Efficient Estimation Using Panel Data

Abstract

IN AN IMPORTANT RECENT PAPER, Hausman and Taylor (1981)-hereafter HT-considered the instrumental-variable estimation of a regression model using panel data, when the individual effects may be correlated with a subset of the explanatory variables. They provided a simple consistent estimator and an efficient estimator. More recently, Amemiya and MaCurdy (1986)-hereafter AM-have suggested an alternative estimator which is more efficient than the HT estimator, under certain conditions and given stronger assumptions than HT made. However, the relationship between the HT and AM papers is less clear than it might be, in part because of notational differences between the two papers. In this paper we clarify the relationship between the HT and AM estimators, and we show that the difference between these estimators lies in the treatment of the time-varying explanatory variables which are uncorrelated with the effects: HT use each such variable as two instruments (means and deviations from means), while AM use such variables as T + 1 instruments (as deviations from means and also separately for each of the T available time periods). This enables us to make clear the conditions under which the AM estimator is more efficient than the HT estimator. We also present each estimator in a form which allows it to be calculated using standard instrumental-variables (two-stage least squares) software. Following the AM path one step further, we then define a third (BMS) estimator which, under yet stronger assumptions, is more efficient than the AM estimator. Both HT and AM use as instruments the deviations from means of the time-varying variables which are correlated with the effects. A more efficient estimator may be obtained by using separately the (T - 1) linearly independent values of these deviations from individual means. Consistency requires that these be legitimate instruments, and whether this is so depends on why these time-varying variables are correlated with the effects. For example, if such correlation arises solely because of a time-invariant component which is removed in taking deviations from individual means, these instruments are legitimate.