DSGE-Models. Calibration and Introduction to Dynare. Institute of Econometrics and Economic Statistics

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1 DSGE-Models Calibration and Introduction to Dynare Dr. Andrea Beccarini Willi Mutschler, M.Sc. Institute of Econometrics and Economic Statistics Summer 2012 Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

2 Calibration and Introduction to Dynare 1 Overview Estimation-Methods 2 Calibration Hints for calibrating a model 3 Exercise 2: Calibration of a RBC-model with monopolistic competition 4 Calibration - Pros & Cons 5 Exercise 3: A simple RBC model - Practicing Dynare Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

3 Overview Estimation-Methods Econometrically, a DSGE-Model is a state-space model of which one has to determine the parameters. Three concepts: 1 Calibration: The parameters are set in such a way, that they closely correspond to some theoretical moment or stylized fact of data. 2 Methods of limited information or weak econometric interpretation: Minimize the distance between theoretical and empirical moments, i.e. General-Method-of-Moments or Indirect Inference. 3 Methods of full information or strong econometric interpretation: The goal is an exact characterization of observed data, i.e. Maximum-Likelihood or bayesian methods. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

4 Calibration Goal: To answer a specific quantitative research question using a structural model. Construct and parameterize the model such, that it corresponds to certain properties of the true economy. Use steady-state-characteristics for choosing the parameters in accordance with observed data. Often: stable long-run averages wages, working-hours, interest rates, inflation, consumption-shares, government-spending-ratios, etc.). You can use micro-studies as well, however, one has to be careful about the aggregation! Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

5 Calibration Hints for calibrating a model Use long-term averages of interest rates, inflation, average growth of productivity, etc. for steady-state values. BUT: Weil 1989) shows, that in models with representative agents there is an overestimation of steady-state interest rates risk-free rate puzzle). It is possible that you get absurd constellation of parameters, like a discount-factor of β > 1. Usual mark-up on prices is around 1.15 Corsetti et al 2012)). Intertemporal elasticity of substitution 1/σ somewhere between σ = 1 and σ = 3 King, Plosser and Rebelo 1988), Rotemberg and Woodford 1992), Lucas 2003)). Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

6 Calibration Hints for calibrating a model Rigidity of prices: For an average price adjustment of months see Keen and Wang 2007). Coefficients of monetary policy: Often Taylor-Rule, you can use the relative coefficients to put more emphasize/weight on the stability of prices or on smoothing the business cycle. Parameters of stochastic processes: Often persistent, small standard-deviations, otherwise you get high oscillations. You can also estimate the production function via OLS Solow-residual). How to choose shocks: Look at similar studies: Christiano, Eichenbaum and Evans 2005), Smets and Wouters 2003), etc.. Ultimately: Try & Error! Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

7 Exercise 2: Calibration of a RBC-model with monopolistic competition Households maximize expected utility over consumption c t and leisure f t = 1 l t, where l t denotes labor: E t t=0 β[logc t ) + ψlog1 l t )], taking account of the following budget constraint: c t + k t+1 = w t l t + r t k t }{{} =y t +1 δ)k t. k t ist the capital stock of the economy, w t the real wage, r t the real interest rate and δ the rate of depreciation. Further, investment is given by: i t = y t c t. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

8 Exercise 2: Calibration of a RBC-model with monopolistic competition In the market for intermediate goods there is monopolistic competition, whereas perfect competition applies to the market for final goods. The production-function of a firm i [0; 1], that sells intermediate goods, is given by: y it = A t kit α l 1 α it, 0 < α < 1, loga t ) = ρloga t 1 ) + ɛ t, ɛ t N0, σ 2 ), with A t denoting the level of technology. Firms cannot influence the real wage w t or the real interest rate r t. However, they have market power over their price p it for their good y it. The intermediate goods are combined into a final good by a Dixit/Stiglitz-type aggregator: 1 y t = 0 y it ) ε ε 1 with ε being the elasticity of substitution. ) ε 1 ε Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24,

9 Exercise 2: Calibration of a RBC-model with monopolistic competition a) Show that, the structural form of the DSGE-model is given by the following equations and interpret these. [ ] 1 1 = βe t 1 + r t+1 δ) 1) c t c t+1 c t w t = ψ 1 l t 2) y t = c t + i t 3) y t = A t kt α lt 1 α 4) w t = 1 α) y t ε 1 l t ε 5) r t = α y t ε 1 k t ε 6) i t = k t+1 1 δ)k t 7) loga t ) = ρloga t 1 ) + ɛ t 8) Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

10 Exercise 2: Calibration of a RBC-model with monopolistic competition b) What are the parameters of the model? c) Write a mod-file for the model and calibrate the vector of parameters µ. Simulate the model for 1000 periods with Dynare. Save the middle 100 observations of c t, y t, i t, w t and r t into an Excel-file as well as into a mat-file. Plot the path of consumption. d) Reformulate the structural equations such that variables are expressed as percentage deviations from steady-state: x t = e logxt) logx)+logx) = xe xt. Write a mod-file for this model. What has changed? Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

11 Calibration Pros Calibration is commonly used in the literature. It gives a first impression, a flavor of the strengths and weaknesses of a model. A good calibration can provide a valuable and precise image of data. Using different calibrations, one can asses interesting implications of different policies: How does the economy react, if the central bank focuses more on smoothing the business cycle than on price stability? What happens to consumption, if the households have a strong intertemporal elasticity of substitution? What if it is low? Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

12 Calibration Cons This Ad-hoc-approach is at the center of criticism of DSGE-models. There is no statistical foundation, it is based upon subjective views, assessments and opinions. Many parameter, such as those of the exogenous processes, leave room for different values and interpretations intertemporal elasticity of substitution, monetary and fiscal parameters, coefficients of rigidity, standard deviations, etc.). Prescott 1986, S. 10) regarding RBC-models: The models constructed within this theoretical framework are necessarily highly abstract. Consequently, they are necessarily false, and statistical hypothesis testing will reject them. This does not imply, however, that nothing can be learned from such a quantitative theoretical exercise. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

13 Exercise 3: A simple RBC model Consider the following model of an economy. Representative agent preferences ) ] 1 t 1 U = E t [log C t ) L1+γ t. 1 + ρ 1 + γ t=1 The household supplies labor and rents capital to the corporate sector. L t is labor services. ρ 0, ) is the rate of time preference. γ 0, ) is a labor supply parameter. C t is consumption. w t is the real wage. r t is the real rental rate. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

14 Exercise 3: A simple RBC model The household faces the sequence of budget constraints K t = K t 1 1 δ) + w t L t + r t K t 1 C t, where K t is capital at the end of period. δ 0, 1) is the rate of depreciation. The production function is given by the expression Y t = A t K α t g) t L t ) 1 α, where g 0, ) is the growth rate and α and β are parameters. A t is a technology shock that follows the process A t = A λ t 1 exp e t ), where e t is an i.i.d. zero mean normally distributed error with standard deviation σ and λ 0, 1) is a parameter. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

15 Exercise 3: A simple RBC model The household problem Lagrangian L = max C t,l t,k t t=1 ) 1 t 1 [ E t 1 + ρ log C t ) L1+γ t 1 + γ µ t K t K t 1 1 δ) w t L t r t K t 1 + C t ) First order conditions ) L 1 t 1 ) 1 = µ t = 0, C t 1 + ρ C t ) L 1 t 1 = L γ t µ t w t ) = 0, L t 1 + ρ ) L 1 t 1 ) 1 t = µ t + E t µ t+11 δ + r t+1)) = 0. K t 1 + ρ 1 + ρ ]. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

16 Exercise 3: A simple RBC model First order conditions Eliminating the Lagrange multiplier, one obtains L γ t = w t C t, 1 = 1 C t 1 + ρ E t ) 1 r t δ). C t+1 Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

17 Exercise 3: A simple RBC model The firm problem max A t K α t g) t ) 1 α L t rt K t 1 w t L t. L t,k t 1 First order conditions: r t = αa t Kt 1 α g) t ) 1 α L t, w t = 1 α)a t Kt 1 α 1 + g) t ) 1 α L α t. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

18 Exercise 3: A simple RBC model Goods market equilibrium K t + C t = K t 1 1 δ) + A t Kt 1 α 1 + g) t ) 1 α L t. }{{} w tl t+r tk t Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

19 Exercise 3: A simple RBC model Dynamic Equilibrium 1 = 1 ) 1 C t 1 + ρ E t r t δ), C t+1 L γ t = w t C t, r t = αa t K α 1 t 1 w t = 1 α)a t K α t g) t L t ) 1 α, 1 + g) t ) 1 α L α t, K t + C t = K t 1 1 δ) + A t K α t g) t L t ) 1 α, loga t ) = λloga t 1 ) + e t. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

20 Exercise 3: A simple RBC model Existence of a balanced growth path Good markets equilibrium for each period t: K t + C t = K t 1 1 δ) + A t K α t g) t L t ) 1 α. So, there must exist growth rates g c and g k such that 1 + g k ) t K g c ) t C 1 = 1 + g k ) t 1 + g k K 1 1 δ) + A t 1 + gk ) t 1 + g k K 1 ) α 1 + g) t L t ) 1 α ) 1 + t gc K 1 + C 1 = 1 + g k ) K α ) 1 K1 1 + g t ) 1 α 1 δ) + A t L t. 1 + g }{{ k 1 + g } k 1 + g k K 0 This is only valid, if g c = g k = g. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

21 Exercise 3: A simple RBC model Stationarized model Let s define Ĉ t = C t /1 + g) t, K t = K t /1 + g) t, ŵ t = w t /1 + g) t. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

22 Exercise 3: A simple RBC model Stationarized model continued) ) 1 Ĉ t 1 + g) = 1 t 1 + ρ E 1 t Ĉ t g)1 + g) r t t δ), L γ t = ŵt1 + g) t Ĉ t 1 + g), t 1 + g) r t = αa t K t t g ŵ t 1 + g) t = 1 α)a t ) α g) t L t ) 1 α, ) α 1 + g) t ) 1 α L α t, 1 + g) K t t g Ĉt) Kt g) t = K 1 + g) t t 1 1 δ) 1 + g 1 + g) + A t K t ) α t g) t ) 1 α L t. 1 + g Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

23 Exercise 3: A simple RBC model Stationarized model continued) ) 1 = 1 Ĉ t 1 + ρ E 1 t Ĉ t g) r t δ), L γ t = ŵt Ĉ t, ) α 1 L 1 α t, Kt 1 r t = αa t 1 + g ) α Kt 1 ŵ t = 1 α)a t L α t, 1 + g K t + Ĉt = K t g 1 δ) + A t loga t ) = λloga t 1 ) + e t. ) α Kt 1 L 1 α t, 1 + g Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24

24 Exercise 3: A simple RBC model Practicing Dynare a) Write a mod-file for this simple RBC-model and use for calibration: α = 0.33, δ = 0.1, ρ = 0.03, λ = 0.97, γ = 0, g = Use initval with these values: C = 1, K = 3, L = 0.9, w = 1, r = 0.15, A = 1. b) Show that the steady-state implies: A = 1, r = 1 + g)1 + δ) + δ 1 ) 1 α r ) r 1 α 1 L = r α δ g, K = 1 + g) L α α) ) K K α C = 1 δ) 1 + g + L 1 α K, w = C 1 + g c) Use this analytical solution for the mod-file, i.e. use steady state model instead of initval. Dynare creates a steady-state m-file. Have a look at it. Willi Mutschler Institute of Econometrics) DSGE-Models Summer / 24