Testing for Latent Factors in Models with Autocorrelation and Heteroskedasticity of Unknows Form

Abstract

1. IntroductionEmpirical research in financial economics frequently suggests the existence of few latent factors driving the systematic component of asset returns. Existence of such latent factors makes it easier to understand the effect on asset returns of the many variables that comprise the systematic component. Results depend on the number and type of assets used and the number and types of instruments, which themselves serve as proxies for the latent factors (for examples, see Campbell 1987; Zhou 1995; and Costa, Gardini, and Paruolo 1997). In econometric terms, the existence of latent factors translates into a reduced rank restriction on the (array of) coefficients in an asset return regression system.The present work considers the problem of testing for latent factors in a broad class of (multivariate linear stationary) time-series models, wherein model errors have autocorrelation and heteroskedasticity of unknown form. The generality of error dynamics is well suited to financial models of bond and stock returns, as in the macro model of Chen, Roll, and Ross (1986). To deal with such generality, we consider heteroskedasticity and autocorrelation consistent (HAC) methods, a HAC version of Hansen's (1982) Generalized Method of Moments (GMM) test and a lesser-known but more user-friendly minimum distance or ratio of asymptotic densities (RAD) test.The primary benefit of using a HAC-type test of economic hypotheses, in time-series models, is a certain kind of increased robustness relative to tests that rely on parametric assumptions about error dynamics. This robustness implies that stated significance levels of HAC tests are frequently closer to their true values, at least in sufficiently large samples. So, for the financial economist who wants to know "Are there really multiple factors driving the link between the macroeconomy and the returns on bonds and stocks?" HAC tests (and the underlying mental exercise regarding error dynamics) give added insight. HAC tests may or may not agree with less robust tests. In our application, the HAC test results agree with the results of simpler (and more popular) implementation of Hansen's (1982) test for reduced rank, but the crucial point is that stated significance levels in the popular version of Hansen's test are not correct, in statistical terms, when the data have dynamics that cause residual serial correlation. Hence, to say that the HAC tests agree with the popular version of Hansen's test is really too liberal an interpretation; more accurate is to say that the nominal (but likely invalid) conclusions from the popular test coincide with that of the autocorrelation-robust tests.The HAC Hansen test and the RAD test each require some special calculation, and for this we do some programming. The computational complexity is due partly to the presence of corrections for residual autocorrelation and heteroskedasticity, and if instead we assumed that model errors were independent and identically distributed, then we could test for reduced rank via Anderson's (1951) convenient likelihood ratio (LR) test (see Reinsel and Velu [1998] for discussion and Zhou [1994] for a related test couched in GMM terms). It is, of course, possible to extend Anderson's LR test to (parametric) probability models with autocorrelated errors (see Reinsel and Velu 1998), but this approach relies on a correctly specified error dynamic. We instead take the nonparametric HAC approach, allowing a wider variety of error dynamics.Is special calculation really necessary for our testing purposes? In application to asset pricing models, Zhou (1994) gives an interesting modification of Hansen's (1982) testing approach, with an analytical (hence computationally convenient) test for latent factors in asset returns. However, to implement his analytical test, Zhou relies on parametric assumptions about the model's error dynamics. Specifically, he considers the case of white noise errors and also the case of errors that are uncorrelated but have a known (parametric) form of conditional heteroskedasticity. …