PANEL Procedure

Amemiya-MaCurdy Estimation (AMACURDY Option)

You perform Amemiya-MaCurdy estimation by specifying the AMACURDY option in the MODEL statement. The Amemiya-MaCurdy (1986) model is similar to the Hausman-Taylor model. Following the development in the section Hausman-Taylor Estimation (HTAYLOR Option), estimation is identical up to the final 2SLS instrumental variables regression. In addition to the set of instruments that the Hausman-Taylor estimator uses, you use the following:

  • bold x Subscript 1 i Baseline 1 Baseline comma bold x Subscript 1 i Baseline 2 Baseline comma ellipsis comma bold x Subscript 1 i upper T Baseline

For each observation in the ith cross section, you use the data on the time-varying exogenous regressors for the entire cross section. Because of the structure of the added instruments, the Amemiya-MaCurdy estimator can be applied only to balanced data.

The Amemiya-MaCurdy model attempts to gain efficiency over the Hausman-Taylor model by adding instruments. This comes at a price of a more stringent assumption on the exogeneity of the bold x 1 variables. Although the Hausman-Taylor model requires only that the cross-sectional means of bold x 1 be orthogonal to nu Subscript i, the Amemiya-MaCurdy estimation requires orthogonality at every point in time; see Baltagi (2013, sec. 7.4).

A Hausman specification test is provided to test the validity of the added assumption. Define bold-italic alpha prime equals left-parenthesis bold-italic beta prime 1 comma bold-italic beta prime 2 comma bold-italic gamma prime 1 comma bold-italic gamma prime 2 right-parenthesis, its Hausman-Taylor estimate as ModifyingAbove bold-italic alpha With caret Subscript normal upper H normal upper T, and its Amemiya-MaCurdy estimate as ModifyingAbove bold-italic alpha With caret Subscript normal upper A normal upper M. Under the null hypothesis, both estimators are consistent and ModifyingAbove bold-italic alpha With caret Subscript normal upper A normal upper M is efficient. The Hausman test statistic is

m equals left-parenthesis ModifyingAbove bold-italic alpha With caret Subscript normal upper H normal upper T Baseline minus ModifyingAbove bold-italic alpha With caret Subscript normal upper A normal upper M Baseline right-parenthesis prime left-parenthesis ModifyingAbove bold upper Sigma With caret Subscript normal upper H normal upper T Baseline minus ModifyingAbove bold upper Sigma With caret Subscript normal upper A normal upper M Baseline right-parenthesis Superscript negative 1 Baseline left-parenthesis ModifyingAbove bold-italic alpha With caret Subscript normal upper H normal upper T Baseline minus ModifyingAbove bold-italic alpha With caret Subscript normal upper A normal upper M Baseline right-parenthesis

where ModifyingAbove bold upper Sigma With caret Subscript normal upper H normal upper T and ModifyingAbove bold upper Sigma With caret Subscript normal upper A normal upper M are variance-covariance estimates of ModifyingAbove bold-italic alpha With caret Subscript normal upper H normal upper T and ModifyingAbove bold-italic alpha With caret Subscript normal upper A normal upper M, respectively. Under the null hypothesis, m follows a chi squared distributed with degrees of freedom equal to the rank of left-parenthesis ModifyingAbove bold upper Sigma With caret Subscript normal upper H normal upper T Baseline minus ModifyingAbove bold upper Sigma With caret Subscript normal upper A normal upper M Baseline right-parenthesis Superscript negative 1.

Last updated: June 19, 2025