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).
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 i 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.
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 W N 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.