LIMIT THEORY FOR COINTEGRATED SYSTEMS WITH MODERATELY INTEGRATED AND MODERATELY EXPLOSIVE REGRESSORS
研究了协整系统中变量具有适度积分或适度爆炸特征时的渐近理论，发现最小二乘回归在适度积分时一致且渐近正态但存在显著偏差，在适度爆炸时极限分布为混合正态且收敛速度爆炸，为推断提供了理论基础。
An asymptotic theory is developed for multivariate regression in cointegrated systems whose variables are moderately integrated or moderately explosive in the sense that they have autoregressive roots of the form ρ ni = 1 + c i / n α , involving moderate deviations from unity when α ∈ (0, 1) and c i ∈ ℝ are constant parameters. When the data are moderately integrated in the stationary direction (with c i < 0), it is shown that least squares regression is consistent and asymptotically normal but suffers from significant bias, related to simultaneous equations bias. In the moderately explosive case (where c i > 0) the limit theory is mixed normal with Cauchy-type tail behavior, and the rate of convergence is explosive, as in the case of a moderately explosive scalar autoregression (Phillips and Magdalinos, 2007, Journal of Econometrics 136, 115–130). Moreover, the limit theory applies without any distributional assumptions and for weakly dependent errors under conventional moment conditions, so an invariance principle holds, unlike the well-known case of an explosive autoregression. This theory validates inference in cointegrating regression with mildly explosive regressors. The special case in which the regressors themselves have a common explosive component is also considered.