Errors in Variables and Properties of Statistical Inference by Lloyd

Lloyd J. Edwards. Errors in Variables and Properties of Statistical Inference (under the direction of Dr. P. K. Sen.) The objective here is to investigate the effects errors in variables have on tests of separate families of linear hypotheses (Le., non-nested linear hypotheses) using the methodology introduced by D. R. Cox (1961, 1962). Two hypotheses are called separate if an arbitrary simple hypothesis in one cannot be obtained as a limit of simple hypotheses in the other. We derive Cox's test of non-nested linear models for stochastic explanatory variables measured with error under the assumption of normality (normal structural models). These previously underived results reflect the effect the error covariance of the stochastic explanatory variables has on Cox's test. The effect of the error covariance is shown to app~ar in the test statistic itself and its variance. We also derive Cox's test of non-nested linear models for fixed explanatory variables measured with error under the assumption of normality (normal functional models). These previously underived results are shown to differ in form from that of Pesaran (1974) for the case of fixed explanatory variables not measured with error. As in the case of the structural model, the effect of the error covariance is shown to appear in the test statistic itself and its variance. Since the "true" explanatory variables are treated as nuisance parameters, the number of parameters in the models increase with the sample size n. We propose a method of reducing the number of parameters in the likelihood function and investigate conditions (practical and theoretical) for selecting an appropriate reduced likelihood function. We apply our results to real data, discuss the effects of applying our results, and discuss plans for future research.