LM Test of Neglected Correlated Random E¤ects and Its Application

Econometric model speci…cation is usually tested by exploiting one of the three principles, including (i) the Wald test, which is based on the asymptotic distribution of parameter estimates; (ii) the Likelihood Ratio (LR) test; and (iii) the Lagrange Multiplier (LM) procedure, which is based on the derivative of the log likelihood (score) imposing the hypothesis.1 The LM procedure seems to dominate the other two in terms of convenience when the objective of the test is to detect neglected heterogeneity in a given econometric model. The LM test statistic is solely based on the parameter estimates under the null hypothesis that there is no neglected heterogeneity and, as such, it eliminates the burden of specifying and estimating the model under the alternative hypothesis that there is some neglected heterogeneity. The possibility of using the LM test as a way of detecting individual heterogeneity was discussed in Breusch and Pagan (1980), Chesher (1984), Lee and Chesher (1986), and Hahn, Newey, and Smith (2014), among others. These tests can be viewed as a version of White’s (1982) Information Matrix test, as discussed in Chesher (1984). Alternatively, they can be understood to be a test of overdispersion as in Cox (1983), which is our interpretation in this paper. Implicit in these tests is the assumption that the neglected heterogeneity (under the alternative hypothesis) is more or less independent of the randomness that generates the main model, although the assumption is made more explicit in Hahn, Newey, and Smith (2014, Lemmas 3.1 and 3.2). In the panel data context, this assumption amounts to the random e¤ects speci…cation, as developed in Balestra and Nerlove (1966) or Maddala (1971). In other 1The relationship among the three tests are reviewed in Engle (1984), e.g.