Estimation and Inference on Dynamic Panel Data Models with Stochastic Volatility

Transcription

1 Estimation and Inference on Dynamic Panel Data Models with Stochastic Volatility Wen Xu Department of Economics & Oxford-Man Institute University of Oxford (Preliminary, Comments Welcome)

2 Theme y it = B(L)y i,t 1 + γx it + µ i + ε it ε it : martingale dierence sequences parameterized through stochastic volatility In a macroeconomic context: T is relatively large (>20); N is relatively small (<100) 2 /32

3 Contribution Ecient estimation by particle lter techniques Frequentist approach: maximization of approximate simulated-likelihood Bayesian approach: particle Metropolis-Hastings sampler Two-step LSDV-QML: fast but less ecient Ignoring the existence of stochastic volatility might induce systematically false rejection in panel unit root tests Monte Carlo studies show that particle-lter based estimators are more precise than other estimators on average in the presence of stochastic volatility and even in the case of homoscedasticity Our methodology is straightforwardly applied to panel VAR models 3 /32

4 Motivation Time-varying volatility in macroeconomic time series: Evidence of conditional heteroskedasticity in the residuals of many estimated dynamic regression models in macroeconomics: Weiss (1984) U.S. real GDP: Kim and Nelson (1999), McConnell and Perez-Quiros (2000), Blanchard and Simon (2001). Other macro variables: Stock and Watson (2003), Fernández-Villaverde and Rubio-Ramírez (2013). 4 /32

5 Motivation Figure : U.S. Real GDP Growth, Absolute Deviations from Mean (from Fernández-Villaverde and Rubio-Ramírez, 2013) 5 /32

6 Motivation Some attempts in modeling stochastic volatility in macroeconomics DSGE: Fernández-Villaverde and Rubio-Ramírez (2007) VAR: e.g., Koop and Korobilis (2010) Hamilton (2010) Hypothesis tests on the mean might be invalid if the variance is misspecied Statistical eciency gains can be obtained by incorporating observed features of time-varying volatility into the estimation of the conditional mean. 6 /32

7 Motivation Dynamic panel data models in macroeconomic applications: study common relationships across countries or regions purchasing power parity mean reversion of interest rates growth convergence Panel unit root tests Levin and Lin (1992) Harris and Tzavalis (1999): N( ˆβ LSDV T + 1 ) d N(0, 3(17T 2 20T + 17) 5(T 1)(T + 1) 3 ) 7 /32

8 Baseline Model y it = βy i,t 1 + µ i + ε it ε it = σ it ɛ it log(σ 2 it) = µ + φ(log(σ 2 i,t 1) µ) + η it ( ɛit η it ) [ ] 1 0 N(0, 0 θ 2 ) 8 /32

9 Particle Filters Nonlinear and non-gaussian state space models y t = h t (x t, ɛ t ) x t = m t (x t 1, η t ) Particle lters: simulation-based ltering techniques to approximate the posterior density p(x 1:t y 1:t, θ) by a discrete distribution made of weighted draws (x (j), ŵ (j) 1:t t ) (j = 1,..., M) termed particles. Algorithms dier mainly in the choices of incremental importance distributions and resampling algorithms which are aimed to improve the level of statistical eciency in terms of Monte Carlo variation. bootstrap lter (Gordon et al., 1993) auxiliary particle lters (Pitt and Shephard, 1999) mixture Kalman lters (Chen and Liu, 2000)... 9 /32

10 Mixture Kalman Filters (Chen and Liu, 2000) y t = Z(α t,1 )α n,2 + ε t α t,2 = T (α t,1 )α t 1,2 + η t ε t N(0, H(α t,1 )) η t N(0, Q(α t,1 )) α t,1 follows a rst order Markov process Particle state variables α t,1 : simulation M draws or particles per period: (α (j) t,1, ŵ (j) ) (j = 1,..., M) i,t t 1 Kalman state variables α t,2 : conditionally normal given α t,1, Kalman lters 10 /32

11 State Space Forms y it = [ ] α it β β 0 0 α it = α i,t σ it ɛ it log(σ 2 it) = µ + φ(log(σ 2 i,t 1) µ) + η it where the particle state variable is σ it and the Kalman state vector β y i,t 1 α it = σ it ɛ it σ i,t 1 ɛ i,t 1. σ it ɛ it Variables of interests particle state variable σ (j) it conditonal means of Kalman state vector given Y i,t 1 and σ 2(j) it : m (j) = E(α it t 1 it Y i,t 1, σ 2(j) it ) conditonal variances of Kalman state vector Y i,t 1 and σ 2(j) it : Σ (j) = Var(α it t 1 it Y i,t 1, σ 2(j) it ) conditonal likelihood of observations log[ˆl( yit Y i,t 1, β, µ, φ, θ)] 11 /32

12 Algorithm For i = 1,..., N, 1. Set the starting values σ 2(j) i2 For t = 2,..., T, 2. Draw the particle state variable, m(j) i2 1 and Σ(j) i2 1. log(σ 2(j) i,t+1 ) = (1 φ)log(µ) + φlog(σ2(j) ) + θη (j) it i,t+1 3. Compute the conditional likelihood for each particle j. That is, l (j) it = 0.5log V (j) it 0.5v (j) it (V (j) ) 1 v (j) it it 12 /32

13 Algorithm 4. Update the particle weight w (j) i,t+1 t = w (j) + it t 1 l (j) it ŵ (j) i,t+1 t = (j) exp(wi,t+1 t ) M (j) j=1 exp(w i,t+1 t ) 5. Resample with replacement M particles σ 2(j) i,t+1, m(j) and Σ(j) it t 1 it t 1 with the weight ŵ (j) every three increments. After doing this, reset i,t+1 t w (j) it t 1 = 0 and ŵ (j) it t 1 = 1 M. 6. Update Kalman lter estimates of m (j) and Σ(j) it t 1 it t Go back to step 2. Simulation-based estimate of the joint log-likelihood M log[ˆl( yit Y i,t 1, β, µ, φ, θ)] = log[ ŵ (j) (j) exp(l it t 1 it )] j=1 13 /32

14 Resampling Weight degeneracy problem As the series grows over time, one particle's normalized importance weight converges to one while the others converge to zero. In other words, the discrete distribution made of weighted draws would become degenerate. The estimator will be imprecise because it is nally a function of a single draw. Resampling: aimed to mitigate the weight degeneracy problem by eliminating the particles which have low importance weights and multiplying the heavily weighted particles. multinomial resampling: Gordon et al. (1993) It draws new particles { σ 2(j) it, m (j) (j) it t 1, Σ it t 1 }M j=1 from the point mass distribution {σ 2(j) it, m (j) it t 1, Σ(j) it t 1, ŵ (j) it t 1 }M j=1 stratied resampling (Kitagawa, 1996) residual resampling (Liu and Chen, 1998) systematic resampling (Carpenter et al., 1999) 14 /32

15 Frequentist approach Direct maximization of the simulated likelihood suers from discontinuity induced from the generalized inverse operation at the resampling stage, which makes invalid the common gradient-based optimization methods. Pitt (2002) overcame the problem of non-smoothness by developing a new resampling method, but his method is valid only when the dimension of state space is one. Expectation Maximization algorithm: numerically stable and computationally cheap but only guaranteed to be locally optimal. Olsson and Rydén (2008): step functions or B-spline interpolation, and showed consistency and asymptotic normality of the estimators which maximize the approximate likelihood under some assumptions. 15 /32

16 Frequentist approach Olsson and Rydén (2008) Discretize the parameter space Ω by a grid Ω {ω g } G g=1 Ω. Let [ω] denote the closest point in the grid to ω Ω log[ˆl( y1,..., y N ω)] log[ˆl( y1,..., y N [ω])] Maximize approximate simulated-likelihood Advantage: theoretical proof of consistency and asymptotic normality, good nite sample performance, fast. Limit: asymptotic proof requires compact state space potentially released given new results of uniform convergence properties in time dimension in the general state space models. 16 /32

17 Bayesian approach Andrieu et al. (2010) combined particle lters with standard MCMC algorithms Particle independent Metropolis-Hastings sampler A random walk proposal is used for log(β), log(µ), log(φ) and log(θ). Noninformative prior f (β, µ, φ, θ) = 1. Probability of accepting β, µ, φ and θ can be written as min { 1, ˆL( y 1,..., y N β, µ, φ, θ )β µ φ θ ˆL( y 1,..., y N β, µ, φ, θ)βµφθ Metropolis-Hastings sampler can work provided the estimated likelihood is unbiased. }. 17 /32

18 LSDV-QML LSDV ˆβ LSDV = N i=1 y i,( 1)Qy i N i=1 y i,( 1)Qy i,( 1) where Q = I ιι /T and ι is a T -dimension vector of ones. 18 /32

19 LSDV-QML Assumptions 1. β < The initial values of y it follow the steady state distribution y i0 = µ i 1 β + t=0 βt ε i, t. 3. The stochastic volatility is stationary: φ < The initial values of log(σ 2 ) follow the steady state distribution it log(σi0 2 ) = µ + t=0 βt η i, t. 5. {ɛ it } (i = 1,..., N; t = 1,... T ) are i.i.d. variables across both time and individuals with E(ɛ it ) = 0 and Var(ɛ it ) = 1 and E(ɛ 4 ) = κ < and independent of µ it i and η it for all i and t. 6. {η it } (i = 1,..., N; t = 1,... T ) are i.i.d. variables normally distributed across both time and individuals with E(η it ) = 0 and Var(η it ) = θ 2 <. 19 /32

20 LSDV-QML (LSDV Estimators: Stationary) Under Assumptions 1-6, as N and T, NT [ ˆβ LSDV (β 1 T (1 + β))] d N(0, B(β, φ, θ)) where B(β, φ, θ) = (1 β 2 ) 2 t=1 [exp( θ2 φ t 1 φ 2 )β 2t 2 ]. Stochastic volatility changes asymptotic variances of the estimator while keeps unchanged asymptotic means. If φ > 0 and θ > 0, B(β, φ, θ) > B(β, 0, 0) which is equal to 1 β 2 same as in Alvarez and Arellano (2003). When β = 0.7, for example, asymptotic variance is approximately 0.51 for the model without stochastic volatility, but can be 1.52 when φ = 0.9 and θ = /32

21 LSDV-QML Assumptions 7. The data generation process has a unit root: β = The initial values y i0 are xed. (LSDV Estimators: Unit Root) Under Assumptions 3-8, as N and T is xed, N( ˆβ LSDV (1 3 T + 1 )) d N(0, B u (φ, θ)) where B u (φ, θ) = 36(2 5T +2T 2 ) 5( 1+T )T (1+T ) 3 exp( θ2 1 φ 2 )κ + T 1 t=1 [exp( θ2 φ t 1 φ 2 )C(t)] in which C(t) = 36( 9t 5 +30t 4 T 5t 3 T (2+11T )+5t 2 T(1+2T +13T 2 ) 2t( 2+5T +5T 2 +20T 4 )+T( 4+10T +5T 2 +9T 4 )) 5( 1+T ) 2 T 2 (1+T ) /32

22 LSDV-QML If φ > 0 and θ > 0, B u (φ, θ) > B u (0, 0) which is equal to 3(17T 2 20T +17) 5(T 1)(T +1) 3 as in Harris and Tzavalis (1999). When T = 20, for example, asymptotic variance is approximately 0.02 for the model without stochastic volatility, but can be 0.07 when φ = 0.9 and θ = 0.5. B u (φ, θ) > B u (0, 0) implies that naively ignoring stochastic volatility would systematically reject Harris and Tzavalis' panel unit root test more frequently than it should be, although the estimates lose only a little statistical eciency, so stochastic volatility corrected variance is needed. 22 /32

23 LSDV-QML Estimate the error terms using the rst-step estimator, i.e., ˆε it = y it ˆβ LSDV y i,t 1 ˆµ i where ˆµ i = 1 T T t=1 (y it ˆβ LSDV y i,t 1 ) is a consistent estimator of the individual eect. Calculate the quasi-likelihood of ˆε it via Kalman lters in a state space form (Harvey and Shephard, 1996) log(ˆε 2 it) = log(σ 2 it) + E(log(ɛ 2 it)) + ξ it log(σ 2 it) = µ + φ(log(σ 2 i,t 1) µ) + η it where E(log(ɛ 2 )) = 1.27 and ξ it it zero and variance is a normal variable with mean 23 /32

24 Panel VAR y it = B(L)y i,t 1 + µ i + ε it ε it = V 1/2 ɛ it it h it = µ i + Φ i (h i,t 1 µ i ) + η it ( ɛit η it ) [ ] Σɛ 0 N(0, ). 0 Σ η 24 /32

25 Panel VAR 2-variable vector y it = By i,t 1 + µ i + ε it [ ] y it = α it β 11 β 12 β 11 0 β β 21 β 22 β 21 0 β [ ] α it = α i,t ɛit,1 1 0 σ it ɛ it, β 11 y i,t 1,1 + β 12 y i,t 1,2 β 21 y i,t 1,1 + β 22 y i,t 1,2 where the state vector α it = σ it ɛ it,1 σ it 1 ɛ i,t 1,1 σ it ɛ it,1. σ it ɛ it,2 σ it 1 ɛ i,t 1,2 σ it ɛ it,2 25 /32

26 Monte Carlo Studies y it = βy i,t 1 + (1 β)µ i + ε it µ i = τ( q i 1 )ς i 2 ε it = σ it ɛ it log(σ 2 it) = µ + φ(log(σ 2 i,t 1) µ) + η it where q i χ 2 1 ; ɛ it, ς i N(0, 1) and η it N(0, θ 2 ); q i, ɛ it, ς i and are all i.i.d. within series and also independent of each others. η it τ = 1: the degree of cross-section to time-series variation. µ = log(0.04): long-run mean of the log volatility. less persistent β = 0.5; persistent β = 0.9; unit root β = 1. persistent stochastic volatility: φ = 0.9, θ = 0.5; homoscedasticity : φ = θ = /32

27 Simulation results of ˆβ when β = 0.5 T N φ LSDV IV GMM SGMM PF (0.095) (0.193) (0.110) (0.190) (0.093) (0.102) (0.176) (0.123) (0.130) (0.095) (0.090) (0.110) (0.062) (0.131) (0.088) (0.322) (0.121) (0.095) (0.100) (0.101) (0.044) (0.075) (0.048) (0.249) (0.044) (0.053) (0.094) (0.058) (0.177) (0.048) (0.035) (0.047) (0.037) (0.106) (0.035) (0.045) (0.071) (0.050) (0.076) (0.041) 27 /32

28 Simulation results of ˆβ when β = 0.9 T N φ LSDV IV GMM SGMM PF (0.134) (2.111) (0.178) (0.168) (0.133) (0.141) (0.803) (0.185) (0.125) (0.128) (0.131) (0.160) (0.138) (0.053) (0.127) (0.138) (0.164) (0.163) (0.084) (0.127) (0.050) (0.102) (0.058) (0.383) (0.045) (0.057) (0.119) (0.065) (0.244) (0.040) (0.046) (0.058) (0.056) (0.108) (0.043) (0.049) (0.088) (0.063) (0.082) (0.034) 28 /32

29 Simulation results of ˆβ when β = 1 T N φ LSDV GMM SGMM PF (0.155) (0.230) (0.119) (0.152) (0.163) (0.247) (0.067) (0.158) (0.154) (0.233) (0.080) (0.149) (0.164) (0.273) (0.055) (0.159) (0.065) (0.078) (0.281) (0.068) (0.069) (0.087) (0.180) (0.055) (0.061) (0.090) (0.094) (0.065) (0.063) (0.100) (0.062) (0.051) 29 /32

30 Other simulation results LSDV-QML PF β T N φ ˆφ ˆθ ˆµ ˆφ ˆθ ˆµ (0.402) (0.360) (0.003) (0.410) (0.165) (0.003) (0.042) (0.133) (0.008) (0.050) (0.121) (0.009) (0.413) (0.302) (0.002) (0.412) (0.089) (0.002) (0.030) (0.087) (0.004) (0.028) (0.066) (0.006) 30 /32

31 Other simulation results Robustness of ˆβ: the shocks have t-distribution with degree of freedom 5. β = 0.5 T N φ LSDV PF t Normal t Normal (0.042) (0.044) (0.039) (0.044) (0.078) (0.053) (0.045) (0.048) (0.036) (0.035) (0.034) (0.035) (0.073) (0.045) (0.041) (0.041) 31 /32

32 Other simulation results y it = βy i,t 1 + αx it + (1 β)µ i + ε it x it = qx i,t 1 + ζ it N = 50, T = 50, β = 0.5, α = 0.7, φ = 0.9, θ = 0.5, µ = log(0.04), q = 0.9, ζ it N(0, 0.01) ˆβ ˆα ˆφ ˆθ ˆµ LSDV-QML (0.038) (0.045) (0.030) (0.087) (0.005) PF (0.042) (0.071) (0.046) (0.070) (0.009) 32 /32