A Simplified Version of a Differencing Specification Test

In a recent paper, Plosser, Schwert and White (1982) proposed a general test for model misspecification based on a comparison of estimates of the model in levels and first-differences. We demonstrate that this test is equivalent to a certain F test for omitted variables. The latter test has several advantages over the former. For example, the new test is exact whenever F tests are exact, and it is easy to compute. As part of our proof, we demonstrate that many tests based on a comparison of ordinary least squares and instrumental variables estimators are equivalent to F tests, and that the OLS estimator based on first differences is equivalent to a certain instrumental variables estimator based on levels.