Nonlinear DSGE Model of the Czech Economy with Time ... ·...
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Transcript of Nonlinear DSGE Model of the Czech Economy with Time ... ·...
Contents Motivation Nonlinearity Model Empirical results Conclusion
Nonlinear DSGE Model of the Czech Economywith Time-varying Parameters
Stanislav Tvrz, Osvald Va²í£ek
Faculty of Economics and Administration, Masaryk University, Brno
Modern Tools for Financial Modeling and Analysis
June 5, 2014, Praha
Contents Motivation Nonlinearity Model Empirical results Conclusion
1 Contents
2 Motivation
3 Nonlinearity
4 Model
5 Empirical results
6 Conclusion
Contents Motivation Nonlinearity Model Empirical results Conclusion
Motivation
Did any structural changes occur during the turbulent periodof recent nancial and economic crisis of 20082009?
Which structural parameters did change? Were the changestemporary or permanent?
How was the behaviour of the economy aected by thesestructural changes?
Which changes are specic for the Czech economy and whichcorrespond to the europe-wide trends?
Contents Motivation Nonlinearity Model Empirical results Conclusion
Time-varying parameters within state-space models
time-varying parameters are dened as unobserved states
θt = (1− αθt ) · θt−1 + αθt · θ + νθt
θ is initial value of parameter θtαθt is a time-varying adhesion parameter (panel)
αθ = 0⇒ random walk,αθ = 1⇒ white noise around θt ,αθ = 0.25⇒ our choice
νθt ∼ N(0, σθν)
⇒ nonlinearity is introduced into the model ⇒ nonlinearstate-space model
xt = g(xt−1,wt−1)
yt = h(xt , vt)
Contents Motivation Nonlinearity Model Empirical results Conclusion
Non-linear ltering methods
Kalman lter is optimal for linear systems
Extended Kalman lter (Jacobian matrix of the state vector)can be used for nonlinear systems but performs poorly forsevere nonlinearities
⇒ Nonlinear lterswith additive Gaussian noise - Extended Kalman lters
Monte Carlo based
Transformation based
with non Gaussian noise - Particle lters
Gaussian particle lter
Unscented particle lter
Contents Motivation Nonlinearity Model Empirical results Conclusion
Non-linear particle lter
1 Initialization: t = 0, set the prior mean x0 and covariancematrix P0 for the state vector xt .
2 Generating particles: Draw a total of N particles x(i)t+1,
i = 1, . . . ,N from distribution p(xt+1) with mean x t+1|t andcovariance matrix Pt+1|t (transition equation). Calculatey t+1|t (measurement equation) and covariance matrices Py ,yand Px ,y .
3 Kalman lter:
Kt+1 = Px ,y (Py ,y )−1,x t+1 = x t+1|t + Kt+1(yt+1 − y t+1|t),
Pt+1 = Pt+1|t − Kt+1Py ,y (Kt)T
4 Time Update: t = t + 1.
Contents Motivation Nonlinearity Model Empirical results Conclusion
Non-linear particle lter, diagram
InitializationGenerating Particles
UnscentedKalmanFilter
TimeUpdate
Contents Motivation Nonlinearity Model Empirical results Conclusion
Non-linear particle lter setting
20 runs of the UPF with 30.000 particles each were calculatedfor the second order approximation of the model.
Initial values of the time-varying parameters (θ) were set tothe posterior means of the Bayesian estimation of the modelwith constant parameters
Standard deviations of time-varying parameter innovations(σθν) were set proportional to the estimated posterior means(10%).
Bayesian Random Walk Metropolis-Hastings estimation: twochains of 1.000.000 draws each, 50% burn-in sample,acceptance rate near 30%.
Contents Motivation Nonlinearity Model Empirical results Conclusion
Model
Overall structure of the DSGE model of a small open economy(SOE) is based on Shaari (2008), who incorporated thenancial accelerator mechanism á la Bernanke et al. (1999)into the basic SOE model of Galí and Monacelli (2005).
The model contains following optimizing representative agents:households, entrepreneurs and domestic and foreign retailers.
The monetary policy of the central bank is modelled with theuse of forward looking Taylor rule.
Foreign sector observables are modelled as SVAR(1) block.
Contents Motivation Nonlinearity Model Empirical results Conclusion
Data
The model is estimated on two sets of data: CZ+EA (blue)and EA+US (red)
Quarterly time series of the period between 1999Q2 and2013Q4, 59 observations
Domestic economy: real aggregate product, real investment,consumer price index , 3-month PRIBOR (3-month EURIBOR)
Foreign economy EA17 (US): real aggregate product, CPIindex and 3-month EURIBOR (3-month T-Bill rate)
CZK/EUR (EUR/USD) real exchange rate
Original time series were transformed so as to expresspercentage deviations from steady state (HP lter, λ = 1600)
Contents Motivation Nonlinearity Model Empirical results Conclusion
Filtered observables (deviations from steady state in per cent)
yobs ... observed real output
2000 2002 2004 2006 2008 2010 2012 2014
−2
0
2
4
robs ... observed interest rate
2000 2002 2004 2006 2008 2010 2012 2014
−1
0
1
2
πobs ... observed CPI inflation
2000 2002 2004 2006 2008 2010 2012 2014
−2
0
2
4
y∗obs ... observed foreign real output
2000 2002 2004 2006 2008 2010 2012 2014
−2
0
2
r∗obs ... observed foreign interest rate
2000 2002 2004 2006 2008 2010 2012 2014
−1
0
1
2
π∗obs ... observed foreign CPI inflation
2000 2002 2004 2006 2008 2010 2012 2014−6
−4
−2
0
2
rerobs ... observed real exchange rate
2000 2002 2004 2006 2008 2010 2012 2014
−10
0
10
invobs ... observed real investment
2000 2002 2004 2006 2008 2010 2012 2014
−5
0
5
10
15
Contents Motivation Nonlinearity Model Empirical results Conclusion
Calibration
Parameter Value
β Discount factor 0.995α Capital share in production 0.350δ Capital depreciation rate 0.025µ Steady-state domestic mark-up 1.200Ω Household's share in labour supply 0.990
Contents Motivation Nonlinearity Model Empirical results Conclusion
Estimation results
Prior CZ Posterior EA PosteriorParameter Distribution Mean Std Mean Std Mean Std
Structural parametersΥ Habit persistence B 0.60 0.05 0.60 0.05 0.68 0.06Ψ Inv. elast. of lab. supply G 2.00 0.50 1.25 0.35 0.88 0.26ψB Debt-elastic risk premium G 0.05 0.02 0.02 0.01 0.02 0.01η Home/foreign elast. subst. G 0.65 0.10 0.52 0.08 0.43 0.02κ Price indexation B 0.50 0.10 0.49 0.09 0.44 0.09γ Pref. bias to foreign goods B 0.40 0.15 0.48 0.07 0.27 0.04θH Home goods Calvo B 0.70 0.10 0.82 0.03 0.80 0.03θF Foreign goods Calvo B 0.70 0.10 0.84 0.02 0.81 0.03ψI Capital adjustment costs G 8.00 3.00 11.5 2.92 15.5 3.35Financial frictionsΓ Leverage ratio ss ratio G 2.00 0.50 1.41 0.24 1.16 0.21ς Bankruptcy rate B 0.025 0.015 0.05 0.02 0.02 0.01χ Financial accelerator G 0.05 0.015 0.04 0.01 0.04 0.01Taylor ruleρ Interest rate smoothing B 0.70 0.10 0.86 0.02 0.74 0.04βπ Ination weight G 1.50 0.20 1.75 0.23 1.76 0.23Θy Output gap weight G 0.50 0.20 0.16 0.05 0.22 0.06
Contents Motivation Nonlinearity Model Empirical results Conclusion
Estimation results, shock parameters
Prior Posterior MeanParameter Distribution Mean Std CZ EA
Autocorrealtion coecientsρY Domestic productivity B 0.50 0.20 0.58 0.47ρUIP Uncovered interest parity B 0.50 0.20 0.63 0.79ρLOP Law of one price B 0.50 0.20 0.93 0.84ρNW Entrepreneurial net worth B 0.50 0.20 0.44 0.40Standard deviationsσY Domestic productivity IG 1.00 ∞ 1.09 0.38σUIP Uncovered interest parity IG 0.50 ∞ 0.27 0.24σLOP Law of one price IG 0.50 ∞ 3.22 5.05σNW Entrepreneurial net worth IG 1.00 ∞ 1.84 1.48σMP Monetary policy IG 0.50 ∞ 0.08 0.10σy∗ Foreign output IG 1.00 ∞ 0.52 0.54σπ∗ Foreign ination IG 0.50 ∞ 0.14 0.30σr∗ Foreign interest rate IG 0.50 ∞ 0.08 0.10
Contents Motivation Nonlinearity Model Empirical results Conclusion
Measurement errors (deviations from steady state in per cent)
yobs − y ... real output error
2002 2004 2006 2008 2010 2012 2014−0.2
0
0.2
robs − r ... interest rate error
2002 2004 2006 2008 2010 2012 2014
−0.2
0
0.2
0.4
πobs − π ... CPI inflation error
2002 2004 2006 2008 2010 2012 2014
−0.05
0
0.05
y∗obs − y∗ ... foreign real output error
2002 2004 2006 2008 2010 2012 2014
−0.2
0
0.2
r∗obs − r∗ ... foreign interest rate error
2002 2004 2006 2008 2010 2012 2014
−0.3
−0.2
−0.1
0
0.1
π∗obs − π∗ ... foreign CPI inflation error
2002 2004 2006 2008 2010 2012 2014
−0.04
−0.02
0
0.02
0.04
rerobs − rer ... real exchange rate error
2002 2004 2006 2008 2010 2012 2014
−0.1
0
0.1
invobs − inv ... real investment error
2002 2004 2006 2008 2010 2012 2014
−0.1
0
0.1
Contents Motivation Nonlinearity Model Empirical results Conclusion
Filtered shock innovations (deviations from steady state in per cent)
εy ... productivity innovations
2002 2004 2006 2008 2010 2012 2014
−1.5
−1
−0.5
0
0.5
εMP ... monetary policy innovations
2002 2004 2006 2008 2010 2012 2014
−0.1
0
0.1
εUIP ... UIP innovations
2002 2004 2006 2008 2010 2012 2014
−0.4
−0.2
0
0.2
0.4
εLOP ... LOP innovations
2002 2004 2006 2008 2010 2012 2014−10
0
10
εNW ... survival rate innovations
2002 2004 2006 2008 2010 2012 2014
−5
0
5
εy∗ ... foreign output innovations
2002 2004 2006 2008 2010 2012 2014
−1.5
−1
−0.5
0
0.5
εr∗ ... foreign interest rate innovations
2002 2004 2006 2008 2010 2012 2014−0.2
−0.1
0
0.1
επ∗ ... foreign inflation innovations
2002 2004 2006 2008 2010 2012 2014
−0.5
0
0.5
Contents Motivation Nonlinearity Model Empirical results Conclusion
Selected ltered variables (deviations from steady state in per cent)
efp ... Ext. fin. premium
2002 2004 2006 2008 2010 2012 2014
−0.5
0
0.5
k ... Capital stock
2002 2004 2006 2008 2010 2012 2014
−0.5
0
0.5
n ... Entrepr. net worth
2002 2004 2006 2008 2010 2012 2014
−10
0
10
Contents Motivation Nonlinearity Model Empirical results Conclusion
Time-varying parameters (deviations from initial value in per cent)
χ ... financial accelerator
2002 2004 2006 2008 2010 2012 2014
−10
−5
0
ς ... bankruptcy rate
2002 2004 2006 2008 2010 2012 2014
0
0.5
1
1.5
ΨI ... capital adj. costs
2002 2004 2006 2008 2010 2012 2014
0
5
10
Γ ... leverage rate
2002 2004 2006 2008 2010 2012 2014
−10
−5
0
γ ... foreign goods preference bias
2002 2004 2006 2008 2010 2012 2014
0
5
10
15
θH ... domestic Calvo param.
2002 2004 2006 2008 2010 2012 2014
−5
0
5
10
βπ ... Taylor rule, inflation
2002 2004 2006 2008 2010 2012 2014
−4
−3
−2
−1
0
Θy ... Taylor rule, output gap
2002 2004 2006 2008 2010 2012 2014
−1
0
1
ρ ... Taylor rule, smoothing
2002 2004 2006 2008 2010 2012 2014
0
5
10
Contents Motivation Nonlinearity Model Empirical results Conclusion
Correlation of time-varying parameters
Parameter Correlation
χ Financial accelerator 0.66ς Bankruptcy rate -0.13ΨI Capital adj. costs 0.71Γ Leverage ratio ss 0.64γ Foreign goods pref. bias 0.19θH Domestic Calvo param. 0.34βπ Taylor rule, ination -0.20Θy Taylor rule, output gap 0.25ρ Tylor rule, smoothing -0.18
Contents Motivation Nonlinearity Model Empirical results Conclusion
Conclusion
Results of the estimation suggest that some structural changesoccurred during recent nancial and economic crisis
The structural changes were probably temporary as theparameters tend to return to their initial values
Some parameters showed only negligible deviations from theirinitial values (elast. of intertemp subst., risk premium elast.,ination indexation)
Some parameters of the nancial sector, openness parameter,Calvo parameters and interest rate smoothing parameterchanged markedly during the recent economic crisis
Contents Motivation Nonlinearity Model Empirical results Conclusion
Conclusion (continued)
Overall, the estimated trajectories show many similaritiesbetween the development in the Czech economy and in theeuro area with some dierences in the magnitude of thedeviations and timing.
The dierences can be attributed to earlier onset and moredramatic course of the nancial crisis in the euro area than inrelatively sheltered Czech economy.
The trajectories of the Taylor rule parameters also showinteresting dierences in the behaviour of the ECB and CNB.
Contents Motivation Nonlinearity Model Empirical results Conclusion
References
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Formulating Monetary Policy: An Empirical Approach. Doctoral thesis,
College of Business and Economics, The Australian National University, 2008.
Van Der Merwe, R., Doucet, A., De Freitas, N., and Wan, E.: TheUnscented Particle Filter. CUED Technical Report 380, Cambridge
University Engineering Department, 2000.
Va²í£ek, O., Tonner, J., and Polanský, J.: Parameter Drifting in a
DSGE Model Estimated on Czech Data. Czech Journal of Economics and
Finance, vol. 61, no. 5, UK FSV, Praha, 2011, 510524.
Yano, K.: Time-varying Analysis of Dynamic Stochastic General
Equilibrium Models Based on Sequential Monte Carlo Methods.
ESRI Discussion Paper No. 231. Economic and Social Research Institute, 2010.
Contents Motivation Nonlinearity Model Empirical results Conclusion
Thank you for your attention!
