Testing for latent factors in models with autocorrelation and heteroskedasticity of unknown form.

• Author: Gilbert, Scott

1. Introduction

   Empirical 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. The method can, in principle, be extended to models with a parametric form of error autocorrelation, and while this is also the case with Anderson's analytical LR test, both methods are necessarily parametric. Because we pursue instead a nonparametric HAC objective, we undertake a computationally harder problem.

   Comparing the HAC robust Hansen-type and RAD tests for reduced rank, the latter is much easier to implement with full flexibility regarding the reduced rank form, that is, selection of linearly independent matrix rows in the reduced-rank coefficient matrix. For this reason, the RAD test is more user friendly than the robust Hansen test and, with it, we obtain conclusions robust to the form of reduced rank as well as the form of error dynamics. Both are calculated by minimizing a quadratic form with the optimal weighting matrix given by the inverse of a relevant covariance matrix. Covariance-matrix estimation is made robust to both heteroskedasticity and autocorrelation via various nonparametric and parametric methods. We consider several kernel-based heteroskedasticity and autocorrelation-consistent (HAC) procedures, with various combinations of kernel, bandwidth selection, and prewhitening filter (see Newey and West 1987, 1994; Andrews 1991; Andrews and Monahan 1992). We also implement a simple parametric procedure with prewhitening advocated by den Haan and Levin (1997). For the Hansen test, we follow Hansen, Heaton, and Yaron (1996), simultaneously iterating over both the weighting matrix and model parameters. For the RAD test, such simultaneous iteration is unnecessary.

   Our empirical study, like the influential work of Chen, Roll, and Ross (1986), looks at the link between asset markets and macroeconomic fundamentals. As dependent variables in our regression system, we choose a set of excess returns broadly characterizing the U.S. bond and stock markets over the last four decades. Specifically, we use monthly excess returns on the Treasury securities of maturities of 90 days and 5 years. To capture main features of the U.S. stock markets, we sort returns by firm size and use excess returns on the Center for Research in Security Prices (CRSP) small capitalization and large capitalization portfolios. There are many candidates for our explanatory variables. Given our dependent variables (a small number of both stock and bond returns), we do not consider size and book-to-market-related portfolios of Fama and French 1993 (henceforth FF) and term and default premia (FF; Chen, Roll, and Ross 1986). The size-related variables are mostly used in studies focusing on stocks; also, in those studies, a large number of assets is involved. The problem with bond-related factors is that we might run into econometric problems with term structure-related dependent and independent variables. (1) Moreover, while the FF factors do a good job in explaining cross-sectional and time-series variation of stock returns, they are somewhat ad hoc. Hence, we focus on macroeconomic variables that can be theoretically linked (at least loosely) to expected returns. We are left with a subset of risk factors similar to Chen, Roll, and Ross (1986), consisting of the stock market, consumption, industrial production, money supply, and the unexpected inflation rate.

   The models of asset returns with macroeconomic explanatory variables are well specified statistically, provided that the used time series are stationary. From the theoretical perspective, the models used can be considered to be examples of the intertemporal capital asset pricing model (CAPM), in which case the test for reduced rank is the test for the number of latent risk factors inherent in this model. (2) If we wanted to interpret the test of our model as a test of the arbitrage pricing theory, all the explanatory variables would have to be excess returns on assets. Interpreting the growth rate of consumption and the expected inflation rate literally as asset returns is problematic but the return on the stock market is obviously an asset return, the industrial production growth rate may be viewed as a return on physical assets, and the growth rate of money as a (negative) return on cash holdings (due to inflation).

   We analyze the rank structure in the bond and stock markets, both separately and jointly. We first document the presence of autocorrelation and heteroskedasticity in residuals of all unrestricted regression systems, using a battery of tests. Then we apply the RAD and the Hansen tests (as a benchmark) with several HAC robust covariance-matrix estimators. We find that the one-factor hypothesis is rejected both in the bond market and the stock market. A joint estimation and tests of the stock and bond markets do not alter these results--statistically, at least four factors are needed for an accurate description of both markets. The sources of differences between bonds of different maturity can be traced to significantly different market and industrial production betas, suggesting (consistently with existing theory; see Campbell [1999] for a survey on stylized facts regarding various premia) that there is a term premium mainly due to a higher sensitivity to the market risk and to business cycles. For stocks of different firm size, such differences can be traced to consumption and monetary betas. This confirms that there are differences between stocks of different sizes but in a sense diverging from the literature because the returns in our sample actually do not statistically differ,3 only the betas do. Presence of consumption in our data is motivated by the consumption-based CAPM and the higher consumption beta suggests that small firms are riskier from this perspective. Differing monetary betas may be due to greater sensitivity to tighter monetary policy of small firms (see Gertler and Gilchrist 1994). The differences between bonds and...