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A Matlab program and user's guide for the fractionally cointegrated

1. Input coef s Matlab structure of coefficients in their usual matrix form k number of lags de r number of cointegrating vectors p number of variables in the system opt object containing the estimation options Output param vector of parameters 5 11 14 SEvec2matU m Listing 43 SEvec2matU m function coeffs SEvec2matU param k r p opt function coeffs SEvec2matU param k r p opt Written by Michal Popiel and Morten Nielsen This version 10 22 2014 Based on Lee Morin Morten Nielsen August 22 2011 DESCRIPTION This function transforms the vectorized model parameters into matrices Input param vector of parameters k number of lags r number of cointegrating vectors p number of variables in the system opt object containing the estimation options Output coef s Matlab structure of coefficients in their usual matrix form 5 11 15 TransformData m Listing 44 TransformData m function Z0 Z1 Z2 Z3 TransformData x k db opt function ZO Z1 Z2 Z3 TransformData x k db opt Written by Michal Popiel and Morten Nielsen This version 10 22 2014 Based on Lee Morin amp Morten Nielsen May 24 2013 DESCRIPTION Returns the transformed data required for regression and reduced rank regression Input x matrix of variables to be included in the system k number of lags 41
2. Lbk x b k function Lbkx Lbk x b k Written by Michal Popiel and Morten Nielsen This version 10 22 2014 Based on Lee Morin amp Morten Nielsen May 24 2013 DESCRIPTION Lbk x b k is a lag polynomial in the fractional lag operator Input x vector or matrix of data de b scalar value at which to calculate the fractional lag k number of lags Output matrix Lb 1 x Lb 2 x Lb k x where Lb 1 1 L 7b The output matrix has the same number of rows as x but k times as many columns Calls the function FracDiff x d 5 11 11 LikeGrid m Listing 40 LikeGrid m function params LikeGrid x k r opt function params LikeGrid x k r opt Written by Michal Popiel and Morten Nielsen This version 11 17 2014 DESCRIPTION This function evaluates the likelihood over a grid of values for d b or phi It can be used when parameter estimates are sensitive to starting values to give a close approximation of the global max which can then be used as the starting value in the numerical optimization in FCVARestn 38 10 TI 12 13 14 15 16 17 Input x matrix of variables to be included in the system k number of lags e r number of cointegrating vectors opt object containing the estimation options Output params row vector of d b and mu if level parameter is selected corresponding
3. 11 12 13 14 15 16 JN oO a B wow WN o 10 11 12 13 14 15 16 17 18 19 20 21 22 23 1 YA db fractional differencing parameters d and b opt object containing the estimation options Output ZO Z1 Z2 and Z3 of transformed data Calls the function FracDiff x d and Lbk x b k 5 11 16 CharPolyRoots m Listing 45 CharPolyRoots m function cPolyRoots CharPolyRoots coeffs opt k r p function cPolyRoots CharPolyRoots coeffs opt k r p Written by Michal Popiel and Morten Nielsen This version 10 22 2014 Based on Lee Morin amp Morten Nielsen May 31 2013 DESCRIPTION CharPolyRoots calculates the roots of the characteristic polynomial and plots them with the unit circle yA transformed for the fractional model see Johansen 2008 input coef s Matlab structure of coefficients opt object containing the estimation options k number of lags r number of cointegrating vectors p number of variables in the system output complex vector cPolyRoots with the roots of the characteristic polynomial No dependencies Note The roots are calculated from the companion form of the VAR where the roots are given as the inverse eigenvalues of the coefficient matrix 5 11 17 GetBounds m Listing 46 GetBounds m function UB LB GetBounds opt function UB LB GetBou
4. gt gt EstOptions EstOptions with properties UncFminOptions 1x1 struct ConFminOptions 1x1 struct dbMax 2 dbMin 0 0100 dbO 1 1 constrained 1 restrictDB 1 N O unrConstant 0 rConstant 0 levelParam 1 C_db c_db UB_db LB_db 29 JI Oo no pu WN o 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 R_psi r_psi 0 R_Alpha r_Alpha R_Beta r_Beta print2screen 1 printGammas 1 printRoots 1 plotRoots 1 gridSearch 1 plotLike 1 progress 1 updateTime 5 progLoc usr bin fdpval CalcSE 1 5 2 FCVARestn m Listing 18 FCVARestn m function results FCVARestn x k r opt function results FCVARestn x k r opt Written by Michal Popiel and Morten Nielsen This version 11 18 2014 DESCRIPTION This function performs estimation of the FCVAR system It is the main function in the program with several nested functions each described below It estimates the model parameters calculates the standard errors and the number of free parameters obtains the residuals and the roots of the characteristic polynomial and prints the output Input x matrix of variables to be included in the system k number of lags r number of cointegrating vectors opt object containing the estimation options Output results a Matlab structure containing estimation r
5. 0 512 SE 1 C 0 157 0 026 0 507 Var 2 0 013 1 115 3 001 SE 2 C 0 345 0 189 1 909 Var 3 0 053 0 008 0 694 SE 3 C 0 022 C 0 005 0 164 Note Standard errors in parenthesis Lag matrix 2 Gamma_2 Variable Var 1 Var 2 Var 3 Var 1 0 570 0 104 0 586 SE 1 C 0 184 C 0 044 0 606 Var 2 0 685 0 371 0 223 SE 2 C 0 508 0 159 2 510 Var 3 0 043 0 020 0 330 SE 3 C 0 032 0 008 0 138 21 Roots of the characteristic polynomial Number Real part Imaginary part Modulus 1 2 710 0 000 2 710 2 1 498 0 000 1 498 3 1 129 0 939 1 469 4 1 129 0 939 1 469 5 1 098 0 000 1 098 6 1 000 0 000 1 000 7 1 000 0 000 1 000 8 0 934 0 281 0 976 9 0 934 0 281 0 976 Multivar 97 665 0 752 Vari l 9 084 0 696 11 267 0 506 Var2 14 931 0 245 9 338 0 674 Var3 10 729 0 552 12 241 0 426 Sometimes it is the case that the model output is not normalized with respect to the user s variable of interest For this reason we also include a code section that normalizes the output i e imposes an identity matrix in the first r x r block of 8 Of course this code section should also be executed if it does not interfere with any restrictions imposed on the model Listing 12 Normalizing output RESTRICTED MODEL OUTPUT print normalized beta and alpha for model mir4 modelRstrct mir4 G inv modelRstrct coeffs betaHat 1 r 1 5 betaHatR modelRstrct coeffs
6. 15 16 N o 0 JD A A w 10 11 12 13 14 15 16 17 JN OQ 0 B wo WN o 10 11 12 Written by Michal Popiel and Morten Nielsen This version 10 22 2014 DESCRIPTION This function calls the program FDPVAL in the terminal and returns the P value based on the user s inputs The function s arguments must be converted to strings in order to interact with the terminal Input q number of variables minus rank de b parameter consT boolean variable indicating whether or not there is constant present testStat value of the test statistic opt object containing estimation options Output pv P value for likelihood ratio test 5 5 HypoTest m Listing 22 HypoTest m function results HypoTest modelUNR modelR function results HypoTest modelUNR modelR Written by Michal Popiel and Morten Nielsen This version 2 24 2015 DESCRIPTION This function performs a likelihood ratio test of the null hypothesis model is modelR against the alternative hypothesis model is modelUNR Input modelUNR structure of estimation results created for unrestricted model modelR structure of estimation results created for restricted model Output results a Matlab structure containing test results de results loglikUNR loglikelihood of unrestricted model results loglikR loglikelihood of restricted model h results df degree
7. 037 0 004 0 009 Var 1 0 345 SE 1 0 069 Var 2 11 481 SE 2 0 548 Var 3 2 872 SE 3 0 033 14 Note Standard errors in parenthesis from numerical Hessian but asymptotic distribution is unknown Variable Var 1 Var 2 Var 3 Var 1 0 276 0 032 0 510 SE 1 C 0 160 0 026 0 513 Var 2 0 148 1 126 3 285 SE 2 0 378 C 0 196 1 975 Var 3 0 052 0 008 0 711 SE 3 C 0 022 0 005 0 170 Note Standard errors in parenthesis Lag matrix 2 Gamma_2 Variable Var 1 Var 2 Var 3 Var 1 0 566 0 106 0 609 SE 1 C 0 182 C 0 045 C 0 612 Var 2 0 493 0 462 0 450 SE 2 0 562 C 0 198 2 627 Var 3 0 039 0 020 0 318 SE 3 C 0 032 C 0 008 0 143 Note Standard errors in parenthesis Roots of the characteristic polynomial Number Real part Imaginary part Modulus 1 2 893 0 000 2 893 2 1 522 0 000 1 522 3 1 010 0 927 1 371 4 1 010 0 927 1 371 5 1 108 0 000 1 108 6 1 000 0 000 1 000 7 1 000 0 000 1 000 8 0 944 0 261 0 980 9 0 944 0 261 0 980 In addition to the coefficient estimates we are also interested in testing the model residuals for serial correlation The results of the white noise tests called in the last line of Listing 6 are shown below For each residual both the Q and LM test statistics and their P values are reported in addition to the multivariate Q test and P value in the first line of the table From the output of this table we can conclude that there does not appe
8. allowed so that there is no need to specify ra As before the restricted model is estimated with results stored in mir2 the residuals are tested for white noise and the model under the null is tested against the unrestricted model mi with results stored in Hbetal Again since the estimation output is similar to the first example we only show the results of the hypothesis test here With a P value close to zero this hypothesis is also strongly rejected Unrestricted log likelihood 451 174 Restricted log likelihood 444 395 Test results df 1 LR statistic 13 557 P value 0 000 Next we move to tests on a 18 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 102 103 104 105 107 108 109 110 Listing 9 Hypothesis 4 h Test restriction that political variable is long run exogenous opti DefaultOpt opti R_Alpha 1 O 0 mir3 FCVARestn x1 k r opti This restricted model is now in the structure mir3 mv_wntest mir3 Residuals order printWNtest Halphai HypoTest m1 mir3 Test the null of mir3 against the alternative ml and store the results in the structure Halphal Again we first reset opt1 to the default options to clear previously imposed restrictions Note that if it were the case that we failed to reject Hy and wanted to leave it imposed while adding a restriction on a we could either omit the first line opti DefaultO
9. be printed to screen by typing xf in the command window For this example the forecast yields the following output xf 0 143650853872867 5 857045698472417 2 636093306380834 0 084879697875892 5 959876267800802 2 654427560476592 0 025316897413510 6 110371912213171 2 673504138913151 0 023638093475328 6 291703273410405 2 692353972906476 0 065949410839033 6 495094604485914 2 710793272160685 0 101481596234305 6 712273626012697 2 728549953757014 0 131039943766450 6 937034957454911 2 745463213504070 0 155109228667221 7 164421426216199 2 761407052913604 0 174199823866847 7 390566368295421 2 776295932082252 0 188774626007768 7 612442288726599 2 790073824966662 0 199276573388833 7 827701550158650 2 802710023527008 0 206126015229692 8 034546986055656 2 814194648028864 24 161 163 164 165 167 168 169 170 172 173 174 175 Figure 3 Forecast of final model 12 steps ahead m Series including forecast 20 15 10 f 0 50 100 150 200 250 300 350 t Equilibrium relation including forecasts 2 5 1 5 0 5 0 50 100 150 200 250 300 350 4 2 Bootstrap hypothesis test Listing 14 demonstrates the use of the wild bootstrap for hypothesis tests on the parameters as developed by Boswijk et al 2013 for the CVAR model The user specifies two sets of options corresponding to two different nested models and the function FCVARboot m returns the results of the wild bootstrap The wild bootstrap is programmed to
10. betaHat G alphaHat is post multiplied by G 1 so that pi a G 1 Gb ab alphaHatR modelRstrct coeffs alphaHat inv G display betaHatR display alphaHatR The user assigns the model of interest to the variable modelRstrct in this case the restricted model is mir4 and executes the cell The output is shown below Since in this case 8 was already equal to 1 the output is the same as the estimation output but with more significant digits betaHatR 1 000000000000000 0 105719248546476 0 182450552211581 alphaHatR 0 187668779488160 0 0 038564004272104 22 As an example of when this feature can be useful consider model 3 A part of the output is shown below where we notice that the cointegrating vector has not been normalized Variable CI equation 1 Vari 1 023 Var2 0 074 Var3 0 044 Variable CI equation 1 Var 1 0 188 SE 1 0 063 Var 2 0 101 SE 2 C 0 186 Var 3 0 000 SE 3 C 0 000 Executing the Restricted Model Output code section for this model mir5 yields the following output betaHatR 1 0000 0 0719 0 0430 alphaHatR 0 1924 0 1028 0 4 Additional examples MoreExamples m To show some additional functionality of the FCVAR software package this section contains several other examples which are based on Jones et al 2014 but are not part of that paper 4 1 Forecasting Listing 13 performs recursive one step ahead forecasts for each of the var
11. correlation tests printResults set 1 to print results to screen Output Q 1xp vector of Q statistics for individual series pvQ 1xp vector of P values for Q test on individual series LM 1xp vector of LM statistics for individual series pvLM 1xp vector of P values for LM test on individual series mvQ multivariate Q statistic pvMVQ P value for multivariate Q statistic using p 2 maxlag df 5 7 1 LMtest Listing 25 LMtest m function LMstat pv LMtest x q Breusch Godfrey Lagrange Multiplier test for serial correlation 5 7 2 Qtest Listing 26 Qtest m function Qstat pv Qtest x maxlag Multivariate Ljung Box Q test for serial correlation see Luetkepohl 2005 New Introduction to Multiple Time Series Analysis p 169 5 8 FCVARboot m The wild bootstrap procedure for hypothesis tests on the parameters is based on the procedure for the I 1 model in Boswijk et al 2013 Listing 27 FCVARboot m function LRbs H mBS mUNR FCVARboot x k r optRES optUNR B function LRbs H mBS mUNR FCVARboot x k r optRES optUNR B Written by Michal Popiel and Morten Nielsen This version 08 06 2015 DESCRIPTION This function generates a distribution of a likelihood ratio test statistic using a Wild bootstrap following the method of Boswijk Cavaliere Rahbek and Taylor 2013 It takes two sets of options as inputs to
12. d b 1 Some of the parameters are well known from the CVAR model and these have the usual interpretations also in the FCVAR model The most important of these are the long run parameters a and 8 which are p x r matrices with 0 lt r lt p The rank r is termed the cointegration or cofractional rank The columns of 8 constitute the r cointegration cofractional vectors such that 6 X are the cointegrating combinations of the variables in the system i e the long run equilibrium relations The parameters in a are the adjustment or loading coefficients which represent the speed of adjustment towards equilibrium for each of the variables The short run dynamics of the variables are governed by the parameters T Ty Tg in the autoregressive augmentation The FCVAR model has two additional parameters compared with the CVAR model namely the fractional parameters d and b Here d denotes the fractional integration order of the observable time series and b determines the degree of fractional cointegration i e the reduction in fractional integration order of P X compared to X itself These parameters are estimated jointly with the remaining parameters This model thus has the same main structure as in the standard CVAR model in that it allows for modeling of both cointegration and adjustment towards equilibrium but is more general since it accommodates fractional integration and cointegration In the next four subsections we briefly describ
13. is now in the structure de mirbs mv_wntest mir5 Residuals order printWNtest Halpha3 HypoTest m1 mir5 Test the null of mir5 against the alternative ml and store the results in the structure Halpha3 Output Unrestricted log likelihood 451 174 Restricted log likelihood 446 184 Test results df 1 LR statistic 9 979 P value 0 002 The only hypothesis that we fail to reject is 4 under which interest rates are long run exogenous Since this is the final restricted model we provide the full estimation output Note from the output that a2 0 as imposed by the restriction Dimension of system 3 Number of observations in sample 316 Number of lags 2 Number of observations for estimation 316 Restricted constant No Initial values 0 Unestricted constant No Level parameter Yes Cointegrating rank 1 AIC 849 715 Log likelihood 450 857 BIC 752 065 log det Omega_hat 11 367 Free parameters 26 Variable CI equation 1 Var1 1 000 Var2 0 106 Var3 0 182 Variable CI equation 1 Var 1 0 188 SE 1 C 0 065 Var 2 0 000 SE 2 C 0 000 Var 3 0 039 20 Variable Var 1 Var 2 Var 3 Var 1 0 188 0 020 0 034 Var 2 0 000 0 000 0 000 Var 3 0 039 0 004 0 007 Var 1 0 310 SE 1 0 067 Var 2 11 538 SE 2 0 553 Var 3 2 873 SE 3 0 033 Note Standard errors in parenthesis from numerical Hessian but asymptotic distribution is unknown Variable Var 1 Var 2 Var 3 Var 1 0 269 0 032
14. oe poss ee ee a a Re ae aa a E eee A del SUMGIAOM s ci be ee a eG Be ke Bah eee Ob Re be ee we goei Software description 51 Bst ptionsam ss ee eee ae ee SR Bee eee ba ee e Bo Pe VAR CSI Arico bee we os Ry hae he SA a wm ee De ps ic taste aaa ee we bh abe dd BAAD OG ee ew ee RE ee Da III DAI NA So Hypo TSE e a al Seah ee he ee Gd Shee Re eee a WOR Qe ee 50 EOVA RISE i i055 ke e be Bit Soe He he we a a ee ee dd Sat AVES wae ek Ra ee RARE BOS RE Ree Oe ee ee eR Re A nc hin ewe aoe te ae ee be ae Ee eee x en eae en we a de amp Ad g2 Otet cocida a AE be eee ee Oe ee a eb we ee be a DS ER NOMEN aa deat et Ae Rh ae Se Ba ae Be a mo FOVAR DOS Ramen a 400 02 2b ho bb PBA MAD REO eee a ea G Ee ES SO PR VAR SIE oie a oe ae Bale hd ae a MADE wR SA BER da a Gil Auxillory fimetions 2 2 ie ee ee A Dae ee ee 5111 POVARSMBSID acces ee ha PRR AR ORE RRR RA AR ee Re BU PC VAR Hess li we ce od hw te Bo Sod Ra BO oe Se GS SiL PC VAR GS Mioa a ee Ge Goce HA A Bo BR a ws BA ee Bs SO ae HS A HA Be POVAR ee Ma uo Bw Roe A os Be Se Rega Bee el ae ae nda de Go DVS ICAC 4 pack a Be dd de ee A a dR Re we ee ee hh de ee e HAIG Preararamsim III Slit Ball C VAT Wee xs cee Ge ae he A Rk i a ee de eh e a SALS GOtraramsBl lt ak we RRR REMADE a RAR AA SE RRA KLE wD 5 119 GotResidualsi 0 ac ee A a Re a a ee Ge ea ew ee as RN eS ha ee ao les wh ee ee ed al a Wa ROR EG ae Ak ee e SALITRE sj aaa A eee ware AS ee we eS 38 BLIP RstrotOpta Switch
15. the Matlab program can run standalone one of the functions RankTests m makes an external system call to a separately installed program fdpval This external program is the C implementation of a Fortran program used to obtain simulated P values from MacKinnon and Nielsen 2014 Tf the user would like P values for the cointegration rank tests to be automatically calculated we recommend obtaining this companion program which is made available by Jason Rhinelander and can be downloaded from https github com jagerman fracdist releases It can be either installed or downloaded in a compressed folder It is important to note where the program is stored or installed because the Matlab program requires the program location as an input in the estimation options For example if the program is stored in the folder usr bin on a Linux system the location variable is defined as follows progLoc usr bin fdpval For details see Sections 5 1 and 5 4 1 1 3 Citation If you use this program or any program or computer codes based on it we ask that you please cite this document For example you could add The results were obtained using the computer program by Nielsen and Popiel 2015 in the main text or in a footnote of your paper and then add the following citation to your list of references Nielsen M and M K Popiel 2015 A Matlab program and user s guide for the fractionally cointe grated VAR model QED working pap
16. the number of variables in the system Note that in this example the restriction d b has been imposed The log likelihood for each lag is shown in column LogL The likelihood ratio test statistic LR is for the null hypothesis 0 with P value reported in column pv This is followed by AIC and BIC information criteria The next set of columns provides P values for white noise tests on the residuals The first P value pmvQ is for the multivariate Q test followed by univarite Q tests as well as LM tests on the p individual residuals that is pQ1 and pLM1 are the P values for the residuals in the first equation pQ2 and pLM2 are for the residuals in the second equation and so on 3 4 Cointegration rank testing The user now chooses the lag order based on the information provided above and can move to the next step which is cointegration rank testing The next code section is shown in listing 5 The user first assigns the lag augmentation k 2 in this case and then calls the function RankTests m Listing 5 Cointegration rank testing kh 5 COINTEGRATION RANK TESTING k 2 rankTestStats RankTests x1 k opt 12 Executing the code in Listing 5 produces the following output Dimension of system 3 Number of observations in sample 316 Number of lags 2 Number of observations for estimation 316 Restricted constant No Initial values 0 Unrestricted constant No Level parameter Yes Rank d b Log like
17. to the global max over the grid of d b or phi This function allows the user to pre estimate to obtain starting values by using a grid search There are four types of estimation that the grid search can perform If d and b are completely unconstrained the grid search is over two dimensions An example of the likelihood obtained in an unconstrained grid search is shown in Figure 6 a Next if d gt b is imposed the computation can be cut in half An example of this likelihood is shown in Figure 6 b If the restriction d b is imposed then the grid search is one dimensional as shown in Figure 6 c Finally if a restriction is imposed on either d or b via Ry and ry in 10 then the grid search is one dimensional An example of this situation is shown in Figrue 6 d Note that the x axis is over the parameter which is unrestricted In this case the fractional parameters are found from Hl Hap 16 where H Ri and h Ri RyRy ry 39 SOD0q 2p2QOsSNv o 10 12 13 14 15 16 17 Figure 6 Grid search a Unconstrained b d gt b Rank 1 Lag 0 Rank 1 Lag 0 log likelihood log likelihood c d b d Restrictions imposed Rank 1 Lag 0 T T T Rank 1 Lag 0 T r T 200 400 600 log likelihood log likelihood 800 1000 1200 1 1 1 1 1 1 1 1 1 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 d b 5 11 12 RstrctOptm_Switch m Listing 41
18. 1 df 20 15 10 5 0 5 10 15 20 25 LR statistic 22 268 P value BS 0 033 P value 0 043 4 4 Simulation Finally Listing 16 shows how to simulate an FCVAR model for a given set of parameters The user provides data for starting values and a Matlab structure containing model parameters for simulation as well as the number of periods to simulate The simulated data is generated using Gaussian errors Listing 16 Simulation hh S2 5 22525 SIMULATION Simulate the final restricted model the same one used for forecasting 0 above Number of periods to simulate T_sim 100 Simulate data xSim FCVARsim x1 modelF T_sim Plot the results figure plot xSim 27 219 Legend Support Unemployment Interest rate For the example above the generated data is shown in Figure 5 Figure 5 Simulated data Support Unemployment Interest rate 5 Software description This section describes the individual components of the software package in detail The main folder contains the following files e data_JNP2014 csv e EstOptions m e FCVARestn m e FCVARforecast m e FCVARboot m e FCVARbootRank m e FCVARsim m e HypoTest m e LagSelect m 28 SO o 10 11 e mv_wntest m e RankTests m e replication_JNP2014 m e MoreExamples m There is one data file data_JNP2014 csv two scripts replication_JNP2014 m and MoreExamples m on
19. 1 was selected in the choice of options in Listing 3 the restriction that d b is already imposed Thus the user needs to only impose an additional restriction that either d or b is equal to one In this example the restriction that d 1 is imposed by setting 17 75 76 77 78 79 80 81 82 83 84 85 opt1 R_psi 1 0 and opt1 r_psi 1 but the result would be the same if b 1 were imposed instead The restricted model is then estimated and the results are stored in the Matlab structure mir1 As before the user can perform a series of white noise tests on the residuals by calling the mv_wntest m function The next step is to perform the actual test With the results structures from the restricted and unrestricted models the user can call the function HypoTest m and perform an LR test This function takes the two model result structures as inputs automatically compares the number of free parameters to obtain the degrees of freedom computes the LR test statistic and displays the output The results of this test are then stored in the Matlab structure Hdb and can be accessed at any time Since the output of the estimated model and the white noise tests are similar to the previous example we only show the output from the hypothesis test Unrestricted log likelihood 451 174 Restricted log likelihood 442 027 Test results df 1 LR statistic 18 295 P value 0 000 The log likelihoods from both models are
20. ED Queen s Economics Department Working Paper No 1330 A Matlab program and user s guide for the fractionally cointegrated VAR model Morten AYrregaard Nielsen MichaAC Ksawery Popiel Queen s University and CREATES Queen s University Department of Economics Queen s University 94 University Avenue Kingston Ontario Canada K7L 3N6 9 2015 A Matlab program and user s guide for the fractionally cointegrated VAR model version 1 3 0 Morten rregaard Nielsen Michat Ksawery Popiel Queen s University and CREATES Queen s University Email monfecon queensu ca Email popielm econ queensu ca September 9 2015 Abstract This manual describes the usage of the accompanying freely available Matlab program for estimation and testing in the fractionally cointegrated vector autoregressive FCVAR model This program replaces an earlier Matlab program by Nielsen and Morin 2014 and although the present Matlab program is not compatible with the earlier one we encourage use of the new program JEL codes C22 C32 Keywords cofractional process cointegration rank computer program fractional autoregressive model fractional cointegration fractional unit root Matlab VAR model We are grateful to Federico Carlini Andreas Noack Jensen S ren Johansen Maggie Jones James MacKinnon Jason Rhinelander and Daniela Osterrieder for comments and to the Canada Research Chairs program the Social Sciences and Humanities R
21. RstrctOptm_Switch m function betaStar alphaHat OmegaHat RstrctOptm_Switch beta0 S00 S01 T p opt function betaStar alphaHat OmegaHat RstrctO0ptm_Switch beta0 S00 S501 S11 T p opt Written by Michal Popiel and Morten Nielsen This version 11 12 2014 DESCRIPTION This function is imposes the switching algorithm of Boswijk and Doornik 2004 page 455 to optimize over free parameters psi and phi directly We translate between psi phi and alpha beta using the relation of R_Alpha vec alpha 0 and A psi vec alpha and R_Beta vec beta r_beta and H phit h vec beta Note the transposes Input betaO unrestricted estimate of beta S00 S01 S11 product moments T number of observations p number of variables opt object containing the estimation options 40 S11 18 19 20 21 TI 12 13 14 15 JN oO nm B wo WN o 10 11 12 13 14 15 Output betaStar estimate of betaStar alphaHat estimate of alpha OmegaHat estimate of Omega 5 11 13 SEmat2vecU m Listing 42 SEmat2vecU m function param SEmat2vecU coeffs k r p opt function param SEmat2vecU coeffs k r p opt Written by Michal Popiel and Morten Nielsen This version 10 22 2014 Based on Lee Morin Morten Nielsen August 22 2011 DESCRIPTION This function transforms the model parameters in matrix form into a vector
22. Wild bootstrap following the method of Cavaliere Rahbek and Taylor 2010 It takes the two ranks as inputs to estimate the model under the null and the model under the alternative Input data if k gt 0 actual data is used for initial values x k number of lags opt estimation options rank under the null rank under the alternative B number of bootstrap samples LRbs B x 1 vector simulated likelihood ratio statistics pv approximate p value for LRstat based on bootstrap distribution Matlab structure containing LR test results it is identical to the output from HypoTest with one addition namely H pvBS which is the Bootstrap P value Output H a mBS mUNR model estimates under the null model estimates under the alternative 5 10 FCVARsim m Listing 29 FCVARsim m f unction xSim function xSim Written by Mi DESCRIPTION FCVARsim data model NumPeriods FCVARsim data model NumPeriods chal Popiel and Morten Nielsen This version 08 06 2015 This function simulates the FCVAR model as specified by input model and starting values specified by data Errors are drawn from a Normal distribution 34 10 11 12 13 Ti 12 13 14 15 16 UT Input data T x p matrix of data model a Matlab structure containing estimation results h NumPeriods number of steps for simulation Output xSim N
23. ar to be any problems with serial correlation in the residuals White Noise Test Results lag 12 Variable Q P val LM P val Multivar 97 868 0 747 Vari l 9 301 0 677 11 238 0 509 Var2 14 443 0 273 8 566 0 739 Var3 10 596 0 564 12 269 0 424 Because opt plotRoots 1 in Listing 3 the roots of the characteristic polynomial is also plotted along with the unit circle and the transformed unit circle Cj see Johansen 2008 The plot is shown in Figure 1 Note that the axes of the plot are fixed and therefore very large roots may not be shown in the plot in this example the real root 2 893 This should not be a problem since such roots will always be well outside the transformed unit circle Figure 1 Roots of characteristic polynomial gt Roots of the characteristic polynomial with the image of the unit circle Furthermore the estimation was performed with the grid search and the plot option selected i e with opt gridSearch 1 and opt plotLike 1 which produces a plot of the log likelihood The plot for this model is shown in Figure 2 16 Figure 2 Plot of log likelihood Rank 1 Lag 2 460 440 420 log likelihood N 3 300 280 260 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 2 The complete results for the unrestricted model are stored in the Matlab structure m1 and can be accessed anytime For instance if the user would like to perform a more careful analysis of the residuals they a
24. ation of that program After all options have been set line 43 stores them in DefaultOpt so that the user can recall them at any point in the estimation This is particularly useful if the user wants to change only a few options in between estimations 3 3 Lag order selection Once the options are set the user moves to the next step which involves choosing the appropriate lag order The relevant information is obtained with a call to LagSelect m shown in Listing 4 which performs estimation of models with lag orders from 0 to kmax The program performs lag selection on the full rank unrestricted model Listing 4 Lag selection hh RRE LAG SELECTION LagSelect x1 kmax p order opt The output generated by this function is shown below Dimension of system 3 Number of observations in sample 316 Order for WN tests 12 Number of observations for estimation 316 Restricted constant No Initial values 0 Unestricted constant No Level parameter Yes k r d b LogL LR pv AIC BIC pmvQ pQ1 pLMi pQ2 pLM2 pQ3 pLM3 3 3 0 256 0 256 461 22 16 91 0 050 842 44 692 21 0 00 0 45 0 40 0 48 0 87 0 63 0 35 2 3 0 581 0 581 452 77 20 59 0 015 843 53 727 11 0 00 0 69 0 45 0 29 0 75 0 54 0 40 1 31 043 1 043 442 47 56 99 0 000 840 94 758 31 0 00 0 75 0 52 0 15 0 58 0 34 0 18 O 3 1 036 1 036 413 97 0 00 0 000 801 95 753 12 0 00 0 01 0 01 0 00 0 08 0 37 0 17 Estimates of d and b are reported for each lag k with rank r set to
25. cifically when b gt 0 5 Johansen and Nielsen 2012 shows that under i i d errors with suitable moment conditions the conditional maximum likelihood parameter estimates d b a Ey sacs ar k are asymptotically Gaussian while p are locally asymptotically mixed normal These results allow asymptotically standard chi squared inference on all parameters of the model including the cointegrating relations and orders of fractionality using quasi likelihood ratio tests As in the CVAR model see Johansen 1995 the same results hold for the same parameters in the full models 2 and 3 whereas the asymptotic distribution theory for the remaining parameters and p is currently unknown 2 3 Cointegration rank tests Letting II af the likelihood ratio LR test statistic of the hypothesis H rank II r against Hp rank II p is of particular interest because it deals with an important empirical question This statistic is often denoted the trace statistic Let 6 d b for model 2 and 9 d b 1 for model 3 denote the parameters for which the likelihood is numerically maximized Then let L 0 r be the profile likelihood function given rank r where a 8 T and possibly p if appropriate have been concentrated out by regression and reduced rank regression see Johansen and Nielsen 2012 and Dolatabadi et al 2014 for details The profile likelihood function is maximized both under the hypothesis H an
26. containing the estimation options Output hessian matrix of second derivatives 5 11 3 FCVARlike m Listing 32 FCVARlike m function like FCVARlike x params k r opt function like FCVARlike x params k r opt Written by Michal Popiel and Morten Nielsen This version 11 10 2014 DESCRIPTION This function adjusts the variables with the level parameter 35 oO N 0 10 12 13 14 11 12 13 14 15 16 17 JN Oa B w Ww o 10 11 12 13 if present and returns the log likelihood given d b Input x matrix of variables to be included in the system params a vector of parameters d b and mu if option selected k number of lags ba number of cointegrating vectors opt object containing the estimation options Output like concentrated log likelihood evaluated at given parameters 5 11 4 FCVARlikMu m Listing 33 FCVARlikeMu m function like FCVARlikeMu y db mu k r opt function like FCVARlikeMu y db mu k r opt Written by Michal Popiel and Morten Nielsen This version 10 22 2014 DESCRIPTION This function evaluates the likelihood for a given set of parameter values It is used by the LikeGrid function to numerically optimize over the level parameter for given values of the fractional parameters Input y matrix of variables to be included in the system db frac
27. corsa a a dd wee ew ee es 40 SUISSE mtv ro a a a a a a a Be HA aa 41 5 11 14 SEvec2mat Uii AREA 41 Sd Transi m stad da a da a a ae ee hE ee ag 41 SALIG CHAPO ROO IE a a Dp A a D a hh Gee a a 42 SILIT GetBounds Ti ia ke eb eee A Bw ee eh aa amp we ae dio ad 42 S LIS FCVARSIMBS AW s gt o a he ae PAAR we Re e ke ae hd ew RR Rae e 42 A Version change log 44 Al Version 100 October 24 2014 occiso a ee ee ee ee 44 A2 Version 1 140 October 30 DOTA we eh ee Bea aha he ee BS eG a he eS 44 A 3 Version 1 2 0 November 12 2014 da a a o a a a e ea d a 44 Ad Version 1 2 1 November 17 2014 e 45 AS Version 1 32 November 19 2014 naa a ka RR RRR arar a A AR 45 AO Version 12 3 July 3L 2005 26 ho ea HAA a a we eS de da 46 A 7 Version 1 3 0 September 9 2015 2 0 0 2 44 044 8k we eR da he eee 46 References 47 1 Obtaining and using the software 1 1 Disclaimer We have done our best to make this program as functional and free from errors as possible but no warranty is given whatsoever We cannot guarantee that we have been 100 successful in eliminating bugs so if you find any please let us know 1 2 Obtaining the Matlab program The Matlab program can be downloaded from the first author s website at Queen s University http www econ queensu ca faculty mon software It is freely available for non commercial academic use For a nearly complete change log please see Appendix A Although
28. ctions on the model parameters can be considered as in Johansen 1995 The most interesting restrictions from an economic theory point of view would likely be restrictions on the adjustment parameters and cointegration vectors 6 We formulate hypotheses as Ryw Taps 10 Ravec a 0 11 Rgvec P rg 12 with 8 6 p and use the switching algorithm in Boswijk and Doornik 2004 p 455 to optimize the likelihood numerically subject to the restrictions The only limitation on the linear restrictions that can be imposed on d b a 8 in 10 12 is that only homogenous restrictions can be imposed on vec a in 11 Otherwise any combination of linear restrictions can be imposed on these parameters For now the remaining parameters cannot be restricted Note that when the restricted constant term p is included in the model restrictions on P and p must be written in the form given by 12 This is without loss of generality The restrictions in 10 12 above can be implemented individually or simultaneously in the Matlab program The next section provides an example session illustrating the use of the program with a step by step description of a typical empirical analysis including several restricted models in Section 3 6 2 5 Forecasting from the FCVAR model Because the FCVAR model is autoregressive the best linear predictor takes a simple form and is relatively straightforward to calculate Consider for exam
29. d under H and the LR test statistic is then LRy q 2log L 0 p L 0 r where L 0p p max L 9 p L 0 r max L 9 r and q p r This problem is qualitatively different from that in Johansen 1995 since the asymptotic distribution of LRr q depends qualitatively and quantitatively on the parameter b In the case with 0 lt b lt 1 2 sometimes known as weak cointegration LRr q has a standard asymptotic distribution see Johansen and Nielsen 2012 Theorem 11 ii namely LRr q 3 x7 0 lt b lt 1 2 8 On the other hand when 1 2 lt b lt d strong cointegration asymptotic theory is nonstandard and LRr q 3 nf arenes f Fwvr yss f rowo b gt 1 2 9 where the vector process dW is the increment of ordinary non fractional vector standard Brownian motion of dimension q p r The vector process F depends on the deterministics in a similar way as in the CVAR model in Johansen 1995 although the fractional orders complicate matters The following cases have been derived in the literature 1 When no deterministic term is in the model F u W u where W u T b fo u s 1dW s is vector fractional Brownian motion of type II see Johansen and Nielsen 2012 Theorem 11 i 2 When only the restricted constant term is included in model 2 F u Wi uy u 1 D y see Johansen and Nielsen 2012 Theorem 11 iv for the result with d b and an earlier working paper version for
30. dels 1 2 and 3 are estimated by conditional maximum likelihood conditional on N initial values by maximizing the function Tp T log Lr log 27 1 5 log det a 5 veo 5 t N 1 where the residuals are defined as k el AX aA Ly 8X p XOTA LXE A d b a b T p 8 6 i 1 for model 2 and hence also for submodels of model 2 such as 1 with the appropriate restrictions imposed on p and For model 3 the residuals are k ex A AUX p ap AT Xi u STAT Li Xp d b a BT u 7 i 1 It is shown in Johansen and Nielsen 2012 and Dolatabadi et al 2014 how for fixed d b the estimation of model 2 reduces to regression and reduced rank regression as in Johansen 1995 In this way the parameters a 8 T p can be concentrated out of the likelihood function and numerical optimization is only needed to optimize the profile likelihood function over the two fractional parameters d and b In model 3 we can similarly concentrate the parameters a 8 T out of the likelihood function resulting in numerical optimization over d b u making the estimation of model 3 slightly more involved numerically than that of model 2 For model 2 with 0 Johansen and Nielsen 2012 shows that asymptotic theory is standard when b lt 0 5 and for the case b gt 0 5 asymptotic theory is non standard and involves fractional Brownian motion of type II Spe
31. e class definition EstOptions m and nine functions These main functions depend on 17 auxiliary functions stored in the subfolder Auxiliary The replication script adds these auxiliary files to the path definition so that the main functions have access to them We remark again that it should not be necessary to modify any files except the scripts i e replication_JNP2014 m or MoreExamples m Only advanced users wishing to modify or extend the actual functionality of the pro grams will need to make any changes to the remaining files The following subsections briefly describe the functionality of each program file 5 1 EstOptions m Listing 17 EstOptions m classdef EstOptions classdef EstOptions Written by Michal Popiel and Morten Nielsen This version 11 10 2014 DESCRIPTION This class defines the estimation options used in the FCVAR estimation procedure and the related programs Assigning this class to a variable stores the default properties defined below in that variable In addition to the properties the methods section includes the function updateRestrictions which performs several checks on the user specified options prior to estimation EstOptions is a class definition which is assigned to an object and used in most of the functions It contains all the model specifications and options that are available to the user Here is an example of the contents when EstOptions is entered in the command line
32. e the accommodation of deterministic terms as well as estimation and testing in the FCVAR model 2 1 Deterministic terms There are several ways to accommodate deterministic terms in the FCVAR model 1 The inclusion of the so called restricted constant was considered in Johansen and Nielsen 2012 and the so called unrestricted constant term was considered in Dolatabadi et al 2014 A general formulation that encompasses both models is k AUX aA La 8 Xi p SOTA LEX Et er 2 i 1 The parameter p is the so called restricted constant term since the constant term in the model is restricted to be of the form ap which is interpreted as the mean level of the long run equilibria when these are stationary i e EB X p 0 The parameter is the unrestricted constant term which gives rise to a deterministic trend in the levels of the variables When d 1 this trend is linear Thus the model 2 contains both a restricted constant and an unrestricted constant In the usual CVAR model i e with d b 1 the former would be absorbed in the latter but in the fractional model they can both be present and are interpreted differently For the representation theory related to 2 and in particular for additional interpretation of the two types of constant terms see Dolatabadi et al 2014 An alternative formulation of deterministic terms was suggested by Johansen and Nielsen 2015 albeit in a simpler model with the aim of reduc
33. er 1330 Queen s University 1 4 Using the Matlab program The use of this program requires a functioning installation of Matlab Any recent version should work Unzip the contents of the zip file into any directory which will be the working directory of the program The next section describes the FCVAR model and the restricted models that can be estimated with this program Section 3 describes the functioning of the main program which is a replication of one of the tables of results in Jones et al 2014 Section 4 describes another example program which demonstrates some additional functionality of the software Importantly these are the only two files that would need to be changed to apply the program for other empirical analyses Section 5 describes how each of the major program files work each in a separate subsection The Appendix contains a version change log 2 The fractionally cointegrated VAR model The fractionally cointegrated vector autoregressive FCVAR model was proposed in Johansen 2008 and analyzed by e g Johansen and Nielsen 2010 2012 For a time series X of dimension p the fractionally cointegrated VAR model is given in error correction form as k AX af At LX XOTA LX s 1 i l where e is p dimensional i i d 0 Q d gt b gt 0 A is the fractional difference operator and L 1 A is the fractional lag operator Model 1 includes the Johansen 1995 CVAR model as the special case
34. esearch Council of Canada SSHRC and the Center for Research in Econometric Analysis of Time Series CRE ATES funded by the Danish National Research Foundation for financial support Corresponding author If you find any bugs or other problems please let us know Contents Obtaining and using the software Led Disclaimer esoo bee a Pea ER a eee A eo 1 2 Obtaining the Matlab program pa oead co te RRR BeOS ee eR Ow e eS G LS Caba ae AAA ee bh r de BA MPR eA dae DOE de 1 4 Using the Matlab program sica e The fractionally cointegrated VAR model 21 Determiimetie terme e soros ede ek Be Pea Beh he we RL ee ee we a 2 2 Maximum likelihood estimation rad aeeie a E a ee Zo Combetration rank tests s cesa ee A ee We ee we Se 2A Restricted models oxide 4444 8 oo OE Pa ee ae dE ke eh ee 2 5 Forecasting from the FOVAR model lt 2 02 2a ca ee eR ee Example session replication_JNP2014 m Ol Importing data cco ce ee ee ee se ee ee A A ee ee g e 22 Choosing OPtIONs sa ra ale Ge Se Schad Ades ag AA Gwe Bee a ad 3 3 Lag Order selection aa ye ee eh a be Re ee ee S4 Contestation rank testing o oir airada DO HR ER eS A Lk he 3 5 Unrestricted madel estimation 6 4 26 442 Phe A y 3 6 Hypothesis testing occasion et eee ad Pe ed Additional examples MoreExamples m QT a A ee Bee a a A EE ok ee ee oe ay a ER A we a 4 2 Bootstrap hypothesis tet mm pa eee ee OR Re eee we a ae ee e RE 43 Boousteap TANK ABS De a
35. estimate the model under the null and the unrestricted model Input x data if k gt 0 actual data is used for initial values k number of lags 33 13 14 15 16 17 18 19 20 21 22 23 24 25 N JNO nm BP w 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 NO a B w optRES options object for restricted model under the null optUNR options object to estimate unrestricted model B number of bootstrap samples Output LRbs B x 1 vector simulated likelihood ratio statistics pv approximate p value for LRstat based on bootstrap distribution H a Matlab structure containing LR test results it is identical to the output from HypoTest with one addition namely H pvBS which is the Bootstrap P value mBS model estimates under the null mUNR model estimates under the alternative 5 9 FCVARbootRank m The wild bootstrap procedure for hypothesis tests on the parameters is based on the procedure for the I 1 model in Cavaliere et al 2010 Listing 28 FCVARbootRank m f unction LRbs H mBS mUNR FCVARbootRank x k opt ri r2 B function LRbs H mBS mUNR FCVARbootRank x k opt ri r2 B Written by Michal Popiel and Morten Nielsen This version 08 06 2015 DESCRIPTION This function generates a distribution of a likelihood ratio test statistic for the rank test using a
36. esults results startVals Starting values used for optimization results options Updated estimation options results like Model log likelihood results coef s Parameter estimates results rankJ Rank of Jacobian for identification condition results fp Number of free parameters results SE Standard errors o results NeglnvHessian Negative of inverse Hessian matrix results Residuals Model residuals results cPolyRoots Roots of characteristic polynomial This function is the central estimation function in the program Calling this function returns a results Matlab structure An example of a typical results structure is shown here mir4 startVals 0 8000 0 8000 0 0938 11 5400 2 8633 options 1x1 EstOptions like 450 8574 coeffs 1x1 struct rankJ 4 30 12 13 14 15 JIN OQ a BP w WN o 11 12 13 14 15 16 17 18 19 20 21 22 23 1 fp 26 SE 1x1 struct NegInvHessian 25x25 double Residuals 316x3 double cPolyRoots 9x1 double 5 3 LagSelect m Listing 19 LagSelect m function LagSelect x kmax r order opt function LagSelect x kmax r order opt Written by Michal Popiel and Morten Nielsen This version 7 21 2015 DESCRIPTION This program takes a matrix of variables and performs lag selection on it by using the likelihood ratio test Output and test results are p
37. grated VAR model with deterministic trends and application to commodity futures markets QED working paper 1327 Queen s University Jensen A N and M Nielsen 2014 A fast fractional difference algorithm Journal of Time Series Analysis 35 428 436 Johansen S 1995 Likelihood Based Inference in Cointegrated Vector Autoregressive Models New York Oxford University Press Johansen S 2008 A representation theory for a class of vector autoregressive models for fractional processes Econometric Theory 24 651 676 Johansen S and M Nielsen 2010 Likelihood inference for a nonstationary fractional autoregressive model Journal of Econometrics 158 51 66 Johansen S and M Nielsen 2012 Likelihood inference for a fractionally cointegrated vector autore gressive model Econometrica 80 2667 2732 Johansen S and M Nielsen 2015 The role of initial values in conditional sum of squares estimation of nonstationary fractional time series models Forthcoming in Econometric Theory Jones M M Nielsen and M K Popiel 2014 A fractionally cointegrated VAR analysis of economic voting and political support Canadian Journal of Economics 47 1078 1130 MacKinnon J G and M Nielsen 2014 Numerical distribution functions of fractional unit root and cointegration tests Journal of Applied Econometrics 29 161 171 Nielsen M and L Morin 2014 FCVARmodel m a Matlab software package for est
38. iables as well as the equilibrium relation Listing 13 Forecasting 128 4h FORECAST 23 Forecast from the final restricted model NumPeriods 12 forecast horizon set to 12 months ahead Assign the model whose coefficients will be used for forecasting modelF mirl1 xf FCVARforecast x1 modelF NumPeriods Series including forecast seriesF x1 xf Equilibrium relation including forecasts equilF seriesF modelF coeffs betaHat Determine the size of the vertical line to delimit data and forecast values T size x1 1 yMaxS max max seriesF yMinS min min seriesF yMaxEq max max equilF yMinEq min min equilF figure subplot 2 1 1 plot seriesF title Series including forecast xlabel t line T T yMinS yMaxS Color k subplot 2 1 2 plot equilF title Equilibrium relation including forecasts xlabel t line T T yMinEq yMaxEq Color k The user specifies the forecast horizon NumPeriods as well as the model in this case modelF mir1 These two inputs along with the data are used in the call to the function FCVARforecast m This function returns xf a NumPeriods by p matrix of forecasted values of X This code section also plots the original series and the equilibrium relation along with the forecasts These plots are shown in Figure 3 The forecasts can
39. ictions on d b are imposed e Transposed d b in case of full identification R_psi has two restrictions this is done to match the way that the rank test results are stored e Fixed the adjustment of UB and LB after grid search because it interfered with the d gt b constraint e Removed the fast inversion of Hessian for calculating standard errors because it wasn t precise e Changed the way that the commutation matrix enters in the translation from vec a to vec a and vice versa in the rank condition for identification 44 FCVARlike m e Adjusted the way that likelihood is calculated when linear restrictions on d b are imposed e Changed how constrained option is imposed regardless of linear restrictions GetBounds m e Added function for calculating upper and lower bounds GetParams m e Changed matrix inversion method to more precise method i e now use inv instead of and in low dimension situations LikeGrid m e Added the variable phi in the loop Now phi goes into FCVARlike and can be either a singleton or 2x1 vector db is reserved for FCVAR1ikeMU which does not make the translation from phi to db automatically It is also used in the output for the waitbar fixing an issue where the waitbar displayed d b phi in the case of a linear restriction RstrctOptm_Switch m e Added new function that replaces RstrctOptm m and rLike m which were called for estimation when either a or P or both were restricted e Th
40. ifferent data sets The set of available options can be broken into several categories numerical optimization model deter ministics and restrictions output grid search and P values for the rank test We recommend that only advanced users make changes to the numerical optimization options Adding deterministics requires setting the variable corresponding to the type of deterministic component to 1 For instance in the present example a model estimated with options opt will include the level parameter u but no restricted or unrestricted con stant Output variables refer to either printing or plotting various information post estimation and usually take values 1 or 0 on or off For example if the user is not interested in the estimates of TP they can be suppressed by setting opt printGammas 0 11 45 46 47 48 49 An important feature in this package is the ability to pre estimate by using a grid search If the user selects this option they can view progress by setting opt progress to 1 waitbar or 2 output in command line The minimum frequency of these updates is set by opt updateTime The user also has the option opt plotLike to view a plot of the likelihood over d and or b after the grid search completes In order to automatically obtain P values for cointegration rank tests when b gt 0 5 the user needs to download and install the necessary program see Section 1 The last option opt progLoc identifies the loc
41. imates of Alpha Beta Rho Pi Gamma and Omega Input x matrix of variables to be included in the system k number of lags r number of cointegrating vectors db value of d and b opt object containing the estimation options Output estimates Matlab structure containing the following estimates db taken directly from the input estimates alphaHat estimates betaHat estimates rhoHat estimates piHat estimates OmegaHat 37 21 22 bo o 0 JD A A w 10 dl 12 13 14 15 A o Mn JJ On 11 12 13 14 15 16 17 estimates GammaHat p x kp matrix GammaHat1 GammaHatk 5 11 9 GetResiduals m Listing 38 GetResiduals m function epsilon GetResiduals x k r coeffs opt function epsilon GetResiduals x k r coeffs opt Written by Michal Popiel and Morten Nielsen This version 10 22 2014 Based on Lee Morin amp Morten Nielsen August 22 2011 DESCRIPTION This function calculates the model residuals Input x matrix of variables to be included in the system k number of lags r number of cointegrating vectors coef s Matlab structure of coefficients opt object containing the estimation options Output epsilon matrix of residuals from model estimation evaluated at the parameter estimates specified in coeffs 5 11 10 Lbk m Listing 39 Lbk m function Lbkx
42. imation and testing in the fractionally cointegrated VAR model QED working paper 1273 Queen s University 47
43. impose restriction d b 1 yes 0 no opt db0 8 8 set starting values for optimization algorithm opt N 0 number of initial values to condition upon opt print2screen 1 print output opt printRoots 1 print roots of characteristic polynomial opt plotRoots 1 plot roots of characteristic polynomial opt gridSearch 1 For more accurate estimation perform the grid search This will make estimation take longer opt plotLike 1 Plot the likelihood if gridSearch 1 opt progress 1 Show grid search progress indicator waitbar opt updateTime 5 How often progress is updated seconds Linux example opt progLoc usr bin fdpval location path with program name of fracdist program if installed Note use both single outside and double quotes inside This is especially important if path name has spaces DefaultOpt opt Store the options for restoring them in between hypothesis tests The first line initializes the object opt and assigns all of the default options set in EstOptions The user can see the full set of options by typing EstOptions or opt after initialization in the command line Listing 3 shows how to easily change any of the default options Defining the program options in this way allows the user to create and store several option objects with different attributes This can be very convenient when for example performing the same hypothesis tests on d
44. ing the impact of pre sample observations of the process This model is k AUX u ap ATPL Xe u SOT AS LXi ys Er 3 i l Both the fractional difference and fractional lag operators are defined in terms of their binomial expansion in the lag operator L Note that the expansion of L has no term in L and thus only lagged disequilibrium errors appear in 1 2In Dolatabadi et al 2014 the constants are included as p 1 and 71 1 where m u denotes coefficients in the binomial expansion of 1 z This is mathematically convenient but makes no difference in terms of the practical implementation which can be derived easily from the unobserved components formulation k Xo u X AtX Leaf X Y T A LX er 4 i l The formulation 3 or equivalently 4 includes the restricted constant which may be obtained as p P p More generally the level parameter y is meant to accommodate a non zero starting point for the first observation on the process i e for X1 It has the added advantage of reducing the bias arising due to pre sample behavior of X at least in simple models even when conditioning on no initial values see below For details see Johansen and Nielsen 2015 2 2 Maximum likelihood estimation It is assumed that a sample of length T N is available on X where N denotes the number of observations used for conditioning for details see Johansen and Nielsen 2015 The mo
45. ing the notation we omit this subscript and let it be understood in the sequel where X se X for s lt t Then forecasts are calculated recursively from 15 for j 1 2 h to generate h step ahead forecasts Roadie Clearly one step ahead and h step ahead forecasts for the model 2 with a restricted constant term and possibly also an unrestricted constant term instead of the level parameter can be calculated entirely analogously oo JD A Fw NY RB o a 3 Example session replication JNP2014 m The main file is replication_JNP2014 m and it serves as an example of what a typical session of estimation testing and forecasting can include This code replicates Table 4 FCVAR results for Model 1 from Jones et al 2014 and follows the empirical procedure developed in that paper This procedure includes the following steps 1 Importing data 2 Choosing estimation options 3 Lag selection 4 Cointegration rank selection 5 Model estimation 6 Hypothesis testing It is important to note that all necessary commands for file execution and option modification can be called from this script All other files contained in the package described in detail in the next section do not require any modification by the user To accommodate the sequential nature of the procedure the main file is broken up into code sections These code sections known as cells in previous versions of Matlab allow the user to execute specific
46. is function uses a switching algorithm from Boswijk and Doornik 2004 and performs much better than the previous algorithm which used unconstrained numerical optimization A 4 Version 1 2 1 November 17 2014 LikeGrid m e Fixed a problem when the grid search finds a non unique maximum of the likelihood RankTests m e Fixed a bug where the wrong value of b was being used to calculate the P value e Changed the output from a matrix to a Matlab struct FCVARforecast m e Changed rhoHatUNR to xiHat to use the same notation throughout FCVARestn m e Adjusted output of roots of characteristic polynomial to line them up properly A 5 Version 1 2 2 November 19 2014 FCVARestn m e Fixed a problem with starting values from the grid search when d b are unrestricted and level param eters are included 45 A 6 Version 1 2 3 July 21 2015 HypoTest m e Fixed a typo in the function description mv_wntest m e Added specified lag to output Lag_select m e Added model specification to output FreeParameters m function nested in FCVARestn m e Set fDB 2instead offDB 1 opt restrictDB because regardless of option restrictDB opt R_psi is updated to incorporate that restriction and this results in a double count This bug probably did not affect previous hypothesis testing as long as tests were nested within a particular d b model Also if restrictions were imposed using R_psi directly instead of restrictDB then the program returned the co
47. lihood LR statistic P value 0 0 643 0 643 440 040 25 454 0 026 1 0 569 0 569 451 174 3 186 0 828 2 0 576 0 576 452 707 0 120 0 948 3 0 581 0 581 452 767 gt The first block of output provides a summary of the model specification The second block provides the test results relevant for selecting the appropriate rank The table is meant to be read sequentially from lowest to highest rank i e from top to bottom Since we can reject the null of rank 0 against the alternative of rank 3 we move to the test of rank 1 against rank 3 This test fails to reject with a P value of 0 828 so this is the appropriate choice in this case 3 5 Unrestricted model estimation With the rank and lag selected the user can now move to the next code section shown in Listing 6 Listing 6 Unrestricted model estimation aa anaana a a a FR O NPO A UNRESTRICTED MODEL ESTIMATION opti DefaultOpt mi FCVARestn x1 k r opti This model is now in the structure ml mv_wntest m1i Residuals order printWNtest Here the user first specifies the choice for the rank based on the previously performed cointegrating rank tests thus setting r 1 in this example Next the default options set in the initialization see Section 3 2 are assigned to opt1 which is used as an argument in the call to the function FCVARestn m This function is the main part of the program since it performs the estimation of the parameters obtain
48. nds opt Written by Michal Popiel and Morten Nielsen This version 11 04 2014 DESCRIPTION This function obtains upper and lower bounds on d b or on phi given by db H phi h performs a likelihood ratio test of the null hypothesis model is modelR against the alternative hypothesis model is modelUNR Input opt object containing estimation options Output UB a 2x1 or 1x1 upper bound for db or phi YA LB a 2x1 or 1x1 upper bound for db or phi 5 11 18 FCVARsimBS m Listing 47 FCVARsimBS m function xBS FCVARsimBS data model NumPeriods 42 oO JD A PB w 10 12 13 14 15 16 17 function xBS Written by Mi DESCRIPTION Input data model NumPe Output xBS FCVARsimBS data model NumPeriods chal Popiel and Morten Nielsen This version 02 09 2015 This function simulates the FCVAR model as specified by input model and starting values specified by data It creates a bootstrap sample by augmenting each iteration with a bootstrap error The errors are sampled from the residuals specified under the model input and have a positive or negative sign with equal probability Rademacher distribution T x p matrix of data a Matlab structure containing estimation results riods number of steps for simulation NumPeriods x p matrix of simulated bootstrap values 43 A Version change log A 1 Version 1 0 0 October 24 2014 Fir
49. of programs html 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 Listing 2 Initialization of local variables hie ES INITIALIZATION p size x1 2 system dimension kmax 535 maximum number of lags for VECM order 12 number of lags for white noise test in lag selection printWNtest 1 to print results of white noise tests post estimation The variable kmax determines the highest lag order for the sequential testing that is performed in the lag selection whereas p is the dimension of the system The other variables are self explanatory The next set of initialization commands shown in Listing 3 assign values to the variables contained in object opt defined by the class EstOptions Listing 3 Choosing estimation options hn ES Choosing estimation options h opt EstOptions Define variable to store Estimation Options object opt dbMin 0 01 lower bound for d b opt dbMax 2 00 upper bound for d b opt unrConstant 0 include an unrestricted constant 1 yes 0 no opt rConstant 0 include a restricted constant 1 yes 0 no opt levelParam 1 include level parameter 1 yes O no opt constrained 0 impose restriction dbMax gt d gt b gt dbMin 1 yes O no opt restrictDB 1
50. parts of a script individually Each of the code sections are delimited by a double comment and the section header 3 1 Importing data The first step is importing the data Executing the code in Listing 1 shown below assigns the data from the file data_JNP2014 csv to a matrix called data Listing 1 Importing data te Inport Data clear all data csvread data_JNP2014 csv 1 skip first row because var names data for each model xi data 1 3 5 x2 data 2 3 5 x3 data 1 2 3 5 x4 data 1 3 4 5 6 x5 data 2 3 4 5 6 x6 data 1 2 3 4 5 6 The columns contain the following variables 1 aggregate support for the Liberal party 2 aggregate support for the Conservative party 3 Canadian 3 month T bill rates 4 US 3 month T bill rates 5 Canadian unemployment rate and 6 US unemployment rate Since each of the models in JNP 2014 contain different combinations of these variables the relevant columns of data for each model are assigned to different matrices of variables named x1 through x6 3 2 Choosing options Once the data is imported the user sets the program options The script contains two sets of options variables set for function arguments in the script itself and model estimation related options Listing 2 shows the first of set of options 4For more information see http www mathworks com help matlab matlab_prog run sections
51. perform under parallel processing If the user has the capability to use multiple processors then computation time can be greatly reduced If not the function can still be performed but the bootstrap iterations will appear out of order since the loop is coded using parfor instead of for Listing 14 Bootstrap hypothesis test ih 225222 BOOTSTRAP HYPOTHESIS TEST Test restriction that political variables do not enter the A cointegrating relation s Define estimation options for unrestricted model alternative optUNR DefaultOpt Define estimation options for restricted model null optRES DefaultOpt optRES R_Beta 1 0 0 Number of bootstrap samples to generate B 999 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 204 Call to open the distributed processing comment out if unavailable matlabpool open 4 LRbs H mBS mUNR FCVARboot x1 k r optRES optUNR B Compare the bootstrap distribution to chi squared distribution Estimate kernel density F XI ksdensity LRbs Plot bootstrap density with chi squared density figure plot XI F XI chi2pdf XI H df legend Bootstrap PDF with num2str B BS samples Chi Squared with num2str H df df An example output Bootstrap results Unre
52. ple the model with level parameter in 3 We first note that A Xeqa u aga w Xiri u AX u Xia w Lal Kegs u and then rearrange 3 as k Xi wt Lal Xi u 08 A Xia u GAL Xia H Et 13 1 Since Ly 1 A is a lag operator so that Li X41 is known at time t for gt 1 this equation can be used as the basis to calculate forecasts from the model We let conditional expectation given the information set at time t be denoted E and the best linear predictor forecast of any variable Z 1 given information available at time t be denoted Acie Ei Z441 Clearly we then have that the forecast of the innovation for period t 1 at time t is 4114 Esler41 0 and Keay is then easily found from 13 Inserting also coefficient estimates based on data available up to time t denoted d b ji 8 T1 P we have that k s ooa Sanai ss airi Riri A LLegar fa GB At Ls Xia AL Xiri A 14 j 1 This defines the one step ahead forecast of X41 given information at time t Multi period ahead forecasts can be generated recursively That is to calculate the h step ahead forecast we first generalize 14 as k Xa je B Lalit OB ATPL s rage A D TAL Ritse 15 l 3To emphasize that these estimates are based on data available at time t they could be denoted by a subscript t However to avoid clutter
53. pt or we could replace it with opt1 m1r2 options The latter assignment is preferred in this case because it is explicit about which model options we are leaving imposed The hypothesis 4 is tested in the exact same way as before only now we are changing the variable Ra instead of Rg The results are shown below and we can see that this hypothesis is also rejected Unrestricted log likelihood 451 174 Restricted log likelihood 446 086 Test results df 1 LR statistic 10 176 P value 0 001 We next move to the remaining long run exogeneity tests 4 and 3 shown in Listings 10 and 11 respectively The results of the tests are shown below each listing Listing 10 Hypothesis 4 h Test restriction that interest rate is long run exogenous opti DefaultOpt opti R_Alpha O 1 0 mir4 FCVARestn x1 k r opti This restricted model is now in the structure mir4 mv_wntest mir4 Residuals order printWNtest Halpha2 HypoTest m1 mir4 Test the null of mir4 against the alternative ml and store the results in the structure Halpha2 Output Unrestricted log likelihood 451 174 Restricted log likelihood 450 857 Test results df 1 LR statistic 0 633 P value 0 426 Listing 11 Hypothesis 3 Test restriction that unemployment is long run exogenous opti DefaultOpt opti R_Alpha 0 0 1 k 2 r 1 19 mir5 FCVARestn x1 k r opti This restricted model
54. re stored in m1 Residuals 3 6 Hypothesis testing We now move into the hypothesis testing section of the code where we can test several restricted models and perform inference For restricted model estimation the grid search option is switched off because computation can be very slow especially in the presence of the level parameter However if the user wishes to verify the accuracy of the results or if estimates are close to the upper or lower bound the grid search option can resolve these issues and give the user additional insight about the behaviour of the likelihood All hypotheses are defined as shown in 10 12 The first hypothesis test is for precise definitions of each hypothesis please see Jones et al 2014 and it is shown in Listing 7 Listing 7 Hypothesis 4 hh Se IMPOSE RESTRICTIONS AND TEST THEM vA DefaultOpt gridSearch 0 turn off grid search for restricted models because it s too intensive hh Test restriction that d b 1 opti DefaultOpt opti R_psi 1 0 opti r_psi 1 miri FCVARestn x1 k r opti This restricted model is now in the structure miri mv_wntest miri Residuals order printWNtest Hdb HypoTest m1 mir1 Test the null of miri against the alternative mi and store the results in the structure Hdb Here we test the CVAR model null hypothesis d b 1 against the FCVAR model alternative hypothesis d b 1 Since opt1 restrictDB
55. reported along with the degrees of freedom the LR test statistic and its P value In this case the test clearly rejects the null hypothesis that the model is a CVAR For more significant digits or to access any of these values from the command window the user can type Hdb The next hypothesis of interest is Hy which is a zero restriction on the first element of the cointegration vector Listing 8 Hypothesis 4 Test restriction that political variables do not enter the cointegrating relation s opti DefaultOpt opti R_Beta 1 0 0 mir2 FCVARestn x1 k r opti This restricted model is now in the structure mir2 mv_wntest mir2 Residuals order printWNtest Hbetal HypoTest mi mir2 Test the null of mir2 against the alternative ml and store the results in the structure Hbetal Since the object opt1 has the restriction d b 1 stored the first step is to reset the options to default The restriction on is then specified as in 12 There are two things to note here First the column length of Rg must equal pir where p p 1 if a restricted constant is present and p p otherwise recall that p is the number of variables in the system and r is the number of cointegrating vectors Second zero restrictions are the default and automatically imposed when rg is empty Therefore the user only needs to specify rg if it includes non zero elements Recall that for restrictions on a only rg 0 is
56. rinted to the screen Input x matrix of variables to be included in the system kmax maximum number of lags Y cointegration rank number of cointegrating vectors h order order of serial correlation for white noise tests opt object containing estimation options Output none only output to screen 5 4 RankTests m Listing 20 RankTests m function rankTestStats RankTests x k opt function rankTestStats RankTests x k opt Written by Michal Popiel and Morten Nielsen This version 11 17 2014 Based on Lee Morin amp Morten Nielsen June 5 2013 DESCRIPTION Performs a sequence of likelihood ratio tests for cointegrating rank The results are printed to screen if the indicator print2screen is 1 input vector or matrix x of data scalar k denoting lag length opt object containing estimation options output rankTestStats structure with results from cointegrating rank h tests containing the following p 1 vectors with i th element corresponding to rank i 1 dHat estimates of d bHat estimate of b LogL maximized log likelihood LRstat LR trace statistic for testing rank r against rank p pv P value of LR trace test or 999 if P value is not available 5 4 1 get_pvalues Listing 21 get_pvalues m function pv get_pvalues q b consT testStat opt 31 JN Oo na B w o 10 12 13 14
57. rrect number of free parameters A 7 Version 1 3 0 September 9 2015 FCVARestn m e Moved all nested functions to Auxiliary folder FCVARforecast m e Deleted all nested functions these are now in Auxiliary folder replication_JNP2014 m e Added line that adds the path of Auxiliary folder with all necessary functions e Removed forecasting subsection e Added variable to store output from rank tests Added the following new files functions FCVARboot m e Performs wild bootstrap for hypothesis tests on parameters FCVARbootRank m e Performs wild bootstrap for rank tests FCVARsim m e Generates samples with errors drawn from Normal distribution FCVARsimBS m Auxillary function e Generates wild bootstrap samples MoreExamples m e Contains examples of forecasting bootstrapping simulation 46 References Boswijk H P G Cavaliere A Rahbek and A M R Taylor 2013 Inference on co integration parameters in heteroskedastic vector autoregressions Department of Economics Discussion Paper 13 13 University of Copenhagen Boswijk H P and J A Doornik 2004 Identifying estimating and testing restricted cointegrated systems An overview Statistica Neerlandica 58 440 465 Cavaliere G A Rahbek and A M R Taylor 2010 Testing for co integration in vector autoregressions with non stationary volatility Journal of Econometrics 158 7 24 Dolatabadi S M Nielsen and K Xu 2014 A fractionally cointe
58. s model residuals and standard errors and calculates many other relevant components such as the number of free parameters and the roots of the characteristic polynomial If opt1 print2screen 1 then in addition to storing all of these results in the Matlab structure m1 the function outputs the estimation results to the command window To see a list of variables stored in m1 the user can type m1 in the command line After the unrestricted model has been estimated this code section concludes with a call to mv_wntest m which performs a series of white noise tests on the residuals and prints the output in the command window The program output is shown below It begins with a table summarizing relevant model specifications and then the coefficients and their standard errors The roots of the characteristic polynomial are displayed at the bottom Dimension of system 3 Number of observations in sample 316 Number of lags 2 Number of observations for estimation 316 Restricted constant No Initial values 0 13 Unrestricted constant No Level parameter Yes Cointegrating rank 1 AIC 848 348 Log likelihood 451 174 BIC 746 943 log det Omega_hat 11 369 Free parameters 27 Variable CI equation 1 Var1 1 000 Var2 0 111 Var3 0 240 Variable CI equation 1 Var 1 0 180 SE 1 C 0 064 Var 2 0 167 SE 2 0 194 Var 3 0 037 SE 3 C 0 014 Variable Var 1 Var 2 Var 3 Var 1 0 180 0 020 0 043 Var 2 0 167 0 019 0 040 Var 3 0
59. s of freedom for the test results LRstat likelihood ratio test statistic results p_LRtest P value for test 5 6 FCVARforecast m Listing 23 FCVARforecast m function xf FCVARforecast data model NumPeriods function xf FCVARforecast data model NumPeriods Written by Michal Popiel and Morten Nielsen This version 11 17 2014 DESCRIPTION This function calculates recursive forecasts It uses y FracDiff and Lbk which are nested below Input data T x p matrix of data h model a Matlab structure containing estimation results NumPeriods number of steps ahead for forecast Output xf NumPeriods x p matrix of forecasted values 5 7 mv_wntest m 32 o 0 JD A A w 10 11 12 13 14 15 16 17 18 19 20 N NO a B w 10 11 12 Listing 24 mv_wntest m function Q pvQ LM pvLM mvQ pvMVQ mv_wntest x maxlag printResults function Q pvQ LM pvLM mvQ pvMVQ mv_wntest x maxlag printResults Written by Michal Popiel and Morten Nielsen This version 7 21 2015 DESCRIPTION This function performs a multivariate Ljung Box Q test for de white noise and univariate Q tests and LM tests for white noise on the columns of x yA The LM test should be consistent for heteroskedastic series Q test is not Input x matrix of variables under test typically model residuals maxlag number of lags for serial
60. st publicly available version A 2 Version 1 1 0 October 30 2014 FCVARestn m e Fixed declaration of number of observations T so that it accounts for initial values opt N This affects AIC BIC calculations and printed number of observations in the output but nothing else e Changed number of significant digits in the printed output for likelihood AIC BIC e Added redundancy check for restrictions in R_psi matrix and restrictDB e Changed estimation method when R_psi together with restrictDB has rank 2 LikeGrid m e Changed output so that actual restricted d b is shown in waitbar terminal e Fixed how the endpoints for are calculated if R_psi is non empty e Changed the way h is calculated less efficient more accurate A 3 Version 1 2 0 November 12 2014 EstOptions m e Fixed typo in warning message e Changed the order in which R_psi matrices are checked for redundancies errors restrictDB with R_psi non empty had to be moved to before 1 1 is imposed e Added a check to make sure that restrictions on w in the model restrictDB are imposed correctly e Added option CalcSE to turn off calculation of standard errors for faster computation LagSelect m e Turned off calculation of standard errors RankTests m e Turned off calculation of standard errors e Fixed typo in output un r estriced FCVARestn m e Fixed typo in output un r estriced e Changed the way that optimization is performed when linear restr
61. stricted log likelihood 451 174 Restricted log likelihood 444 395 Test results df 1 LR statistic 13 557 P value 0 000 P value BS 0 017 The user might also be interested in comparing the bootstrap likelihood ratio test statistic distribution to the asymptotic one The second part of Listing 14 performs this comparison by producing a plot of the two distributions shown in Figure 4 4 3 Bootstrap rank test Listing 15 shows how to perform a wild bootstrap rank test following the methodology of Cavaliere et al 2010 for the CVAR model This procedure works in much the same way as the bootstrap hypothesis test described in Section 4 2 The difference is that instead of providing two sets of estimation options the user specifies two different ranks for comparison Listing 15 Bootstrap rank test hh gt BOOTSTRAP RANK TEST 7 Test rank 0 against rank 1 ri 0 r2 1 LR_Rnk H_Rnk mBSri mBSr2 FCVARbootRank x1 k DefaultOpt ri r2 B Compare to P value based on asymptotic distribution fprintf P value Xt 1 3f n rankTestStats pv 1 Close distributed processing comment out if unavailable matlabpool close The results are printed as Bootstrap results Unrestricted log likelihood 451 174 Restricted log likelihood 440 040 Test results 26 208 Figure 4 Density of bootstrap LR test statistic Bootstrap PDF with 999 BS samples Chi Squared with
62. the general result 3 In model 3 the same result as in bullet 2 holds because P u p is the restricted constant and y has no influence on the asymptotic distribution in a similar way to Xy in a random walk 4 When both the restricted and unrestricted constants are included in model 2 with d 1 1 Fi u Waali f Waaluldu 1 91 0 1 F u u f udu u 1 b 1 0 1 Fa 1 u ue ub du u 1 b 0 see Dolatabadi et al 2014 Importantly the asymptotic distribution 9 of the test statistic LRr q depends on both b and q p r The dependence on the unknown true value of the scalar parameter b complicates empirical analysis compared to the CVAR model Generally the distribution 9 would need to be simulated on a case by case basis However for model 1 and for model 2 with d b and 0 and hence also for model 3 with d bin light of bullet 3 above computer programs for computing asymptotic critical values and asymptotic P values for the LR cointegration rank tests based on numerical distribution functions are made available by MacKinnon and Nielsen 2014 Their computer programs are incorporated in the present program for the relevant cases models as discussed and illustrated below 2 4 Restricted models Note that a reduced rank restriction has already been imposed on models 1 3 where the coefficient matrix II a has been restricted to rank r lt p Other restri
63. the rank condition is used to count the free parameters in those two variables Input x matrix of variables to be included in the system k number of lags r number of cointegrating vectors opt object containing the estimation options Output fp number of free parameters 5 11 7 FullFCVARIlike m Listing 36 FullFCVARlike m function like FullFCVARlike x k r coeffs beta rho opt function like FullFCVARlike x k r coeffs beta rho opt Written by Michal Popiel and Morten Nielsen This version 10 22 2014 Based on Lee Morin amp Morten Nielsen August 22 2011 DESCRIPTION This function returns the value of the log likelihood evaluated at the parameters provided as inputs Input x matrix of variables to be included in the system k number of lags VA r number of cointegrating vectors coef s Matlab structure of coefficients beta value of beta rho value of rho opt object containing the estimation options Output like value of the log likelihood 5 11 8 GetParams m Listing 37 GetParams m function estimates GetParams x k r db opt function estimates GetParams x k r db opt Written by Michal Popiel and Morten Nielsen This version 11 10 2014 Based on Lee Morin amp Morten Nielsen August 22 2011 DESCRIPTION This function uses FWL and reduced rank regression to obtain the est
64. tional parameters d b mu level parameter k number of lags r number of cointegrating vectors opt object containing the estimation options Output like log likelihood evaluated at specified parameter values 5 11 5 FracDiff m Listing 34 FracDiff m function dx FracDiff x d function dx FracDiff x d Andreas Noack Jensen Morten Nielsen May 24 2013 FracDiff x d is a fractional differencing procedure based on the fast fractional difference algorithm of Jensen amp Nielsen 2014 JTSA input x vector or matrix of data d scalar value at which to calculate the fractional difference output vector or matrix 1 L d x of same dimensions as x The function FracDiff m is the implementation of the fast fractional difference algorithm by Jensen and Nielsen 2014 5 11 6 FreeParams m Listing 35 FreeParams m function fp FreeParams k r p opt rankJ function fp FreeParams k r p Opt rankJ Written by Michal Popiel and Morten Nielsen This version 10 22 2014 36 om N O 0 10 11 12 13 14 15 12 13 14 15 16 17 A o 00 NN Oa 10 12 13 14 15 16 I 18 19 20 DESCRIPTION This function counts the number of free parameters based on the number of coefficients to estimate minus the total number of vA restrictions When both alpha and beta are restricted
65. umPeriods x p matrix of simulated values 5 11 Auxillary functions 5 11 1 FCVARsimBS m Listing 30 FCVARsim m function xBS FCVARsimBS data model NumPeriods function xBS FCVARsimBS data model NumPeriods Written by Michal Popiel and Morten Nielsen This version 08 06 2015 DESCRIPTION This function simulates the FCVAR model as specified by input model and starting values specified by data It creates a bootstrap sample by augmenting each iteration with a bootstrap error The errors are sampled from the residuals specified under the model input and have a positive or negative sign with equal probability Rademacher distribution Input data T x p matrix of data model a Matlab structure containing estimation results NumPeriods number of steps for simulation Output xBS NumPeriods x p matrix of simulated bootstrap values 5 11 2 FCVARhess m Listing 31 FCVARhess m function hessian FCVARhess x k r coeffs opt function hessian FCVARhess x k r coeffs opt Written by Michal Popiel and Morten Nielsen This version 10 22 2014 DESCRIPTION This function calculates the Hessian matrix of the log likelihood numerically Input x matrix of variables to be included in the system k number of lags de r number of cointegrating vectors coef s coefficient estimates around which estimation takes place opt object

A Matlab program and user's guide for the fractionally cointegrated

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