Building and simulating DSGE models part 2

Transcription

1 Building and simulating DSGE models part II DSGE Models and Real Life I. Replicating business cycles A lot of papers consider a smart calibration to replicate business cycle statistics (e.g. Kydland Prescott). To compare DSGE models second moment statistics with data, we need to get transformed time series. Let s get time series from OECD stats Output, prices and interest rate (and population). 1

2 I. Replicating business cycles Quarterly time series: GDP (spending), millions of national currency, current prices GDP deflator. Money market rate (annual basis) Working population aged I. Replicating business cycles First step: having the GDP per capita and in real terms: = Finally we get the growth rate: =log 1

3 I. Replicating business cycles Concerning prices, inflation is computed via: =log 1 I. Replicating business cycles And the rate we set it on a quarterly basis: = /4 3

4 I. Replicating business cycles Statistics: d std 1,3 1,7,7 Comparing with the model? I. Replicating business cycles Lets first edit dynare and add the log growth rate of the GDP as an equation: dy = y-y(-1); And then simulate the business cycles statistics only. stoch_simul(order=1,irf=,nocorr) dy pi r; 4

5 I. Replicating business cycles Calling dynare: THEORETICAL MOMENTS VARIABLE MEAN STD. DEV. VARIANCE dy pi r d std 1,3 1,7,7 We clearly overestimate the volatility of all macroeconomic variables. I. Replicating business cycles Let s lower for instance the shock volatility: shocks; var e_b; stderr.; var e_p; stderr.4; var e_r; stderr.1; end; stoch_simul(order=1,irf=,nocorr) dy pi r; 5

6 I. Replicating business cycles THEORETICAL MOMENTS VARIABLE MEAN STD. DEV. VARIANCE dy pi r d std 1,3 1,7,7 The model is now able to replicate fairly well inflation and the interest rate, but clearly fails for output growth. Calibration could be improved. II. Bayesian estimations A bridge between calibration and maximum likelihood. Uncertainty and a priori knowledge about the model and its parameters are described by prior probabilities. Confrontation to the data, through the likelihood function, leads to a revision of these probabili?es ( posterior probabili?es). 6

7 II. Bayesian estimations A bridge between calibration and maximum likelihood. Uncertainty and a priori knowledge about the model and its parameters are described by prior probabilities. Confrontation to the data, through the likelihood function, leads to a revision of these probabili?es ( posterior probabili?es). II. Bayesian estimations Notations: (taken from Wouter Den Haan) Likelihood: (,Ψ) are the observations until period T. Ψ is a specific (linear) model represents parameters of the model Ψ ( Ψ) is the prior density, it describes a priori beliefs before observing the data. 7

8 II. Bayesian estimations In a nutshell, dynare : Calculates the likelihood ( Ψ) Calculates the posterior: (,Ψ),Ψ ( Ψ) Calculates the mode Calculate preliminary info about the posterior using assumption of normality. Use MCMC to: trace the shape of (,Ψ) calculate confidences intervals Draws IRFs and graphs II. Bayesian estimations To calculate the posterior and mode (,Ψ),Ψ ( Ψ) Dynare provides 6 different optimization routines which aim at finding the value of that maximizes,ψ 8

9 II. Bayesian estimations Once we have the maximum of,ψ, We want to calculate mean, standards errors, confidence intervals, these are all integrals. To compute them, we generate a sequence for Ψ by implementing MCMC (Metropolis Hastings). Data have trends and bayesian estimation only works with stationary data. Solutions: Use first differences of model variables Use HP filter Put stochastic trend in the model 9

10 Here we use first difference. We transformed 3 times series and we removed trends in GDP, we had!",!" and!". We have to link theses transformed series with our 3-equation model using measurement equations. Assuming that#, # and are the log-linear variables in our model and!",!" and!" the observables. Measurement equations are:!" =# #!" =#!" = In dynare s model block, we add measurement equations: % observable y_obs = y - y(-1); pi_obs = pi; r_obs = r; 1

11 Choosing priors in dynare: Name Distribution & parameters range Normal_pdf '((,)) R Gamma_pdf +((,),, ) [,,+ ] Beta_pdf 1((,),,, ) [,, ] Inv_gamma_pdf 3+((,)) R 4 Uniform_pdf 5(,, ) [,, ] Posterior results are very sensitive to the priors: priors distributions should be chosen wisely by using previous papers or justified. Choosing priors for shocks: 6 7 = ,: =;,<,= To be stationnary, 8 7 should be in [,1) -> this is the range of a Beta distribution1((,)). Then a low opinated prior could be at equal distance between min and max( =,5 and a very wide std) =, (like Smets & Wouters, 7). 11

12 Choosing priors for shocks: 6 7 = ,: =;,<,= Where9 7 ~',) 7 A we estimate) 7. ) 7, then aninversegamma distribution meets our requirements 3+((,)) We calibrate( =,1 and ) = as in Smets and Wouters (7). (Alternative: Darracq-Pariès 5 chooses an uniform distribution with a positive support). 1

13 In dynare, our priors write: estimated_params; stderr e_b,.58,,, INV_GAMMA_PDF,.1, ; stderr e_p,.1,,, INV_GAMMA_PDF,.1, ; stderr e_r,.4,,, INV_GAMMA_PDF,.1, ; rho_b,.85,,, beta_pdf,.5,.; rho_p,.85,,, beta_pdf,.5,.; rho_r,.6,,, beta_pdf,.5,.; sigmac,.67,,, gamma_pdf, 1.5,.37; sigmal,.57,,, gamma_pdf,,.75; phi_pi,.,,, normal_pdf,,.15; phi_dy,.1,,, normal_pdf,.5,.1; phi_y,.1,,, normal_pdf,.5,.1; rho,.86,,, beta_pdf,.7,.1; theta,.6,,, beta_pdf,.5,.1; end; 13

14 SE_e_b SE_e_p SE_e_r 4 phi_dy 4 phi_y 4 rho rho_b.5 1 rho_p.5 1 rho_r theta sigmac sigmal phi_pi Then, we have to specify to dynare which variables in the model can be found in our datafile: varobs y_obs pi_obs r_obs; 14

15 Finally setting the command to ask dynare to estimate the model: estimation( datafile=data,xls_sheet=dataset,first_obs=1,nobs=79,xls_range=b1:d8,mh_drop=.,mh_replic=5,mh_jscale=.6,conf_sig=.9,prefilter =1,lik_init=,mh_nblocks=) y_obs pi_obs r_obs; datafile: contains observables data.mat or data.m or data.xls nobs: number of observations used (default all) first_obs: first observation (default is first) mh_drop: share of first draws to be dropped. mh_replic: number of observations in each MCMC sequence mh_nblocks: number of MCMC sequences mh_jscale: variance of the jumps in Ψ in MCMC chain : a higher value of mh_jscale = bigger steps through the domain of Ψ & lower acceptance ratio acceptance ratio should be between, and,3. mh_init_scale: variance of initial draw (important to make sure that the different MCMC sequences start in different points) 15

16 Concerning the acceptance ratio: it can be seen here and when the estimation is done. If this value is too low: increase mh_jscale and viceversa. Estimation::mcmc: Current acceptance ratio per chain: Chain 1: % Chain : % Is my MCMC good? -> MCMC should generate a sequence that should replicate(,ψ). How to check? The distribution generated by MCMC should be the same: For different parts of the same sequence Across sequences (if you have more than one). Dynare automatically generates MCMC diagnostics as in Brooks & Gelman

17 We need: Red and blue line to get close Red line to cool down. 8 Interval m x m3 x x 1 4 Dynare output: first we obtain the mode RESULTS FROM POSTERIOR ESTIMATION parameters prior mean mode s.d. prior pstdev rho_b beta. rho_p beta. rho_r beta. sigmac gamm.37 sigmal gamm.75 phi_pi norm.15 phi_dy norm.1 phi_y norm.1 rho beta.1 theta beta.1 standard deviation of shocks prior mean mode s.d. prior pstdev e_b invg. e_p invg. e_r invg. Log data density [Laplace approximation] is

18 Dynare output: after MCMC, dynare computes de posterior mean parameters prior mean post. mean 9% HPD interval prior pstdev rho_b beta. rho_p beta. rho_r beta. sigmac gamma.37 sigmal gamma.75 phi_pi norm.15 phi_dy norm.1 phi_y norm.1 rho beta.1 theta beta.1 standard deviation of shocks prior mean post. mean 9% HPD interval prior pstdev e_b invg. e_p invg. e_r invg. MH output: prior and posterior distributions 1 SE_e_b 1 SE_e_p 1 SE_e_r 5 phi_dy 1 5 phi_y 5 rho..4 rho_b rho_p rho_r theta sigmac sigmal phi_pi

19 Prior and posterior distributions should be different, otherwise it reveals an identification issue: data are not informative to identify the parameter. Solution: calibration rather than estimation; increasing the number of series or the number of periods (in the sample); Running identification diagnostics as Saltelli et al. (8), Andrle (1) and Iskrev (1), etc. Here for instance 4 parameters are not well identified. Parameters regarding the utility function are known not to be identifiable. Parameters in the monetary policy rule are not well identified because the smoothing coefficient in the taylor rule captures most of the information, see An and Schorfheide (7). 19

20 Dynare also plots the smoothed shocks. In the fit exercise, for each shock, a sequence of shocks is estimated to generate the observed time series..5 e_b e_p e_r Combining all the shocks at the same time allows us to generate the time series: (black is observed, red is model generated) 1 y_obs pi_obs r_obs

21 Finally, like VARs, dynare generate the bayesian impulse response with a 9% interval band Demand y_obs pi_obs r_obs 1 3 Supply y_obs pi_obs r_obs 1 3 MP. y_obs -.5 pi_obs.6.4 r_obs

22 II. Empirical applications A. Second moments statistics First we can analyze from the estimated model the second moment statistics and the variance decomposition After loading estimated parameters after estimation() command: stoch_simul(irf=)y_obs pi_obs r_obs; II. Empirical applications A. Second moments statistics Standard deviation comparison: VARIABLE MEAN STD. DEV. VARIANCE y_obs pi_obs r_obs d std 1,3 1,7,7 The model underestimates the std of the interest rate. Another friction maybe necessary to improve the model

23 Other statistics: II. Empirical applications A. Second moments statistics VARIANCE DECOMPOSITION (in percent) e_b e_p e_r y_obs pi_obs r_obs MATRIX OF CORRELATIONS Variables y_obs pi_obs r_obs y_obs pi_obs r_obs COEFFICIENTS OF AUTOCORRELATION Order y_obs pi_obs r_obs II. Empirical applications B. Historical variance decomposition As Smets and Wouters (7) or Christiano, Motto, Rostagno (9) we can decompose a time serie according to the contribution of each shock. 3

24 II. Empirical applications B. Historical variance decomposition 6 4 Initial values Monetary Policy - Demand -4 Supply II. Empirical applications B. Historical variance decomposition 8 6 Initial values 4 Monetary Policy - Demand Supply 4

25 II. Empirical applications B. Forecast error variance decomposition We can also ask dynare to compute the driving forces of variables for various horizons. After loading estimated parameters: stoch_simul(irf=,conditional_variance_decomp osition=[1,5,1,5,1]) y_obs pi_obs r_obs; We can set different time horizons. II. Empirical applications B. Forecast error variance decomposition CONDITIONAL VARIANCE DECOMPOSITION (in percent) Period 1: e_b e_p e_r y_obs pi_obs r_obs Period 5: e_b e_p e_r y_obs pi_obs r_obs Period 1: e_b e_p e_r y_obs pi_obs r_obs Period 5: e_b e_p e_r y_obs pi_obs r_obs Period 1: e_b e_p e_r y_obs pi_obs r_obs g 5

26 II. Empirical applications B. Forecasting Finally, dynare can also perform forecasting exercises estimation (,forecast=1) y_obs pi_obs r_obs; Will perform a one period forecast after the end of the sample used in the estimation. II. Empirical applications B. Forecasting y_obs.5 pi_obs 1 r_obs

27 II. Empirical applications B. Forecasting According to our estimates: The polish GDP growth rate is going to decrease next quarter Inflation is going to be slightly higher than its trend. The money market rate is going to stay at the same value. III. Improving the model From the estimated previous model, we can fi some issues: Identification by calibrating unidentified parameters. We could also improve the model fit by adding backward price, consumption habits, etc. We could also add trends. 7

28 III. Improving the model Let s compare our model with a backward looking one: households are subject to consumption habits in proportion h: 5 E he,f Sticky price: firms are constrained to index their prices on the previous level of inflation: : = G H : III. Improving the model The log linear model writes: # = h 1+h # h # 4 1 h )(1+h) I # 4 +6 J # =K L # +MI # M )+N # +6 O = P Q # +P R # +P SR (# # )+6 T 8

29 III. Improving the model In dynare: The model equations are : % IS curve y = h/(1+h)*y(-1) + 1/(1+h)*y(+1) - (1-h)/(sigmaC*(1+h))*(r-pi(+1)) + s_b; % AS curve pi = xi_p*pi(-1) + beta*pi(+1) + ((1-theta)*(1-beta*theta)/theta)*(sigmaC+sigmaL)*y + s_p; % Monetary Policy Rule r = rho*r(-1) + (1-rho)*( phi_pi*pi + phi_y*y ) + phi_dy*(y-y(-1)) + s_r ; In dynare: III. Improving the model I add two estimated parameters using SW 7 priors: xi_p,.5,,, beta_pdf,.5,.; h,.5,,, beta_pdf,.7,.1; 9

30 III. Improving the model The two parameters are well identified. 5 phi_dy 4 phi_y rho theta.5 1 xi_p.5 1 h III. Improving the model Convergence looks fine as well for these parameters.1 theta (Interval) 1 x 1-4 theta (m) 6 x 1-5 theta (m3) x 1 4 xi_p (Interval) 6 4 x x 1-4 xi_p (m) 4 x x 1-5 xi_p (m3) x 1 4 h (Interval) 4 x x 1-3 h (m) 4 x x 1-3 h (m3) x x x 1 4 3

31 III. Improving the model Bayesian IRF (the rate uncertainty is reduced) Demand y_obs pi_obs r_obs y_obs pi_obs r_obs Supply MP y_obs pi_obs r_obs 1 3 III. Improving the model Log data density is parameters prior mean post. mean 9% HPD interval prior pstdev rho_b beta. rho_p beta. rho_r beta. sigmac gamma.37 sigmal gamma.75 phi_pi norm.15 phi_dy norm.1 phi_y norm.1 rho beta.1 theta beta.1 xi_p beta. h beta.1 standard deviation of shocks prior mean post. mean 9% HPD interval prior pstdev e_b invg. e_p invg. e_r invg. 31

32 III. Improving the model Which model is better backward vs non-backward? We can analyze this through the log data density Bechmark: Log data density is Backward looking model: Log data density is Surprisingly, the benchmark is better (the best model maximizes the log data density). III. Improving the model Second way to check: Posterior odds ratio 3

33 III. Improving the model Another way to improve the model is to estimate trends. In the previous estimations, we assumed that the average growth rate of the GDP was (same for the interest rate and inflation). Now we can ask dynare to estimate a trend, put differently, we estimate at which rate the quarterly GDP increases each period. III. Improving the model What makes the GDP increase steadily? Labour productivity. The production function is now: =(1+Υ) F Υ represents the labour-augmenting deterministic growth rate in the economy. If Υ=, then the economy is now growing in steady state. If Υ>, then the GDP increases by Υ% each period. Detrending all variables: = /(1+Υ), V =V /(1+Υ) 33

34 III. Improving the model Thus the measurement equation is now:!" =Υ+ For inflation:!" =ΠX+ For interest rate:!" =RX+ We estimate trends Υ, ΠX and RX III. Improving the model But ΠX and RX have side effects on the steady state. We must compute the steady state for each new value ofπx,rx: In dynare: #beta = (1+b/1)^-1; #cr = (((1+cgamma/1)^sigmaC)*(1+cpi/1)/beta-1)*1; We estimate an auxiliary parameter b which will affect beta the discount factor and RX. 34

35 Priors: III. Improving the model cgamma, 1,,, gamma_pdf, 1,.5; cpi, 1.,,, gamma_pdf, 1,.5; b,.5,,,gamma_pdf,.5,.1; III. Improving the model Posteriors are in our paper. 35