24. GMM for Linear Factor Models

GMM tools from He (2019a) Chapter 13. Moment restriction (24.1): \(\mathbb E[\mathbf F(\mathbf X,\boldsymbol\beta)]=\mathbf 0_{r\times1}\), generally \(r\times1\), \(\mathbf X\) from data, \(\boldsymbol\beta\) the \(k\times1\) coefficients of interest to estimate; these \(r\) restrictions come from the structural model and hold exactly at the true \(\boldsymbol\beta\).

Identification: \(r\ge k\) gives just/over-identification, assumed to have a unique solution; \(runder-identification, possibly multiple solutions.

Selection matrix: an \(r\times k\) matrix \(\mathbf A\) applied to (24.1) gives \(\mathbf A'\mathbb E[\mathbf F(\mathbf X,\boldsymbol\beta)]=\mathbf 0_{k\times1}\) (24.2), combining \(r\) equations into \(k\). Asymptotically one generally cannot find \(\mathbf b_N\) with \(\mathbb E[\mathbf F(\mathbf X,\mathbf b_N)]=\mathbf 0\) for all \(r\) equations (random errors). Different \(\mathbf A\) give different estimators \(\mathbf b^{\mathbf A}_N\). The optimal selection matrix \(\mathbf A^\star\) minimizes asymptotic variance: \(\sqrt N(\mathbf b^{\mathbf A}_N-\boldsymbol\beta)\xrightarrow{d}\mathcal N(\mathbf 0,\text{Cov}(\mathbf A))\), with \(\text{Cov}(\mathbf A)-\text{Cov}(\mathbf A^\star)\) positive semi-definite. He (2019a) Chapter 13 proves (24.3):

where the \(r\times r\) matrix \(\mathbf V\) is defined by \(\mathbf F(\mathbf X,\boldsymbol\beta)\sim\mathcal N(\mathbf 0,\mathbf V)\) (24.4), and the \(r\times k\) matrix \(\mathbf D\equiv\mathbb E[\frac{\partial\mathbf F(\mathbf X,\boldsymbol\beta)}{\partial\boldsymbol\beta'}]\) (24.5). The optimal estimator satisfies \(\sqrt N(\mathbf b^\star_N-\boldsymbol\beta)\xrightarrow{d}\mathcal N(\mathbf 0,(\mathbf D'\mathbf V^{-1}\mathbf D)^{-1})\). Hypothesis testing (He 2019a (13.34)) (24.6):

Following §23.3's excess return space \(\mathcal Z_t\) (spanned by the \(N\times1\) vector \(\mathbf z_t\)). Motivated by (23.7), suppose a linear SDF (24.7):

\(\mathbf b\) is \(k\times1\) to estimate; \(\mathbf f_t\) is the \(k\times1\) vector of observable factors specified by us (not estimated, so \(m^\star\) is only a function of \(\mathbf b\) and \(\boldsymbol\mu_{\mathbf f}\)); \(\boldsymbol\mu_{\mathbf f}\equiv\mathbb E[\mathbf f_t]\) is the factor mean to estimate, sample mean \(\bar{\mathbf f}=\frac1T\sum_{t=1}^T\mathbf f_t\); \(\boldsymbol\Sigma_{\mathbf f}\) the variance-covariance matrix of \(\mathbf f_t\). Assumption: \(\mathbf f_t\) and \(\boldsymbol\varepsilon_t\) (see 24.14) are orthogonal and i.i.d. across time (not cross-section).

24.3 Traded Factors

24.3.1 Beta Pricing Representation

The SDF prices \(\mathbf z_t\): \(\mathbb E[m^\star_t\mathbf z_t]=\mathbf 0\), rearranging to (24.8): \(\mathbb E[\mathbf z_t]=\boldsymbol\Sigma_{\mathbf z\mathbf f'}\mathbf b\), \(\boldsymbol\Sigma_{\mathbf z\mathbf f'}=\text{Cov}(\mathbf z_t,\mathbf f_t')\). Define \(\boldsymbol\beta\equiv\boldsymbol\Sigma_{\mathbf z\mathbf f'}\boldsymbol\Sigma_{\mathbf f}^{-1}\) (24.9), \(\boldsymbol\lambda\equiv\boldsymbol\Sigma_{\mathbf f}\mathbf b\) (24.10), giving beta pricing (24.11):

\(\boldsymbol\beta'=\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\Sigma_{\mathbf f\mathbf z'}\) is the coefficient matrix of regressing \(\mathbf z_t-\boldsymbol\mu_{\mathbf z}\) on \(\mathbf f_t-\boldsymbol\mu_{\mathbf f}\) (i.e. \(z_{i,t}\) on 1 and \(\mathbf f_t\)); the \(i\)th column of \(\boldsymbol\beta'\) is \(z_{i,t}\)'s factor loading; \(\boldsymbol\lambda\) is each factor's risk premium. Since \(\mathbf f_t\) are also excess returns, beta pricing (24.11) also holds with \(\mathbf z_t\to\mathbf f_t\), where \(\boldsymbol\beta\to\boldsymbol\Sigma_{\mathbf f}\boldsymbol\Sigma_{\mathbf f}^{-1}=\mathbf I\) and \(\boldsymbol\lambda\) unchanged, so \(\mathbb E[\mathbf f_t]=\boldsymbol\lambda\), i.e. \(\boldsymbol\mu_{\mathbf f}=\boldsymbol\lambda\) (24.12). Combined with (24.11), (24.13): \(\mathbb E[\mathbf z_t]=\boldsymbol\beta\boldsymbol\mu_{\mathbf f}\), and (24.14): \(\mathbf z_t=\boldsymbol\beta\mathbf f_t+\boldsymbol\varepsilon_t\) (\(\mathbb E[\boldsymbol\varepsilon_t]=\mathbf 0\)). \(\boldsymbol\mu_{\mathbf f}=\boldsymbol\lambda\) with (24.10) gives (24.15): \(\mathbf b=\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\mu_{\mathbf f}\).

24.3.2 Moment Restrictions

Suppose all \(\mathbf f_t\) trade in the market as excess returns. Three sets of restrictions (\(N+2k\) total):

but only \(2k\) coefficients (\(\mathbf b\) and \(\boldsymbol\mu_{\mathbf f}\)) to identify, so this is over-identification.

24.3.3 Calculations

The general-notation \(\boldsymbol\beta\) in (24.1) maps here to \((\mathbf b',\boldsymbol\mu_{\mathbf f}')'\). Computing \(\mathbf D\) (24.5) gives (24.16), and \(\mathbf V\) (24.4) gives (24.17)–(24.23):

Using \(\mathbf z_t=\boldsymbol\beta\mathbf f_t+\boldsymbol\varepsilon_t\) and \(\mathbf f_t\perp\boldsymbol\varepsilon_t\) i.i.d. (24.18–24.22), substituting into \(\mathbf V\) (24.17) gives (24.23).

Optimal selection matrix \(\mathbf A^\star=\mathbf V^{-1}\mathbf D\) (24.3). Guess the blocks of \(\mathbf V^{-1}\) (24.24–24.26): \(\mathbf V_{11}=(1+\boldsymbol\mu_{\mathbf f}'\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\mu_{\mathbf f})^{-1}\boldsymbol\Sigma_{\boldsymbol\varepsilon}^{-1}\), \(\mathbf V_{12}=-(1+\boldsymbol\mu_{\mathbf f}'\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\mu_{\mathbf f})^{-1}\boldsymbol\Sigma_{\boldsymbol\varepsilon}^{-1}\boldsymbol\beta\), \(\mathbf V_{13}=\mathbf 0_{N\times k}\); after verification, \(\mathbf A^\star=\mathbf V^{-1}\mathbf D\) (24.27).

24.3.4 Estimation

Key result: the first \(N\) rows of \((\mathbf A^\star)'\) are all zeros (24.29) — i.e. zero weight on the first \(N\) moment restrictions \(\mathbb E[m^\star\mathbf z_t]=\mathbf 0\), with the remaining \(2k\) giving just-identification.

The i.i.d. sample analogue (24.30), estimators (24.31)/(24.32):

((24.32) is the sample analogue of \(\mathbf b=\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\mu_{\mathbf f}\) (24.33), derived from the middle \(k\) rows.)

24.3.5 Hypothesis Testing

The estimators satisfy the last \(2k\) rows, so can only fail the first \(N\); denote the first-\(N\)-row errors \(\hat{\boldsymbol\alpha}\). The first \(N\) rows \(\Leftrightarrow\boldsymbol\mu_{\mathbf z}-\boldsymbol\Sigma_{\mathbf z\mathbf f'}\mathbf b=\mathbf 0\) (24.34), sample analogue (24.35), plugging \(\hat{\mathbf b}\) gives (24.36): \(\hat{\boldsymbol\alpha}=\bar{\mathbf z}-\hat{\boldsymbol\Sigma}_{\mathbf z\mathbf f'}\hat{\mathbf b}\). By (24.6) (\(\chi^2_{N+2k-2k}=\chi^2_N\)), only the first \(N\) rows nonzero, giving the test statistic (24.40):

where \(\hat{\boldsymbol\Sigma}_{\boldsymbol\varepsilon},\hat{\boldsymbol\beta},\hat{\boldsymbol\Sigma}_{\mathbf f}\) are in (24.39), used to decide whether to reject the null that "the SDF in (24.7) satisfies all \(N+2k\) moment restrictions".

GMM vs time-series regression: the optimal GMM estimate of (24.14), \(\hat{\boldsymbol\beta}'=\hat{\boldsymbol\Sigma}_{\mathbf f}^{-1}\hat{\boldsymbol\Sigma}_{\mathbf f\mathbf z'}\) (24.39), is the time-series regression coefficient of \(\mathbf z_t-\boldsymbol\mu_{\mathbf z}\) on \(\mathbf f_t-\boldsymbol\mu_{\mathbf f}\). Since \(\boldsymbol\mu_{\mathbf z}=\boldsymbol\beta\boldsymbol\mu_{\mathbf f}\) (24.13), \(\mathbf z_t=\hat{\boldsymbol\beta}\mathbf f_t+\tilde{\boldsymbol\varepsilon}_t\) (24.41), i.e. under i.i.d. optimal GMM = time-series OLS. Ideally \(\tilde{\boldsymbol\varepsilon}\) has zero sample mean, but not guaranteed → \(\mathbf z_t=\hat{\boldsymbol\alpha}+\hat{\boldsymbol\beta}\mathbf f_t+\hat{\tilde{\boldsymbol\varepsilon}}_t\) (24.42), \(\hat{\boldsymbol\alpha}\) being used for the test statistic.

Interpreting the test statistic (24.40): \(\hat{\boldsymbol\alpha}'(\hat{\boldsymbol\Sigma}_{\boldsymbol\varepsilon}^{-1})\hat{\boldsymbol\alpha}\) is the maximized squared Sharpe ratio of the zero-exposure portfolio (zero exposure to \(\mathbf f_t\)) built from \(\mathbf z_t\) (\(\hat{\boldsymbol\alpha}=\mathbb E[\mathbf z_t-\boldsymbol\beta\mathbf f_t]\) is the mean after removing factor exposure; if the model is correct \(\hat{\boldsymbol\alpha}=\mathbf 0\), and the closer to zero the better). The adjustment \((1+\hat{\boldsymbol\mu}_{\mathbf f}'\hat{\boldsymbol\Sigma}_{\mathbf f}^{-1}\hat{\boldsymbol\mu}_{\mathbf f})^{-1}\) corrects for factor-coefficient estimation error: higher \(\hat{\boldsymbol\mu}_{\mathbf f}\) → smaller statistic, less likely to reject (factors explaining more excess return should be harder to reject); higher \(\hat{\boldsymbol\Sigma}_{\mathbf f}\) → larger statistic, more likely to reject (more volatile factors should have more explaining power, so a higher testing standard). \(\hat{\boldsymbol\alpha}'(\hat{\boldsymbol\Sigma}_{\boldsymbol\varepsilon}^{-1})\hat{\boldsymbol\alpha}\) is closely related to the Hansen-Jagannathan distance (§5.1.7), justifying model comparison.

Comparing two factor models: use \(\hat{\boldsymbol\alpha}'_A(\hat{\boldsymbol\Sigma}_{\boldsymbol\varepsilon,A}^{-1})\hat{\boldsymbol\alpha}_A\) and \(\hat{\boldsymbol\alpha}'_B(\hat{\boldsymbol\Sigma}_{\boldsymbol\varepsilon,B}^{-1})\hat{\boldsymbol\alpha}_B\) (maximized squared Sharpe of the zero-exposure portfolio). The test assets \(\mathbf z_t\) are irrelevant to this comparison; the smaller one gives a better linear SDF with all tradable factors. For comparison, drop the factor-variance normalization \((1+\boldsymbol\mu_{\mathbf f}'\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\mu_{\mathbf f})^{-1}\).

Cross-sectional \(R^2\) for model comparison (general, not restricted to tradable factors): - Way 1: compute the implied mean return \(\widehat{\mathbb E[\mathbf z_t]}\). The first \(N\) restrictions → \(\mathbb E[\mathbf z_t]=\mathbb E[\mathbf b'(\mathbf f_t-\boldsymbol\mu_{\mathbf f})\mathbf z_t]\) (24.43), sample analogue \(\widehat{\mathbb E[\mathbf z_t]}=\frac1T\sum_t\hat{\mathbf b}'(\mathbf f_t-\hat{\boldsymbol\mu}_{\mathbf f})\mathbf z_t\); actual sample mean \(\bar{\mathbf z}\). Run the cross-sectional regression \(\bar z_i=\widehat{\mathbb E[z_i]}+\epsilon_i\) (24.44) (restricting constant 0, coefficient 1), report \(R^2=1-\frac{\overline{\text{Var}}(\epsilon_i)}{\overline{\text{Var}}(\bar z_i)}\) (24.45). Since \(\bar\epsilon\) need not be zero, this \(R^2\) need not be positive. - Way 2: Fama-MacBeth — Step 1 time-series regression gives \(\hat{\boldsymbol\beta}_i\); Step 2 cross-sectionally regress the time mean \(\bar y_i\) on \(\hat{\boldsymbol\beta}_i\) to estimate the risk premium \(\hat{\boldsymbol\phi}\), report \(R^2\).

Different people define \(R^2\) differently with no unifying framework, so be careful which \(R^2\) is meant.

The factors \(\mathbf f_t\) are now non-tradable, so (24.12)–(24.15) (all based on traded factors) no longer hold. But \(\mathbb E[\mathbf z_t]=\boldsymbol\beta\boldsymbol\lambda\) (24.46) still holds, with \(\boldsymbol\beta\equiv\boldsymbol\Sigma_{\mathbf z\mathbf f'}\boldsymbol\Sigma_{\mathbf f}^{-1}\) (24.47), \(\boldsymbol\lambda\equiv\boldsymbol\Sigma_{\mathbf f}\mathbf b\) (24.48) (same as 24.9/24.10). The model (24.49):

\(\boldsymbol\beta_0\) is a natural result of factors being non-tradable, not a pricing error \(\boldsymbol\alpha\) (unlike 24.42). The important later estimation equation (24.50): \(\mathbf b=\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\lambda\), and in this case we no longer have \(\boldsymbol\lambda=\boldsymbol\mu_{\mathbf f}\).

24.4.2 Moment Restrictions

Same as §24.3's traded case, except factors are not directly tradable, giving \(k\) restrictions on pricing the \(k\) factors. \(N+k\) restrictions: \(\mathbb E[m^\star_t\mathbf z_t]=\mathbf 0_{N\times1}\) (\(N\)), \(\mathbb E[\mathbf f_t-\boldsymbol\mu_{\mathbf f}]=\mathbf 0_{k\times1}\) (\(k\)). \(2k\) coefficients (\(\mathbf b,\boldsymbol\mu_{\mathbf f}\)), \(N>k\), still over-identification.

24.4.3 Calculations

Similar to §24.3 minus \(k\) rows. \(\mathbf D_{(N+k)\times2k}\) (24.51): like (24.16) without the middle \(k\) rows, \(\mathbf D=[[-\boldsymbol\beta\boldsymbol\Sigma_{\mathbf f},\boldsymbol\beta\boldsymbol\mu_{\mathbf f}\mathbf b'],[\mathbf 0,-\mathbf I]]\). \(\mathbf V\) (24.52) without middle rows/columns. \(\mathbf A^\star=\mathbf V^{-1}\mathbf D\), using \(\mathbf V_{11}=(1+\boldsymbol\lambda'\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\lambda)^{-1}\boldsymbol\Sigma_{\boldsymbol\varepsilon}^{-1}\) (24.53), \(\boldsymbol\lambda=\boldsymbol\Sigma_{\mathbf f}\mathbf b\). But the first \(N\) columns of \(\mathbf A^\star\) are no longer all zeros — the test assets \(\mathbf z_t\) enter estimation.

24.4.4 Estimation

Moment restrictions (24.54), sample analogue (24.55), estimators (24.56)–(24.58):

The first \(N\) rows imply \(\boldsymbol\Sigma_{\mathbf z\mathbf f'}\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\Sigma_{\mathbf f}\mathbf b=\boldsymbol\mu_{\mathbf z}\) (i.e. \(\boldsymbol\beta\boldsymbol\lambda=\boldsymbol\mu_{\mathbf z}\)) (24.59), sample analogue (24.60): \(\hat{\mathbf b}=\hat{\boldsymbol\Sigma}_{\mathbf f}^{-1}(\hat{\boldsymbol\beta}'\hat{\boldsymbol\Sigma}_{\boldsymbol\varepsilon}^{-1}\hat{\boldsymbol\beta})^{-1}\hat{\boldsymbol\beta}'\hat{\boldsymbol\Sigma}_{\boldsymbol\varepsilon}^{-1}(\frac1T\sum\mathbf z_t)\), \(\hat{\boldsymbol\beta}=\hat{\boldsymbol\Sigma}_{\mathbf z\mathbf f'}\hat{\boldsymbol\Sigma}_{\mathbf f}^{-1}\) (24.61). The \(\hat{\boldsymbol\lambda}\) in (24.60) is the coefficient of a cross-sectional GLS regression of \(\hat{\boldsymbol\mu}_{\mathbf z}\) on \(\hat{\boldsymbol\beta}\) (He 2019a §4.5.4). GLS is the most efficient linear estimator under heteroskedasticity, so it is given by the optimal selection matrix \(\mathbf A^\star\).

24.4.5 Factor Mimicking Portfolios

We can interpret the GLS estimator \(\boldsymbol\lambda\) in (24.60) as the mean vector of the mimicking portfolio built from a linear combination of \(\mathbf z_t\). Mimicking portfolios track factor movement best; cross-sectional GLS regression of \(\boldsymbol\mu_{\mathbf z}\) on \(\boldsymbol\beta\) is precisely the best way to find the mean of each factor mimicking portfolio. The weight matrix \(\mathbf W=(\boldsymbol\beta'\boldsymbol\Sigma_{\boldsymbol\varepsilon}^{-1}\boldsymbol\beta)^{-1}\boldsymbol\beta'\boldsymbol\Sigma_{\boldsymbol\varepsilon}^{-1}\), sample \(\hat{\mathbf W}\), its \(k\)th row the weight vector for the (non-tradable) \(k\)th factor mimicking portfolio (tradable). The mimicking portfolio vector \(\tilde{\mathbf f}_t=\mathbf W\mathbf z_t\), sample (24.62); its mean \(\tilde{\boldsymbol\mu}_{\mathbf f}=\mathbf W\boldsymbol\mu_{\mathbf z}=\boldsymbol\lambda\), sample (24.63): \(\hat{\tilde{\boldsymbol\mu}}_{\mathbf f}=\hat{\mathbf W}\hat{\boldsymbol\mu}_{\mathbf z}=\hat{\boldsymbol\lambda}\), the average of the regression coefficient vector \(\tilde{\mathbf f}\) in (24.62) (corresponding to FM Step 2; Step 1 estimates \(\hat{\boldsymbol\beta}\) by time-series OLS). (24.63) is exactly the \(\boldsymbol\lambda\) in (24.60), so (24.60) rewrites as \(\mathbf b=\boldsymbol\Sigma_{\mathbf f}^{-1}\tilde{\boldsymbol\mu}_{\mathbf f}\), the same as the tradable case (24.32) — because mimicking portfolios are tradable and satisfy all conditions of §24.3 (\(\mathbf b=\boldsymbol\Sigma_{\mathbf f}^{-1}\tilde{\boldsymbol\mu}_{\mathbf f}\), not \(\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\mu_{\mathbf f}\)).

24.4.6 Hypothesis Testing

The estimators satisfy the last \(k\) rows, so can only fail the first \(N\); denote the errors \(\hat{\boldsymbol\alpha}\). By (24.6) (\(\chi^2_{N+k-2k}=\chi^2_{N-k}\)), only the first \(N\) rows nonzero, giving the test statistic (24.67):

where \(\hat{\boldsymbol\Sigma}_{\boldsymbol\varepsilon},\hat{\boldsymbol\beta},\hat{\boldsymbol\Sigma}_{\mathbf f}\) are in (24.66), used to decide whether to reject the null that "the SDF in (24.7) satisfies all \(N+k\) moment restrictions".