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.)
* Two anonymous referees and Badi Baltagi provided helpful, timely comments on the first draft.
Andrews, D. W.K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817-58.
Baltagi, B. H. (1995). Econometric Analysis of Panel Data. New York: Wiley.
Bera, A. K., and C. M. Jarque (1982). Model specification tests: A simultaneous approach. Journal of Econometrics 20, 59-82.
Bierens, H. J. (1990). A conditional moment test of functional form. Econometrica 58, 1443-58.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307-27.
Breusch, T. S. (1978). Testing for autocorrelation in dynamic linear models. Australian Economic Papers 17, 334-55.
Breusch, T. S., and A. R. Pagan (1979). A simple test for heteroskedasticity and random coefficient variation. Econometrica 50, 987-1007.
Brock, W., W. D. Dechert, B. LeBaron, and J. Scheinkman (1996). A test for independence based on the correlation dimension. Econometric Reviews 15, 197-235.
Davidson, R., and J. G. MacKinnon (1981). Several tests of model specification in the presence of alternative hypotheses. Econometrica 49, 781-93.
Davidson, R., and J. G. MacKinnon (1985). Heteroskedasticity-robust tests in regression directions. Annales de l’INSEE 59/60, 183-218.
Davidson, R., and J. G. MacKinnon (1987). Implicit alternatives and the local power of test statistics. Econometrica 55, 1305-29.
Davidson, R., and J. G. MacKinnon (1993). Estimation and Inference in Econometrics. New York: Oxford University Press.
de Jong, R. M. (1996). The Bierens test under data dependence. Journal of Econometrics 72, 1-32.
Durbin, J., and G. S. Watson (1950). Testing for serial correlation in least squares regressions I. Biometrika 37, 409-28.
Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987-1008.
Engle, R. F. (1984). Wald, likelihood, ratio, and Lagrange multiplier tests in econometrics. In Z. Griliches and M. Intriligator, eds., Handbook of Econometrics, vol. 2, Amsterdam: North-Holland.
Fan, Y., and Q. Li (1996). Consistent model specification tests: Omitted variables and semiparametric functional forms. Econometrica 64, 865-90.
Godfrey, L. G. (1978). Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables. Econometrica 46, 1303-10.
Godfrey, L. G. (1988). Misspecification Tests in Econometrics. Cambridge: Cambridge University Press.
Gourieroux, C., A. Monfort, and C. Trognon (1984). Pseudo-maximum likelihood methods: Theory. Econometrica 52, 681-700.
Hong, Y., and H. White (1995). Consistent specification testing via nonparametric series regression. Econometrica 63, 1133-59.
Hsiao, C. (1986). Analysis of Panel Data. Cambridge: Cambridge University Press.
Lewbel, A. (1995). Consistent nonparametric hypothesis tests with an application to Slutsky symmetry. Journal of Econometrics 67, 379-401.
Messer, K., and H. White (1984). A note on computing the heteroskedasticity consistent covariance matrix using instrumental variable techniques. Oxford Bulletin of Economics and Statistics 46, 181-4.
Mizon, G. E., and J.-F. Richard (1986). The encompassing principle and its application to testing non-nested hypotheses. Econometrica 54, 657-78.
Moon, C.-G. (1988). Simultaneous specification test in a binary logit model: Skewness and heteroskedasticity. Communications in Statistics 17, 3361-87.
Newey, W. K. (1985). Maximum likelihood specification testing and conditional moment tests. Econometrica 53, 1047-70.
Newey, W. K., and K. D. West (1987). A simple positive semi-definite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703-8.
Pagan, A. R., and F. Vella (1989). Diagnostic tests for models based on individual data: A survey. Journal of Applied Econometrics 4, S29-59.
Porter, R. D., and A. K. Kashyap (1984). Autocorrelation and the sensitivity of RESET. Economics Letters 14, 229-33.
Ramsey, J. B. (1969). Tests for specification errors in the classical linear least squares regression analysis. Journal of the Royal Statistical Society Series B 31, 350-71.
Stoker, T. M. (1992). Lectures on Semiparametric Econometrics. Louvain-la-Neuve, Belgium: CORE Lecture Series.
Tauchen, G. (1985). Diagnostic testing and evaluation of maximum likelihood models. Journal of Econometrics 30, 415-43.
Thursby, J. G. (1979). Alternative specification error tests: A comparative study. Journal of the American Statistical Association 74, 222-5.
Thursby, J. G. (1989). A comparison of several specification error tests for a general alternative. International Economic Review 30, 217-30.
White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48, 817-38.
White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica 50, 1-26.
White, H. (1994). Estimation, Inference and Specification Analysis. Cambridge: Cambridge University Press.
Wooldridge, J. M. (1990a). A unified approach to robust, regression-based specification tests. Econometric Theory 6, 17-43.
Wooldridge, J. M. (1990b). An encompassing approach to conditional mean tests with applications to testing nonnested hypotheses. Journal of Econometrics 45, 331-50.
Wooldridge, J. M. (1991a). On the application of robust, regression-based diagnostics to models of conditional means and conditional variances. Journal of Econometrics 47, 5-46.
Wooldridge, J. M. (1991b). Specification testing and quasi-maximum likelihood estimation. Journal of Econometrics 48, 29-55.
Wooldridge, J. M. (1992a). Some alternatives to the Box-Cox regression model. International Economic Review 33, 935-55.
Wooldridge, J. M. (1992b). A test for functional form against nonparametric alternatives.
Econometric Theory 8, 452-75.
Wooldridge, J. M. (1994). Estimation and inference for dependent processes. In R. F. Engle and D. L. McFadden (eds.) Handbook of Econometrics, Volume 4. pp. 2639-2738. Amsterdam: North-Holland.
Wooldridge, J. M. (1997). Quasi-likelihood methods for count data. In M. H. Pesaran and P. Schmidt (eds.) Handbook of Applied Econometrics, Volume 2. pp. 352-406. Oxford: Blackwell.
Wooldridge, J. M. (1999). Asymptotic properties of some specification tests in linear models with integrated processes. In R. F. Engle and H. White (eds.) Cointegration, Causality, and Forecasting. Oxford: Oxford University Press.
Zheng, J. X. (1996). A consistent test of functional form via nonparametric estimation techniques. Journal of Econometrics 75, 263-89.