Hypothesis Testing in Demand Systems: Some Examples of Size Corrections Using Edgeworth Approximations
研究了在需求分析中，使用Edgeworth近似对Wald、似然比和拉格朗日乘子检验的临界值进行小样本修正，发现修正效果在方程数较少时较好，但计算成本高且极端情况下修正不足。
A size correction to adjust the critical values of asymptotic χ 2 tests in a SUR context has been developed by Rothenberg and verified by Phillips. The adjustment centers the distribution of the observed test statistic so that it coincides more closely with its hypothetical distribution. The correction itself is an approximation to O ( T −1 ), and while the adjustment should be superior to the asymptotic test (no adjustment) in small-sample situations, its performance can be expected to deteriorate as the sample size decreases. The Edgeworth corrections, using formulas provided by Phillips, are calculated for the Wald, likelihood ratio, and Lagrange multiplier tests in the context of symmetry testing in demand analysis using the Laitinen–Meisner simulated data set. The sample consists of 31 observations and the corrections were “adequate to useful” in the context of 5, 8, and 11 equations. However, in the extreme case of 14 equations the corrections typically only took up 60% of the adjustment required. The computational cost of calculating the Edgeworth correction for large SUR systems with a large number of restrictions also turns out to be quite heavy. The conclusion reached is that while Edgeworth corrections are not totally satisfactory, they are easy to include in a program and provide a useful critical value correction.