Asymptotic normality is established for a class of statistics which includes as special cases weighted sum of independent and identically distributed (i.i.d.) random variables, unsigned linear rank statistics, signed rank statistics, linear combination of functions of order statistics, and linear function of concomitants of order statistics. The results obtained unify as well as extend a number of known results.

This paper presents conditions under which a quadratic form based on a g-inverted weighting matrix converges to a chi-square distribution as the sample size goes to infinity. Subject to fairly weak underlying conditions, a necessary and sufficient condition is given for this result. The result is of interest because it is needed to establish asymptotic significance levels and local power properties of generalized Wald tests (i.e., Wald tests with singular limiting covariance matrices). Included in this class of tests are Hausman specification tests and various goodness-of-fit tests, among others. The necessary and sufficient condition is relevant to procedures currently in the econometrics literature because it illustrates that some results stated in the literature only hold under more restrictive assumptions than those given.

We compare the distributional properties of the four predictors commonly used in practice. They are based on the maximum likelihood, two types of the least squared, and the Yule-Walker estimators. The asymptotic expansions of the distribution, bias, and mean-squared error for the four predictors are derived up to O(T−1), where T is the sample size. Examining the formulas of the asymptotic expansions, we find that except for the Yule-Walker type predictor, the other three predictors have the same distributional properties up to O(T−1).

We give an expression to order O(T-1), where T is the sample size, for bias to the estimated coefficient on a lagged dependent variable when all other regressors are exogenous. The general expression is a nonlinear function of the coefficient on the lagged dependent variable, the autoregressive structure of the exogenous variables, and the coefficients on the exogenous variables. The maximum bias that can arise is a linear function of the number of exogenous regressors in the estimating equation.

The distributions of the test statistics are investigated in the context of an AR(1) model where the root is unity or near unity and where the exogenous process is a stable process, a random walk or a time trend. The finite sample distributions are estimated by Monte Carlo methods assuming normal disturbances. The sensitivity of the distributions to both the values of the parameters of the AR(1) model and the process generating the exogenous time series is examined. The Monte Carlo results motivate several theorems which describe the exact sampling behavior of the test statistics. The analytical and empirical results present a mixed picture with respect to the accuracy of the relevant asymptotic approximations.