Generalized Method of Moments for Samples of Unequal Length

This paper extends the generalized method of moments technique of Hansen (1982) to cases where moment conditions are observed over different sample periods. Many applications in financial economics use data series that have different starting dates, or, more rarely, different ending dates. Common practice is to take the intersection of the sample periods over which the data are observed; the intersection then becomes the sample period for the study and the rest of the data are ignored. This paper describes an alternative that allows the researcher to make use of all of the data available for each moment condition. We describe two asymptotically equivalent estimators that are consistent, asymptotically normal, and more efficient asymptotically than standard GMM. The first uses sample averages over the full sample to estimate the moments for which full-sample data are available, and sample averages over the short sample to estimate moments for which only the short-sample data are available, and then adjusts the short-sample moment using coefficients from a regression of the short-sample moments on the full-sample moments. The second uses the non-overlapping segment of the data available for the full-sample moments to form an additional set of moment conditions. We extend both of these estimators to settings with more general patterns of missing data. We show that the extended estimators are asymptotically equivalent, consistent, asymptotically normal, and asymptotically more e±cient than estimators that ignore a portion of the sample, whether or not it is observed for all series. By implication, the extended estimators are more efficient than standard GMM.