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 {Σt} and satisfying supt=1,..., nE{v′tvtI(v′tvt>a) |zt, vt-1, zt-1, ...} → P 0 as a → ∞. The sample covariance of the independent variables z1, ..., zn, is assumed to have a probability limit M, constant and nonsingular; maxt=1,...,nz′tzt/n → P 0. If (1/n)Σt=1nΣt → PΣ, constant, then √nvec(B̂n-B) → LN(0,M-1⊗Σ) and Σ̂n → PΣ. 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)Σt=max(r,s)+1(Σt⊗vt-1-rv′t-1-s) → P δrs(Σ⊗Σ), where δrs is the Kronecker delta. Then √nvec(B̂n-B) → LN(0,Γ-1⊗Σ), where Γ = Σs = 0∞BsΣ(B′)s. © 1992.