The limiting behavior of residuals from measurement error regressions

Multivariate measurement error regression models with normal errors are investigated and residuals, analogous to those of ordinary least squares, are defined. The limiting behavior of test statistics based on the residuals is determined;The residuals, properly standardized, are represented as a linear combination of two independent random vectors. This representation is used to show that the empirical process based on the standardized residuals converge to a unique Gaussian process, where the limit process is that of a normal sample standardized with estimated mean and variance. It is shown that many goodness-of-fit tests for normality based on the standardized residuals have the same limiting distribution as that of tests based on a sample of iid normal random vectors;Tests for outliers, for autocorrelation, and for homogeneity of variance are investigated. A test for autocorrelation is constructed by regressing the residuals on their lagged values and testing for zero coefficients. A test for homogeneity of variance is constructed by regressing the squared residuals on estimated values of the true independent variables and testing for zero coefficients. It is shown that the regression t-statistics and F-statistics for the autocorrelation test and for the homogeneity test converge to N(0, 1) and chi-square random variables, respectively;Monte Carlo studies are conducted to examine the adequacy of the asymptotic approximations in small samples. The large sample approximations are judged adequate even for models with only twenty degrees of freedom