Estimating the arbitrage pricing theory with observed macro factors
Introduction
Estimation of the arbitrage pricing theory (APT) with observed macroeconomic factors has long been hampered by the large number of parameters to be estimated and the non-linearities inherent in the model. McElroy et al. (1985) (MBW) first detailed an approach to jointly estimate all the parameters of the APT using Gallant (1987) method of Iterated Non-linear Seemingly Unrelated Regressions (ITNLSUR), but many authors still use the two-step regression method based on Fama, Macbeth (1973) because the approach as presented by MBW is difficult to implement, requiring numerical optimization over a very large number of parameters.
This paper presents analytical results which greatly simplifies joint estimation of the APT by either the method of ITNLSUR or quasi full information maximum likelihood (QFIML).1 The intuition behind the procedure is as follows: if the “prices of risk” common to all securities are known a priori, then the APT is a system of Seemingly Unrelated Regressions (SUR), with closed form solutions for estimates of the “factor loadings” and the covariances. Using these closed form solutions, we derive the objective functions under QFIML and ITNLSUR as functions of the prices of risk only. For a model with 50 assets and 5 factors, this reduces the objective function under ITNLSUR from a dimension of 255 to a dimension of 5. We also derive analytical expressions for the standard errors under ITNLSUR and QFIML.
Section snippets
Model
We set-up the APT following McElroy, Burmeister (1988). Let yt be an N×1 vector of excess returns; let ft be a K×1 vector of factor realizations; let B be an N×K matrix of factor sensitivities; and let λ be a K×1 vector of risk prices common to all securities. It is assumed thatwhere Et is the expectation operator conditional on the information set at the beginning of time t, which includes returns dated t−1 and earlier. In general, H is diagonal,
Full information maximum likelihood estimation
If yt is distributed multivariate normal with mean Bxt(λ) and variance H, then the sample log-likelihood is the sum of the log of the conditional densities of y1, y2,…, yT and is given byThe full information maximum likelihood estimates (MLEs) are those values of λ, B and H that maximize the sample log-likelihood (2). In general, analytic optimization is not possible because the normal equations are non-linear, and so
Non-linear seemingly unrelated regressions
MBW detail for the APT the ITNLSUR estimator of Gallant (1987), which, unlike maximum likelihood, does not require a specific model of the error distribution. By using the analytical results above, estimation by ITNLSUR can also be greatly simplified. The NLSUR estimator minimizeswhere H0 is the estimated covariance matrix obtained from the residuals of an OLS regression of the returns yt on the factors ft and a constant. The first order conditions of
Inference under maximum likelihood
Under maximum likelihood inference can be carried out by the usual procedures. That is, if the standard regularity conditions are satisfied, then Ξ*−Ξ0≈N(0, T−1J−1), where Ξ≡[λ|vec(B)], Ξ* is the maximum likelihood estimate of the true parameter vector Ξ0 and J is Fisher's information matrix. Two asymptotically equivalent estimates of the information matrix are the mean outer-product of the score of the sample log-likelihoodand the Hessian of the sample
Inference under NLSUR
Under certain regularity conditions (see, e.g., Gallant (1987)), the ITNLSUR estimator is strongly consistent and asymptotically normal withOur experience has been that standard errors calculated by any of the four methods are not substantially different.
Acknowledgements
The author thanks T. Wake Epps for helpful comments and suggestions.
References (6)
- et al.
Two Estimators for the APT When Factors are Measured
Economic Letters
(1985) - Fama, Macbeth, 1973. Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy 81,...
- Gallant, A.R., 1987. Nonlinear Statistical Models. Wiley, New...
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