuck in an odd area of the parameter subspace two things should happen First results within any of the however many iterations of Metropolis Hastings simulation should be similar And second results between the various chains should be close This is the idea of what the MCMC diagnostics track More specifically the red and blue lines on the charts represent specific measures of the parameter vectors both within and between chains For the results to be sensible these should be relatively constant although there will always be some variation and they should converge Dynare reports three measures interval being constructed from an 8096 confidence inter
44. ct For correlated shocks the variance decomposition is computed as in the VAR literature through a Cholesky decomposition of the covariance matrix of the exogenous variables When the shocks are correlated the variance decomposition depends upon the order of the variables in the varexo command 3 8 SELECTING A COMPUTATION 29 on your results since they get averaged between each Monte Carlo trial and in the limit should sum to zero given their mean of zero Note that in the case of a second order approximation Dynare will return the actual sample moments from the simulations For first order linearizations Dynare will in stead report theoretical moments In both cases the return to steady state is asymptotic TIP thus you should make sure to specify sufficient periods in your IRFs such that you actually see your graphs return to steady state Details on implementing this appear below If you re interested to peer a little further into what exactly is going on behind the scenes of Dynare s computations have a look at Chapter 7 Here instead we focus on the application of the command and reproduce below the most common options that can be added to stoch simul For a complete list of options please see the Reference Manual Options following the stoch simul command e ar INTEGER Order of autocorrelation coefficients to compute and to print default 5 e dr algo Q or 1 specifies the algorithm used for computing the qu
45. d Bayesian estimation 85 8 3 1 Rewriting the solution to the DSGE model 85 8 3 2 Estimating the likelihood function of the DSGE model 86 8 3 3 Finding the mode of the posterior distribution 8T 8 3 4 Estimating the posterior distribution 87 8 4 Comparing models using posterior distributions 90 9 Optimal policy under commitment 93 10 Troubleshooting 95 vi List of Figures Bibliography 97 List of Figures 1 1 Dynare a bird s eyeview LL 3 2l Sructure of the mod fle a s 04 fA bw ES Ai owe me RU 16 6 1 CIA model illustration 2o oz ono EORR dew Oe Ee 62 6 2 Steps of model estimation E e a 74 8 1 Illustration of the Metropolis Hastings algorithm 89 Work in Progress This is the second version of the Dynare User Guide which is still work in progress This means two things First please read this with a critical eye and send me comments Are some areas unclear Is anything plain wrong Are some sections too wordy are there enough examples are these clear On the contrary are there certain parts that just click particularly well How can others be improved I m very interested to get your feedback The second thing that a work in progress manuscript comes with is a few internal notes These are mostly placeholders for future work notes to myself or others of the Dynare development team or at times notes to you our read ers to highlig
46. d file must be termi nated by a semicolon although a single instruction can span two lines if you need extra space just don t put a semicolon at the end of the first line TIP You can also comment out any line by starting the line with two forward slashes or comment out an entire section by starting the section with and ending with For example varyckilylwrz varexo e parameters beta psi delta alpha rho sigma epsilon the above instruction reads over two lines the following section lists Several parameters which were calibrated by my co author Ask her all the difficult questions alpha 0 33 beta 0 99 delta 0 023 psi 1 75 rho 0 95 18 CHAPTER 3 SOLVING DSGE MODELS BASICS sigma 0 007 1 alpha epsilon 10 3 5 Specifying the model 3 5 1 Model in Dynare notation One of the beauties of Dynare is that you can input your model s equa tions naturally almost as if you were writing them in an academic paper This greatly facilitates the sharing of your Dynare files as your colleagues will be able to understand your code in no time There are just a few conventions to follow Let s first have a look at our model in Dynare notation and then go through the various Dynare input conventions What you can already try to do is glance at the model block below and see if you can recognize the equations from the earlier example See how easy it is to read Dynare code model
47. d window and type for instance dynare ExSolStoch to execute your mod file Running these mod files should take at most 30 seconds As a result you should get two forms of output tabular in the Matlab command window and graphical in one or more pop up windows Let s review these results 3 10 1 Results stochastic models The tabular results can be summarized as follows 1 Model summary a count of the various variable types in your model endogenous jumpers etc 2 Eigenvalues should be displayed and you should see a confirmation of the Blanchard Kahn conditions if you used the command check in your mod file 34 CHAPTER 3 SOLVING DSGE MODELS BASICS 3 Matrix of covariance of exogenous shocks this should square with the values of the shock variances and co variances you provided in the mod file 4 Policy and transition functions Solving the rational exectation model E f yiii Yt Yt 1 Ut 0 means finding an unkown function Ut g yi 1 Ut that could be plugged into the original model and satisfy the implied restrictions the first order conditions A first order approx imation of this function can be written as yr Y gyfjt 1 Juut with U yt y and y being the steadystate value of y and where gx is the partial derivative of the g function with respect to variable x In other words the function g is a time recursive approximated representation of the model that can generate timeseries t
48. dded to the endval block and serves the same functionality as described earlier namely of telling Dynare to start and or end at a steady state close to the values you entered If you do not use steady after endval and the latter does not list exact steady state values you may impose on your system that it does not return to steady state This is unusual In this case your problem would become a so called two boundary problem which when solved requires that the path of your endogenous variables pass through the steady state closest to your endval values In our example we make use of the second steady since the actual terminal steady state values are bound to be somewhat different from those entered above which are noth ing but the initial values for all variables except for technology In the above example the value of technology would move to 0 1 in pe riod 1 tomorrow and thereafter But of course the other variables the endogenous variables will take longer to reach their new steady state values TIP If you instead wanted to study the effects of a permanent but future shock anticipated as usual you would have to add a shocks block after the endval block to undo the first several periods of the permanent shock For instance suppose you wanted the value of technology to move to 0 1 but only in period 10 Then you would follow the above endval block with shocks var z 3 8 SELECTING A COMPUTATION 27 periods 1 9
49. ded in section 3 6 3 below You can either enter exact steady state values into your mod file or just approximations and let Dynare find the exact steady state which it will do using numerical methods based on your approximations In either case these values are entered in the initval block as in the following fashion initval 22 CHAPTER 3 SOLVING DSGE MODELS BASICS ve we we ON BKRS Paw Il OO ON o o Then by using the command steady you can control whether you want to start your simulations or impulse response functions from the steady state or from the exact values you specified in the initval block Adding steady just after your initval block will instruct Dynare to consider your initial values as mere approximations and start simulations or impulse response functions from the exact steady state On the contrary if you don t add the command Steady your simulations or impulse response functions will start from your initial values even if Dynare will have calculated your model s exact steady state for the purpose of linearization For the case in which you would like simulations and impulse response functions to begin at the steady state the above block would be expanded to yield steady TIP If you re dealing with a stochastic model remember that its lin ear approximation is good only in the vicinity of the steady state thus it is strongly recommended that you start your simulations from a steady stat
50. do just as well to ask Dynare to compute your model s steady state except maybe if you want to run loops to vary your parameter values for instance in which case writing a Matlab program may be more handy But you may also be interested in the second possibility described above namely of specifying shocks in an external file to simulate a model based on shocks from a prior estimation for instance You could then retrieve the ex ogenous shocks from the oo file by saving them in a file called datafile mat Finally you could simulate a deterministic model with the shocks saved from the estimation by specifying the source file for the shocks using the shocks shocks file datafile mat command But of course this is a bit of a workaround since you could also use the built in commands in Dynare to generate impulse response functions from estimated shocks as described in chapter 5 42 FILES CREATED BY DYNARE 43 4 1 5 Infinite eigenvalues If you use the command check in your mod file Dynare will report your sys tem s eigenvalues and tell you if these meet the Blanchard Kahn conditions At that point don t worry if you get infinite eigenvalues these are are firmly grounded in the theory of generalized eigenvalues They have no detrimental influence on the solution algorithm As far as Blanchard Kahn conditions are concerned infinite eigenvalues are counted as explosive roots of modulus larger than one 4 2 Files created by Dy
51. e this means either using the command steady or entering exact steady state values 3 6 SPECIFYING STEADY STATES AND OR INITIAL VALUES 23 3 6 2 Deterministic models and initial values Deterministic models do not need to be linearized in order to be solved Thus technically you do not need to provide a steady state for these model But practically most researchers are still interested to see how a model reacts to shocks when originally in steady state In the deterministic case the initval block serves very similar functions as described above If you wanted to shock your model starting from a steady state value you would enter approximate or exact steady state values in the initval block followed by the command steady Otherwise if you wanted to begin your solution path from an arbi trary point you would enter those values in your initval block and not use the steady command An illustration of the initval block in the determin istic case appears further below 3 6 3 Finding a steady state The difficulty in the above of course is calculating actual steady state val ues Doing so borders on a form of art and luck is unfortunately part of the equation Yet the following TIPS may help As mentioned above Dynare can help in finding your model s steady state by calling the appropriate Matlab functions But it is usually only successful if the initial values you entered are close to the true steady state If you have trouble
52. e be useful when the economy is far away from steady state when linearization offers a poor approximation Stochastic models instead have the following characteristics 1 These types of models tend to be more popular in the literature Exam ples include most RBC models or new keynesian monetary models 2 In these models shocks hit today with a surprise but thereafter their expected value is zero Expected future shocks or permanent changes in the exogenous variables cannot be handled due to the use of Taylor approximations around a steady state 3 Note that when these models are linearized to the first order agents behave as if future shocks where equal to zero since their expectation is null which is the certainty equivalence property This is an often overlooked point in the literature which misleads readers in supposing their models may be deterministic 3 2 Introducing an example The goal of this first section is to introduce a simple example Future sections will aim to code this example into Dynare and analyze its salient features under the influence of shocks both in a stochastic and a deterministic envi ronment Note that as a general rule the examples in the basic chapters 3 and 5 are kept as bare as possible with just enough features to help illustrate Dynare commands and functionalities More complex examples are instead presented in the advanced chapters The model introduced here is a basic RBC model
53. e found in Hamilton 1994 chapter 12 8 2 1 Bayesian estimation somewhere between calibration and maximum likelihood estimation an example Suppose a data generating process y 4 t for t 1 T where amp N 0 1 is gaussian white noise Then the likelihood is given by P Yr p 21 7 3 e 8 Lis 8 3 DSGE MODELS AND BAYESIAN ESTIMATION 85 We know from the above that fiir n p y y and that V iur T i In addition let our prior be a gaussian distribution with expectation uo and variance d Then the posterior density is defined up to a constant by p u Yr x man 3e 7 x 27 Be 3n n y _ El nu Or equivalently p u Ym cce VW with and OR MLT 0 Ho 5 op El From this we can tell that the posterior mean is a convex combination of the prior mean and the ML estimate In particular if o oo ie we have no prior information so we just estimate the model then E u gt f mL r the maximum likelihood estimator But if o 0 ie we re sure of ourselves and we calibrate the parameter of interest thus leaving no room for estimation then E u uo the prior mean Most of the time we re somewhere in the middle of these two extremes 8 3 DSGE models and Bayesian estimation 8 3 1 Rewriting the solution to the DSGE model Recall from chapter 7 that any DSGE model which is really a collection of first order and equilibrium c
54. e model setup above 68 CHAPTER 6 ESTIMATING DSGE MODELS ADVANCED TOPICS 6 1 5 Linking stationary variables to the data And finally we must make a decision as to our non stationary observa tions We could simply stationarize them by working with rates of growth which we know are constant In the case of output the observable variable would become Y Y i We would then have to relate this observable call it gy obs to our stationary model s variables Y by using the definition that Y Y A Thus we add to the model block of the mod file gy obs dA y y 1 where the y of the mod file are the stationary Di But we could also work with non stationary data in levels This complicates things somewhat but illustrates several features of Dynare worth highlighting we therefore follow this path in the remainder of the example The result is not very different though from what we just saw above The goal is to add a line to the model block of our mod file that relates the non stationary observables call them Yobs to our stationary output Y We could simply write You Y A But since we don t have an A variable but just a dA we we write the above relationship in ratios To the mod file we there fore add Y obs Y obs 1 dA y y 1 We of course do the same for prices our other observable variable except that we use the relationship Poos P M A as noted earlier The details of the correct transformations fo
55. e of stochastic models the command stoch simul is appropriate This command instructs Dynare to compute a Taylor approxi mation of the decision and transition functions for the model the equations listing current values of the endogenous variables of the model as a func tion of the previous state of the model and current shocks impulse response functions and various descriptive statistics moments variance decomposition correlation and autocorrelation coefficients Impulse response functions are the expected future path of the endogenous variables conditional on a shock in period 1 of one standard deviation TIP If you linearize your model up to a first order impulse response functions are simply the algebraic forward iteration of your model s policy or decision rule If you instead linearize to a second order impulse response functions will be the result of actual Monte Carlo simulations of future shocks This is because in second order linear equations you will have cross terms involving the shocks so that the effects of the shocks depend on the state of the system when the shocks hit Thus it is impossible to get algebraic average values of all future shocks and their impact The technique is instead to pull fu ture shocks from their distribution and see how they impact your system and repeat this procedure a multitude of times in order to draw out an average response That said note that future shocks will not have a significant impa
56. e steady state solution by hand in Dynare But of course this procedure could be time consuming and bothersome especially if you want to alter parameter values and thus steady states to undertake robustness checks The alternative is to write a Matlab program to find your model s steady state Doing so has the clear advantages of being able to incorporate your Matlab program directly into your mod file so that running loops with differ ent parameter values for instance becomes seamless NOTE When doing so your matlab m file should have the same name as your mod file followed by steadystate For instance if your mod file is called example mod your Matlab file should be called example steadystate m and should be saved in the same directory as your mod file Dynare will automatically check the di rectory where you ve saved your mod file to see if such a Matlab file exists If so it will use that file to find steady state values regardless of whether you ve provided initial values in your mod file Because Matlab does not work with analytical expressions though unless you re working with a particular toolbox you need to do a little work to write your steady state program It is not enough to simply input the equations as you ve written them in your mod file and ask Matlab to solve the system You will instead need to write your steady state program as if you were solv ing for the steady state by hand That is you need to inpu
57. e the menu File Set Path to add the path to the Dynare matlab subdirectory Following our example this would corre spond to c dynare v4 x matlab 6 Save these changes in Matlab and you re ready to go 2 4 Matlab particularities A question often comes up what special Matlab toolboxes are necessary to run Dynare In fact no additional toolbox is necessary for running most of Dynare except maybe for optimal simple rules see chapter 9 but even then remedies exist see the Dynare forums for discussions on this or to ask your particular question But if you do have the optimization toolbox installed you will have additional options for solving for the steady state solve_algo option and for searching for the posterior mode mode_compute option both of which are defined later As of writing this Guide Dynare is being developed on Matlab version 7 Nonetheless great care is taken not to introduce features that would not work with reasonably recent versions of Matlab However Dynare requires at least the Matlab feature set of version 6 5 1 released September 22 2003 Chapter 3 Solving DSGE models basics This chapter covers everything that leads to and stems from the solution of DSGE models a vast terrain That is to say that the term solution in the title of the chapter is used rather broadly You may be interested in simply finding the solution functions to a set of first order conditions stemming from your
58. each one in Dynare such as Ej 124 4 2xt4 IE km Chapter 5 Estimating DSGE models basics As in the chapter 3 this chapter is structured around an example The goal of this chapter is to lead you through the basic functionality in Dynare to estimate models using Bayesian techniques so that by the end of the chapter you should have the capacity to estimate a model of your own This chapter is therefore very practically oriented and abstracts from the underlying com putations that Dynare undertakes to estimate a model that subject is instead covered in some depth in chapter 8 while more advanced topics of practical appeal are discussed in chapter 6 5 1 Introducing an example The example introduced in this chapter is particularly basic This is for two reasons First we did not want to introduce yet another example in this sec tion there s enough new material to keep you busy Instead we thought it would be easiest to simply continue working with the example introduced in chapter 3 with which you are probably already quite familiar Second the goal of the example in this chapter is really to explain features in context but not necessarily to duplicate a real life scenario you would face when writing a paper Once you feel comfortable with the content of this chapter though you can always move on to chapter 6 where you will find a full fledged replication of a recent academic paper featuring a
59. ed as a capital accumulation equation after bringing c to the right hand side and noticing that wil rikt total payments to factors equals y or aggregate output by the zero profit condition As a consequence if we define investment as i yr cz we obtain the intuitive result that 4 ki 1 6 ky or that investment replenishes the capital stock thereby countering the effects of depreciation In any given period the consumer therefore faces a tradeoff between consuming and in vesting in order to increase the capital stock and consuming more in following periods as we will see later production depends on capital Maximization of the household problem with respect to consumption leisure and capital stock yields the Euler equation in consumption capturing the intertemporal tradeoff mentioned above and the labor supply equation linking labor positively to wages and negatively to consumption the wealth ier the more leisure due to the decreasing marginal utility of consumption These equation are 1 MN GE 1 ri 6 Ct Ct 1 and a t zi Vr Wt The firm side of the problem is slightly more involved due to monopolistic competition but is presented below in the simplest possible terms with a little hand waiving involved as the derivations are relatively standard There are two ways to introduce monopolistic competition We can ei ther assume that firms sell differentiated varieti
60. ed in chapter 3 the steady state block will look exactly the same TIP During estimation in finding the posterior mode Dynare recalcu lates the steady state of the model at each iteration of the optimization rou tine more on this later based on the newest round of parameters available Thus by providing approximate initial values and relying solely on the built in Dynare algorithm to find the steady state a numerical procedure you will significantly slow down the computation of the posterior mode Dynare will end up spending 60 to 7096 of the time recalculating steady states It is much more efficient to write an external Matlab steady state file and let Dynare use that file to find the steady state of your model by algebraic procedure For more details on writing an external Matlab file to find your model s steady state please refer to section 3 6 3 of chapter 3 5 6 Declaring priors Priors play an important role in Bayesian estimation and consequently de serve a central role in the specification of the mod file Priors in Bayesian estimation are declared as a distribution The general syntax to introduce priors in Dynare is the following 50 CHAPTER 5 ESTIMATING DSGE MODELS BASICS estimated params PARAMETER NAME PRIOR SHAPE PRIOR MEAN PRIOR STANDARD ERROR PRIOR 3 4 PARAMETER PRIOR 4 PARAMETER end where the following table defines each term more clearly PRIOR SHAPE Corresponding distribution Range
61. ences workshops and seminars that may be of interest 1 4 NOMENCLATURE 5 1 4 Nomenclature To end this introduction and avoid confusion in what follows it is worthwhile to agree on a few definitions of terms Many of these are shared with the Reference Manual 1 5 Integer indicates an integer number Double indicates a double precision number The following syntaxes are valid 1 1e3 1 1E3 1 1E 3 1 1d3 1 1D3 Expression indicates a mathematical expression valid in the underlying language e g Matlab Variable name indicates a variable name NOTE These must start with an alphabetical character and can only contain other alphabetical characters and digits as well as underscores All other characters including accents and spaces are forbidden Parameter name indicates a parameter name which must follow the same naming conventions as above Filename indicates a file name valid in your operating system Note that Matlab requires that names of files or functions start with alpha betical characters this concerns your Dynare mod files Command is an instruction to Dynare or other program when specified Options or optional arguments for a command are listed in square brackets unless otherwise noted If for instance the option must be specified in parenthesis in Dynare it will show up in the Guide as option Typewritten text indicates text as it should appear in Dynare code v4 what s new and bac
62. end 32 CHAPTER 3 SOLVING DSGE MODELS BASICS steady check shocks var e sigma 2 end stoch simul periods 2100 3 9 2 The deterministic model case of temporary shock varyckilylwr varexo Z parameters beta psi delta alpha sigma epsilon alpha 0 33 beta 0 99 delta 0 023 psi 1 75 sigma 0 007 1 alpha epsilon 10 model 1 c beta 1 c 1 1 r 1 delta psi c 1 1 w cti y y k 1 Yalpha exp z 1 1 alpha w y Cepsilon 1 epsilon 1 alpha 1 r y epsilon 1 epsilon alpha k 1 i k 1 delta k 1 yl y l end 3 10 FILE EXECUTION AND RESULTS 33 check shocks var z periods 1 9 values 0 1 end simul periods 2100 3 10 File execution and results To see this all come to life let s run our mod file which is conveniently installed by default in the Dynare examples directory the mod file cor responding to the stochastic model is called RBC_Monop_JFV mod and that corresponding to the deterministic model is called RBC_Monop_Det mod note this may not be the case when testing the beta version of Matlab version 4 To run a mod file navigate within Matlab to the directory where the example mod files are stored You can do this by clicking in the current di rectory window of Matlab or typing the path directly in the top white field of Matlab Once there all you need to do is place your cursor in the Matlab comman
63. equations appear in the following section 6 1 2 Declaring variables and parameters This block of the mod file follows the usual conventions and would look like varm Pc eWR kd n 1 Y obs P obs y dA varexo ea em parameters alp bet gam mst rho psi del where the choice of upper and lower case letters is not significant the first set of endogenous variables up to l are as specified in the model setup above and where the last five variables are defined and explained in more details in the section below on declaring the model in Dynare The exogenous variables are as expected and concern the shocks to the evolution of technology and money balances 6 1 3 The origin of non stationarity The problem of non stationarity comes from having stochastic trends in tech nology and money The non stationarity comes out clearly when attempting to solve the model for a steady state and realizing it does not have one It can be shown that when shocks are null real variables grow with A except for labor N which is stationary as there is no population growth nominal vari ables grow with M and prices with M A Detrending therefore involves 6 1 ALTERNATIVE AND NON STATIONARY EXAMPLE 67 the following operations where hats over variables represent stationary vari ables for real variables d q Az where q yr Ct it kt 1 For nominal variables Q Q M where Q d lj Wi And for prices P P Ai Mi 6 1
64. er 3 in section 3 6 3 NOTE You will unfortunately have to slightly amend any old steady state files you may have written Speed Several large scale improvements have been implemented to speed up Dynare This should be most noticeable when solving de terministic models but also apparent in other functionality Chapter 2 Installing Dynare 2 1 Dynare versions Three versions of Dynare exist one for Matlab one for Scilab and one for Gauss The first benefits from ongoing development and is the most popular Development of the Scilab version stopped after Dynare version 3 02 and that for Gauss after Dynare version 1 2 This User Guide will exclusively focus on the Matlab version of Dynare For the installation procedure for the Scilab or Gauss versions of the program please see the Reference Manual Note though that the Dynare syntax re mains mostly unchanged across the Matlab Scilab or Gauss versions for those features common to the three versions You may also be interested by another version of Dynare developed in par allel Dynare This is a standalone C version of Dynare specialized in computing k order approximations of dynamic stochastic general equilibrium models See the Dynare webpage for more information 2 2 System requirements Dynare can run on Windows as well as Unix like operating systems such as any Linux distribution Solaris and of course Mac OS X If you have ques tions about the support of a p
65. es of a good to consumers who 3 2 INTRODUCING AN EXAMPLE 13 aggregate these according to a CES index Or we can postulate that there is a continuum of intermediate producers with market power who each sell a different variety to a competitive final goods producer whose production function is a CES aggregate of intermediate varieties If we follow the second route the final goods producer chooses his or her optimal demand for each variety yielding the Dixit Stiglitz downward sloping demand curve Intermediate producers instead face a two pronged decision how much labor and capital to employ given these factors perfectly competi tive prices and how to price the variety they produce Production of intermediate goods follows a CRS production function de fined as ya ker where the i subscript stands for firm i of a continuum of firms between zero and one and where a is the capital elasticity in the production function with 0 a 1 Also z captures technology which evolves according to Zi pz4 1 et where p is a parameter capturing the persistence of technological progress and ei N 0 0 The solution to the sourcing problem yields an optimal capital to labor ratio or relationship between payments to factors a kate we l a The solution to the pricing problem instead yields the well known con stant markup pricing condition of monopolistic competition Dit map e 1 where pi is firm s s
66. estimated params bet normal_pdf 1 0 05 end TIP Finally another useful command to use is the estimated params init command which declares numerical initial values for the optimizer when these are different from the prior mean This is especially useful when redoing an estimation if the optimizer got stuck the first time around or needing a greater number of draws in the Metropolis Hastings algorithm and wanting to enter the posterior mode as initial values for the parameters instead of a prior The Reference Manual gives more details as to the exact syntax of this command Coming back to our basic example we would write estimated params alpha beta pdf 0 35 0 02 beta beta pdf 0 99 0 002 delta beta pdf 0 025 0 003 psi gamma pdf 1 75 0 02 rho beta pdf 0 95 0 05 epsilon gamma pdf 10 0 003 stderr e inv_gamma pdf 0 01 inf end 52 CHAPTER 5 ESTIMATING DSGE MODELS BASICS 5 7 Launching the estimation To ask Dynare to estimate a model all that is necessary is to add the com mand estimation at the end of the mod file Easy enough But the real complexity comes from the options available for the command to be entered in parentheses and sequentially separated by commas after the command estimation Below we list the most common and useful options and en courage you to view the Reference Manual for a complete list 1 datafile FILENAME the datafile a m file a mat file or an xls file
67. example below illustrates how you would introduce this into Dynare Actually the example provided is somewhat more complete than strictly necessary This is to give you an alternative full blown example to the one described in chapter 3 The model The model is a simplified standard RBC model taken from Collard and Juil lard 2003 which served as the original User Guide for Dynare The economy consists of an infinitely living representative agent who values consumption c and labor services according to the following utility function 5577 tog e E at t 1 T t where as usual the discount factor 0 lt 8 lt 1 the disutility of labor 0 gt 0 and the labor supply elasticity w gt 0 A social planner maximizes this utility function subject to the resource constraint ctu Yt where i is investment and y output Consumers are therefore also owners of the firms The economy is a real economy where part of output can be consumed and part invested to form physical capital As is standard the law of motion of capital is given by kt 1 exp be it 1 0 ke with 0 lt 6 lt 1 where 6 is physical depreciation and b a shock affecting incorporated technological progress We assume output is produced according to a standard constant returns to scale technology of the form Ye exp aj ke hl 41 DYNARE FEATURES AND FUNCTIONALITY 39 with a being the capital elasticity in the production func
68. external or internal to Matlab It will then be read by Matlab by first navigating within Matlab to the directory where the mod file is stored and then by typing in the Matlab command line Dynare filename mod although actually typing the extension mod is not necessary But before we get into executing a mod file let s start by writing one It is convenient to think of the mod file as containing four distinct blocks illustrated in figure 3 3 e preamble lists variables and parameters e model spells out the model e steady state or initial value gives indications to find the steady state of a model or the starting point for simulations or impulse response functions based on the model s solution e shocks defines the shocks to the system e computation instructs Dynare to undertake specific operations e g forecasting estimating impulse response functions Our exposition below will structured according to each of these blocks 3 4 Filling out the preamble The preamble generally involves three commands that tell Dynare what are the model s variables which are endogenous and what are the parameters The commands are e var starts the list of endogenous variables to be separated by commas e varexo starts the list of exogenous variables that will be shocked e parameters starts the list of parameters and assigns values to each In the case of our example let s differentiate between the stochastic and de terministic ca
69. f models based on fit Indeed the posterior distribution corresponding to competing models can easily be used to determine which model best fits the data This procedure as other topics mentioned above is discussed more technically in the subsection below 8 2 THE BASIC MECHANICS OF BAYESIAN ESTIMATION 83 8 2 The basic mechanics of Bayesian estimation This and the following subsections are based in great part on work by and discussions with St phane Adjemian a member of the Dynare development team Some of this work although summarized in presentation format is available in the conferences and workshops page of the Dynare website Other helpful material includes An and Schorfheide 2006 which includes a clear and quite complete introduction to Bayesian estimation illustrated by the application of a simple DSGE model Also the appendix of Schorfheide 2000 contains details as to the exact methodology and possible difficulties encountered in Bayesian estimation You may also want to take a glance at Hamilton 1994 chapter 12 which provides a very clear although somewhat outdated introduction to the basic mechanics of Bayesian estimation Finally the websites of Frank Schorfheide and Jesus Fernandez Villaverde contain a wide variety of very helpful material from example files to lecture notes to related papers Finally remember to also check the open online examples of the Dynare website for examples of mod files touching o
70. finding the steady state of your model you can begin by playing with the options following the steady command These are e solve algo 0 uses Matlab Optimization Toolbox FSOLVE e solve algo 1 uses Dynare s own nonlinear equation solver e solve algo 2 splits the model into recursive blocks and solves each block in turn e solve algo 3 uses the Sims solver This is the default option if none are specified For complicated models finding suitable initial values for the endogenous variables is the trickiest part of finding the equilibrium of that model Often it is better to start with a smaller model and add new variables one by one But even for simpler models you may still run into difficulties in finding your steady state If so another option is to enter your model in linear terms In this case variables would be expressed in percent deviations from steady state Thus their initial values would all be zero Unfortunately if any of your original non linear equations involve sums a likely fact your 24 CHAPTER 3 SOLVING DSGE MODELS BASICS linearized equations will include ratios of steady state values which you would still need to calculate Yet you may be left needing to calculate fewer steady state values than in the original non linear model Alternatively you could also use an external program to calculate ex act steady state values For instance you could write an external Maple file and then enter th
71. gh is that the equations are non linear in the deep parameters Yet they are linear in the endogenous and exogenous variables so that the likelihood may be evaluated with a linear prediction error algorithm like the Kalman filter This is exactly what Dynare does As a reminder here s what the Kalman filter recursion does For t 1 T and with initial values y and P given the recursion follows vu yY Mi Nr F MP M V Ki geo d i 9g Kivi Pa QyPi gy KM 9 0 For more details on the Kalman filter see Hamilton 1994 chapter 13 From the Kalman filter recursion it is possible to derive the log likelihood given by T Tk 1 NR In 0 Y7 7 In 27 5 gt F suk ly where the vector 0 contains the parameters we have to estimate 0 V 0 and Q 0 and where Yf expresses the set of observable endogenous variables y found in the measurement equation The log likelihood above gets us one step closer to our goal of finding the posterior distribution of our parameters Indeed the log posterior kernel can be expressed as In K 0 Y7 ln 0 Y 7 In p 0 8 3 DSGE MODELS AND BAYESIAN ESTIMATION 87 where the first term on the right hand side is now known after carrying out the Kalman filter recursion The second recall are the priors and are also known 8 3 3 Finding the mode of the posterior distribution Next to find the mode of the posterior distribution a key parameter and an i
72. hat will approximatively sat isfy the rational expectation hypothesis contained in the original model In Dynare the table Policy and Transition function contains the el ements of gy and gu Details on the policy and transition function can be found in Chapter 6 5 Moments of simulated variables up to the fourth moments 6 Correlation of simulated variables these are the contemporaneous correlations presented in a table T Autocorrelation of simulated variables up to the fifth lag as spec ified in the options of stoch simul The graphical results instead show the actual impulse response func tions for each of the endogenous variables given that they actually moved These can be especially useful in visualizing the shape of the transition func tions and the extent to which each variable is affected TIP If some variables do not return to their steady state either check that you have included enough periods in your simulations or make sure that your model is stationary i e that your steady state actually exists and is stable If not you should detrend your variables and rewrite your model in terms of those variables 3 10 2 Results deterministic models Automatically displayed results are much more scarce in the case of deter ministic models If you entered steady you will get a list of your steady state results If you entered check eigenvalues will also be displayed and you should receive a statemen
73. he model equations residuals are stored in a vector named residuals The model jacobian is put in gl matrix Second resp third derivatives are in g2 matrix resp g3 If the use_dll option has been specified in the model decla ration the pre processor will output a C file with c extension rather than a matlab file It is then compiled to create a library DLL file Us ing a compiled C file is supposed to give better computing performance in model simulation estimation 44 CHAPTER 4 SOLVING DSGE MODELS ADVANCED TOPICS e filename _static m a matlab file containing the stationarized version of the model i e where lagged variables are replaced by current variables with its jacobian Used to compute the steady state Same notations than the dynamic file Replaced by a C file when use_dll option is specified 4 3 Modeling tips 4 3 1 Stationarizing your model Models in Dynare must be stationary such that you can linearize them around a steady state and return to steady state after a shock Thus you must first stationarize your model then linearize it either by hand or by letting Dynare do the work You can then reconstruct ex post the non stationary simulated variables after running impulse response functions For deterministic models the trick is to use only stationary variables in t 1 More generally if y is I 1 you can always write y 1 as yr dy 1 where dy yt yi 1 Of course you need
74. ht a feature not yet fully stable Any such notes are marked with two stars Thanks very much for your patience and good ideas Please write either direclty to myself tommaso mancini stanfordalumni org or preferably on the Dynare Documentation Forum available in the Forum section of the Dynare website vii Contacts and Credits Dynare was originally developed by Michel Juillard in Paris France Cur rently the development team of Dynare is composed of e St phane Adjemian stephane adjemian AT ens fr e Michel Juillard michel juillard AT ens fr e Ferhat Mihoubi ferhat mihoubi AT univ evry fr e Ondra Kamenik ondra kamenik AT volny cz e Marco Ratto marco ratto AT jrc it e S bastien Villemot sebastien villemot AT ens fr Several parts of Dynare use or have strongly benefitted from publicly avail able programs by F Collard L Ingber P Klein S Sakata F Schorfheide C Sims P Soederlind and R Wouters Finally the development of Dynare could not have come such a long ways withough an active community of users who continually pose questions re port bugs and suggest new features The help of this community is gratefully acknowledged The email addresses above are provided in case you wish to contact any one of the authors of Dynare directly We nonetheless encourage you to first use the Dynare forums to ask your questions so that other users can benefit from them as well remember a
75. ically normal The algorithm in the words of An and Shorfheide constructs a Gaussian approximation around the posterior mode and uses a scaled version of the asymptotic covariance matrix as the covariance matrix for the proposal distribution This allows for an efficient exploration of the posterior distribution at least in the neighbor hood of the mode An and Schorfheide 2006 p 18 More precisely the Metropolis Hastings algorithm implements the following steps 1 Choose a starting point 0 where this is typically the posterior mode and run a loop over 2 3 4 2 Draw a proposal 0 from a jumping distribution J 0 0 1 N 0 1 cm CHAPTER 8 ESTIMATING DSGE MODELS BEHIND THE SCENES 88 OF DYNARE where Y is the inverse of the Hessian computed at the posterior mode 3 Compute the acceptance ratio PUY K 8 Yr pO r KO Yr 4 Finally accept or discard the proposal 0 according to the following rule and update if necessary the jumping distribution gt 0 with probability min r 1 0 7 otherwise Figure 8 3 4 tries to clarify the above In step 1 choose a candidate paramter 0 from a Normal distribution whose mean has been set to 6 this will become clear in just a moment In step 2 compute the value of the posterior kernel for that candidate parameter and compare it to the value of the kernel from the mean of the drawing distribution In step 3 decide whether or not to hold on
76. in which case the order is automati cally set to 1 periods INTEGER specifies the number of periods to use in simu lations default 0 TIP A simulation is similar to running impulse response functions with a model linearized to the second order in the way that both sample shocks from their distribution to see how the system reacts but a simulation only repeats the process once whereas impulse response functions run a multitude of Monte Carlo trials in order to get an average response of your system qz_criterium INTEGER or DOUBLE value used to split stable from unstable eigenvalues in reordering the Generalized Schur decomposition used for solving 1st order problems default 1 000001 replic INTEGER number of simulated series used to compute the IRFs default 1 if order 1 and 50 otherwise simul seed INTEGER or DOUBLE or EXPRESSION specifies a seed for the random number generator so as to obtain the same random sample at each run of the program Otherwise a different sample is used for each run default seed not specified If you linearized to a second order Dynare will actually undertake Monte Carlo simulations to generate moments of your variables Because of the simulation results are bound to be slightly different each time you run your program except if you fix the seed for the random number generator TIP If you do decide to fix the seed you should at least try to run your program without using si
77. kward compatibility The current version of Dynare for which this guide is written is version 4 With respect to version 3 this new version introduces several important features as well as improvements optimizations of routines and bug fixes The major new features are the following Analytical derivatives are now used everywhere for instance in the Newton algorithm for deterministic models and in the linearizations nec essary to solve stochastic models This increases computational speed significantly The drawback is that Dynare can now handle only a lim ited set of functions although in nearly all economic applications this should not be a constraint CHAPTER 1 INTRODUCTION Variables and parameters are now kept in the order in which they are declared whenever displayed and when used internally by Dynare Recall that in version 3 variables and parameters where at times in their order of declaration and at times in alphabetical order NOTE This may cause some problems of backward compatibility if you wrote programs to run off Dynare v3 output The names of many internal variables and the organization of output variables has changed These are enumerated in details in the relevant chapters The names of the files internally generated by Dynare have also changed more on this when explaining internal file structure TBD The syntax for the external steady state file has changed This is cov ered in more details in chapt
78. le 61 6 1 1 Introducing the example 61 6 1 2 Declaring variables and parameters 66 6 1 3 The origin of non stationarity 66 6 4 4 Stationarizing variables 67 6 1 5 Linking stationary variables to the data 68 6 1 6 The resulting model block of the mod file 68 6 1 7 Declaring observable variables 69 6 1 8 Declaring trends in observable variables 69 6 1 9 Declaring unit roots in observable variables 70 6 1 10 Specifying the steady state 71 6 1 1 Declaring priors se sp so taponi bee ee ee as 71 6 1 12 Launching the estimation 71 6 1 18 The complete mod file 72 6 1 14 Summing if Wp 2 ios a x99 x m 9o oko one 74 6 2 Comparing models based on their posterior distributions 74 6 3 Where is your output stored 2l 75 7 Solving DSGE models Behind the scenes of Dynare 77 Tel Introduction L3 ck ahh de e e E39 SR RO 77 7 2 What is the advantage of a second order approximation 77 7 3 How does dynare solve stochastic DSGE models 78 8 Estimating DSGE models Behind the scenes of Dynare 81 8 1 Advantages of Bayesian estimation 00 4 81 8 2 The basic mechanics of Bayesian estimation 83 8 2 1 Bayesian estimation somewhere between calibration and maximum likelihood estimation an example 84 8 3 DSGE models an
79. lmost no question is specific enough to interest just one person and yours is not the exception ix Chapter 1 Introduction Welcome to Dynare 1 1 About this Guide approach and structure This User Guide aims to help you master Dynare s main functionalities from getting started to implementing advanced features To do so this Guide is structured around examples and offers practical advice To root this un derstanding more deeply though this Guide also gives some background on Dynare s algorithms methodologies and underlying theory Thus a secondary function of this Guide is to serve as a basic primer on DSGE model solving and Bayesian estimation This Guide will focus on the most common or useful features of the pro gram thus emphasizing depth over breadth The idea is to get you to use 90 of the program well and then tell you where else to look if you re interested in fine tuning or advanced customization This Guide is written mainly for an advanced economist like a pro fessor graduate student or central banker needing a powerful and flexible program to support and facilitate his or her research activities in a variety of fields The sophisticated computer programmer on the one hand or the specialist of computational economics on the other may not find this Guide sufficiently detailed We recognize that the advanced economist may be either a beginning or intermediate user of Dynare This Guide is written t
80. ls with an Application to a Nonlinear Phillips Curve Model Computational Economics 17 2 3 125 39 2003 Stochastic simulations with DYNARE A practical guide CEPREMAP mimeo DURBIN J AND S KOOPMAN 2001 Time Series Analysis by State Space Methods Oxford University Press Oxford U K FERNANDEZ VILLAVERDE J AND J F RUBIO RAMIREZ 2004 Compar ing dynamic equilibrium models to data a Bayesian approach Journal of Econometrics 123 1 153 187 GEWEKE J 1999 Using Simulation Methods for Bayesian Econometric Models In ference Development and Communication Econometric Re view 18 1 1 126 HAMILTON J D 1994 Time Series Analysis Princeton University Press Princeton NJ IRELAND P N 2004 A method for taking models to the data Journal of Economic Dynamics and Control 28 6 1205 1226 JUILLARD M 1996 Dynare a program for the resolution and simulation of dynamic models with forward variables through the use of a relaxation algorithm CEPREMAP working papers 9602 CEPREMA P 97 98 BIBLIOGRAPHY LuBIK T AND F SCHORFHEIDE 2003 Do Central Banks Respond to Exchange Rate Movements A Structural Investigation Economics Work ing Paper Archive 505 The Johns Hopkins University Department of Eco nomics 2005 A Bayesian Look at New Open Economy Macroeco nomics Economics Working Paper Archive 521 The Johns Hopkins Uni versity Depar
81. mportant output of Dynare we simply maximize the above log posterior kernel with respect to 0 This is done in Dynare using numerical methods Recall that the likelihood function is not Gaussian with respect to 0 but to functions of 0 as they appear in the state equation Thus this maximization problem is not completely straightforward but fortunately doable with mod ern computers 8 3 4 Estimating the posterior distribution Finally we are now in a position to find the posterior distribution of our parameters The distribution will be given by the kernel equation above but again it is a nonlinear and complicated function of the deep parameters 0 Thus we cannot obtain an explicit form for it We resort instead to sampling like methods of which the Metropolis Hastings has been retained in the literature as particularly efficient This is indeed the method adopted by Dynare The general idea of the Metropolis Hastings algorithm is to simulate the posterior distribution It is a rejection sampling algorithm used to generate a sequence of samples also known as a Markov Chain for reasons that will become apparent later from a distribution that is unknown at the outset Remember that all we have is the posterior mode we are instead more often interested in the mean and variance of the estimators of 0 To do so the algorithm builds on the fact that under general conditions the distribution of the deep parameters will be asymptot
82. mul seed just to check the robustness of your results Going back to our good old example suppose we were interested in print ing all the various measures of moments of our variables want to see impulse response functions for all variables are basically happy with all default op tions and want to carry out simulations over a good number of periods We would then end our mod file with the following command stoch_simul periods 2100 3 9 THE COMPLETE MOD FILE 31 3 9 The complete mod file For completion s sake and for the pleasure of seeing our work bear its fruits here are the complete mod files corresponding to our example for the de terministic and stochastic case You can find the corresponding files in the models folder under UserGuide in your installation of Dynare The files are called RBC_Monop_JFV mod for stochastic models and RBC Monop Det mod for deterministic models 3 9 1 The stochastic model varyckilylwrz varexo e parameters beta psi delta alpha rho sigma epsilon alpha 0 33 beta 0 99 delta 0 023 psi 1 75 rho 0 95 sigma 0 007 1 alpha epsilon 10 model 1 c beta 1 c 1 1 r 1 delta psi c 1 1 w cti y y k 1 alpha exp z 1 1 alpha w y epsilon 1 epsilon 1 alpha 1 r y epsilon 1 epsilon alpha k 1 i k 1 delta k 1 yl y i z rho z 1 e end initval O 0 we oO NH zz no N HW o O ono we
83. n Bayesian estimation At its most basic level Bayesian estimation is a bridge between calibra tion and maximum likelihood The tradition of calibrating models is inherited through the specification of priors And the maximum likelihood approach en ters through the estimation process based on confronting the model with data Together priors can be seen as weights on the likelihood function in order to give more importance to certain areas of the parameter subspace More tech nically these two building blocks priors and likelihood functions are tied together by Bayes rule Let s see how First priors are described by a density function of the form p 0 4 A where A stands for a specific model 0 4 represents the parameters of model A p e stands for a probability density function pdf such as a normal gamma shifted gamma inverse gamma beta generalized beta or uniform function Second the likelihood function describes the density of the observed data given the model and its parameters 0AlY v A p Yr 04 A where Yr are the observations until period T and where in our case the likelihood is recursive and can be written as T P Yr 0 A A P Yo 0 4 A lH 04 A t 1 CHAPTER 8 ESTIMATING DSGE MODELS BEHIND THE SCENES 84 OF DYNARE We now take a step back Generally speaking we have a prior density p 0 on the one hand and on the other a likelihood p Y7 In the end we are interested in p 0
84. n stationary variables 6 1 11 Declaring priors We expand our mod file with the following information estimated params alp beta pdf 0 356 0 02 bet beta pdf 0 993 0 002 gam normal pdf 0 0085 0 003 mst normal pdf 1 0002 0 007 rho beta pdf 0 129 0 223 psi beta pdf 0 65 0 05 del beta pdf 0 01 0 005 stderr e a inv gamma pdf 0 035449 inf stderr em inv gamma pdf 0 008862 inf end 6 1 12 Launching the estimation We add the following commands to ask Dynare to run a basic estimation of our model 72 CHAPTER 6 ESTIMATING DSGE MODELS ADVANCED TOPICS estimation datafile fsdat nobs 192 loglinear mh replic 2000 mode compute 4 mh nblocks 2 mh drop 0 45 mh jscale 0 65 NOTE As mentioned earlier we need to instruct Dynare to log linearize our model since it contains non linear equations in non stationary variables A simple linearization would fail as these variables do not have a steady state Fortunately taking the log of the equations involving non stationary variables does the job of linearizing them 6 1 13 The complete mod file We have seen each part of the mod separately it s now time to get a picture of what the complete file looks like For convenience the file also appears in the models folder under UserGuide in your Dynare installation The file is called s2000ns mod varm Pc eWR kd n 1 Y_obs P obs y dA varexo e a em parameters alp bet gam mst rho psi del
85. nare At times you may get a message that there is an error in a file with a new name or you may want to have a closer look at how Dynare actually solves your model out of curiosity or maybe to do some customization of your own You may therefore find it helpful to get a brief overview of the internal files that Dynare generates and the function of each one The dynare pre processors essentially does three successive tasks i pars ing of the mod file it checks that the mod file is syntactically correct and its translation into internal machine representation in particular model equa tions are translated into expression trees ii symbolic derivation of the model equations up to the needed order depending on the computing needs iii outputting of several files which are used from matlab If the mod file is filename mod then the pre processor creates the following files e filename m a matlab file containing several instructions notably the parameter initializations and the matlab calls corresponding to comput ing tasks e filename dynamic m a matlab file containing the model equations and their derivatives first second and maybe third Endogenous vari 6x7 ables resp exogenous variables parameters are contained in a y resp x params vector with an index number depending on the declaration order The y vector has as many entries as their are vari able lag pairs in the declared model T
86. nd deterministic models To work with a temporary shock you are free to set the duration and level of the shock To specify a shock that lasts 9 periods on z for instance you would write shocks var Z periods 1 9 values 0 1 end Given the above instructions Dynare would replace the value of z spec ified in the initval block with the value of 0 1 entered above If variables were in logs this would have corresponded to a 1096 shock Note that you can also use the mshocks command which multiplies the initial value of an exogenous variable by the mshocks value Finally note that we could have entered future periods in the shocks block such as periods 5 10 in order to study the anticipatory behavior of agents in response to future shocks 3 7 2 Deterministic models permanent shocks To study the effects of a permanent shock hitting the economy today such as a structural change in your model you would not specify actual shocks but would simply tell the system to which steady state values you would like it to move and let Dynare calculate the transition path To do so you would use the endval block following the usual initval block For instance you may specify all values to remain common between the two blocks except for the value of technology which you may presume changes permanently The corresponding instructions would be 26 CHAPTER 3 SOLVING DSGE MODELS BASICS end steady where steady can also be a
87. nt equations of our model yit e 1 Ure z and vit e 1 kat To end we aggregate the production of each individual firm to find an aggregate production function On the supply side we factor out the capital to labor ratio k l which is the same for all firms and thus does not depend on i On the other side we have the Dixit Stiglitz demand for each variety By equating the two and integrating both side and noting that price dispersion is null or that as hinted earlier pi pi we obtain aggregate production Tt Q y ALD which can be shown is equal to the aggregate amount of varieties bought by the final good producer according to a CES aggregation index and in turn equal to the aggregate output of final good itself equal to household con sumption Note to close that because the ratio of output to each factor is the same for each intermediate firm and that firm specific as well as aggre gate production is CRS we can rewrite the above two equations for tw and rz without the i subscripts on the right hand side This ends the exposition of the example Now let s roll up our sleeves and see how we can input the model into Dynare and actually test how the model will respond to shocks 3 3 DYNARE MOD FILE STRUCTURE 15 3 3 Dynare mod file structure Input into Dynare involves the mod file as mentioned loosely in the intro duction of this Guide The mod file can be written in any editor
88. o accommodate both If you re new to Dynare we recommend starting with chapters 3 and 5 which introduce the program s basic features to solve including running im pulse response functions and estimate DSGE models respectively To do 2 CHAPTER 1 INTRODUCTION so these chapters lead you through a complete hands on example which we recommend following from A to Z in order to learn by doing Once you have read these two chapters you will know the crux of Dynare s functionality and hopefully feel comfortable using Dynare for your own work At that point though you will probably find yourself coming back to the User Guide to skim over some of the content in the advanced chapters to iron out details and potential complications you may run into If you re instead an intermediate user of Dynare you will most likely find the advanced chapters 4 and 6 more appropriate These chapters cover more advanced features of Dynare and more complicated usage scenarios The presumption is that you would skip around these chapters to focus on the top ics most applicable to your needs and curiosity Examples are therefore more concise and specific to each feature these chapters read a bit more like a ref erence manual We also recognize that you probably have had repeated if not active ex posure to programming and are likely to have a strong economic background Thus a black box solution to your needs is inadequate To hopefully address this i
89. onditions can be written in the form E f ye 1 Yt Ye 1 ut 0 taking as a solution equations of the type y g yr 1 ut which we call the decision rule In more appropriate terms for what follows we can rewrite the solution to a DSGE model as a system in the following manner y Myw 0 c Mi N 0 mi m gy 0 0i 1 gu 0 E mm V 0 E wu Q 0 where are variables in deviations from steady state y is the vector of steady state values and 0 the vector of deep or structural parameters to be esti mated Other variables are described below CHAPTER 8 ESTIMATING DSGE MODELS BEHIND THE SCENES 86 OF DYNARE The second equation is the familiar decision rule mentioned above But the equation expresses a relationship among true endogenous variables that are not directly observed Only y is observable and it is related to the true variables with an error m Furthermore it may have a trend which is captured with N 0 x to allow for the most general case in which the trend depends on the deep parameters The first and second equations above therefore naturally make up a system of measurement and transition or state equations respec tively as is typical for a Kalman filter you guessed it it s not a coincidence 8 3 2 Estimating the likelihood function of the DSGE model The next logical step is to estimate the likelihood of the DSGE solution system mentioned above The first apparent problem thou
90. out to one seeGeweke 1999 for more details This suggests the following estimator of the marginal density Oe DA B p 00 Dp Yr 0 T Yr Z b where each drawn vector 07 comes from the Metropolis Hastings iterations and where the probability density function f can be viewed as a weights on the posterior kernel in order to downplay the importance of extreme values of 0 Geweke 1999 suggests to use a truncated Gaussian function leading to what is typically referred to as the Modified Harmonic Mean Estimator Chapter 9 Optimal policy under commitment Chapter 10 Troubleshooting To make sure this section is as user friendly as possible the best is to compile what users have to say Please let me know what your most common problem is with Dynare how Dynare tells you about it and how you solve it Thanks for your precious help 95 Bibliography AN S AND F SCHORFHEIDE 2006 Bayesian Analysis of DSGE Models Econometric Review Forthcoming CLARIDA R J GALI AND M GERTLER 1999 The Science of Monetary Policy A New Keynesian Perspective Journal of Economic Literature XXXVII 1661 1707 COLLARD F AND M JUILLARD 2001a Accuracy of stochastic pertur bation methods The case of asset pricing models Journal of Economic Dynamics and Control 25 6 7 979 999 2001b A Higher Order Taylor Expansion Approach to Simulation of Stochastic Forward Looking Mode
91. pecific price mc is real marginal cost and p is the aggre gate CES price or average price An additional step simplifies this expression symmetric firms implies that all firms charge the same price and thus pi pi we therefore have mc e 1 e But what are marginal costs equal to To find the answer we combine the optimal capital to labor ratio into the production function and take advantage of its CRS property to solve for the amount of labor or capital required to produce one unit of output The real cost of using this amount of any one factor is given by wj riki where we substitute out the payments to the 14 CHAPTER 3 SOLVING DSGE MODELS BASICS other factor using again the optimal capital to labor ratio When solving for labor for instance we obtain 1 ds 1 B 1 l a a me a Y dem which does not depend on i it is thus the same for all firms Interestingly the above can be worked out by using the optimal capital to labor ratio to yield w 1 a yit lt or wi at which is the definition of marginal cost the cost in terms of labor input of producing an additional unit of output This should not be a surprise since the optimal capital to labor ratio follows from the maximization of the production function minus real costs with respect to its factors Combining this result for marginal cost as well as its counterpart in terms of capital with the optimal pricing condition yields the final two importa
92. possible you will certainly want to browse other material for help as you learn about new features struggle with adapting examples to your own work and yearn to ask that one question whose answer seems to exist no where At your disposal you have the following additional sources of help Reference Manual this manual covers all Dynare commands giving a clear definition and explanation of usage for each The User Guide will often introduce you to a command in a rather loose manner mainly through examples so reading corresponding command descriptions in the Reference Manual is a good idea to cover all relevant details Official online examples the Dynare website includes other examples usually well documented of mod files covering models and method ologies introduced in recent papers Open online examples this page lists mod files posted by users covering a wide variety of examples Dynare forums this lively online discussion forum allows you to ask your questions openly and read threads from others who might have run into similar difficulties Frequently Asked Questions FAQ this section of the Dynare site emphasizes a few of the most popular questions in the forums DSGE net this website run my members of the Dynare team is a resource for all scholars working in the field of DSGE modeling Besides allowing you to stay up to date with the most recent papers and possi bly make new contacts it conveniently lists confer
93. preciation of your currency 3 8 Selecting a computation So far we have written an instructive mod file but what should Dynare do with it What are we interested in In most cases it will be impulse re sponse functions IRFs due to the external shocks Let s see which are the appropriate commands to give to Dynare Again we will distinguish between deterministic and stochastic models 28 CHAPTER 3 SOLVING DSGE MODELS BASICS 3 8 1 For deterministic models In the deterministic case all you need to do is add the command simul at the bottom of your mod file Note that the command takes the option periods INTEGER The command simul triggers the computation a numerical simulation of the trajectory of the model s solution for the number of periods set in the option To do so it uses a Newton method to solve simultaneously all the equations for every period see Juillard 1996 for de tails Note that unless you use the endval command the algorithm makes the simplifying assumption that the system is back to equilibrium after the specified number of periods Thus you must specify a large enough number of periods such that increasing it further doesn t change the simulation for all practical purpose In the case of a temporary shock for instance the tra jectory will basicaly describe how the system gets back to equilibrium after being perturbed from the shocks you entered 3 8 2 For stochastic models In the more common cas
94. quation of motion of technology is a stationary AR 1 with an autoregressive parameter p less than one The model s variables would therefore be stationary and we can proceed without complications The al ternative scenario with non stationary variables is more complicated and dealt with in chapter 6 in the additional example In the stationary case our model block would look exactly as in chater 3 model 1 c beta 1 c 1 1 r 1 delta psi c 1 1 w cti y y k 1Yalpha exp z 1 1 alpha w y epsilon 1 epsilon 1 alpha 1 r y epsilon 1 epsilon alpha k 1 i k 1 delta k 1 yl y i z rho z 1 e 5 4 DECLARING OBSERVABLE VARIABLES 49 end 5 4 Declaring observable variables This should not come as a surprise Dynare must know which variables are observable for the estimation procedure NOTE These variables must be available in the data file as explained in section 5 7 below For the moment we write varobs Y 5 5 Specifying the steady state Before Dynare estimates a model it first linearizes it around a steady state Thus a steady state must exist for the model and although Dynare can calcu late it we must give it a hand by declaring approximate values for the steady state This is just as explained in details and according to the same syntax outlined in chapter 3 covering the initval steady and check commands In fact as this chapter uses the same model as that outlin
95. quations written almost as in an academic paper This not only facilitates the inputting of a model but also enables you to easily share your code as it is straightfor ward to read by anyone Figure 1 2 gives you an overview of the way Dynare works Basically the model and its related attributes like a shock structure for instance is writ ten equation by equation in an editor of your choice The resulting file will be called the mod file That file is then called from Matlab This initiates the Dynare pre processor which translates the mod file into a suitable input for the Matlab routines more precisely it creates intermediary Matlab or C files which are then used by Matlab code used to either solve or estimate the model Finally results are presented in Matlab Some more details on the internal files generated by Dynare is given in section 4 2 in chapter 4 Each of these steps will become clear as you read through the User Guide but for now it may be helpful to summarize what Dynare is able to do 1 3 CHAPTER 1 INTRODUCTION compute the steady state of a model compute the solution of deterministic models compute the first and second order approximation to solutions of stochas tic models estimate parameters of DSGE models using either a maximum likelihood or a Bayesian approach compute optimal policies in linear quadratic models Additional sources of help While this User Guide tries to be as complete and thorough as
96. r prices are left as an exercise and can be checked against the results below 6 1 6 The resulting model block of the mod file model dA exp gamte a log m 1 rho log mst rho log m 1 em P c 1 P 1 m bet P 1 alp exp alp gamtlog e 1 K alp 1 n 1 1 alp 1 del exp gamtlog e 1 c 2 P 2 m 1 0 W 1 n psi 1 psi c P 1 n 1 n 0 R P 1 alp exp alp gamte a k 1 alp n alp W 1 c P bet P 1 alp exp alp gamte a k 1 alp n 1 alp m 1 c 1 P 1 0 c k exp alp gamte a k 1 alp n 1 alp 1 del 6 1 ALTERNATIVE AND NON STATIONARY EXAMPLE 69 exp gam e a k 1 P c m m itd 1 e exp e_a y k 1 alp mn 1 alp exp alp gamte_a Y_obs Y_obs 1 dA y y 1 P_obs P_obs 1 p p 1 m 1 dA end where of course the input conventions such as ending lines with semicolons and indicating the timing of variables in parentheses are the same as those listed in chapter 3 TIP In the above model block notice that what we have done is in fact relegated the non stationarity of the model to just the last two equations concerning the observables which are after all non stationary The problem that arises though is that we cannot linearize the above system in levels as the last two equations don t have a steady state If we first take logs though they become linear and it doesn t matter anymore where we calculate their derivati
97. ro in expectations they drop out when taking expectations of the linearized equations This is technically why certainty equivalence holds CHAPTER 7 SOLVING DSGE MODELS BEHIND THE SCENES OF 80 DYNARE in a system linearized to its first order The second thing to note is that we have two unknown variables in the above equation g and g each of which will help us recover the policy function g Since the above equation holds for any 7 and any u each parenthesis must be null and we can solve each at a time The first yields a quadratic equation in gy which we can solve with a series of algebraic trics that are not all imme diately apparent but detailed in Michel Juillard s presentation Incidentally one of the conditions that comes out of the solution of this equation is the Blanchard Kahn condition there must be as many roots larger than one in modulus as there are forward looking variables in the model Having recov ered gy recovering g is then straightforward from the second parenthesis Finally notice that a first order linearization of the function g yields Ut Y Gy Juu And now that we have gy and gy we have solved for the approximate policy or decision function and have succeeded in solving our DSGE model If we were interested in impulse response functions for instance we would simply iterate the policy function starting from an initial value given by the steady state The second order solution uses the
98. s variables as in stoch simul the posterior distribution of the variance decomposition is also in cluded will be implemented shortly if not already in Dynare version 4 20 filtered_vars triggers the computation of the posterior distribution of filtered endogenous variables and shocks See the note below on the difference between filtered and smoothed shocks will be implemented shortly if not already in Dynare version 4 21 smoother triggers the computation of the posterior distribution of smoothed endogenous variables and shocks Smoothed shocks are a reconstruction of the values of unobserved shocks over the sample using all the informa tion contained in the sample of observations Filtered shocks instead are built only based on knowing past information To calculate one pe riod ahead prediction errors for instance you should use filtered not smoothed variables 5 8 THE COMPLETE MOD FILE 55 22 forecast INTEGER computes the posterior distribution of a forecast on INTEGER periods after the end of the sample used in estimation The corresponding graph includes one confidence interval describing un certainty due to parameters and one confidence interval describing un certainty due to parameters and future shocks Note that Dynare cannot forecast out of the posterior mode You need to run Metropolis Hastings iterations before being able to run forecasts on an estimated model Fi nally running a forecast i
99. s very similar to an IRF as in bayesian irf except that the forecast does not begin at a steady state but simply at the point corresponding to the last set of observations The goal of undertaking a forecast is to see how the system returns to steady state from this starting point Of course as observation do not exist for all variables those necessary are reconstructed by sampling out of the posterior distribution of parameters Again repeating this step of ten enough yields a posterior distribution of the forecast will be implemented shortly if not already in Dynare version 4 TIP Before launching estimation it is a good idea to make sure that your model is correctly declared that a steady state exists and that it can be sim ulated for at least one set of parameter values You may therefore want to create a test version of your mod file In this test file you would comment out or erase the commands related to estimation remove the prior estimates for parameter values and replace them with actual parameter values in the preamble remove any non stationary variables from your model add a shocks block make sure you have steady and possibly check following the initval block if you do not have exact steady state values and run a simulation using stoch simul at the end of your mod file Details on model solution and sim ulation can be found in Chapter 3 Finally coming back to our example we could choose a standard option es
100. same perturbation methods as above the notion of starting from a function you can solve like a steady state and iterating forward yet applies more complex algebraic techniques to re cover the various partial derivatives of the policy function But the general approach is perfectly isomorphic Note that in the case of a second order approximation of a DSGE model the variance of future shocks remains after taking expectations of the linearized equations and therefore affects the level of the resulting policy function Chapter 8 Estimating DSGE models Behind the scenes of Dynare This chapter focuses on the theory of Bayesian estimation It begins by mo tivating Bayesian estimation by suggesting some arguments in favor of it as opposed to other forms of model estimation It then attempts to shed some light on what goes on in Dynare s machinery when it estimates DSGE models To do so this section surveys the methodologies adopted for Bayesian estima tion including defining what are prior and posterior distributions using the Kalman filter to find the likelihood function estimating the posterior function thanks to the Metropolis Hastings algorithm and comparing models based on posterior distributions 8 1 Advantages of Bayesian estimation Bayesian estimation is becoming increasingly popular in the field of macro economics Recent papers have attracted significant attention some of these include Schorfheide 2000 which u
101. ses First we lay these out then we discuss them 16 CHAPTER 3 SOLVING DSGE MODELS BASICS Structure of the mod file Define variables amp parameters Spell out equations of model Indicate steady state or Ask to undertake specific operations Figure 3 1 The mod file contains five logically distinct parts 3 4 1 The deterministic case The model is inherited exactly as specified in the earlier description except that we no longer need the e variable as we can make z directly exogenous Thus the preamble would look like var yckilylwy varexo Z parameters beta psi delta alpha sigma epsilon alpha 0 33 beta 0 99 delta 0 023 psi 1 75 sigma 0 007 1 alpha epsilon 10 3 4 2 The stochastic case In this case we go back to considering the law of motion for technology con sisting of an exogenous shock e With respect to the above we therefore 3 4 FILLING OUT THE PREAMBLE 17 adjust the list of endogenous and exogenous variables and add the parameter p Here s what the preamble would look like varyckilylwrz varexo e parameters beta psi delta alpha rho sigma epsilon alpha 0 33 beta 0 99 delta 0 023 psi 1 75 rho 0 95 sigma 0 007 1 alpha epsilon 10 3 4 3 Comments on your first lines of Dynare code As you can tell writing a mod file is really quite straightforward Two quick comments NOTE Remember that each instruction of the mo
102. ses Bayesian methods to compare the fit of two competing DSGE models of consumption Lubik and Schorfheide 2003 which investigates whether central banks in small open economies re spond to exchange rate movements Smets and Wouters 2003 which ap plies Bayesian estimation techniques to a model of the Eurozone lreland 2004 which emphasizes instead maximum likelihood estimation Fernandez Villaverde and Rubio Ramirez 2004 which reviews the econometric proper ties of Bayesian estimators and compare estimation results with maximum likelihood and BVAR methodologies Lubik and Schorfheide 2005 which ap plies Bayesian estimation methods to an open macro model focussing on issues of misspecification and identification and finally Rabanal and Rubio Ramirez 2005 which compares the fit based on posterior distributions of four com peting specifications of New Keynesian monetary models with nominal rigidi 8l CHAPTER 8 ESTIMATING DSGE MODELS BEHIND THE SCENES 82 OF DYNARE ties There are a multitude of advantages of using Bayesian methods to esti mate a model but five of these stand out as particularly important and general enough to mention here First Bayesian estimation fits the complete solved DSGE model as op posed to GMM estimation which is based on particular equilibrium relation ships such as the Euler equation in consumption Likewise estimation in the Bayesian case is based on the likelihood generated by the DSGE
103. some of the inner workings of Dynare The goal is to provide a brief explanation of the files that are created by Dynare to help you in troubleshooting or provide a starting point in case you actually want to customize the way Dynare works The third section of the chapter focusses on modeling tips optimized for Dynare but possibly also helpful for other work 4 1 Dynare features and functionality 4 1 1 Other examples Other examples of mod files used to generate impulse response functions are available on the Dynare website In particular Jesus Fernandez Villaverde has provided a series of RBC model variants from the most basic to some including variable capacity utilization indivisible labor and investment spe cific technological change You can find these along with helpful notes and explanations in the Official Examples section of the Dynare website Also don t forget to check occasionally the Open Online Examples page to see if any other user has posted an example that could help you in your 3T 38 CHAPTER 4 SOLVING DSGE MODELS ADVANCED TOPICS work or maybe you would like to post an example there yourself 4 1 2 Alternative complete example The following example aims to give you an alternative example to the one in chapter 3 to learn the workings of Dynare It also aims to give you exposure to dealing with several correlated shocks Your model may have two or more shocks and these may be correlated to each other The
104. ssue the User Guide goes into some depth in covering the theoreti cal underpinnings and methodologies that Dynare follows to solve and estimate DSGE models These are available in the behind the scenes of Dynare chapters 7 and 8 These chapters can also serve as a basic primer if you are new to the practice of DSGE model solving and Bayesian estimation Finally besides breaking up content into short chapters we ve introduced two different markers throughout the Guide to help streamline your reading e TIP introduces advice to help you work more efficiently with Dynare or solve common problems e NOTE is used to draw your attention to particularly important infor mation you should keep in mind when using Dynare 1 2 What is Dynare Before we dive into the thick of the trees let s have a look at the forest from the top just what is Dynare Dynare is a powerful and highly customizable engine with an intuitive front end interface to solve simulate and estimate DSGE models 1 2 WHAT IS DYNARE 3 Matlab environment c gt mod Dynare pre Matlab file Output processor routines Figure 1 1 The mod file being read by the Dynare pre processor which then calls the relevant Matlab routines to carry out the desired operations and display the results In slightly less flowery words it is a pre processor and a collection of Mat lab routines that has the great advantages of reading DSGE model e
105. system rather than the more indirect discrepancy between the implied DSGE and VAR im pulse response functions Of course if your model is entirely mis specified estimating it using Bayesian techniques could be a disadvantage Second Bayesian techniques allow the consideration of priors which work as weights in the estimation process so that the posterior distribution avoids peaking at strange points where the likelihood peaks Indeed due to the stylized and often misspecified nature of DSGE models the likelihood often peaks in regions of the parameter space that are contradictory with common observations leading to the dilemma of absurd parameter estimates Third the inclusion of priors also helps identifying parameters Unfortu nately when estimating a model the problem of identification often arises It can be summarized by different values of structural parameters leading to the same joint distribution for observables More technically the problem arises when the posterior distribution is flat over a subspace of parameter values But the weighting of the likelihood with prior densities often leads to adding just enough curvature in the posterior distribution to facilitate numerical max imization Fourth Bayesian estimation explicitly addresses model misspecification by including shocks which can be interpreted as observation errors in the struc tural equations Sixth Bayesian estimation naturally leads to the comparison o
106. t that the rank condition has been satisfied if all goes well Finally you will see some intermediate output the errors at each iteration of the Newton solver used to estimate the solution to your model TIP You should see these errors decrease upon each iteration if not your model will probably not converge If so you may want to try to increase the periods for the transition to the new steady state the number of simulations 3 10 FILE EXECUTION AND RESULTS 35 periods But more often it may be a good idea to revise your equations Of course although Dynare does not display a rich set of statistics and graphs corresponding to the simulated output it does not mean that you cannot cre ate these by hand from Matlab To do so you should start by looking at section 4 1 3 of chapter 4 on finding saving and viewing your output Chapter 4 Solving DSGE models advanced topics This chapter is a collection of topics not all related to each other that you will probably find interesting or at least understandable if you have read and or feel comfortable with the earlier chapter 3 on the basics of solving DSGE models To provide at least some consistency this chapter is divided into three sections The first section deals directly with features of Dynare such as dealing with correlated shocks finding and saving your output using loops referring to external files and dealing with infinite eigenvalues The second section overviews
107. t your expressions sequentially whereby each left hand side variable is written in terms of known parameters or variables already solved in the lines above For example the steady state file corresponding to the above example in the stochastic case would be example file to be added shortly 3 6 4 Checking system stability TIP A handy command that you can add after the initval or endval block following the steady command if you decide to add one is the check com mand This computes and displays the eigenvalues of your system which are used in the solution method As mentioned earlier a necessary con dition for the uniqueness of a stable equilibrium in the neighborhood of the steady state is that there are as many eigenvalues larger than one in modulus as there are forward looking variables in the system If this condition is not 3 7 ADDING SHOCKS 25 met Dynare will tell you that the Blanchard Kahn conditions are not satisfied whether or not you insert the check command 3 7 Adding shocks 3 7 1 Deterministic models temporary shocks When working with a deterministic model you have the choice of introducing both temporary and permanent shocks The distinction is that under a tem porary shock the model eventually comes back to steady state while under a permanent shock the model reaches a new steady state In both cases though the shocks are entirely expected as explained in our original discus sion on stochastic a
108. the default case returns the variance decomposition where each col umn captures the independent contribution of each shock to the variance of each variable Furthermore if you decide to run impulse response functions you will find a global variable oo_ irfs comprising of vectors named endogenous variable exogenous variable like y e reporting the values of the endoge nous variables corresponding to the impulse response functions as a result of the independent impulse of each exogenous shock To save your simulated variables you can add the following command at the end of your mod file dynasave FILENAME variable names separated by commas If no variable names are specified in the optional field Dynare will save all endogenous variables In Matlab variables saved with the dynasave command can be retrieved by using the Matlab command load mat FILENAME 4 1 4 Referring to external files You may find it convenient to refer to an external file either to compute the steady state of your model or when specifying shocks in an external file The former is described in section 3 6 of chapter 3 when discussing steady states The advantage of using Matlab say to find your model s steady state was clear with respect to Dynare version 3 as the latter resorted to numerical approximations to find steady state values But Dynare version 4 now uses the same analytical methods available in Matlab For most usage scenarios you should therefore
109. timation datafile simuldataRBC nobs 200 first obs 500 mh replic 2000 mh nblocks 2 mh drop 0 45 mh jscale 0 8 This ends our description of the mod file 5 8 The complete mod file To summarize and to get a complete perspective on our work so far here is the complete mod file for the estimation of our very basic model You can find the corresponding file in the models folder under UserGuide in your in stallation of Dynare The file is called RBC Est mod varyckilylwrz 56 CHAPTER 5 ESTIMATING DSGE MODELS BASICS varexo e parameters beta psi delta alpha rho epsilon model 1 c beta 1 c 1 1 r 1 delta psi c 1 1 w cti y y k 1 Yalpha exp z 1 1 alpha w y Cepsilon 1 epsilon 1 alpha 1 r y epsilon 1 epsilon alpha k 1 i k 1 delta k 1 yd z rho z 1 e end varobs Y steady check estimated params alpha beta pdf 0 35 0 02 beta beta pdf 0 99 0 002 delta beta pdf 0 025 0 003 psi gamma pdf 1 75 0 02 rho beta pdf 0 95 0 05 epsilon gamma pdf 10 0 003 stderr e inv gamma pdf 0 01 inf end estimation datafile simuldataRBC nobs 200 first obs 500 mh replic 2000 mh nblocks 2 mh drop 0 45 mh jscale 0 8 5 9 INTERPRETING OUTPUT 57 5 9 Interpreting output As in the case of model solution and simulation Dynare returns both tabular and graphical output On the basis of the options entered in the example mod file above Dyn
110. tion with 0 lt a lt 1 and where a represents a stochastic technological shock or Solow residual Finally we specify a shock structure that allows for shocks to display persistence across time and correlation in the current period That is aloe di T p bia Vi where p 7 lt 1 and p 7 lt 1 to ensure stationarity we call p the coeffi cient of persistence and 7 that of cross persistence Furthermore we assume E e 0 E 1 0 and that the contemporaneous variance covariance matrix of the innovations e and r is given by a VO Oy Oc c2 and where corr evs 0 corr e e 0 and corr vivs 0 for all t Z s This system probably quite similar to standard RBC models you have run into yields the following first order conditions which are straightforward to reproduce in case you have doubts and equilibrium conditions drawn from the description above Note that the first equation captures the labor supply function and the second the intertemporal consumption Euler equation coh 1 a y 1 6E e coat 1 5 14 exp bi41 C41 1 yr expla kf hy kt 1 exp bi i 1 ke at pat 1 Tbt 1 b Tati pbi 1 vi The mod file To translate the model into a language understandable by Dynare we would follow the steps outlined in chapter 3 We will assume that you re comfort able with these and simply present the final
111. tment of Economics Nason J M AND T CoGLEY 1994 Testing the Implications of Long Run Neutrality for Monetary Business Cycle Models Journal of Applied Econometrics 9 S S37 70 RABANAL P AND J F RUBIO RAMIREZ 2005 Comparing New Key nesian models of the business cycle A Bayesian approach Journal of Monetary Economics 52 6 1151 1166 SCHMITT GROHE S AND M URIBE 2004 Solving dynamic general equi librium models using a second order approximation to the policy function Journal of Economic Dynamics and Control 28 4 755 775 SCHORFHEIDE F 2000 Loss function based evaluation of DSGE models Journal of Applied Econometrics 15 6 645 670 SMETS F anD R WOUTERS 2003 An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area Journal of the European Economic Association 1 5 1123 1175 ZELLNER A 1971 An Introduction to Bayesian Inference in Econometrics John Wiley amp Sons Inc New York
112. to the prior distribution black vs grey lines In addition on the posterior distribution Dynare plots a green line which represents the posterior mode These allow you to make statements about your data other than simply concerning the mean and variance of the parameters you can also 5 9 INTERPRETING OUTPUT 59 discuss the probability that your parameter is larger or smaller than a certain value TIP These graphs are of course especially relevant and present key results but they can also serve as tools to detect problems or build additional confi dence in your results First the prior and the posterior distributions should not be excessively different Second the posterior distributions should be close to normal or at least not display a shape that is clearly non normal Third the green mode calculated from the numerical optimization of the posterior kernel should not be too far away from the mode of the posterior distribution If not it is advisable to undertake a greater number of Metropolis Hastings simulations The last figure returns the smoothed estimated shocks in a useful illustra tion to eye ball the plausibility of the size and frequency of the shocks The horizontal axis in this case represents the number of periods in the sample One thing to check is the fact that shocks should be centered around zero That is indeed the case for our example Chapter 6 Estimating DSGE models advanced topics This chapter
113. to your candidate parameter If the acceptance ratio is greater than one then definitely keep your candidate Otherwise go back to the candidate of last period this is true in very coarse terms notice that in fact you would keep your candidate only with a probability less than one Then do two things Update the mean of your drawing distribution and note the value of the parameter your retain After having repeated these steps often enough in the final step build a histogram of those retained val ues Of course the point is for each bucket of the histogram to shrink to zero This smoothed histogram will eventually be the posterior distribution after sufficient iterations of the above steps But why have such a complicated acceptance rule The point is to be able to visit the entire domain of the posterior distribution We should not be too quick to simply throw out the candidate giving a lower value of the posterior kernel just in case using that candidate for the mean of the drawing distri bution allows us to to leave a local maximum and travel towards the global maximum Metaphorically the idea is to allow the search to turn away from taking a small step up and instead take a few small steps down in the hope of being able to take a big step up in the near future Of course an important parameter in this searching procedure is the variance of the jumping distri bution and in particular the scale factor If the scale factor is
114. val around the parameter mean m2 being a measure of the variance and m3 based on third moments In each case Dynare reports both the within and the between chains measures The figure entitled multivariate diagnos tic presents results of the same nature except that they reflect an aggregate measure based on the eigenvalues of the variance covariance matrix of each parameter In our example above you can tell that indeed we obtain convergence and relative stability in all measures of the parameter moments Note that the horizontal axis represents the number of Metropolis Hastings iterations that have been undertaken and the vertical axis the measure of the parame ter moments with the first corresponding to the measure at the initial value of the Metropolis Hastings iterations TIP If the plotted moments are highly unstable or do not converge you may have a problem of poor priors It is advisable to redo the estimation with different priors If you have trouble coming up with a new prior try starting with a uniform and relatively wide prior and see where the data leads the posterior distribution Another approach is to undertake a greater number of Metropolis Hastings simulations The first to last figure figure 6 in our example displays the most inter esting set of results towards which most of the computations undertaken by Dynare are directed the posterior distribution In fact the figure compares the posterior
115. ve when taking a Taylor expansion of all the equations in the system Thus when dealing with non stationary observations you must log linearize your model and not just linearize it this is a point to which we will return later 6 1 7 Declaring observable variables We begin by declaring which of our model s variables are observables In our mod file we write varobs P obs Y obs to specify that our observable variables are indeed P obs and Y obs as noted in the section above NOTE Recall from earlier that the number of observed variables must be smaller or equal to the number of shocks such that the model be estimated If this is not the case you should add measurement shocks to your model where you deem most appropriate 6 1 8 Declaring trends in observable variables Recall that we decided to work with the non stationary observable variables in levels Both output and prices exhibit stochastic trends This can be seen explicitly by taking the difference of logs of output and prices to compute growth rates In the case of output we make use of the usual by now 70 CHAPTER 6 ESTIMATING DSGE MODELS ADVANCED TOPICS relationship Y Yos Taking logs of both sides and subtracting the same equation scrolled back one period we find Ahy Aln Y 4 4 ea emphasizing clearly the drift term y whereas we know Aln VA is stationary in steady state In the case of prices we apply the same manipulations to show that Aln
116. which would defeat the purpose of running several blocks of Metropolis Hastings chains mode file FILENAME name of the file containing previous value for the mode When computing the mode Dynare stores the mode xparam1 and the hessian hh in a file called MODEL NAME mode This is a particularly helpful option to speed up the estimation process if you have already undertaken initial estimations and have values of the posterior mode mode compute INTEGER specifies the optimizer for the mode com putation 0 the mode isn t computed mode file must be specified 1 uses Matlab fmincon see the Reference Manual to set options for this command 2 uses Lester Ingber s Adaptive Simulated Annealing 3 uses Matlab fminunc 4 default uses Chris Sim s csminwel mode check when mode check is set Dynare plots the minus of the posterior density for values around the computed mode for each esti mated parameter in turn This is helpful to diagnose problems with the optimizer A clear indication of a problem would be that the mode is not at the trough bottom of the minus of the posterior distribution 54 CHAPTER 5 ESTIMATING DSGE MODELS BASICS 15 load mh file when load mh file is declared Dynare adds to previous Metropolis Hastings simulations instead of starting from scratch Again this is a useful option to speed up the process of estimation 16 nodiagnostic doesn t compute the convergence diagnostics for Metropolis
117. with monopolistic com petition used widely in the literature Its particular notation adopted below is drawn mostly from notes available on Jesus Fernandez Villaverde s very instructive website this is a good place to look for additional information on any of the following model set up and discussion Note throughout this model description that the use of expectation signs is really only relevant in a stochastic setting as per the earlier discussion We will none the less illustrate both the stochastic and the deterministic settings on the basis of this example Thus when thinking of the latter you ll have to use a bit of imagination on top of that needed to think you have perfect foresight to ignore the expectation signs 12 CHAPTER 3 SOLVING DSGE MODELS BASICS Households maximize utility over consumption c and leisure 1 l where l is labor input according to the following utility function E B log c v log 1 1 t 0 and subject to the following budget constraint Ct kt 1 wih riki 1 6 k Vt gt 0 where k is capital stock w real wages r real interest rates or cost of capital and the depreciation rate The above equation can be seen as an accounting identity with total ex penditures on the left hand side and revenues including the liquidation value of the capital stock on the right hand side Alternatively with a little more imagination the equation can also be interpret
118. y but yesterday recall that it is a function of yesterday s investment and capital stock it is what we call in the jargon a predetermined variable Thus even though in the example presented above we wrote k 4 i 1 kt as in many papers we would translate this equation into Dynare as k i 1 delta k 1 20 CHAPTER 3 SOLVING DSGE MODELS BASICS e As another example consider that in some wage negociation models wages used during a period are set the period before Thus in the equation for wages you can write wage in period t when they are set but in the labor demand equation wages should appear with a one period lag A slightly more roundabout way to explain the same thing is that for stock variables you must use a stock at the end of the period concept It is investment during period that sets stock at the end of period t Be careful a lot of papers use the stock at the beginning of the period convention as we did on purpose to highlight this distinction in the setup of the example model above 3 5 5 Conventions specifying non predetermined variables e A 1 next to a variable tells Dynare to count the occurrence of that variable as a jumper or forward looking or non predetermined variable Blanchard Kahn conditions are met only if the number of non predetermined variables equals the number of eigenvalues greater than one If this con dition is not met Dynare will put up a warning
119. y of zero over a certain domain in your prior you will necessarily also find a probability of zero in your pos terior distribution Then think about the shape of your prior distribution Should it be symmetric Skewed If so on which side You may also go one step further and build a distribution for each of your parameters in your mind Ask yourself for instance what is the probability that your parameter is bigger than a certain value and repeat the exercise by incrementally de creasing that value You can then pick the standard distribution that best fits 5 6 DECLARING PRIORS 51 your perceived distribution Finally instead of describing here the shapes and properties of each standard distribution available in Dynare you are instead encouraged to visualize these distributions yourself either in a statistics book or on the Web TIP It may at times be desirable to estimate a transformation of a pa rameter appearing in the model rather than the parameter itself In such a case it is possible to declare the parameter to be estimated in the parameters statement and to define the transformation at the top of the model section as a Matlab expression by adding a pound sign at the beginning of the corresponding line For example you may find it easier to define a prior over the discount factor 8 than its inverse which often shows up in Euler equa tions Thus you would write model sig 1 bet c sig c 1 mpk end

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