12. Regression-Based Tests of Linear Factor Models

12.1 Time-Series Regressions

When the factor is itself an excess return (e.g. the CAPM's \(R^{em}=R^m-R^f\)) and the test assets are all excess returns, run OLS time-series regressions \(R^{ei}_t=\alpha_i+\beta_i f_t+\varepsilon^i_t\). The model \(E(R^{ei})=\beta_iE(f)\) has just one implication: all intercepts \(\alpha_i\) should be zero (the intercept is the pricing error). Black-Jensen-Scholes (1972) strategy: run the time-series regressions, the factor risk premium is the factor's sample mean \(\hat\lambda=E_T(f)\), then \(t\)-test the \(\alpha\) with OLS standard errors.

To test all \(\alpha\) jointly zero we need the joint distribution of the intercepts (errors are correlated across assets, \(E(\varepsilon^i\varepsilon^j)\ne0\)). The classic (no autocorrelation/homoskedasticity) χ² test:

where \(\Sigma\) is the residual covariance matrix. Accounting for the sampling variation of \(\Sigma\) gives a finite-sample F distribution — the famous Gibbons-Ross-Shanken (GRS) test (requires normal, uncorrelated, homoskedastic errors; exact in finite samples):

Derive via GMM, free of i.i.d. assumptions. Write the N time-series regressions together as \(R^e_t=\alpha+\beta f_t+\varepsilon_t\), use the OLS moments \(g_T=[E_T(\varepsilon_t);\ E_T(f_t\varepsilon_t)]=0\) (exactly identified, \(a=I\)), and plug into the GMM formula \(\operatorname{var}(\hat\alpha\ \hat\beta)=\tfrac1T d^{-1}Sd^{-1\prime}\) (12.7). Under "errors uncorrelated over time, independent of factors, homoskedastic," \(S\) simplifies and you recover the pretty (12.3); but now you need none of those assumptions — just compute (12.7) for robust standard errors. This also reminds us GMM and \(p=\mathbb E(mx)\) are not paired: one can do GMM on an expected return-beta model too.

12.2 Cross-Sectional Regressions

Cross-sectional regression works even when the factor is not a return. The central economic question is why average returns vary across assets: high β (high exposure to high-premium factors) should mean high average returns. In the figure, each dot is an asset, and the model says average returns are proportional to β.

Figure 12.1　Cross-sectional regression. The horizontal axis is each asset's \(\beta_i\), the vertical its average excess return \(E(R^{ei})\); the fitted line's slope is the factor risk premium \(\lambda\), and each point's vertical distance \(\alpha_i\) to the line is the pricing error.

Two-pass. First estimate β by time-series regression, then regress across assets \(E_T(R^{ei})=\beta_i'\lambda+\alpha_i\) — β are the regressors, \(\lambda\) the coefficients, the residuals \(\alpha_i\) the pricing errors. Theory says the zero-beta excess return (constant) should be zero, which you can impose or estimate and check is small. The OLS cross-sectional estimates are \(\hat\lambda=(\beta'\beta)^{-1}\beta'E_T(R^e)\), \(\hat\alpha=E_T(R^e)-\hat\lambda\beta\). Test \(\hat\alpha'\operatorname{cov}(\hat\alpha)^{-1}\hat\alpha\sim\chi^2_{N-K}\) (since \(\lambda\) was estimated, the covariance is singular, df reduced by K, use a generalized inverse).

GLS cross-sectional regression. Since the cross-sectional residuals are correlated, standard advice is GLS (with error covariance \(\tfrac1T(\Sigma+\beta\Omega_f\beta')\)): \(\hat\lambda=(\beta'\Sigma^{-1}\beta)^{-1}\beta'\Sigma^{-1}E_T(R^e)\), and the test simplifies to \(T\hat\alpha'\Sigma^{-1}\hat\alpha\sim\chi^2_{N-1}\). But the §11.5 caveat applies: GLS is more efficient yet \(\Sigma\) is hard to estimate and invert (especially when N is large), so OLS's robustness may be preferable. GLS can be read as a transformation of the return space onto the statistically most informative portfolios \(CR^e\) — which often contain extreme long-short positions.

Time series vs. cross section. The key difference: when the factor is not a return, only cross-sectional regression works (time-series needs \(\hat\lambda=E_T(f)\)). When the factor is a return the two differ: the time-series regression forces the factor's pricing error to zero, drawing a line through the origin and the factor while ignoring the other points; OLS cross-sectional picks slope and intercept to minimize the squared errors of all points. A pretty result: when the factor is a return and is included among the test assets, the GLS cross-sectional regression is identical to the time-series regression (the factor's time-series residual is zero, so GLS puts all weight on it, giving \(\hat\lambda=E_T(f)\)). Why does the "efficient" technique ignore the other assets' pricing errors and use only the factor mean to estimate the premium? Because in \(R^e_t=a+\beta f_t+\varepsilon_t\) each asset's sample mean return = β times the factor's sample mean plus the residual; "signal plus noise" carries no information beyond the signal itself.

12.3 The Fama-MacBeth Procedure

Fama-MacBeth (1973) is a historically important, computationally simple, still-widely-used alternative: first estimate β by time-series regression, then run a cross-sectional regression at each date \(R^{ei}_t=\beta_i'\lambda_t+\alpha_{it}\), and average over time \(\hat\lambda=\tfrac1T\sum_t\hat\lambda_t\), \(\hat\alpha_i=\tfrac1T\sum_t\hat\alpha_{it}\). The clever part: use the time-series standard deviation of the cross-sectional estimates for the standard errors, \(\sigma^2(\hat\lambda)=\tfrac1{T^2}\sum_t(\hat\lambda_t-\hat\lambda)^2\). Intuition: sampling error is how a statistic varies across samples; unable to re-sample, cut the sample in half, quarters, …, and deduce cross-sample variation from how \(\hat\lambda_t\) varies over time (the \(\sigma^2/T\) idea for a sample mean).

Summary

Testing linear factor models reduces to two routes: time-series regression (factor must be a return, test intercepts \(\alpha=0\), GRS gives an exact F-test, geometrically "is the factor on the mean-variance frontier") and cross-sectional regression (factor may be a non-return, two-pass fit of \(E(R^e)=\beta\lambda\), OLS/GLS, Shanken correction for estimated β). The two coincide via GLS when the factor is a return included among the test assets. Fama-MacBeth is an equivalent, intuitive way to compute cross-correlation-corrected standard errors, but does not correct for estimated β and needs care about time correlation in pooled data. The next chapter handles the same problem with GMM in \(p=\mathbb E(mx)\) form.

Problems

1. When the CAPM is written in excess-return form, can the test assets be differences of risky assets \(R^i-R^j\)? Can the market excess return on the right also use a risky asset, or must it be relative to a risk-free rate? (Hint: start from \(E(R^i)-R^f=\beta_{i,m}[E(R^m)-R^f]\); betas must be regression coefficients.)
2. Can you run the GRS test on a model using industrial production growth as a factor, \(E(R^{ei})=\beta_{i,ip}\lambda_{ip}\)?
3. Fama and French (1997) report that pricing errors are correlated with betas in a factor-model test on industry portfolios. How is this possible?
4. We saw the GLS cross-sectional regression of the CAPM passes through the market and risk-free rate by construction, while OLS does not. Show that if the market return is an equally-weighted portfolio of the test assets, an OLS cross-sectional regression with an estimated intercept passes through the market by construction. What if you force the intercept to zero?
5. Factor models with non-return factors are usually estimated by cross-sectional regressions, but a time-series regression can be used too: impose \(a=\beta(\lambda-E(f))\) (see (12.24)). Write moment conditions to estimate this model and test the restriction, and describe how you would estimate the parameters and use \(\sigma(\hat b),\operatorname{cov}(g_T),J_T\) to construct a test.