Dynamic stochastic general equilibrium models in Stata 15

Transcription

1 Dynamic stochastic general equilibrium models in Stata 15 David Schenck Senior Econometrician Stata October 11, 2017 Schenck (Stata) DSGE October 11, / 25

2 Motivation Models used in macroeconomics for policy analysis Schenck (Stata) DSGE October 11, / 25

3 Motivation Models used in macroeconomics for policy analysis Suppose you want to understand the effect of some policy Schenck (Stata) DSGE October 11, / 25

4 Motivation Models used in macroeconomics for policy analysis Suppose you want to understand the effect of some policy You need a model Schenck (Stata) DSGE October 11, / 25

5 Motivation Models used in macroeconomics for policy analysis Suppose you want to understand the effect of some policy You need a model Dynamic Schenck (Stata) DSGE October 11, / 25

6 Motivation Models used in macroeconomics for policy analysis Suppose you want to understand the effect of some policy You need a model Dynamic Stochastic Schenck (Stata) DSGE October 11, / 25

7 Motivation Models used in macroeconomics for policy analysis Suppose you want to understand the effect of some policy You need a model Dynamic Stochastic General equilibrium Schenck (Stata) DSGE October 11, / 25

8 Motivation Models used in macroeconomics for policy analysis Suppose you want to understand the effect of some policy You need a model Dynamic Stochastic General equilibrium and you need to estimate the model parameters Schenck (Stata) DSGE October 11, / 25

9 Here s a model in words Consumers Consume and save output Take inflation and interest rate as given Schenck (Stata) DSGE October 11, / 25

10 Here s a model in words Consumers Consume and save output Take inflation and interest rate as given Producers Produce output and set prices Take demand as given Schenck (Stata) DSGE October 11, / 25

11 Here s a model in words Consumers Consume and save output Take inflation and interest rate as given Producers Produce output and set prices Take demand as given Central bank Sets interest rate Adjusts interest rate in response to inflation Schenck (Stata) DSGE October 11, / 25

12 Here s a model in words Consumers Consume and save output Take inflation and interest rate as given Producers Produce output and set prices Take demand as given Central bank Sets interest rate Adjusts interest rate in response to inflation In equilibrium, this is a model that simultaneously determines output, inflation, and the interest rate Schenck (Stata) DSGE October 11, / 25

13 Here s a model in equations I Consumers demand output (x t ), given inflation (π t ), the interest rate (r t ), and latent factors (z t ): x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) Schenck (Stata) DSGE October 11, / 25

14 Here s a model in equations I Consumers demand output (x t ), given inflation (π t ), the interest rate (r t ), and latent factors (z t ): x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) Producers set prices, given output demand: π t = βe t (π t+1 ) + κx t Schenck (Stata) DSGE October 11, / 25

15 Here s a model in equations I Consumers demand output (x t ), given inflation (π t ), the interest rate (r t ), and latent factors (z t ): x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) Producers set prices, given output demand: π t = βe t (π t+1 ) + κx t Central bank sets interest rate, given inflation r t = 1 β π t + u t Schenck (Stata) DSGE October 11, / 25

16 Here s a model in equations II The model s endogenous variables are characterized by equations: x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) π t = βe t (π t+1 ) + κx t r t = 1 β π t + u t Schenck (Stata) DSGE October 11, / 25

17 Here s a model in equations II The model s endogenous variables are characterized by equations: x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) π t = βe t (π t+1 ) + κx t r t = 1 β π t + u t The model is completed by adding equations for the state variables: z t+1 = ρ z z t + ξ t+1 u t+1 = ρ u u t + ε t+1 Schenck (Stata) DSGE October 11, / 25

18 Here s a model in Stata The model equations: x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) π t = βe t (π t+1 ) + κx t r t = 1 β π t + u t z t+1 = ρ z z t + ξ t+1 u t+1 = ρ u u t + ε t+1 Schenck (Stata) DSGE October 11, / 25

19 Here s a model in Stata The model equations: x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) π t = βe t (π t+1 ) + κx t r t = 1 β π t + u t In Stata: z t+1 = ρ z z t + ξ t+1 u t+1 = ρ u u t + ε t+1. dsge (x = E(F.x) - (r - E(F.p) - z), unobserved) /// (p = {beta}*e(f.p) + {kappa}*x) /// (r = 1/{beta}*p + u) /// (F.z = {rhoz}*z, state) /// (F.u = {rhou}*u, state) Schenck (Stata) DSGE October 11, / 25

20 Data and data management We will use data on inflation and interest rates Available from the Federal Reserve Economic Database (FRED) repository. import fred GDPDEF FEDFUNDS, aggregate(quarterly, eop). generate dateq = qofd(daten). tsset dateq, quarterly. generate lprice = ln(gdpdef). generate p = 400*D.lprice. rename FEDFUNDS r Schenck (Stata) DSGE October 11, / 25

21 Data figure q3 1970q3 1985q3 2000q3 2015q3 Date (quarters) Growth rate of prices (GDPDEF) Federal funds rate (FEDFUNDS) Schenck (Stata) DSGE October 11, / 25

22 Parameter estimation. dsge (x = E(F.x) - (r - E(F.p) - z), unobserved) /// > (p = {beta}*e(f.p) + {kappa}*x) /// > (r = 1/{beta}*p + u) /// > (F.z = {rhoz}*z, state) /// > (F.u = {rhou}*u, state), nolog DSGE model Sample: 1955q1-2015q4 Number of obs = 244 Log likelihood = OIM Coef. Std. Err. z P> z [95% Conf. Interval] /structural beta kappa rhoz rhou sd(e.z) sd(e.u) Schenck (Stata) DSGE October 11, / 25

23 Policy questions What is the effect of an unexpected increase in interest rates? Estimated DSGE model provides an answer to this question. We can subject the model to a shock, then see how that shock feeds through the rest of the system. Schenck (Stata) DSGE October 11, / 25

24 Impulse response setup First we set up the impulse response file:. irf set dsge_irf. irf create model1 Schenck (Stata) DSGE October 11, / 25

25 Impulse response setup First we set up the impulse response file:. irf set dsge_irf. irf create model1 Then we draw the graph:. irf graph irf, impulse(u) response(x p r u) byopts(yrescale) yline(0) Schenck (Stata) DSGE October 11, / 25

26 Impulse responses from the estimated model 0 model1, u, p.6 model1, u, r model1, u, u 0 0 model1, u, x step Graphs by irfname, impulse, and response 95% CI impulse response function (irf) Schenck (Stata) DSGE October 11, / 25

27 Solving a DSGE Model Solution to a model is the key to estimation and generating impulse responses Schenck (Stata) DSGE October 11, / 25

28 Solving a DSGE Model Solution to a model is the key to estimation and generating impulse responses Solution expresses endogenous variables as a function of state variables alone Schenck (Stata) DSGE October 11, / 25

29 Viewing solution matrices You may wish to view the solution matrices Use the postestimation commands estat policy and estat transition. estat policy Policy matrix Delta-method Coef. Std. Err. z P> z [95% Conf. Interval] x p r z u z u z u Schenck (Stata) DSGE October 11, / 25

30 Predictions of observed variables We can use the DSGE model to generate one-step-ahead predictions of the observed variables. predict p_pred r_pred (option xb assumed; fitted values) Schenck (Stata) DSGE October 11, / 25

31 Prediction of inflation rate, p q3 1970q3 1985q3 2000q3 2015q3 Date (quarters) Growth rate of prices (GDPDEF) xb prediction, p, onestep Schenck (Stata) DSGE October 11, / 25

32 Prediction of interest rate, r q3 1970q3 1985q3 2000q3 2015q3 Date (quarters) Federal funds rate (FEDFUNDS) xb prediction, r, onestep Schenck (Stata) DSGE October 11, / 25

33 Predictions of latent states We can use the DSGE model to generate estimates of the latent states. predict z u, state Schenck (Stata) DSGE October 11, / 25

34 Prediction of latent demand state, z xb states prediction, z, onestep q3 1970q3 1985q3 2000q3 2015q3 Date (quarters) Schenck (Stata) DSGE October 11, / 25

35 Prediction of latent monetary state, u xb states prediction, u, onestep q3 1970q3 1985q3 2000q3 2015q3 Date (quarters) Schenck (Stata) DSGE October 11, / 25

36 Using constraints to fix some parameters You might want to fix some parameters and estimate others. x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) π t = βe t (π t+1 ) + κx t r t = ψπ t + u t z t+1 = ρ z z t + ξ t+1 u t+1 = ρ u u t + ε t+1 (Output demand) (Pricing) (Interest rate) (Demand shock) (Monetary shock) New: the ψ parameter in the third equation. New: β no longer appears in multiple equations. Assume: you wish to fix β and estimate the remaining parameters conditional on your choice of β. Schenck (Stata) DSGE October 11, / 25

37 Constrained model in Stata. constraint 1 _b[beta]=0.96. dsge (x = E(F.x) - (r - E(F.p) - z), unobserved) /// (p = {beta}*e(f.p) + {kappa}*x) /// (r = {psi=1.5}*p + u) /// (F.z = {rhoz}*z, state) /// (F.u = {rhou}*u, state), constraint(1) nolog Schenck (Stata) DSGE October 11, / 25

38 Parameter estimation with constraints DSGE model Sample: 1955q1-2015q4 Number of obs = 244 Log likelihood = ( 1) [/structural]beta =.96 OIM Coef. Std. Err. z P> z [95% Conf. Interval] /structural beta.96 (constrained) kappa psi rhoz rhou sd(e.z) sd(e.u) Schenck (Stata) DSGE October 11, / 25

39 Conclusion dsge estimates the parameters of DSGE models Impulse response functions trace the effect of a shock on the model View the model s estimated reduced-form matrices Predictions of observable variables and latent states Constrained estimation Other features: solve a model at a particular parameter point (calibration), identification and stability diagnostics Schenck (Stata) DSGE October 11, / 25

40 Thank You! Schenck (Stata) DSGE October 11, / 25