Test incorrect functional form when residuals have non-normal distribution

J. B. Ramsey (in "Tests for specification errors in classical linear least-squares regression analysis." Journal of the Royal Statistical Society. 1969) says that the RESET test assumes that the residuals are normally distributed. If one wants to test the incorrect functional form of a model but the residuals have a non-normal distribution, how can it be done? Ramsey also says that "the cases where mis-specification leads to a non-normal distribution of û [residuals] are to be discussed in a later paper". Does any one know which paper is this?

• regression
• hypothesis-testing
• misspecification

$\begingroup$ Sapra, S. (2005) A regression error specification test (RESET) for generalized linear models. Economic Bulletin 3(1), pp. 1-6. $\endgroup$

1 Answer 1

The Ramsey Test relies on the assumption of normally distributed residuals to justify the use of the F-test for exact finite sample inference for testing nested models (the linear model versus the saturated polynomial model). When the normality assumption is violated, few if any other distributions yield cogent, testable ratio statistics. This is a general result for inference from linear models.

A workaround is to use the asymptotic test. When the sample size is decently large, the chi-square approximation to the likelihood ratio statistic performs decently well. The asymptotic test has good power and is of correct size in all sample sizes, so its use is justified over a variety of scenarios.

Yet another approach that is justified here is to make use of robust standard errors to test nested models. Inference using robust standard errors does not rely on either homogeneity or identical distribution of error terms. This is an important consideration because, if model misspecification is present, then the apparent "error" term is a mixture of the inherent error (which may or may not follow a known distribution) and the unmeasured contribution of uncaptured variation in Y. It seems to me that's the issue that's alluded to in the cited "forthcoming" paper: but I find it hard to believe that any useful results could be stated in general since the extent of model misspecification could generate data following almost any form. A downside to using these types of "sandwich" standard errors is that they require a larger sample size than in the exact finite sample inference case.