Bootstrap Bartlett Adjustment in Seemingly Unrelated Regression
研究了在看似不相关回归中使用对数似然比检验时，小样本下渐近分布不准确的问题，提出用自助法计算巴特利特调整因子以提高检验的显著性水平准确性。
Abstract Seemingly unrelated regression (SUR) is a method introduced by Zellner (1962) for estimating several regression equations simultaneously, a common and important problem in econometrics. This article considers hypothesis testing in SUR using the log-likelihood ratio (LLR) test. Although the asymptotic distribution of this statistic is well known, substantial departures occur in samples of the size commonly employed by economists. Significance levels given by the use of this asymptotic distribution are too low, sometimes by an order of magnitude, leading to the rejection of too many true null hypotheses. A common approach to this problem is to use a Bartlett (1937) adjustment to the LLR statistic, in which the test statistic is multiplied by a factor derived from second-order asymptotics before it is referred to the asymptotic χ2(q) distribution. This article proposes that the Bartlett adjustment factor be computed using Efron's bootstrap (1979, 1982). In the cases examined, this leads to more accurate significance levels. In SUR, regression equations y j = X j b j + e j (j = 1, 2, …, K) are given, where each regression is over time period t = 1, 2, …, T. It is assumed that the errors for different time periods are independent, but that the K-dimensional vector of errors for all of the equations at any given time has an unknown covariance matrix Σ. Using ordinary least squares (OLS) for each equation is inefficient and leads to invalid inferences unless Σ = I or all X j are identical. Consistent estimates asymptotically equivalent to the normal maximum likelihood estimates (MLE) can be developed by beginning with the OLS estimates, estimating Σ, and then using generalized least squares (GLS) to reestimate the parameters. If this process is iterated, the MLE results (Sec. 1). If a restriction of dimension q is placed on the parameter vector b, then the LLR statistic is asymptotically distributed as χ2(q). Simulation results show that this approximation is not very accurate in small samples. The Bartlett adjustment resulting from the second-order asymptotic method of Rothenberg (1984a) is both difficult to apply and effective only in certain cases; the Bartlett adjustment determined by the bootstrap is considerably more accurate (Sec. 2). An example is given in Section 3 in which the use of the bootstrap Bartlett adjustment would substantially increase the plausibility of a hypothesis.