Program GAP∗- Version 2.2 User Manual May 2003
Contents
1. 1st series 1st series cycle 0 to 1 lag e Differenced 1st series 1st series cycle 0 lag e Differenced 1st series e ist series cycle 0 to 4 lags e ist series cycle 0 to 3 lags e ist series cycle 0 to 2 lags e ist series cycle 0 to 1 lag e ist series cycle 0 lag e None 3 Estimation Parameter estimation is made using the Diffuse Kalman Filter in its collapsed version see also Durbin and Koopman 2001 Time Series Analysis by State Space Methods Oxford University Press pp 115 120 All details are given in the Technical Annex Likelihood maximisation can be made either via Simulating Annealing Estimate via simulated annealing or via a Newton like optimization method Estimate via E04UCF mk 19 Estimation is performed when pressing RUN Because it explores the parameter space without considering the objective func tion gradient the simulated annealing algorithm is less likely than the Newton like algorithm to isolate a local maximum The price to pay is that the simulated an nealing algorithm needs much more computing time than the Newton like technique In most cases the gradient based algorithm is satisfactory and we would recommend the simulated annealing algorithm only in case of estimation problem 3 1 Simulating annealing parameters If the simulated annealing algorithm is selected then users can tune several param eters in the simulated annealing box These parameters a2. ND and CYCLE estimates for the first series These estimates are those obtained via fixed point smoother e RESID TXT contains the residuals for the two equations e RMSE TXT contains the Root Mean Square Errors of the smoothed unobserved component estimates e If Recursive parameter estimates is used the program produces two addi tional file named PAR TXT that contains the model parameters estimates for the different sample lengths and REV TXT containing smoothed estimates of the cycle computed by real time parameters in PAR TXT e PAR TXT contains the model parameters and 2log likelihood when the model is estimated recursively 6 Graphs When the program has run you can see plots of the decomposition of the variable in the first equation by pushing UPDATE GRAPHS in the worksheet Graph More over users may select the magnitude of confidence bounds for the estimated short term component by entering a number in cell A3 in worksheet Graphs For instance a value of 1 96 corresponds to a 95 confidence bound
3. PROGRAM GAP VERSION 2 2 USER MANUAL May 2003 CHRISTOPHE PLANAS and ALESSANDRO ROSSI Emails christophe planas jrc it alessandro rossi yrc it European Commission Joint Research Centre Ispra Italy Please read disclaimer in worksheet About of the Excel interface Contents 1 Data input output 2 Model specification 2 1 General setting 2 2 Model type 2 3 Canonical trend 2 4 ARorders 2 5 MA order in Phillips curve equation 0 2 6 Exogenous regressors 2 7 Endogenous regressors in Phillips curve equation 3 Estimation 3 1 Simulating annealing parameters 2 20004 3 2 Recursive estimation 4 Parameter constraints 5 Output of the program 6 Graphs DAD nnn wow w N 1 Data input output Before starting the analysis the user must store the data 1st 2nd and exogenous series if any in worksheet Data starting from the third row Columns B and C are reserved to the endogenous variables Ten regressors on the Phillips curve equation can be introduced in columns D to P while 3 other columns Q to S are reserved for the exogenous variable on the first variable typically unemployment or GDP e Box A insert the location and name of the file where the 1st series will be stored e g C data filenamel e Box B insert the location and name of the file where the 2nd series will be stored e g C data filename2 e Box C ins
4. The canonical trend is specified as D XP m Dam That specification removes all orthogonal noise from the long term component and assigns it to the short term component Hence it can offer a solution to the case where the variance of the innovations on the trend level are estimated at zero Notice that if the short term component was described as a AR 2 cycle it becomes an ARMA 2 2 If instead it was a noise then it remains as such 2 4 AR orders Users are free to enter the autoregressive orders of equation 4 describing the first series cycle and the autoregressive order of the Phillips curve 5 The values accepted are 0 1 and 2 2 5 MA order in Phillips curve equation Users can enter the order of the MA term in 5 The values accepted are 0 1 2 and 3 2 6 Exogenous regressors The users must choice the number of exogenous regressors that will enter in equation 1 for the first series and in the Phillips curve equation 5 These regressors must be loaded in the worksheet data see section 1 The maximum number of regressors accepted in equation 1 is 3 and is 10 for equation 5 2 7 Endogenous regressors in Phillips curve equation For the endogenous terms in the right hand side of 5 users can choose among e Differenced 1st series Ist series cycle 0 to 4 lags e Differenced 1st series Ist series cycle 0 to 3 lags e Differenced 1st series Ist series cycle 0 to 2 lags e Differenced
5. ert the location and name of the file where the exogenous variable will be stored e g C data filename3 optional e Box D insert the location where all outputs will be stored e g C results As the program does not make any type of data transformation users may wish to apply logarithm and or difference operations to price series 2 Model specification 2 1 General setting The most general model that can be specified is a bivariate model such that see also Kuttner 1994 Journal of Business Economics and Statistics 12 3 pp 361 368 A short overview is developed below full details can be read in the Technical Annex available in the worksheet Help Equation 1 The observed series is made up of a long term component or trend X a short term component or cycle X7 and a set of regressors according to Nex1 Xit XR x a Qi Zit 1 i l where Z is a vector of Nex1 x 1 exogenous variables with Nex1 less or equal to 3 The trend can be specified as a second order random walk 1 L X put ane 2 where L is the lag operator The slope is such that 1 L par Gut 3 Both anz and apt are niid innovations with variances Vy and V Notice that if Var a 0 then 2 reduces to a random walk plus drift The short term compo nent is such that 1 iL pL X as 4 The variables as are nitd innovations with variance Vg Equation 2 in agreement with the Phillips curve f
6. ramework series 1 is related to a second series according to Nex2 4 Xa pot 5 a Yu 9 1 L Xiu 1 DoR i 1 i 01 Xo4 1 3X2 4 1 OL O21 4 3L Ont 5 where Y is a vector of Nex2 x 1 exogenous variables with Nex2 less or equal to 10 and at is a nid innovation with variance V that can be correlated with as That correlation is denoted by p Warnings 1 the second series and the regressors on the second equation should be covariance stationary 2 if X is included in the Phillips curve equation 2nd equation the innovations ar and ag must be uncorrelated so the program imposes p 0 when bo 0 Without that restriction the innovation a and the regressor X in the second equation would not be orthogonal and the corresponding estimator would not be consistent 2 2 Model type e The users have the possibility to choose among e Bivariate AR RW with drift equations 1 2 3 4 with restriction Var apt 0 e Bivariate AR 2nd order RW equations 1 2 3 4 e Univariate AR RW with drift equations 1 2 3 with restriction Var ay 0 e Univariate AR 2nd order RW equations 1 2 3 e Hodrick Prescott filter This is a facility for a univariate detrending of the first series The box Hodrick Prescott parameter allows users to enter the inverse signal to noise ratio 2 3 Canonical trend That specification is only possible when the trend is a random walk plus drift
7. re e T The initial temperature e RT The temperature reduction factor The suggested value is 85 e EPS Error tolerance for termination e NS Number of cycles e NT Number of iterations before temperature reduction After NT x NS x Npar function evaluations temperature T is changed by the factor RT The suggested value is MAX 100 5x Number of model parameters e MAX EVL The maximum number of function evaluations e VM The initial step length vector e Intermediate results are produced if the user selects Y in the so called box 3 2 Recursive estimation If Y is selected in the cell next to Rolling estimates the program performs a recursive estimation of the model parameters starting at the Starting date that is entered by the user By selecting Y in the cell next to Start at previous optimum the maximization procedure starts from the previous maximum Otherwise the starting values are obtained by a regression type analysis A file PAR TXT is created in the output directory that displays on every line all model parameters estimates for a given sample size The order of the parameters in the PAR TXT file is Q1 Q2 UN Hr Y B1 01 02 3 Vin Vs P Vr Q21 W210 1 1 1 3 Ba Qi gt b3 ba bo A second file entitled REV TXT is produced It contains the smoothed estimates of the cycle computed in real time using the parameters in PAR TXT REV TXT is a t x k matrix where
8. the rows t Starting point nobs 11 where t represents the period for which the component is estimated and the columns are numerated as k 1 2 12 with k such that k 1 is the number of additional observations available after time t So concurrent estimates are on the first column and estimates with t 11 data are on the twelfth column 4 Parameter constraints For every parameter the two boxes under the parameter constraints title offer the possibility to set lower and upper bounds For the innovation variances the lower and upper bounds are subject to several constraints e For the variance of the innovations in the first series unobserved components upper bounds must be less or equal to 1 2 times the variance of the differenced first series For the second equation residuals variance the maximum upper bound allowed is 1 2 times the variance of the second series In both cases the minimum upper bound is 0 the lower bound The program automatically resets the upper bound to the maximum authorized value if user enter a non acceptable value 5 Output of the program When running the program produces in the output directory Box D the following files e SOL TXT contains the model parameters together with their standard devia tions and 2 log likelihood plus the Ljung Box statistics on the residuals This file can be displayed by pressing EDIT RESULTS in worksheet Graphs e STATE TXT contains the TRE
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