A COMPANION TO Theoretical Econometrics
Omnibus tests on the errors in time series regression models
Omnibus tests on the errors in time series regression models have recently become popular. A good example is the so-called BDS test (Brock, Dechert, LeBaron, and Scheinkman, 1996), which has been viewed as a general test for "nonlinearity." The null hypothesis for the BDS test is that the errors are independent and identically distributed, and the test has power against a variety of departures from the iid assumption, including neglected nonlinearities in the conditional mean, dynamic forms of heteroskedasticity, and even dynamics in higher order moments. Unfortunately, no economic theory implies that errors are iid. In many applications, especially in finance, it is often easy to reject the iid assumption using a simple test for dynamic heteroskedasticity, such as ARCH.
As with other omnibus tests, BDS gives equal weight to hypotheses that have very different practical importance. Finding that the errors in, say, an asset pricing equation are serially correlated - which usually means a violation of the efficient markets hypothesis - is more important than finding dynamic heteroskedasticity, which in turn is more important than finding, say, a nonconstant conditional fourth moment.
2 Final Comments
Our focus in this chapter has been on the most common setting for diagnostic tests, namely, in univariate parametric models of conditional means and conditional variances. Recently, attention has turned to testing when some aspect of the estimation problem is nonparametric. For example, we might wish to construct a test of a parametric model that has unit asymptotic power against all alternatives that satisfy fairly weak regularity conditions. Bierens (1990), Wooldridge (1992b), Hong and White (1995), de Jong (1996), Fan and Li (1996), and Zheng (1996) are some examples. Or, the estimated model may be semiparametric in nature, depending on an infinite dimensional parameter in addition to a finite dimensional parameter (Stoker, 1992; Fan and Li, 1996). The alternative is an infinite dimensional parameter space. In some cases the null model may be fully nonparametric, in which case the alternative is also nonparametric (e. g. Lewbel, 1995; and Fan and Li, 1996).
For diagnostic testing in time series contexts, we assumed that the underlying stochastic processes were weakly dependent. Currently, there is no general theory of diagnostic testing when the processes are not weakly dependent. Wooldridge (1999) considers a particular class of diagnostic tests in linear models with integrated processes and shows that, when the misspecification indicator is cointegrated, in a generalized sense, with the included explanatory variables, LM-type statistics have asymptotic chi-square distributions.
Another important topic we have omitted is diagnostic testing for panel data models. Panel data raises some additional important considerations, most of which revolve around our ability to control, to some extent, for time-constant heterogeneity. Strict exogeneity assumptions on the regressors, especially conditional on the unobserved effect, are important. Dynamic models with unobserved effects raise even more issues for estimation and diagnostic testing. (See Hsiao (1986) and Baltagi (1995) for discussions of these issues.)
Note
* Two anonymous referees and Badi Baltagi provided helpful, timely comments on the first draft.
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