13. GMM for Linear Factor Models in Discount Factor Form

13.1 GMM on Pricing Errors Gives a Cross-Sectional Regression

Treating the constant \(a\) as a constant factor, the model is \(E(p)=E(xf')b\) (\(p,x\) are \(N\times1\), \(f,b\) are \(K\times1\)). The natural moments are the pricing errors \(g_T(b)=E_T(xf'b-p)\) (a choice, not a necessity — GMM and \(p=\mathbb E(mx)\) are not paired). The first-order condition of \(\min_b g_T'Wg_T\) is \(d'Wg_T=0\), where \(d'=\partial g_T'/\partial b=E_T(fx')\) is the second moment of payoffs and factors. The linear model solves analytically:

13.2 The Case of Excess Returns

Linear factor models are most often applied to excess returns, but then the mean of \(m\) is unidentified (\(E(mR^e)=0\Rightarrow E((2m)R^e)=0\); analogous to no information on the zero-beta rate level in beta models). Writing \(m=a-b'f\), \(a\) and \(b\) cannot be separately identified, so one picks a normalization (purely for convenience — unidentification precisely means the pricing errors don't depend on the choice).

• Normalization \(a=1\): \(g_T(b)=-E_T(R^e)+E(R^ef')b\), \(d=E(R^ef')\) (the second moment of returns and factors). The GMM estimate is a cross-sectional regression of mean excess returns on the return-factor second moments.
• Normalization \(a=1+b'E(f)\) (i.e. \(m=1-b'[f-E(f)]\), \(E(m)=1\)): \(d=E(R^e\tilde f')\) becomes the covariance matrix. The GMM estimate is a cross-sectional regression of expected excess returns on the return-factor covariances — the model says \(E(R^e)=-\operatorname{cov}(R^e,f')b\), with covariances replacing betas. This is almost identical to the Ch 12 cross-sectional regression.

The one fly in the ointment: \(E(f)\) is estimated, and the distribution theory should account for this sampling variation (analogous to β being a generated regressor in Ch 12). An appropriate block-diagonal weighting \(a_T\) handles it, giving standard errors resembling the Shanken correction (small in practice, but easy to do right once understood). Note: this "second stage" regression is not the truly efficient GMM estimate — the efficient one uses \(a_T=d'S^{-1}\) (not block-diagonal), which lets some moments deviate from sample values to bring others closer to zero.

13.3 Horse Races

One often tests whether one set of factors drives out another (Chen-Roll-Ross 1986 ask whether five macro factors price assets so well that even the market return can be ignored). Estimate a general model \(m=b_1'f_1+b_2'f_2\) and ask whether \(b_2=0\): ① a Wald test \(\hat b_2'\operatorname{var}(\hat b_2)^{-1}\hat b_2\sim\chi^2_{\#b_2}\); ② a χ²-difference test: estimate the restricted model \(m=b_1'f_1\) (same \(S\)), \(TJ_T(\text{restricted})-TJ_T(\text{unrestricted})\sim\chi^2(\#\text{restrictions})\) (like a likelihood-ratio test).

13.4 Testing for Priced Factors: \(\lambda\) or \(b\)?

13.5 Mean-Variance Frontier and Performance Evaluation

A factor model holds \(\iff\) a linear combination of the factors (or their mimicking portfolios) is on the mean-variance frontier, so any factor-model test can be read as a test of "is a given return on the frontier." A GMM test of \(m=a-bR^p\) tests whether \(R^p\) is on the test assets' frontier (e.g. international diversification, fund-manager performance evaluation: real skill or luck?). But to test whether a set of assets \(R^d\) spans the combined frontier of \(R^d\) and \(R^i\), take care: with no risk-free rate, the \(R^d\) frontier may merely intersect the combined frontier rather than span it (Figure 13.1).

Figure 13.1　Two mean-variance frontiers might intersect rather than coincide. Testing that \(m=a-b'R^d\) prices both \(R^d\) and \(R^i\) for a single value of \(a\) only checks intersection; to test spanning/coincidence, it must hold for two prespecified values of \(a\) simultaneously.

De Santis (1993) and Chen-Knez (1995, 1996) test spanning vs. intersection: for intersection, \(m=a-b_d'R^d\) prices both for just one value of \(a\) (i.e. \(E(m)\)); if the frontiers coincide/span, it prices both for any \(a\). So test spanning by checking \(m=a-b'R^d\) prices \(R^d,R^i\) for two fixed values \(a_1,a_2\) simultaneously (a \(J_T\) test on four sets of moments).

13.6 Testing for Characteristics

One often checks whether a model drives out a characteristic (size, book/market, volatility). A good model should let betas explain average returns, with the pricing errors no longer associated with the characteristic. Put the characteristic \(y^i\) (e.g. a size rank) into the moments: \(g^i_T=E_T(m_{t+1}(b)x^i_{t+1}-p^i_t-\gamma y^i_t)\); the estimate of \(\gamma\) tells you how the characteristic relates to the model's pricing errors, and the standard GMM formulas give the standard error of \(\gamma\) or a χ²-difference test for \(\gamma=0\) (including the fact that \(E(y)\) is estimated).

Summary

Writing a linear factor model as \(p=\mathbb E(mx),m=b'f\) and using the pricing errors as moments for GMM, the first/second-stage estimates are precisely OLS/GLS cross-sectional regressions — converging with Chapter 12. With excess returns the mean of \(m\) is unidentified and needs a normalization (\(a=1+b'E(f)\) gives a regression on covariances). Horse races use Wald or χ²-difference tests. The most important conceptual distinction is \(b\) vs \(\lambda\): to decide "whether to include a factor" test \(b_j=0\) (the multiple-regression coefficient), not \(\lambda_j=0\) (the single-regression coefficient = whether it is priced). One can also test frontier spanning (needs two \(a\)'s) and whether characteristics are driven out. The next chapter turns to maximum likelihood.

Problems

1. Derive the GMM distribution theory for the first-stage estimate of \(m=1-b'[f-E(f)]\) (excess returns), recognizing that \(E(f)\) is estimated. Use \(g_T=[E_T(R^e-R^e(f'-Ef')b);\ E_T(f-Ef)]\), \(a_T=\operatorname{diag}(E_T(\tilde f R^{e\prime}),I_K)\), parameters \(b,Ef\). You should get a standard error for \(b\) resembling the Shanken correction (12.19) and an unchanged \(J_T\) test.
2. Show that with single-regression betas, the corresponding \(\lambda\) can test the marginal importance of factors; but then \(\lambda\) are no longer the expected returns of factor-mimicking portfolios.
3. From Ken French's website take the Fama-French three factors and 25 size/book-market portfolios; evaluate the three-factor model with the following methods, reporting coefficients, standard errors, the \(\alpha'V^{-1}\alpha\) test, RMS pricing errors, an actual-vs-predicted mean-return plot and its \(R^2\): (a) time-series regression (OLS+GRS / GMM 0 lags); (b) cross-sectional OLS/GLS (no Shanken / Shanken / GMM); (c) Fama-MacBeth; (d) DF form \(m=1-b'f\) first/second-stage GMM; (e) \(m=1-(f-Ef)'b\) (returns on covariances); (f) can you drop the market factor? Drop SMB?