11. GMM: General Formulas and Applications

11.1 General GMM Formulas

Write the model as \(E[f(x_t,b)]=0\) (\(f\) an \(L\)-vector of moments, \(x_t\) data, \(b\) an \(N\)-vector of parameters, generalizing \(u_t(b)\)). The GMM estimate: pick \(\hat b\) so some linear combination \(a_Tg_T(\hat b)=0\), where \(g_T(b)=\tfrac1T\sum_t f(x_t,b)\). The last chapter's minimization is a special case: the first-order condition of \(\min_b g_T'Wg_T\) is \(a_T=\partial g_T'/\partial b\,W\).

\(a=d'S^{-1}\) is "efficient" because the coefficient variance for any \(a\) (11.4) equals the efficient variance (11.8) plus a positive semidefinite matrix — smallest among estimators that set linear combinations to zero (estimators based on other moments may be more efficient still). The χ² test can use the nonsingular \(S^{-1}\) directly: \(Tg_T'S^{-1}g_T\to\chi^2\) (only for the optimal weights; other weights need the general formulas (11.4), (11.5) and a pseudo-inverse of the singular covariance).

Model comparison. If one model is a "restricted" case of another, using the same \(S\) (usually the unrestricted model's), the restricted \(J_T\) must rise; if the restricted model is true it should not rise "much":

This is Newey-West's (1987a) "χ²-difference test" (D-test), much like a likelihood-ratio test.

11.2 Testing Moments

To see how a model does on particular assets/pricing errors (e.g. the famous "small-firm effect": an unconditional CAPM prices the smallest-cap portfolio badly). The elements of \(g_T\) are the pricing errors, and their sampling distribution (11.5) gives the standard error — a \(t\)-test for a single \(g_T\), a χ² test for a group, or error bars on a "predicted vs. actual mean return" plot. Or use a χ²-difference test: zero out the moments to be tested, re-estimate with the same \(S\), and the difference in objectives \(\sim\chi^2(\#\text{eliminated moments})\). A trap: do not pick the largest of 10 pricing errors and note it is over two standard deviations from zero — the distribution of the largest is far wider than that of a single one; you must pick which to test before looking at the data.

11.3 Standard Errors of Anything by the Delta Method

When the quantity is a nonlinear function of sample means \(b=\phi[E(x_t)]=\phi(\mu)\), (11.2) reduces to

Very intuitive: the bracket is the covariance of the sample mean, and the derivatives linearize \(\phi\) near the true value. This yields standard errors for correlation coefficients, impulse responses, variance decompositions, and more.

11.4 Using GMM for Regressions

Mapping OLS into GMM yields standard errors corrected for autocorrelation and conditional heteroskedasticity. The OLS first-order condition is "the residual is orthogonal to the right-hand variable," \(g_T(\beta)=E_T[x_t(y_t-x_t'\beta)]=0\) (exactly identified, \(a=I\)). Substituting into (11.2) (\(d=-E(x_tx_t')\), \(f=x_t\varepsilon_t\)) gives

It reduces in special cases: serially uncorrelated + homoskedastic returns to \(\sigma_\varepsilon^2(X'X)^{-1}\); dropping homoskedasticity gives White heteroskedasticity-robust standard errors \(\tfrac1T E(xx')^{-1}E(\varepsilon^2xx')E(xx')^{-1}\); overlapping long-horizon regressions (e.g. six-month returns on monthly data) give Hansen-Hodrick standard errors (keeping only \(|j|not GLS — when errors violate OLS assumptions, OLS is still consistent and often more robust than GLS, but its standard errors need correction.

11.5 Prespecified Weighting Matrices and Moment Conditions

Fixed-\(W\) formulas: \(\operatorname{var}(\hat b)=\tfrac1T(d'Wd)^{-1}d'WSWd(d'Wd)^{-1}\), \(\operatorname{var}(g_T)=\tfrac1T(I-d(d'Wd)^{-1}d'W)S(I-Wd(d'Wd)^{-1}d')\) (reducing to the efficient form when \(W=S^{-1}\)). \(S\) still appears in all standard errors — using a prespecified \(W\) is not the same as ignoring error correlation.

Figure 11.1　True (ex ante) and sample (ex post) mean-variance frontier. The sample frontier is typically much wider; its "minimum-variance portfolio" largely reflects luck and has little to do with the true one — exactly what efficient GMM tends to overweight.

Two useful prespecified weights. ① Second-moment matrix \(W=E(xx')^{-1}\) (Hansen-Jagannathan 1997): its objective \(g_T'E(xx')^{-1}g_T\) is exactly the distance from a candidate discount factor \(y\) to the space of true discount factors (the HJ distance). As in Figure 11.2, \(\|y-\text{nearest }m\|=\|\operatorname{proj}(y|X)-x^*\|=[E(yx)-p]'E(xx')^{-1}[E(yx)-p]\). It is invariant to the initial choice of assets (a property it shares with \(S\)), ensuring "a better model is better because its pricing errors are smaller, not because it blew up the weighting matrix." ② Identity matrix \(W=I\): weights the initial assets equally and sidesteps inverting a near-singular \(S\) — many OLS cross-sectional regressions are exactly first-stage GMM with identity weights.

Figure 11.2　Hansen-Jagannathan distance: the distance from a candidate discount factor \(y\) to the nearest valid \(m\) equals the distance between \(\operatorname{proj}(y\mid X)\) and \(x^*\).

11.6 Estimating on One Group of Moments, Testing on Another

One can force the system to estimate on one set of moments and test on another — the macro RBC literature does this extensively (calibrate on "first moments," evaluate on "second moments"), or estimate parameters on one set of assets (stocks/domestic/first nine size deciles) and see "out of sample" how the model does on another (bonds/foreign/small-firm/mutual funds), with the test's distribution theory incorporating the sampling uncertainty of the parameters. Implementation is simple: use an appropriate \(a_T\) or \(W\) — e.g. with the first \(N\) moments estimating \(N\) parameters and the rest testing, take \(a_T=[I_N\ 0_{N+M}]\) (or a \(W\) that is identity in the estimation block, zero elsewhere).

11.7 Estimating the Spectral Density Matrix

Estimating \(S=\sum_j E(u_tu_{t-j}')\) (sample counterparts for population moments): (1) use a sensible first-stage \(W\) (often \(W=I\)) or transform data so moments have comparable units (e.g. \(R\times(1+d/p)\)); (2) remove means (Hansen-Singleton 1982, avoiding the added singular \(E(u)E(u)'\)); (3) downweight higher-order correlations — you cannot use all autocorrelations (a sample of 100 cannot estimate \(E(u_tu_{t+101}')\)); use the Newey-West/Bartlett estimator \(S=\sum_{j=-k}^{k}\tfrac{k-|j|}{k}\tfrac1T\sum_t u_tu_{t-j}'\) (only up to lag \(k\), downweighted, positive definite in sample); \(k\) must grow with the sample but slower (\(k\to\infty,k/T\to0\)), and in practice takes judgment; (4) consider parametric structures for autocorrelation/heteroskedasticity (e.g. AR(1): \(S=\sigma_u^2\tfrac{1+\rho}{1-\rho}\), more reliable than a long list of autocorrelations); (5) use the null to limit correlations — in asset pricing \(E_t(u_{t+1})=0\) implies all autocorrelation terms of \(S\) vanish, so one can estimate just \(S=\tfrac1T\sum u_tu_t'\) (but if returns may be predictable, keep a few terms); (6) size problems — when moments exceed ~1/10 the data points, \(S\) becomes unstable/singular, so second-stage GMM is usually limited to few assets and instruments, and one might impose a factor structure on \(S\); (7) alternatives to two-stage: iterate (repeatedly update \(S\) with \(\hat b_j\) toward a fixed point, better small-sample per Ferson-Foerster 1994) or pick \(b,S\) simultaneously (\(\min_b g_T'S^{-1}(b)g_T\), asymptotically equivalent, Hansen-Heaton-Yaron 1996).

Summary

GMM's full power is condensed into two formulas (coefficient and moment covariance) and a selection matrix \(a\): changing \(a\) lets you do parameter tests, individual pricing-error tests, robust OLS standard errors, delta-method standard errors of any nonlinear quantity, the HJ distance, and χ²-difference model comparisons. The key practical wisdom: the "efficient" \(S^{-1}\) is not always best — for robustness, near-singular \(S\), and fair model comparison, prespecified weighting matrices (\(W=I\) or HJ's \(E(xx')^{-1}\)) are often preferable, just as OLS is to GLS in regressions. Estimating \(S\) takes care (remove means, downweight higher-order correlations, use parametric structure or the null when warranted). The next chapter applies these tools to regression-based tests of linear factor models.

Problems

1. Use the delta-method version of the GMM formulas to derive the sampling variance of an autocorrelation coefficient.
2. (a) Write a formula for the standard error of OLS regression coefficients that corrects for autocorrelation but not heteroskedasticity. (b) Show that in this case conventional standard errors are fine if the \(x\)'s are uncorrelated over time, even if the errors \(\varepsilon\) are correlated over time.
3. If the GMM errors come from an asset pricing model, \(u_t=m_tR_t-1\), can you ignore lags in the spectral density matrix? What if you know returns are predictable? What if the error is formed from an instrument/managed portfolio \(u_tz_{t-1}\)?