Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is yt = Bzt + vt. The unobserved error sequence {vt} is a sequence of martingale differences with conditional covariance matrices {[Sigma]t} and satisfying supt=1,..., n3{v'tvtI(v'tvt>a) zt, vt-1, zt-1, ...} 0 0 as a --> [infinity]. The sample covariance of the independent variables z1, ..., zn, is assumed to have a probability limit M, constant and nonsingular; maxt=1,...,nz'tzt/n0 0. If (1/n)[Sigma]t=1n[Sigma]t0[Sigma], constant, then [radical sign]nvec(Bn-B)0N(0,M-1[circle times operator][Sigma]) and [Sigma]n0[Sigma]. The autoregression model is xt = Bxt - 1 + vt with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {vt}, and (1/n)[Sigma]t=max(r,s)+1([Sigma]t[circle times operator]vt-1-rv't-1-s)0 [delta]rs([Sigma][circle times operator][Sigma]), where [delta]rs is the Kronecker delta. Then [radical sign]nvec(Bn-B)0N(0,[Gamma]-1[circle times operator][Sigma]), where [Gamma] = [Sigma]s = 0[infinity]Bs[Sigma](B')s.