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Dynamic Panel Data: Challenges and Estimation Amine Ouazad Ass. Prof. of Economics

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Outline 1.Problemo: Bias of dynamic fixed effect models – Within estimator – First differenced estimator 2.Consistent estimators 1.Hsiao estimator 2.Arellano-Bond estimator

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PROBLEMO

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Models of the dynamics of investment Where I it is investment, K it is capital. c t is the year-specific constant of the equation, and y it =I it /K it is the investment rate (= growth of capital – depreciation rate).

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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.

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Autoregressive model i is an individual effect, potentially correlated with the yi. Covariates xi can be added to this specification.

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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.

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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.

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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.

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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.

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CONSISTENT ESTIMATORS: HSIAO AND ARELLANO-BOND

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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.

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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.

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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).

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Moment conditions Matrix of instruments. And moment conditions. With:

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GMM estimator The asymptotically efficient consistent estimator of the model minimizes the GMM criterion. Where W N is the inverse of the variance- covariance matrix of the moments. Estimated as:

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Implementation

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CONCLUSIONS

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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.

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