25. Dynamic Factor Models

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 conditional linear SDF (25.1):

\(\mathbf b_t\) is \(k\times1\) to estimate; \(\mathbf f_t\) the specified \(k\times1\) observable factors (not estimated); \(\boldsymbol\mu_{\mathbf f,t}=\mathbb E_t[\mathbf f_t]\) the factor mean to estimate; \(\boldsymbol\Sigma_{\mathbf f,t}\) the variance-covariance matrix of \(\mathbf f_t\). \(\mathbf b_t,\boldsymbol\mu_{\mathbf f,t},\boldsymbol\Sigma_{\mathbf f,t}\) can be time-varying, in the time-\(t\) information set \(\mathcal F_t\).

Crucial difference from Chapter 24: here the pricing equation (5.13) holds conditionally at any time \(t\), whereas Chapter 24 only assumes the unconditional version. By the law of iterated expectations (LIE), conditional moment restrictions imply the unconditional version. Crucially, we allow \(\mathbf b_t,\boldsymbol\mu_{\mathbf f,t},\boldsymbol\Sigma_{\mathbf f,t}\) to be time-varying (all constant in Chapter 24). Without loss of generality, assume factors \(\mathbf f_t\) are tradable.

25.2 Conditional Moment Restrictions

25.2.1 Separability by Linearity in Stochastic Discount Factor

By (25.1), the conditional moment restriction (25.2): \(\mathbb E_t[m^\star_{t+1}\mathbf z_{t+1}]=\mathbf 0\). Substituting (25.1), rewrite (25.3):

The LHS of (25.3) splits into two parts: Part 1 \(\mathbb E_t[\mathbf z_{t+1}(\mathbf f_{t+1}-\boldsymbol\mu_{\mathbf f,t})']\) (estimable by statistical models); Part 2 \(\mathbf b_t\) (focusing on the relationship between \(\mathbf b_t\) and underlying state variables). Separability is helpful since the two parts can be worked out in very different ways. But if the SDF is nonlinear in factors (cannot be equivalently rewritten as linear), separability fails.

25.2.2 Dynamically Managed Portfolio

To test (25.2), multiply the restriction by a vector of time-\(t\) variables \(\mathbf h_t\) (25.4): \(\mathbf h_t\otimes\mathbb E_t[m^\star_{t+1}\mathbf z_{t+1}]=\mathbf 0\Rightarrow\mathbb E_t[m^\star_{t+1}(\mathbf h_t\otimes\mathbf z_{t+1})]=\mathbf 0\), \(\otimes\) the Kronecker product (\(\mathbf h_t\otimes\mathbf z_{t+1}=(h_{1,t}\mathbf z_{t+1},\dots,h_{m,t}\mathbf z_{t+1})'\)).

• If we don't add \(\mathbf h_t\), the conditional restriction \(\mathbb E_t[m^\star\mathbf z_{t+1}]\) reduces by LIE to the unconditional \(\mathbb E[m^\star\mathbf z_{t+1}]\) — only testing the unconditional restriction, implicitly degenerating the conditional problem to unconditional.
• But after adding the time-\(t\) variable \(\mathbf h_t\), the conditional version \(\mathbb E_t[m^\star(\mathbf h_t\otimes\mathbf z_{t+1})]=\mathbf 0\) does not degenerate to unconditional, because \(\mathbb E[m^\star(\mathbf h_t\otimes\mathbf z_{t+1})]=\mathbf 0\) does not necessarily hold (unless the conditional version holds).

\(\mathbf h_t\otimes\mathbf z_{t+1}\) can be interpreted as a dynamically managed portfolio: the portfolio weights \(\mathbf h_t\) are adjusted every period \(t\), but the managed portfolio still must be priced by the dynamic SDF \(m^\star_{t+1}\) every period, giving an economic interpretation to the dynamic moment restrictions (25.4).

25.3 Calculations

The same idea and algebra as §24.3, except everything here is conditional on time \(t\). The SDF prices \(\mathbf z_t\) → \(\mathbb E_t[\mathbf z_{t+1}]=\boldsymbol\Sigma_{\mathbf z\mathbf f',t}\mathbf b_t\) (25.5), \(\boldsymbol\Sigma_{\mathbf z\mathbf f',t}=\text{Cov}_t(\mathbf z_{t+1},\mathbf f_{t+1}')\). Define \(\boldsymbol\beta_t\equiv\boldsymbol\Sigma_{\mathbf z\mathbf f',t}\boldsymbol\Sigma_{\mathbf f,t}^{-1}\) (25.6), \(\boldsymbol\lambda_t\equiv\boldsymbol\Sigma_{\mathbf f,t}\mathbf b_t\) (25.7), giving beta pricing (25.8):

\(\boldsymbol\beta_t'=\boldsymbol\Sigma_{\mathbf f,t}^{-1}\boldsymbol\Sigma_{\mathbf f\mathbf z',t}\) is the coefficient of regressing \(\mathbf z_t-\boldsymbol\mu_{\mathbf z,t}\) on \(\mathbf f_t-\boldsymbol\mu_{\mathbf f,t}\); \(\boldsymbol\lambda_t\) is the time-\(t\) risk premium. Since \(\mathbf f_t\) are also excess returns, (25.8) holds with \(\mathbf z\to\mathbf f\) (\(\boldsymbol\beta\to\mathbf I\), \(\boldsymbol\lambda_t\) unchanged), so \(\mathbb E_t[\mathbf f_{t+1}]=\boldsymbol\lambda_t\), i.e. \(\boldsymbol\mu_{\mathbf f,t}=\boldsymbol\lambda_t\) (25.9). Combined with (25.8), (25.10): \(\mathbb E_t[\mathbf z_{t+1}]=\boldsymbol\beta_t\boldsymbol\mu_{\mathbf f,t}\), and (25.11): \(\mathbf z_{t+1}=\boldsymbol\beta_t\mathbf f_{t+1}+\boldsymbol\varepsilon_{t+1}\). \(\boldsymbol\mu_{\mathbf f,t}=\boldsymbol\lambda_t\) with (25.7) gives (25.12): \(\mathbf b_t=\boldsymbol\Sigma_{\mathbf f,t}^{-1}\boldsymbol\mu_{\mathbf f,t}\).

25.4 Converting Dynamic Pricing to Static Pricing

The expressions in §25.3 hold generally without further assumptions. Way 1: directly work on \(\mathbb E_t[m^\star_{t+1}\mathbf z_{t+1}]=\mathbf 0\) using statistical models for conditional moments. Way 2: make one more assumption on \(\mathbf b_t\) (25.12) to degenerate dynamic pricing into the static pricing of §24.3.

Assumption (25.13): \(\mathbf b_t=\boldsymbol\phi_0+\boldsymbol\phi_h h_t\), \(\boldsymbol\phi_0,\boldsymbol\phi_h\) being \(k\times1\) constants, \(h_t\) a scalar state variable at \(t\). For simplicity, consider a single state variable (WLOG, extendable to multi-state \(\mathbf h_t\)). The state \(h_t\) must be assumed observable (else the model cannot be tested).

Since the SDF for excess returns can be multiplied by any (time-\(t\)) scalar and still prices, (consistent with (25.1)) the SDF (25.14): \(M_{t+1}=k_t-\mathbf b_t'(\mathbf f_{t+1}-\boldsymbol\mu_{\mathbf f,t})\), where \(\mathbf b_t\) is a rescaled version of \(\mathbf b_t\) in (25.1), and \(k_t\) is defined by (25.15):

\(\boldsymbol\mu_{\mathbf f}\) is the unconditional mean of \(\mathbf f_t\), \(\boldsymbol\mu_{\mathbf f,t}\) the conditional mean at \(t\). Denote \(\boldsymbol\phi=(\boldsymbol\phi_0',\boldsymbol\phi_h')'\), \(\mathbf g_{t+1}=(\mathbf f_{t+1}',(\mathbf f_{t+1}h_t)')'\), \(\boldsymbol\mu_{\mathbf g}=\mathbb E[\mathbf g_{t+1}]\). Plugging (25.13) into (25.14), after algebra (25.16):

(25.16) has constant \(\boldsymbol\phi\) and \(\boldsymbol\mu_{\mathbf g}\), of exactly the same form as Chapter 24's (24.7). So under the crucial assumption (25.13), we successfully degenerate the time-varying-parameter dynamic SDF (25.14) into the constant-parameter static SDF (25.16), and all of Chapter 24's estimation methods (GMM, Fama-MacBeth, testing, model comparison) for \(\boldsymbol\phi\) and \(\boldsymbol\mu_{\mathbf g}\) apply directly.