Asymptotics of forced-in variables from screening processes

Freedman, Navidi and Peters (1988) had shown that when a two-stage variable selection procedure is used to screen linear regression equations, the type I error rate of the t-test for a forced-in-first estimator is inflated if the ratio of the number of parameters to number of data points is large. In this paper, we investigate the asymptotic distribution of the “force-in-last” estimator, and show that this estimator is bias toward zero asymptotically and its asymptotic standard error is close to the nominal standard error under certain conditions. To remedy the bias and reduce the inflation of the type I error rate, two estimators involved with propensity score (Rubin and Rosenbaum, 1983) were introduced.