23. Stochastic Discount Factor Extraction

By Theorem 4.2, under Portfolio Formation & the Law of One Price there is a unique SDF \(M_{t+1}\in\mathcal X_{t+1}\), where \(\mathbf x_{t+1}=(x_{1,t+1},\dots,x_{N,t+1})'\) are \(N\) random payoffs and \(\mathcal X_{t+1}\) their linear span.

23.1.1 Extraction without Risk-Free Payoff

Theorem 4.2 gives the unique SDF without a risk-free asset, \(m_{t+1}=\mathbf x_{t+1}'\mathbb E[\mathbf x_{t+1}\mathbf x_{t+1}']^{-1}\mathbf p(\mathbf x_{t+1})\). Normalizing payoffs by their prices into gross returns \(\mathbf R_{t+1}\) (\(\mathbf p(\mathbf R_{t+1})=\mathbf 1\)) gives (23.1):

This is estimable, but since we use the \(N\) returns themselves to build the SDF, the factor dimension is as high as \(N\) and out-of-sample performance may be poor.

23.1.2 Extraction with Risk-Free Payoff

Theorem 5.1 gives the unique SDF with a risk-free asset. After normalization it rearranges to (23.2):

where \(R_f\) is the risk-free gross return and \(\boldsymbol\Sigma_{\mathbf R_{t+1}}\) the variance-covariance matrix of the \(N\) risky payoffs. By Proposition 5.2, the SDF is a linear function of the mean-variance frontier tangency portfolio gross return. Still \(N\) factors, possibly poor out-of-sample.

23.1.3 Extraction with Excess Return

Recall the excess return space (5.33), with time subscript (23.3)/(23.4): \(\mathcal Z_{t+1}=\{z_{t+1}\in\mathcal X_{t+1}:p(z_{t+1})=0\}\), spanned by \(\mathbf z_{t+1}=\mathbf R_{t+1}-R_f\mathbf 1\). Apply the SDF (23.2) to \(\mathbf z_{t+1}\) (23.5):

This shows scaling is irrelevant for pricing excess returns: any \(m^\star_{t+1}=\alpha m_{t+1}\) (\(\alpha\) constant) still prices \(\mathbf z_{t+1}\) and any excess returns. So rewrite (23.2) with \(m^\star_{t+1}\equiv m_{t+1}R_f\) to get (23.6):

where \(\boldsymbol\mu_z=\mathbb E[\mathbf R_{t+1}-R_f\mathbf 1]\). Still \(N\)-dimensional since we use the \(N\) returns to build the SDF, so out-of-sample performance may be bad.

Naturally, assume stock return moments are functions of firm characteristics, and use this to reduce the \(N\) factors to \(J\) characteristics. Let the \(N\times J\) characteristic matrix \(\mathbf X_t\) have \(N\) rows for \(N\) firms, each row a firm's characteristics (size, book-to-market, profitability, etc.), including a constant column (a column of ones), with \(J\le N\). The conditional version of (23.6) is (23.7):

where \(\boldsymbol\mu_{z,t}=\mathbb E[\mathbf z_{t+1}\mid\mathbf X_t]\), \(\boldsymbol\Sigma_t\) is the conditional variance-covariance matrix of \(\mathbf R_{t+1}\), and \(\mathbb E[(\mathbf z_{t+1}-\boldsymbol\mu_{z,t})(\mathbf z_{t+1}-\boldsymbol\mu_{z,t})']=\boldsymbol\Sigma_t\) (23.8).

Crucial assumption (23.9):

\(\boldsymbol\phi\) is a \(J\times1\) vector. The linear form is quite general: it doesn't stop us from including nonlinear functions of characteristics as one of the \(J\) characteristics, and \(\mathbf X_t\) may include time-series predictors. Plugging into (23.7), the conditional SDF becomes (23.10):

\(\mathbf W_t=\mathbf X_t'\boldsymbol\Sigma_t^{-1}\) is \(J\times N\), its \(j\)th row being factor \(j\)'s loadings on each excess return; the \(J\times1\) vector \(\tilde{\mathbf f}_{t+1}\equiv\mathbf X_t'\boldsymbol\Sigma_t^{-1}\mathbf z_{t+1}\) is the \(J\)-dimensional factor. But \(\tilde{\mathbf f}_{t+1}\) has a time-varying conditional mean \(\mathbf X_t'\boldsymbol\Sigma_t^{-1}\mathbf X_t\boldsymbol\phi\). Premultiplying by \((\mathbf X_t'\boldsymbol\Sigma_t^{-1}\mathbf X_t)^{-1}\) builds a new factor (23.11):

with constant conditional mean \(\boldsymbol\mu_{\mathbf f}=\boldsymbol\phi\) (23.12). Since each entry of \(\mathbf f_{t+1}\) is a linear combination of excess returns, \(\boldsymbol\mu_{\mathbf f}=\boldsymbol\phi\) is the risk premium of the factor vector. Its conditional variance-covariance matrix (23.13):

Plugging \(\mathbf f_{t+1}\) into (23.10) gives (23.14): \(m^\star_{t+1}=1-\mathbf b'(\mathbf f_{t+1}-\boldsymbol\mu_{\mathbf f})\), where \(\mathbf b\equiv(\mathbf X_t'\boldsymbol\Sigma_t^{-1}\mathbf X_t)\boldsymbol\phi\). By (23.13), (23.12):

\(\mathbf b\) is the factors' Sharpe ratio (normalized risk premium), i.e. the price of risk vector.

Connecting \(\boldsymbol\mu_{\mathbf f}=\boldsymbol\phi\) with (23.9): \(\boldsymbol\mu_{z,t}=\mathbf X_t\boldsymbol\mu_{\mathbf f}\) (23.16) — different firm characteristics lead to different cross-sectional expected excess returns.

23.2.2 Factors As Cross-Sectional Regression Coefficients

By (23.11) with (23.8) (\(\boldsymbol\Sigma_t=\boldsymbol\Sigma_{z,t}\)) and the GLS coefficient formula (He 2019a §4.5.4), \(\mathbf f_{t+1}\) is the coefficient vector of a cross-sectional GLS regression of excess return \(z_{i,t+1}\) on the \(J\) characteristics in \(\mathbf x_{i,t}\) (23.17):

The obtained coefficient \(\mathbf f_{t+1}\) is the risk premium common to all firms for each characteristic loading. \(\mathbf f_{t+1}\) is deterministic given \(t+1\) info but random given \(t\) info (randomness from \(\varepsilon_{i,t+1}\)), so it is still the random part of the SDF from \(t\) to \(t+1\); its time-\(t\) conditional expectation is \(\boldsymbol\mu_{\mathbf f}=\boldsymbol\phi\).

Fama-French (2018) use OLS instead of GLS for (23.17), yielding \(\mathbf f^{\text{OLS}}_{t+1}=(\mathbf X_t'\mathbf X_t)^{-1}\mathbf X_t'\mathbf z_{t+1}\), only equivalent to GLS when \(\boldsymbol\Sigma_t=\alpha\mathbf I\) (i.i.d. returns). This holds only if all returns are i.i.d., so OLS regression is always problematic.

23.2.3 Fama-MacBeth Method

Fama-MacBeth (1973), two-step:

• Step 1: for each firm \(i\), run time-series OLS \(y_{i,t+1}=\mathbf h_{t+1}'\boldsymbol\beta_i+\varepsilon_{i,t+1}\) (\(\mathbf h\) includes a constant) to get \(\hat{\boldsymbol\beta}_i\) (23.18).
• Step 2: for each \(t+1\), run the cross-sectional regression \(y_{i,t+1}=\alpha_{t+1}+\hat{\boldsymbol\beta}_i'\boldsymbol\phi_{t+1}+\varepsilon_{i,t+1}\) (23.19) to get \(\hat{\boldsymbol\phi}_t\); then the time mean \(\bar{\boldsymbol\phi}=\frac1T\sum_t\hat{\boldsymbol\phi}_t\), with standard deviation the per-entry std divided by \(\sqrt T\).

Comments: although Step 1 is simple OLS with possibly off standard errors, \(\hat{\boldsymbol\beta}_i\) is still unbiased and consistent as long as \(\mathbf h_{t+1}\perp\varepsilon_{i,t+1}\) (idiosyncratic risk orthogonal to systematic risk, reasonable); Step 2 is almost a pooled OLS, where time fixed effects partially solve autocorrelation but firm-time fixed effects or double clustering are needed for firm-level error autocorrelation. FM Step 2 resembles (23.17) (taking \(\hat{\boldsymbol\beta}_i\) as \(\mathbf x_{i,t}\)); the characteristic regression (23.17) is more general than FM — \(\mathbf x_{i,t}\) can include more characteristics (ROA, dividend yield, debt/equity) than just betas; 3/5-factor models are ad hoc, since there is evidence that hundreds of firm characteristics predict returns yet they pick only 3 or 5 betas.

23.2.4 Stochastic Discount Factor Extraction

As long as we choose \(\mathbf X_t\) well, use (23.11) to build \(\mathbf f_{t+1}\), then (23.14) \(m^\star_{t+1}=1-\mathbf b'(\mathbf f_{t+1}-\boldsymbol\mu_{\mathbf f})\) with \(\mathbf b=(\mathbf X_t'\boldsymbol\Sigma_t^{-1}\mathbf X_t)\boldsymbol\phi\), \(\boldsymbol\mu_{\mathbf f}=\boldsymbol\phi\). Estimation: \(\mathbf X_t\) directly specified (financial-statement terms) or estimated (Step-1 betas); \(\boldsymbol\Sigma_t\) from \(\mathbf R_{t+1}\) data; \(\boldsymbol\phi\) from the Step-2 regression (23.19) with \(\hat{\boldsymbol\beta}_i\) replaced by the more general \(\mathbf x_{i,t}\).

Characteristic reduction is not enough — \(J\) is still in the hundreds. Kozak et al. (2018), via a more technical method and the insight of absence of near arbitrage, reduce the SDF factor dimension far more dramatically.

23.3.1 Absence of Near Arbitrage

Absence of near arbitrage requires that no portfolio have too-high Sharpe ratio \(\frac{\bar z_p}{\sigma_p}\), otherwise we call \(\frac{\bar z_p}{\sigma_p}\) a near arbitrage opportunity.

[!example] Example 23.1 (Absence of near arbitrage) One-period investment, \(N\) stocks, half with mean \(\mu+m\) and half \(\mu-m\) (\(m>0\)), each with variance \(\sigma^2\). The investor uses a long-short strategy with weights \(\boldsymbol\omega_p=(\frac1N,\dots,\frac1N,-\frac1N,\dots,-\frac1N)'\), giving portfolio mean \(\mu_p=m\). - Case 1 (i.i.d., \(\boldsymbol\Sigma=\sigma^2\mathbf I\)): \(\sigma_p^2=\frac1N\sigma^2\to0\), Sharpe ratio explodes — no absence of near arbitrage. - Case 2 (perfectly positively correlated): \(\sigma_p^2=0\), Sharpe explodes. - Case 3 (perfectly correlated within type, same mean, uncorrelated across types): \(\sigma_p^2=\frac12\sigma^2\), Sharpe \(=\frac{\sqrt2 m}{\sigma}\) finite — absence of near arbitrage holds.

Comparing the three: absence of near arbitrage requires that stocks with similar mean levels comove and be non-diversifiable.

23.3.2 Eigen-Decomposition

Suppose the variance-covariance matrix \(\boldsymbol\Sigma\) of excess returns \(\mathbf z_t\) is constant. \(\boldsymbol\Sigma=\mathbf Q\boldsymbol\Lambda\mathbf Q^{-1}\). Since \(\boldsymbol\Sigma\) is symmetric and real, \(\mathbf Q\) is orthogonal (\(\mathbf Q\mathbf Q'=\mathbf I\), \(\mathbf Q'=\mathbf Q^{-1}\)), so (23.20):

We can rearrange so \(\lambda_1\ge\lambda_2\ge\dots\ge\lambda_N\ge0\). \(\boldsymbol\Lambda\) is diagonal with eigenvalues, \(\mathbf Q=[\mathbf q_1\ \cdots\ \mathbf q_N]\) contains eigenvectors.

23.3.3 Principal Component Factors

By (23.20), \(\mathbf Q'\boldsymbol\Sigma\mathbf Q=\boldsymbol\Lambda\), i.e. \(\mathbf q_k'\boldsymbol\Sigma\mathbf q_j=0\) (\(k\neq j\)) (23.21), \(\lambda_k=\mathbf q_k'\boldsymbol\Sigma\mathbf q_k\) (23.22). So \(\mathbf q_k\) is the portfolio weight on the \(N\) excess returns \(\mathbf z_t\) to build factor \(k\). Factor \(k\)'s excess return (23.23): \(F_{k,t}=\mathbf q_k'\mathbf z_t\), variance \(\sigma_k^2=\mathbf q_k'\boldsymbol\Sigma\mathbf q_k=\lambda_k\), factors orthogonal \(\text{Cov}(F_k,F_j)=0\) (23.24).

Any portfolio of \(\mathbf z_t\) (including each excess return) is constructible from the \(N\) factor portfolios \(\{F_1,\dots,F_N\}\): \(\boldsymbol\omega\equiv\mathbf Q^{-1}\mathbf W\) (23.25), \(\mathbf W=\mathbf Q\boldsymbol\omega=\sum_k\omega_k\mathbf q_k\) (23.26). Portfolio mean \(z_{p,t}=\mathbf W'\mathbf z_t=\sum_k\omega_k F_{k,t}\), variance \(\sigma_p^2=\mathbf W'\boldsymbol\Sigma\mathbf W=\sum_k\omega_k^2\lambda_k\). Interpretation of \(\mathbf q_k\): \(\text{Var}(F_k)^{-1}\text{Cov}(F_k,\mathbf z_t')=\mathbf q_k'\), so the \(i\)th entry of \(\mathbf q_k'\) is the regression coefficient of \(z_i\) on \(F_k\) (asset \(i\)'s exposure to factor \(k\)); the \(i\)th row of \(\mathbf Q\) is the regression coefficient vector of \(z_i\) on \(\mathbf F_t\) (exposures).

23.3.4 Level Eigen-Factor and Long-Short Eigen-Factors

One empirical property is the fast decay of eigenvalues (near zero from \(\lambda_3\) or \(\lambda_4\) on, see Figure 23.1).

Another empirical property: the first eigenvector \(\mathbf q_1\) has almost identical entries, i.e. \(\mathbf q_1\) assigns almost equal weights to all entries of \(\mathbf z_t\), so the first factor \(F_{1,t}\) can be interpreted as a level factor (Figure 23.2).

So assume (23.27): \(\mathbf q_1=(\frac1{\sqrt N},\dots,\frac1{\sqrt N})'\), ensuring equal weights and \(\mathbf q_1'\mathbf q_1=1\). This implies \(\mathbf q_k'\mathbf 1=0\) (\(k\ge2\)) (23.29), i.e. all other factors are long-short factors.

23.3.5 Stochastic Discount Factor Extraction

Define normalized weights \(\mathbf w_k\equiv\frac1{\sqrt N}\mathbf q_k\), so \(\mathbf 1'\mathbf w_1=1\), \(\mathbf 1'\mathbf w_k=0\) (\(k\ge2\)). New (equivalent) factor (23.30): \(\tilde F_{k,t}=\mathbf w_k'\mathbf z_t=\frac1{\sqrt N}F_{k,t}\), mean \(\tilde\mu_k=\mathbf w_k'\boldsymbol\mu_z\), variance \(\tilde\sigma_k^2=\mathbf w_k'\boldsymbol\Sigma\mathbf w_k=\frac{\lambda_k}N\). The \(\{\tilde F_{1,t},\dots,\tilde F_{N,t}\}\) are called latent factors, mutually orthogonal (23.31).

SDF (23.32): \(m^\star_{t+1}=1-\boldsymbol\mu_z'\boldsymbol\Sigma^{-1}(\mathbf z_{t+1}-\boldsymbol\mu_z)\). Eigen-decomposition rewrites it as a linear combination of latent factors (23.34):

Key insight: after a chain of algebra (see the collapsible derivation below), the SDF variance \(\text{Var}(m^\star_{t+1})\) can be written as (23.42), containing \(\sum_{k=2}^N\frac{(\overline{\text{Corr}}(\mu_{z,i},q_{k,i}))^2}{\lambda_k}\). By Hansen-Jagannathan (1991) (5.16), the absolute Sharpe ratio bound (23.43):

We can therefore define the reduced-dimension SDF (23.44):

\(K\) is usually small (5 or 6), and \(\tilde m_{t+1}\approx m^\star_{t+1}\). Absence of near arbitrage reduces the SDF dimension to a very small \(K\). All quantities are estimable: \(\tilde\mu_k=\mathbf w_k'\boldsymbol\mu_z\), \(\tilde\sigma_k^2=\frac{\lambda_k}N\) (from eigen-decomposition), \(\tilde F_{k,t}=\mathbf w_k'\mathbf z_t\). Kozak et al. (2018) show empirically that a small number of PC factors price especially well out-of-sample.

Relation with APT: in approximate APT (§6.1.3), idiosyncratic risk is diversified to small but nonzero, \(R^e-R_f=\boldsymbol\mu^p+\boldsymbol\beta_p'\tilde{\mathbf f}+\varepsilon^p\) (23.45), \(\mathbb E[\varepsilon^p]=0\), \(\sigma_{\varepsilon p}^2\le\delta^2\) small. The APT-factor SDF \(\tilde m_{t+1}=a+\mathbf b'\tilde{\mathbf f}\) also prices excess returns, \(\tilde m_{t+1}=m^\star_{t+1}+\tilde\varepsilon_{t+1}\) (5.12). Ideally \(\tilde\varepsilon_{t+1}\perp\mathcal Z_{t+1}\) (\(\tilde\varepsilon\perp\varepsilon^p\)) gives perfect pricing; otherwise there is a pricing error, and (23.46): \(|\mathbb E[\tilde m_{t+1}\varepsilon^p]|\le\tilde\sigma_\varepsilon\bar\sigma\). Ross (1976) critiqued that \(\tilde\sigma_\varepsilon\)'s size is unclear; absence of near arbitrage bounds \(\tilde\sigma_\varepsilon\) via (23.44) (\(\tilde\varepsilon\) contains only dropped high-\(k\) factors, orthogonal to \(\tilde m\) → \(\tilde\sigma_\varepsilon^2\le\text{Var}(m^\star)\), bounded by max Sharpe), so the APT pricing error is bounded, rescuing APT.

Kelly et al. (2019) connect firm characteristics with PCA, assuming the covariance between latent factors is based on the observable characteristic matrix. With \(T\) periods, the \(N\times1\) excess return (23.47): \(\mathbf z_t=\boldsymbol\beta_t\mathbf F_t+\boldsymbol\varepsilon_t\), \(\boldsymbol\beta_t\) being \(N\times K\), \(\mathbf F_t\) a \(K\times1\) latent factor (the first \(K\) of PCA, the other \(N-K\) folded into \(\boldsymbol\varepsilon_t\)). \(\boldsymbol\varepsilon_t\perp\mathbf F_t\) (by 23.31), \(\text{Var}(\boldsymbol\varepsilon_t)=\sigma^2\mathbf I\) (a strong assumption, as (23.31) gives homoskedasticity but not necessarily this). Crucial assumption (23.49):

\(\mathbf X_t\) is \(N\times J\), \(\boldsymbol\Gamma\) is \(J\times K\). Kelly et al. focus on small \(K\), large \(J\): using many firm characteristics as instruments to build fewer latent factors.

23.4.2 Minimization: Special Case with Constant Beta

Suppose \(\boldsymbol\beta_t=\boldsymbol\beta\) constant. Minimize SSR (23.50): \(\min_{\boldsymbol\beta,\mathbf F_1,\dots,\mathbf F_T}\sum_{t=1}^T(\mathbf z_t-\boldsymbol\beta\mathbf F_t)'(\mathbf z_t-\boldsymbol\beta\mathbf F_t)\). FOC: w.r.t. \(\mathbf F_t\) gives \(\mathbf F_t=(\boldsymbol\beta'\boldsymbol\beta)^{-1}\boldsymbol\beta'\mathbf z_t\) (23.51); w.r.t. \(\boldsymbol\beta\) gives \(\boldsymbol\beta'=(\sum_t\mathbf F_t\mathbf F_t')^{-1}(\sum_t\mathbf F_t\mathbf z_t')\) (23.52) (\(\boldsymbol\beta'\) is the OLS coefficient matrix of regressing \(\mathbf z_t\) on \(\mathbf F_t\)). Relation to PC factors: in this special case (23.51)/(23.52) are consistent with PC factors — plugging in the first \(K\) PC factors \(\boldsymbol\beta=\mathbf Q_K\), \(\mathbf F_t=\mathbf Q_K'\mathbf z_t\) satisfies them. This only says the first \(K\) PC factors are one solution to (23.50) (not unique). Time-varying \(\boldsymbol\beta\) means letting \(\mathbf Q_K\) vary over time.

23.4.3 Minimization: General Case

Substituting \(\boldsymbol\beta_t=\mathbf X_t\boldsymbol\Gamma\), solve (23.53): \(\min_{\boldsymbol\Gamma,\mathbf F_1,\dots,\mathbf F_T}\sum_{t=1}^T(\mathbf z_t-\mathbf X_t\boldsymbol\Gamma\mathbf F_t)'(\mathbf z_t-\mathbf X_t\boldsymbol\Gamma\mathbf F_t)\). The FOC w.r.t. \(\mathbf F_t\) gives (23.54): \(\mathbf F_t=(\boldsymbol\Gamma'\mathbf X_t'\mathbf X_t\boldsymbol\Gamma)^{-1}\boldsymbol\Gamma'\mathbf X_t'\mathbf z_t\); w.r.t. \(\boldsymbol\Gamma\), using the vectorization identity \(\text{vec}(\mathbf A\mathbf X\mathbf B)=(\mathbf B'\otimes\mathbf A)\text{vec}(\mathbf X)\), \(\text{vec}(\boldsymbol\Gamma')=[\sum_t(\mathbf X_t\otimes\mathbf F_t')'(\mathbf X_t\otimes\mathbf F_t')]^{-1}[\sum_t(\mathbf X_t\otimes\mathbf F_t')'\mathbf z_t]\).

23.4.4 IPCA vs. PCA

Assume further that \(\mathbf X_t'\mathbf X_t=\boldsymbol\Psi\) is constant (firm characteristics not varying over time, reasonable). Consider \(J\) managed portfolios with \(N\times J\) weight matrix \(\mathbf X_t\boldsymbol\Psi^{-\frac12}\), whose payoff (23.55): \(\mathbf J_t=(\mathbf X_t\boldsymbol\Psi^{-\frac12})'\mathbf z_t\), which by (23.47)/(23.49) becomes (23.56): \(\mathbf J_t=\boldsymbol\Psi^{\frac12}\boldsymbol\Gamma\mathbf F_t+\boldsymbol\Psi^{-\frac12}\mathbf X_t'\boldsymbol\varepsilon_t\), with residual variance \(\text{Var}(\bar{\boldsymbol\varepsilon}_t)=\mathbf I_{J\times J}\). So \(\mathbf J_t=\tilde{\boldsymbol\beta}\mathbf F_t+\bar{\boldsymbol\varepsilon}_t\), \(\tilde{\boldsymbol\beta}=\boldsymbol\Psi^{\frac12}\boldsymbol\Gamma\) constant. By §23.4.2, the optimal \(\tilde{\boldsymbol\beta}=\tilde{\mathbf Q}_K\), \(\mathbf F_t=\tilde{\mathbf Q}_K'\mathbf J_t\) (\(\tilde{\mathbf Q}\) from the eigen-decomposition of \(\mathbf J_t\)'s variance-covariance matrix \(\tilde{\boldsymbol\Sigma}\)).

That is: when \(\mathbf X_t'\mathbf X_t=\boldsymbol\Psi\), IPCA factors = PCA on the \(J\) managed portfolios: (1) form \(J\) managed portfolios per (23.55); (2) do standard PCA on them; (3) take the first \(K\) PC factors to build the SDF. IPCA's crucial assumption \(\boldsymbol\beta_t=\mathbf X_t\boldsymbol\Gamma\) is equivalent to an assumption on \(\mathbf z_t\)'s variance-covariance matrix \(\boldsymbol\Sigma_t\) (23.57)–(23.59): \(\boldsymbol\Sigma_t=\mathbf X_t\boldsymbol\Gamma(\boldsymbol\Gamma'\mathbf X_t'\mathbf X_t\boldsymbol\Gamma)^{-1}\boldsymbol\Gamma'\mathbf X_t'\boldsymbol\Sigma_t\mathbf X_t\boldsymbol\Gamma(\boldsymbol\Gamma'\mathbf X_t'\mathbf X_t\boldsymbol\Gamma)^{-1}\boldsymbol\Gamma'\mathbf X_t'+\sigma^2\mathbf I_{N\times N}\), which is not naturally true.

So choosing between PCA and IPCA is really choosing between two assumptions: PCA (Kozak 2018) \(\mathbf X_t'\mathbf X_t=\boldsymbol\Psi\) (constant managed-portfolio beta \(\tilde{\boldsymbol\beta}\)); IPCA (Kelly 2019) \(\boldsymbol\Sigma_t\) of the form (23.59).

Out-of-sample performance is one reason to restrict the number of SDF factors: with too many factors fitting in-sample pricing, if estimation has errors (it does in practice), errors get absorbed by the overfitting SDF and out-of-sample is terrible. So parsimonious (non-overfitting) specifications are preferred. Following §23.3's characteristic-factor notation, by (23.7) and (23.9) (here \(\mathbb E[\mathbf z\mid\mathbf X]=\mathbf X_t\mathbf b\)), the SDF without time subscript (23.60):

where \(\mathbf f=\mathbf X_t'\boldsymbol\Sigma_t^{-1}\mathbf z=(f_1,\dots,f_K)'\) contains all \(K\) characteristic factors (Kozak et al. 2020 call them 50 anomalies, \(K=50\), see Table 23.1). \(\mathbf f\) is a linear function of \(\mathbf z\) by (23.9). We care about \(\mathbf b\). Denote \(\boldsymbol\Sigma_{\mathbf f}\equiv\text{Var}(\mathbf f)\), \(\boldsymbol\mu_{\mathbf f}\equiv\mathbb E[\mathbf f]\), \(\bar{\mathbf f}\equiv\frac1T\sum_t\mathbf f_t\).

\(m^\star\) must price \(\mathbf f\) (by (23.5)), i.e. \(\mathbb E[m^\star\mathbf f]=\mathbf 0\) (23.61), expanding \(\boldsymbol\mu_{\mathbf f}-\boldsymbol\Sigma_{\mathbf f}\mathbf b=\mathbf 0\) to (23.62): \(\mathbf b=\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\mu_{\mathbf f}\). MVE portfolio interpretation: (5.9) Proposition 5.2 shows \(\mathbf b\) is proportional to the tangency portfolio weight \(\boldsymbol\omega_z\) (\(\mathbf b=\alpha\boldsymbol\omega_z\), since \(\boldsymbol\mu_{\mathbf f}\) is the mean of \(\mathbf f\), a linear combination of excess returns with \(R_f\) taken out); the MVE frontier spanned by \(\mathbf f\) (\(K\)-dim) lies inside the frontier spanned by \(\mathbf z\) (\(N\)-dim). GLS interpretation: (23.62) rewrites as (23.63): \(\mathbf b=(\boldsymbol\Sigma_{\mathbf f}\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\Sigma_{\mathbf f})^{-1}\boldsymbol\Sigma_{\mathbf f}\boldsymbol\Sigma_{\mathbf f}^{-1}\boldsymbol\mu_{\mathbf f}\), the coefficient of a cross-sectional GLS regression of each entry \(\mu_{f,i}\) of \(\boldsymbol\mu_{\mathbf f}\) on each row \(\boldsymbol\lambda_i\) of \(\boldsymbol\Sigma_{\mathbf f}\): \(\mu_{f,i}=\boldsymbol\lambda_i'\mathbf b+\varepsilon_i\).

23.5.3 Shrinkage Estimator

The GLS estimator (23.64): \(\min_{\mathbf b}(\boldsymbol\mu_{\mathbf f}-\boldsymbol\Sigma_{\mathbf f}\mathbf b)'\boldsymbol\Sigma_{\mathbf f}^{-1}(\boldsymbol\mu_{\mathbf f}-\boldsymbol\Sigma_{\mathbf f}\mathbf b)\), with sample analogue (23.65): \(\hat{\mathbf b}=\arg\min_{\mathbf b}(\bar{\mathbf f}-\boldsymbol\Sigma_{\mathbf f}\mathbf b)'\boldsymbol\Sigma_{\mathbf f}^{-1}(\bar{\mathbf f}-\boldsymbol\Sigma_{\mathbf f}\mathbf b)\) (assuming \(\boldsymbol\Sigma_{\mathbf f}\) known, since high-frequency data estimates variance precisely but not the mean, see Chapter 39).

First penalty (ridge): use a Bayesian framework. Likelihood \(\mathbf f_t\mid\boldsymbol\mu_{\mathbf f}\sim\mathcal N(\boldsymbol\mu_{\mathbf f},\boldsymbol\Sigma_{\mathbf f})\); prior \(\boldsymbol\mu_{\mathbf f}\sim\mathcal N(\mathbf 0,\kappa\boldsymbol\Sigma_{\mathbf f}^2)\) (\(\kappa>0\)), equivalent by (23.62) to \(\mathbf b\sim\mathcal N(\mathbf 0,\kappa\mathbf I_{K\times K})\). The posterior MAP estimator \(\hat{\mathbf b}^{\text{MAP}}=(\boldsymbol\Sigma_{\mathbf f}+\frac1{\kappa T}\mathbf I_{K\times K})^{-1}\bar{\mathbf f}\) motivates (23.67). So add a ridge penalty to (23.65), giving (23.66): \(\mathbf b=\arg\min_{\mathbf b}(\bar{\mathbf f}-\boldsymbol\Sigma_{\mathbf f}\mathbf b)'\boldsymbol\Sigma_{\mathbf f}^{-1}(\bar{\mathbf f}-\boldsymbol\Sigma_{\mathbf f}\mathbf b)+\gamma_1\mathbf b'\mathbf b\), \(\gamma_1=\frac1{\kappa T}\) (larger \(T\) = less penalty; with abundant data, impose less). Its FOC gives (23.67): \(\mathbf b=(\boldsymbol\Sigma_{\mathbf f}+\gamma_1\mathbf I_{K\times K})^{-1}\bar{\mathbf f}\).

Interpreting (23.67): eigen-decompose \(\boldsymbol\Sigma_{\mathbf f}=\mathbf Q_{\mathbf f}\boldsymbol\Lambda_{\mathbf f}\mathbf Q_{\mathbf f}'\), the PC factors \(\mathbf q_{\mathbf f,j}'\mathbf f\) orthogonal, \(\boldsymbol\Sigma_{\mathbf f,\text{PC}}=\boldsymbol\Lambda_{\mathbf f}\) (23.68), rewriting (23.60) as \(m^\star=1-\mathbf b_{\text{PC}}'(\mathbf f_{\text{PC}}-\boldsymbol\mu_{\mathbf f,\text{PC}})\). Then (23.69): \(\mathbf b_{\text{PC}}=(\boldsymbol\Lambda_{\mathbf f}+\gamma_1\mathbf I)^{-1}\bar{\mathbf f}_{\text{PC}}\), \(k\)th entry (23.70):

For large \(\lambda_{k,f}\) the shrinkage adjustment \(\approx1\) (barely shrinks); for small \(\lambda_{k,f}\) it \(\approx0\) (shrinks low-eigenvalue PC factor weights almost to zero) — exactly what we want.

Second penalty (lasso): the ridge penalty only pulls \(\mathbf b\) toward 0, not exactly 0. Add a lasso penalty (23.71): \(\hat{\mathbf b}=\arg\min(\bar{\mathbf f}-\boldsymbol\Sigma_{\mathbf f}\mathbf b)'\boldsymbol\Sigma_{\mathbf f}^{-1}(\bar{\mathbf f}-\boldsymbol\Sigma_{\mathbf f}\mathbf b)+\gamma_1\mathbf b'\mathbf b+\gamma_2\sum_{k=1}^K|b_k|\), trapping some \(b_k\) at exactly 0. The two penalties together are called the elastic net. Lasso corresponds to a Laplace prior \(\mathbf b\sim\text{Laplace}(\mathbf 0,\tau)\) (Bayesian posterior (23.72) → objective with \(\frac2{T\tau}\sum|b_k|\), exactly justifying lasso (23.71)).

23.5.4 Estimation

Split the sample in three: two pieces train \(\hat{\mathbf b}\), the third evaluates out-of-sample \(R^2\) (explaining the third's average return): \(R^2=1-\frac{(\bar{\mathbf f}_3-\boldsymbol\Sigma_{\mathbf f,3}\mathbf b)'(\bar{\mathbf f}_3-\boldsymbol\Sigma_{\mathbf f,3}\mathbf b)}{\bar{\mathbf f}_3'\bar{\mathbf f}_3}\). Minimizing (23.71) is equivalent to minimizing the distance between \(\mathbb E[m^\star\mathbf f]\) and \(\mathbf 0\) (pricing performance). Rotate the three pieces, average the \(R^2\).

The instrument weight matrix \(\mathbf W_t=\mathbf X_t'\boldsymbol\Sigma_t^{-1}\) uses a rank transformation of actual characteristics: \(\mathbf f_{t+1}=\mathbf W_t\mathbf z_{t+1}\) (23.73). Rank transformation: each period normalize characteristic values by rank \(\frac{\text{rank}(C^k_{i,t})}{n_t+1}\) into \(R^k_{i,t}\), then de-mean and standardize \(X^k_{i,t}=\frac{R^k_{i,t}-\bar R^k_t}{\sum_i|R^k_{i,t}-\bar R^k_t|}\). Pros: reduces extreme outliers' impact, and each factor has the same total market loading both cross-sectionally and across time; cons: loss of actual-variation-magnitude information.

Results (Figure 23.3, out-of-sample \(R^2\) shown by color): both penalty coefficients \(\gamma\) (pinned by \(\kappa\)) should be just right (not too high/low) for the best out-of-sample performance; a small number of factors (<10) gives very bad out-of-sample (purely characteristic-based parsimonious factor models cannot perform well); PC factors of characteristics do better than the characteristics themselves (Figure 23.3b beats 23.3a). Table/Figure 23.4 reports the best-performing raw and PC factors with their out-of-sample regression coefficients and \(t\)-statistics.