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Dynamic Panel Data: Challenges and Estimation. Amine Ouazad Ass. Prof. of Economics. Outline. Problemo: Bias of dynamic fixed effect models Within estimator First differenced estimator C onsistent estimators Hsiao estimator Arellano-Bond estimator. problemo.
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Dynamic Panel Data:Challenges and Estimation Amine Ouazad Ass. Prof. of Economics
Outline • Problemo:Bias of dynamic fixed effect models • Within estimator • First differenced estimator • Consistent estimators • Hsiao estimator • Arellano-Bond estimator
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.
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 variance-covariance matrix of the moments. • Estimated as:
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 Arellano-Bond 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.
