Test of Random vs Fixed E¤ects with Small Within Variation

Comparisons of within and between estimators using the conventional Hausman test may be subject to statistical problems if the within variation is not su¢ ciently large. Adopting an alternative asymptotic approximation, we propose a modi…cation of Hausman test that is valid whether the within variation is small or large. 1 Introduction With the advent of many panel data sets, researchers are commonly estimating a textbook panel data model with individual e¤ects yit = i + x 0 it + "it: In carrying out the estimation, the primary concern of many researchers is whether i can be treated as uncorrelated with xit. As is well known, random e¤ects estimation will produce an e¢ ciency gain over …xed e¤ects estimation if i is uncorrelated with xit; however, if this condition does not hold, only …xed e¤ects estimation will produce consistent estimates. Hausman (1978) provided a test of random e¤ects versus …xed e¤ects which in principle resolves the dilemma for researchers. However, if the within variation is small, the …xed e¤ects estimates may not be asymptotically normal, potentially invalidating the basic premise of the Hausman test. This problem often arises in empirical work, and when the within variation is likely to be small, researchers almost always use the random e¤ects speci…cation without using the Hausman test as a diagnostic,1 perhaps because they are concerned that it may not be appropriate in their case. In this paper we …rst show that this intuition is theoretically valid in the sense that it is not appropriate to use the conventional Hausman test when some or all of the explanatory variables have little within-person variation. Next, we provide a valid version of the Hausman test of between versus …xed e¤ects for this case. Finally, we show that a version of the bootstrap in fact provides a valid critical value for this test. See, e.g., Kearney (2005), Sawangfa (2007) and Ham, Li and Shore-Sheppard (2009).