Dynamic Panel Data: Challenges and Estimation Amine Ouazad Ass. Prof. of Economics Outline 1. Problemo: Bias of dynamic fixed effect models – Within estimator – First differenced estimator 2. Consistent estimators 1. Hsiao estimator 2. Arellano-Bond estimator PROBLEMO Models of the dynamics of investment • Where Iit is investment, Kit is capital. • ct is the year-specific constant of the equation, and yit=Iit/Kit is the investment rate (= growth of capital – depreciation rate). Dataset • 703 publicly traded UK firms for which there is consecutive annual data from published company accounts for a minimum of 4 years between 1987 and 2000. Autoregressive model • hi is an individual effect, potentially correlated with the yi. • Covariates xi can be added to this specification. First-differenced estimator • The first-differenced specification does not satisfy A3. • Indeed, there is a negative correlation between lagged changes in y and changes in v (the residual). • This is called “mean reversion.” Individuals that are lucky in one period will see a decline in y in the next period. • Downward bias in the estimator of a. Within-estimator • The within-transformed specification also does not satisfy A3 because the within transformation of the lagged dependent is correlated with the within-transformation of the residual. • Simulation results indicate that in general the within estimator is biased downward. OLS with dummies • We assume throughout that T is small and N is going to infinity. • In this case, the vector of coefficients in OLS with dummies is increasing in size, thus OLS with dummies is not a consistent estimator of the coefficients. • Positive correlation between the fixed effect and the lagged dependent variable. Notes • Random effects models are not affected by the bias. • With random effects, the OLS estimator, or any WLS/GLS gives a consistent estimator of the coefficients. CONSISTENT ESTIMATORS: HSIAO AND ARELLANO-BOND Assumptions • The residuals vit are not correlated across time. Hence the residuals do not have an AR(1) structure. • Corr(vit,vit’)=0 if t is diff. from t’. • Assume that we have at least T>=3 time periods. Hsiao approach • Any instrument correlated with Dyit-1 and uncorrelated with vit will give a consistent 2SLS estimator. • A candidate is yit-2. • With T>3, there are more candidates: twice, k-th time lagged dependent, difference of the lagged dependent. Arellano-Bond • Acknowledge that – there are more than one instrument for T>3. – there is serial correlation of the residuals of the first-differenced equation. • Hence 2SLS is not efficient. • GMM estimator of Holtz-Eakin, Newey and Rosen (1988), and Arellano and Bond (1991). Moment conditions • Matrix of instruments. • And moment conditions. • With: GMM estimator • The asymptotically efficient consistent estimator of the model minimizes the GMM criterion. • Where WN is the inverse of the variancecovariance matrix of the moments. • Estimated as: Implementation CONCLUSIONS Conclusions • A negative effect of the lagged dependent variable can rise suspicion that ‘mean reversion’ is explaining your statistical results. • A practical approach is to assume that the residuals are uncorrelated across time, and either use the (i) Hsiao approach or (ii) the ArellanoBond approach. • The Hsiao approach may yield large confidence intervals. • The AB approach uses a large number of moment conditions and should therefore allow you to get significant coefficient estimates.