Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances (Q1185832)

A central limit theorem for martingale differences is developed to obtain asymptotic normality of statistics for regression and autoregression. A form of the Lindeberg condition appropriate for martingale differences is used. The regression model is \(y_ t=Bz_ t+v_ t\). The unobserved error sequence \(\{v_ t\}\) is a sequence of martingale differences with conditioned covariance matrices. The sample covariance of the independent variables \(\{z_ t\}_{t=1(1)n}\) is assumed to have a probability limit \(M\), constant and nonsingular. The autoregression model is \(x_ t=Bx_{t-1}+v_ t\) with the maximum absolute value of the characteristic roots of \(B\) less than one. The estimations \(\hat B_ n\) of the matrix \(B\) of regression coefficients are considered. E.g., it is shown that the least squares estimator of \(B\) has an asymptotic normal distribution.