Dynamic Panel Data:Challenges and Estimation

Amine OuazadAss. Prof. of Economics

Outline

1. Problemo:Bias of dynamic fixed effect models– Within estimator– First differenced estimator

2. Consistent estimators1. Hsiao estimator2. 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 variance-covariance 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 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.