Estimating dynamic stochastic general equilibrium models in Stata

Transcription

1 Estimating dynamic stochastic general equilibrium models in Stata David Schenck Senior Econometrician Stata 2017 Canadian Stata User Group Meeting June 9, 2017 Schenck (Stata) DSGE June 9, / 21

2 Motivation Models used in macroeconomics for policy analysis Schenck (Stata) DSGE June 9, / 21

3 Motivation Models used in macroeconomics for policy analysis Dynamic Schenck (Stata) DSGE June 9, / 21

4 Motivation Models used in macroeconomics for policy analysis Dynamic Stochastic Schenck (Stata) DSGE June 9, / 21

5 Motivation Models used in macroeconomics for policy analysis Dynamic Stochastic General equilibrium Schenck (Stata) DSGE June 9, / 21

6 Here s a model in words Households Consume and save output Take inflation and interest rate as given Schenck (Stata) DSGE June 9, / 21

7 Here s a model in words Households Firms Consume and save output Take inflation and interest rate as given Produce output and set prices Take demand as given Schenck (Stata) DSGE June 9, / 21

8 Here s a model in words Households Firms Consume and save output Take inflation and interest rate as given Produce output and set prices Take demand as given Central bank Sets interest rate Adjusts interest rate in response to inflation Schenck (Stata) DSGE June 9, / 21

9 Here s a model in words Households Firms Consume and save output Take inflation and interest rate as given 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 June 9, / 21

10 Here s a model in equations I Households demand output, given inflation and interest rates: x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) Schenck (Stata) DSGE June 9, / 21

11 Here s a model in equations I Households demand output, given inflation and interest rates: x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) Firms set prices, given output demand: π t = βe t (π t+1 ) + κx t Schenck (Stata) DSGE June 9, / 21

12 Here s a model in equations I Households demand output, given inflation and interest rates: x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) Firms 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 June 9, / 21

13 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 June 9, / 21

14 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 June 9, / 21

15 Here s a model in Stata I 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 June 9, / 21

16 Here s a model in Stata I 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 June 9, / 21

17 Here s a model in Stata II. 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 June 9, / 21

18 Here s a model in Stata II. 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) There are some rules Equations are bound in parentheses. Parameters are bound in braces. Each variable appears on the left-hand side of one equation. State equations are written in terms of their one-period-ahead value. Schenck (Stata) DSGE June 9, / 21

19 Here s a model in Stata II. 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) There are some rules Equations are bound in parentheses. Parameters are bound in braces. Each variable appears on the left-hand side of one equation. State equations are written in terms of their one-period-ahead value. Data: US inflation rate and nominal interest rate, quarterly Schenck (Stata) DSGE June 9, / 21

20 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 June 9, / 21

21 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 June 9, / 21

22 Impulse response functions Q: What is the effect of a shock to u t on the model variables? Recall our model: 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 (Demand) (Pricing) (Interest rate) z t+1 = ρ z z t + ξ t+1 u t+1 = ρ u u t + ε t+1 (Natural rate of interest) (Monetary policy) Schenck (Stata) DSGE June 9, / 21

23 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 June 9, / 21

24 Solving a DSGE Model I Solution to a model is the key to estimation and generating impulse responses Schenck (Stata) DSGE June 9, / 21

25 Solving a DSGE Model I 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 June 9, / 21

26 Solving a DSGE model II Recall the structural equation for output: x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) Schenck (Stata) DSGE June 9, / 21

27 Solving a DSGE model II Recall the structural equation for output: x t = E t (x t+1 ) (r t E t (π t+1 ) z t ) The reduced form for output is: x t = g 1 z t + g 2 u t g 1 and g 2 are coefficients whose values are functions of the structural parameters Schenck (Stata) DSGE June 9, / 21

28 Solving a DSGE model III What about expectations? Roll the solution forward one period, x t+1 = g 1 z t+1 + g 2 u t+1 Schenck (Stata) DSGE June 9, / 21

29 Solving a DSGE model III What about expectations? Roll the solution forward one period, x t+1 = g 1 z t+1 + g 2 u t+1 And take expectations, E t (x t+1 ) = g 1 E t (z t+1 ) + g 2 E t (u t+1 ) Schenck (Stata) DSGE June 9, / 21

30 Solving a DSGE model III What about expectations? Roll the solution forward one period, x t+1 = g 1 z t+1 + g 2 u t+1 And take expectations, E t (x t+1 ) = g 1 E t (z t+1 ) + g 2 E t (u t+1 ) Then roll the state variables back one period E t (x t+1 ) = g 1 ρ z z t + g 2 ρ u u t Schenck (Stata) DSGE June 9, / 21

31 Solving and Estimating a DSGE Model Compactly, the solution to a model is y t = Gz t z t+1 = Hz t + Me t+1 where y t is a vector of control variables, z t is a vector of state variables, and e t is a vector of shocks. Schenck (Stata) DSGE June 9, / 21

32 Solving and Estimating a DSGE Model Compactly, the solution to a model is y t = Gz t z t+1 = Hz t + Me t+1 where y t is a vector of control variables, z t is a vector of state variables, and e t is a vector of shocks. This is a state-space model whose parameters can be estimated by maximum likelihood. Schenck (Stata) DSGE June 9, / 21

33 Solving and Estimating a DSGE Model Compactly, the solution to a model is y t = Gz t z t+1 = Hz t + Me t+1 where y t is a vector of control variables, z t is a vector of state variables, and e t is a vector of shocks. This is a state-space model whose parameters can be estimated by maximum likelihood. Each entry in G and H is a function of the model s structural parameters. Schenck (Stata) DSGE June 9, / 21

34 Solving and Estimating a DSGE Model Compactly, the solution to a model is y t = Gz t z t+1 = Hz t + Me t+1 where y t is a vector of control variables, z t is a vector of state variables, and e t is a vector of shocks. This is a state-space model whose parameters can be estimated by maximum likelihood. Each entry in G and H is a function of the model s structural parameters. Diagonal elements of matrix M hold the standard deviations of the shocks. Schenck (Stata) DSGE June 9, / 21

35 Impulse Responses, again y t = Gz t z t+1 = Hz t + Me t+1 An impulse is a specific sequence of values (1, 0, 0, 0,... ) given to one shock. Schenck (Stata) DSGE June 9, / 21

36 Impulse Responses, again y t = Gz t z t+1 = Hz t + Me t+1 An impulse is a specific sequence of values (1, 0, 0, 0,... ) given to one shock. From the sequence of shocks and the state transition equation, you can obtain the sequence of state variables. Schenck (Stata) DSGE June 9, / 21

37 Impulse Responses, again y t = Gz t z t+1 = Hz t + Me t+1 An impulse is a specific sequence of values (1, 0, 0, 0,... ) given to one shock. From the sequence of shocks and the state transition equation, you can obtain the sequence of state variables. From the sequence of state variables and the policy matrix, you can obtain the sequence of control variables. Schenck (Stata) DSGE June 9, / 21

38 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 New: the ψ parameter in the third equation. New: β no longer appears in multiple equations. (Demand) (Pricing) (Interest rate) (Natural rate of interest) (Monetary policy) Assume: you wish to fix β and estimate the remaining parameters conditional on your choice of β. Schenck (Stata) DSGE June 9, / 21

39 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 June 9, / 21

40 Parameter estimation 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 June 9, / 21

41 Conclusion dsge estimates the parameters of DSGE models Impulse response functions trace the effect of a shock on the model Other features: estat commands to view the model s state-space matrices, predictions of latent states, identification and stability diagnostics Schenck (Stata) DSGE June 9, / 21

42 Thank You! Schenck (Stata) DSGE June 9, / 21