30. Investor's Decision-Making in High Dimensional Environments

Molavi et al. (2020) study the asset pricing implications of model misspecification: agents in the economy can only use models with at most \(k\) factors (not necessarily consistent with the true factors driving the fundamental). Constrained-rational expectations is the misspecified counterpart to rational expectations — agents form expectations based on the \(k\)-factor model that best fits their observations. The resulting returns are predictable at several horizons (departing from rational-expectations theory), used to analyze UIP violations and momentum/reversal.

A discrete-time economy with a unit mass of homogeneous agents. A sequence of univariate exogenous fundamentals (e.g. asset dividends or interest rate differentials) \(\{x_t\}_{t=-\infty}^\infty\) is driven by a stationary \(n\)-factor model, \(\boldsymbol\theta^\star=(\mathbf A^\star,\mathbf b^\star,\boldsymbol\Sigma_\epsilon^\star)\) (30.1)/(30.2):

\(\{x_t\}\) is observable, \(\mathbf z_t\in\mathbb R^n\) the vector of true factors not observed/known by agents (no redundancy, i.e. no fewer factors can generate the fundamental), \(\mathbf A^\star\in\mathbb R^{n\times n}\) governs factor evolution, \(\boldsymbol\epsilon_t^\star\in\mathbb R^n\) i.i.d. \(\mathcal N(\mathbf 0,\boldsymbol\Sigma_\epsilon^\star)\), \(\mathbf b^\star\in\mathbb R^n\) the fundamental's factor loadings. The endogenous price sequence \(\{y_t\}\) is determined by observed \(\{x_t\}\) and agents' subjective beliefs (30.3): \(y_t=x_t+\delta\tilde{\mathbb E}_t[y_{t+1}]\), \(\tilde{\mathbb E}_t[\cdot]\) the time-\(t\) subjective expectation, \(\delta\in[0,1]\) the discount rate, equivalently (30.4): \(y_t=\sum_{\tau=0}^\infty\delta^\tau\tilde{\mathbb E}_t[x_{t+\tau}]\) (\(\mathbb E^\star[\cdot]\) the objective expectation). Excess return (30.5): \(rx_{t+1}=\delta y_{t+1}+x_t-y_t\), rewritten by (30.3) as the discounted difference between realized and expected price (30.6): \(rx_{t+1}=\delta(y_{t+1}-\tilde{\mathbb E}_t[y_{t+1}])\).

Behavioral assumption: agents use models with at most \(k\) factors (when \(k

30.1.3 Return Predictability

Under the rational-expectations benchmark future returns are unpredictable, so return predictability is a natural measure of how far agents depart from rational expectation. Two families of linear regressions:

The Fama regression (30.8) measures how current fundamental predicts future excess returns at different horizons \(h\ge1\); the momentum regression (30.9) measures how current excess return predicts future excess returns. If rational expectation holds, returns are unpredictable, i.e. \(\beta_h^{\text{Fama}}=0\) and \(\beta_h^{\text{mom}}=0\) \(\forall h\). Proposition 30.1 gives expressions for \(\beta_h^{\text{Fama}}\) (30.10) and \(\beta_h^{\text{mom}}\) (30.11–30.13) (derivation in the collapsible proof). Proposition 30.1 conclusions: under rational expectation (\(\tilde{\mathbb E}_t=\mathbb E^\star\)) \(\alpha_h^{\text{Fama}}=\alpha_h^{\text{mom}}=0\), \(\beta_h^{\text{Fama}}=\beta_h^{\text{mom}}=0\), returns unpredictable at any horizon; when rational expectation fails the slope coefficients are generally nonzero and differ by \(h\).

30.1.4 Constrained Rational Expectations

At \(t\), \(\{x_{\tau}\}_{\tau\le t}\) is observable, but \(\mathbf z_t\) and \((\mathbf A^\star,\mathbf b^\star,\boldsymbol\Sigma_\epsilon^\star)\) are not. Agents use a \(k\)-factor model (30.14)/(30.15): \(\boldsymbol\omega_t=\mathbf A\boldsymbol\omega_{t-1}+\boldsymbol\epsilon_t\), \(x_t=\mathbf b'\boldsymbol\omega_t\), \(\boldsymbol\omega_t\in\mathbb R^k\), \(\mathbf A\in\mathbb R^{k\times k}\), \(\boldsymbol\epsilon_t\) i.i.d. \(\mathcal N(\mathbf 0,\boldsymbol\Sigma_\epsilon)\), \(\mathbf b\in\mathbb R^k\). \(\boldsymbol\theta=(\mathbf A,\mathbf b,\boldsymbol\Sigma_\epsilon)\) summarizes the \(k\)-factor model, \(\Theta_k\) the set of all \(k\)-factor models (\(\Theta_k\subseteq\Theta_{k+1}\); \(\boldsymbol\theta^\star\in\Theta_k\) when \(k\ge n\), \(\boldsymbol\theta^\star\notin\Theta_k\) when \(k

CREE can be achieved by three methods (Theorem 30.2): (1) minimizing the mean-squared error of one-step-ahead predictions; (2) maximum likelihood; (3) Bayesian updating. Formally: \(\boldsymbol\Theta_k^{\text{CREE}}=\arg\min_{\boldsymbol\theta\in\Theta_k}\mathbb E^\star[(x_{t+1}-\tilde{\mathbb E}_t[x_{t+1}])^2]\) (30.23/30.24); since \(\mathbb E^\star[x_{t+1}^2]\) is determined by the true process, minimizing the MSE ⟺ minimizing \(H(\mathbf M,\mathbf u)\) (30.22) ⟺ obtaining CREE.

[!example] Example 30.1 (Single factor, \(k=1\)) True model \(z_{i,t+1}=a_i^\star z_{i,t}+\epsilon_{i,t+1}^\star\), \(x_t=\sum_{i=1}^n b_i^\star z_{i,t}\) (factors mutually independent). Agents use one factor \(\omega_{t+1}=a^{\text{CREE}}\omega_t+\epsilon_t\), \(\omega_t=x_t\), \(a^{\text{CREE}}\) minimizing \(\mathbb E^\star[(x_{t+1}-\tilde{\mathbb E}_t[x_{t+1}])^2]\), f.o.c. giving \(a^{\text{CREE}}=\frac{\sum_i b_i^{\star2}a_i^\star\mathbb E^\star[z_{i,t}^2]}{\sum_i b_i^{\star2}\mathbb E^\star[z_{i,t}^2]}\) (30.25), and by \(\mathbb E^\star[z_{i,t}^2]=\frac1{1-a_i^{\star2}}\) gives (30.26): \(a^{\text{CREE}}=\frac{\sum_i\frac{b_i^{\star2}a_i^\star}{1-a_i^{\star2}}}{\sum_i\frac{b_i^{\star2}}{1-a_i^{\star2}}}\). Observations: \(a^{\text{CREE}}\) puts higher weight on more persistent factors; if any factor is near unit-root (\(|a_i^\star|\approx1\)), then \(|a^{\text{CREE}}|\approx1\), and the single-factor model approximates that unit-root feature — even losing much information, it captures the key feature of the true process.

30.1.5 Asset Pricing under CREE

Propositions 30.2/30.3 (single factor \(k=1\)): \(\mathbf M=m\), \(\mathbf u=1\), \(\phi_\tau=m^\tau\), giving \(\beta_h^{\text{Fama}}=\frac\delta{1-\delta m}(\xi_h^\star-m\xi_{h-1}^\star)\) (30.39), \(\beta_h^{\text{mom}}=\frac{(1+\xi_1^{\star2})\xi_h^\star-\xi_1^\star(\xi_{h-1}^\star+\xi_{h+1}^\star)}{1-\xi_1^{\star2}}\) (30.40). Proposition 30.3: when \(\xi_1^\star>0\) (short-run momentum) there exist \(h,h'\) with \(\frac{\partial\beta_h^{\text{mom}}}{\partial\xi_h^\star}<0\) (reversal) — short-run momentum and long-run reversal can occur simultaneously.

Heterogeneous agents extension: a \(1-\lambda\) fraction of agents are subject to the \(k\)-factor constraint, a \(\lambda\) fraction can use any model. The subjective expectation in price (30.3) \(\tilde{\mathbb E}_t[\cdot]=\lambda\mathbb E^\star[\cdot]+(1-\lambda)\tilde{\mathbb E}_t^c[\cdot]\); since \(\tilde{\mathbb E}_t\) doesn't satisfy LIE, (30.7) cannot be simplified. Proposition 30.4: the Fama coefficient \(\beta_h^{\text{Fama}}(\lambda)=(1-\lambda)\sum_{s=0}^\infty(\delta\lambda)^s\beta_{h+s}^{\text{Fama}}(0)\) (30.41) (\(\beta_h^{\text{Fama}}(0)\) being (30.39)).

30.1.7 Application: Uncovered Interest Rate Parity

UIP: net interest rate \(i_{\text{US}}\) earned in the US over \(k\) periods, \(i_{\text{F}}\) in a foreign country (say China), spot exchange rate \(S_t\) at \(t\), expected \(\mathbb E_t[S_{t+k}]\) (both as Y/USD). UIP says \(\frac{S_t}{\mathbb E_t[S_{t+k}]}(1+i_{\text{F}})^k=(1+i_{\text{US}})^k\). Two puzzles: (1) forward discount puzzle (Fama 1984) — in short horizons (a week to a quarter), high-interest-rate currencies tend to have positive excess returns; (2) predictability reversal puzzle (Bacchetta-Van Wincoop 2010, Engel 2016) — in long horizons high-interest-rate currencies tend to have negative excess returns. Mapped into the model: fundamental \(x_t=i_t^\star-i_t\) (foreign-minus-US log nominal interest rate differential), \(y_t\) the log foreign-exchange rate in USD; with \(\delta=1\), (30.3) is UIP and (30.5) is \(rx_{t+1}=y_{t+1}-y_t+(i_t^\star-i_t)\). Data: Datastream 1985–2020 for Australia, Canada, Euro, New Zealand, Japan, UK (both exchange rate and rate differential are import-plus-import-share-of-total-trade weighted), with the rate differential computed by covered interest rate parity \(i_t^\star-i_t=y_t-f_t\) (\(f_t\) the forward rate). Results (Fig 30.1 Fama coefficient \(\beta_h^{\text{Fama}}\) vs \(h\)): in 1–3 years high-differential currencies earn higher risk premium (forward discount puzzle), in 4–7 years high-differential currencies predict lower risk premium (reversal puzzle), and after 7 years \(\beta_h^{\text{Fama}}\approx0\). Single-factor model: first compute the autocorrelation \(\{\xi_\tau^\star\}\) from data, then use (30.32)/(30.35) to compute the model-implied Fama coefficient (without exchange rate or excess return data) — the model-implied coefficients (Fig 30.2 red line) closely match the real data (black line), so the single-factor model can be simultaneously consistent with both puzzles. For a larger cross-section of countries, the model-implied and statistically estimated Fama coefficients are positively and significantly correlated at both short and long horizons (Fig 30.3).

Multi-factor (\(k=2,3\)): same method, computed via (30.32) (\(k>1\) has no closed form, numerical), also computing the model-implied autocorrelation function (ACF, a function of only \(\mathbf M,\mathbf u\)). Results (Fig 30.4 ACF and Fama coefficients): more factors → better ACF estimate → closer to rational expectation (almost no predictability); since data clearly show predictability (the two puzzles), agents are likely not that sophisticated (using only 1 or 2 factors). Heterogeneous agents: use (30.41) to compute the Fama coefficient with \(\lambda=90\%\) (90% rational agents); both homogeneous and heterogeneous models replicate the data and look similar (Fig 30.5), so even a small share of behavioral investors is enough to generate the anomalies.

30.1.8 Application: Momentum and Reversal

Momentum and reversal: momentum — short-term equity excess returns positively autocorrelated; reversal — long-term negatively autocorrelated. Data: Datastream 1971–2020 MSCI price and total return indices for Australia, Belgium, Canada, France, Germany, Italy, Japan, the Netherlands, Sweden, Switzerland, UK, US (excess returns scaled by lagged volatility, Moskowitz et al. 2012). Fundamental \(x_t\) the dividend/price ratio, \(rx_t\) the equity excess return. Results: the average return autocorrelations across equity indices (Fig 30.6) are mostly positive short-term and mostly negative long-term (see the table below, Cutler et al. 1991); a pooled panel regression \(rx_{s,t+h}=\alpha_h^{\text{mom}}+\beta_h^{\text{mom}}rx_{s,t}+\epsilon\) (Moskowitz et al. 2012) shows a positive-then-negative pattern in \(t\)-statistics for \(h=1,\dots,40\) (Fig 30.7); the single-factor model (using momentum regression 30.36 instead of Fama) computes the model-implied momentum coefficient, and across the cross-section the model-implied and statistically estimated momentum coefficients are positively and significantly correlated at both short and long horizons (Fig 30.8).

30.1.9 / 30.1.10 Contribution & Discussion

Contribution: the low-dimension assumption is not crazy but mirrors real investment; the new CREE framework is useful; a single framework reconciles the two exchange-rate puzzles and explains equity momentum and reversal. Discussion: no need to specify \(k\), so no economic interpretation of the modeling process, and using 3 factors isn't necessarily simpler than 10 (if we don't know what the factors are); the (30.5) excess return definition isn't the typical finance one; Proposition 30.1 is wrongly stated (the authors wrongly claim rational expectation implies only \(\beta=0\), but it also requires \(\alpha=0\)); it focuses only on return predictability with no real SDF/asset pricing (constant discount rate assumed); the \(\mathbf g\) expression in the Theorem 30.1 proof is wrong, should be \(\frac{\hat{\boldsymbol\Sigma}_\omega\mathbf b}{\mathbf b'\hat{\boldsymbol\Sigma}_\omega\mathbf b}\), \(\bar{\boldsymbol\omega}_t=(\mathbf A-\mathbf A\mathbf g\mathbf b')\bar{\boldsymbol\omega}_{t-1}+\mathbf g x_t\); Theorem 30.1 assumes \(\hat{\boldsymbol\Sigma}_\omega\) is constant and not updated, and \(\mathbf M,\mathbf u\) take values in closed intervals, not necessarily true; Theorem 30.2's (2)(3) depend on a stationary data-generating process, but CREE in this model cannot be supported by ML/Bayesian under non-stationarity; why agents never update their belief of \(k\) with more observations is a weird assumption; the (30.41) expectation operator should be \(\tilde{\mathbb E}_{t+\tau-1}\tilde{\mathbb E}_{t+\tau}\) (\(\lambda\)'s exponent should be \(\tau\), not \(\tau+1\)); cross-sectional predictability is not addressed at all. Future: price risks and study the SDF in a non-constant-discount framework; study how dimension restriction responds to MIT shocks; extensions with regime shifting (e.g. Hong et al. 2007); sensitivity analysis for different \(\lambda\).

Martin-Nagel (2020) test market efficiency in a big-data environment (thousands of potentially observable variables): \(N\) assets in the economy, with cash flows linear functions of \(J\) firm characteristics (unknown coefficients). Homogeneous risk-neutral Bayesian investors use shrinkage (ridge/lasso) and sparsity (lasso) to estimate the \(J\) coefficients for pricing. Introducing regularization has a trade-off: regularization downweights some predictors' information, reducing the first asset-pricing distortion from coefficient estimation error, but the downweighting itself introduces a second distortion (some information underweighted). Findings: when \(J\) is comparable to \(N\), an econometrician analyzing data ex-post finds cross-sectional returns predictable in-sample (even without \(p\)-hacking); standard in-sample market-efficiency tests reject the no-predictability null even when investors optimally use information in real time; but out-of-sample returns are unpredictable when investors optimally use information in real time. Takeaway: in the big-data age, out-of-sample unpredictability is the proper definition of market efficiency; finding new cross-sectional return predictors is not surprising.

Discrete time \(t=1,2,\dots\), \(N\) assets. \(\mathbf X\) an \(N\times J\) matrix (row \(i\) = firm \(i\)'s \(J\) characteristics), \(\mathbf y_t\) an \(N\times1\) vector of each firm's dividend in period \(t\) (\(\Delta\mathbf y_t=\mathbf y_t-\mathbf y_{t-1}\)), \(\mathbf p_t\) the price of a claim on the \(t+1\) dividend strip, \(\mathbf r_t=\mathbf y_t-\mathbf p_{t-1}\) the realized return. Each period is considered very long (say a decade).

Assumption 1: firms have constant characteristics; dividend growth is partially predictable by characteristics (30.42): \(\Delta\mathbf y_t=\mathbf X\mathbf g+\mathbf e_t\), \(\mathbf e_t\sim\mathcal N(\mathbf 0,\boldsymbol\Sigma_e)\), \(\mathbf g\in\mathbb R^J\) unknown to investors (who know \(\boldsymbol\Sigma_e\)); \(\mathbf X\) has full rank, \(JAssumption 2: investors are risk-neutral and the interest rate is zero (avoiding the joint-hypothesis problem — risk-neutral agents need no model to determine the discount rate).

30.2.3 Framework

Price (30.44): \(\mathbf p_t=\tilde{\mathbb E}_t[\mathbf y_{t+1}]=\mathbf y_t+\tilde{\mathbb E}_t[\mathbf X\mathbf g+\mathbf e_{t+1}]\). Under rational expectation investors know \(\mathbf g\), \(\mathbf p_t=\mathbf y_t+\mathbf X\mathbf g\), realized return \(\mathbf r_{t+1}=\Delta\mathbf y_{t+1}-\mathbf X\mathbf g=\mathbf e_{t+1}\) (unpredictable by characteristics). This paper: investors don't know \(\mathbf g\), learning from realized dividend growth \(\{\Delta\mathbf y_s\}_{s=1}^t\), \(\mathbf p_t=\mathbf y_t+\mathbf X\bar{\mathbf g}_t\) (30.45), \(\bar{\mathbf g}_t\) the time-\(t\) posterior mean of \(\mathbf g\). Prior (Assumption 3) (30.46): \(\mathbf g\sim\mathcal N(\mathbf 0,\boldsymbol\Sigma_g)\) (investors a priori know \(\mathbf g\)'s magnitude isn't too big but not which characteristics have predictive power). Posterior (30.47): \(\bar{\mathbf g}_t=(\frac1t\boldsymbol\Sigma_g^{-1}+\mathbf X'\boldsymbol\Sigma_e^{-1}\mathbf X)^{-1}\mathbf X'\boldsymbol\Sigma_e^{-1}\overline{\Delta\mathbf y_t}\) (\(\overline{\Delta\mathbf y_t}=\frac1t\sum_{s=1}^t\Delta\mathbf y_s\)) — i.e. generalized least squares with a ridge penalty (GLS+ridge).

Consider \(N,J\to\infty\) with \(\frac JN\to\psi>0\). Assumption 4 (30.48): \(\boldsymbol\Sigma_e=\mathbf I\), \(\boldsymbol\Sigma_g=\frac\theta J\mathbf I\) (\(\theta>0\)). Meaning of \(\theta\): the average variance of the predictable part \(\mathbf X\mathbf g\) of dividend growth \(\frac1N\sum_i(\mathbf X'\boldsymbol\Sigma_g\mathbf X)_{ii}=\theta\) (by 30.43). Assumption 5: the eigenvalues of \(\frac1N\mathbf X'\mathbf X=\mathbf Q\boldsymbol\Lambda\mathbf Q'\) (30.49) satisfy \(\lambda_j>\varepsilon\) (\(\varepsilon>0\) constant, ensuring the \(J\)-factor structure doesn't degenerate). Under Assumption 4 the posterior mean (30.47) is \(\bar{\mathbf g}_t=\boldsymbol\Gamma_t(\mathbf X'\mathbf X)^{-1}\mathbf X'\overline{\Delta\mathbf y_t}\) (30.50), \(\boldsymbol\Gamma_t=\mathbf Q(\mathbf I_{J\times J}+\frac J{Nt\theta}\boldsymbol\Lambda^{-1})^{-1}\mathbf Q'\) the shrinkage coefficient matrix (its \(j\)th diagonal \(\frac{\lambda_j}{\lambda_j+\frac J{Nt\theta}}\), stronger shrinkage when \(t,\theta\) small or \(\frac JN\) large or \(\lambda_j\) small — the less informative the observed data relative to the prior, the stronger the shrinkage). Asset returns (30.51) (derivation in the collapsible proof):

In-sample predictability: an econometrician ex-post runs OLS cross-sectionally regressing \(r_{i,t+1}\) on \(\mathbf X\)'s \(i\)th row \(\mathbf x_i\), \(r_{i,t+1}=\mathbf x_i'\mathbf h_{t+1}+\eta_{i,t+1}\), OLS coefficient \(\mathbf h_{t+1}=(\mathbf X'\mathbf X)^{-1}\mathbf X'\mathbf r_{t+1}\) (30.52). Substituting (30.51) gives (30.53): \(\mathbf h_{t+1}=(\mathbf I_{J\times J}-\boldsymbol\Gamma_t)\mathbf g-\boldsymbol\Gamma_t(\mathbf X'\mathbf X)^{-1}\mathbf X'\bar{\mathbf e}_t+(\mathbf X'\mathbf X)^{-1}\mathbf X'\mathbf e_{t+1}\). Under the econometrician's rational-expectations null (\(\mathbf r_{t+1}=\mathbf e_{t+1}\)), \(\mathbf h_{t+1}\sim\mathcal N(\mathbf 0,(\mathbf X'\mathbf X)^{-1})\) (30.54), constructing a scaled test statistic \(T_{re}=\frac{\mathbf h_{t+1}'(\mathbf X'\mathbf X)\mathbf h_{t+1}-J}{\sqrt{2J}}\), which under the RE null \(\to\mathcal N(0,1)\) as \(N,J\to\infty\), \(\frac JN\to\psi>0\); but under the Bayesian-learner model \(T_{re}\)'s actual distribution depends on \(\mathbf g,\mathbf e_t\). Result 1: if returns are actually generated by (30.51), the rejection probability of the RE-null predictability test \(\to1\) (\(N,J\to\infty\), \(\frac JN\to\psi>0\)) — when both the numbers of factors and assets are large and comparable, in-sample tests always show predictability. Result 2: the in-sample trading strategy with weights \(\boldsymbol\omega_{IS,t}=\frac1N\mathbf X\mathbf h_{t+1}\) (using \(t+1\) information, so not real-time implementable) has realized return whose expectation converges to the constant \(\psi\mu>0\), with Sharpe ratio \(SR_{IS}\) growing at rate \(\sqrt N\) (asymptotically exploding).

Absence of out-of-sample predictability: the real-time implementable strategy with weights \(\boldsymbol\omega_{OOS,t}=\frac1N\mathbf X\mathbf h_t\) (using only time-\(t\) information) earns zero expected return in any other period, \(\mathbb E_s[\mathbf r_s'\boldsymbol\omega_{OOS,t}]=0\) (\(\forall s\neq t\)) — natural, since investors are Bayesian and the econometrician cannot beat them with the same information set; holds for both \(st\), so both forward and backward out-of-sample tests give non-predictability.

30.2.5 / 30.2.6 Simulation & Empirical Analysis

Simulation (\(N=1000\), \(J=1,\dots,1000\)): baseline (\(\mathbf g\) normal prior) in-sample \(R^2\) fits well (Fig 30.9), the rejection probability of the no-predictability null always rejected for large enough \(J\) (Fig 30.10, 5% significance); lasso (Laplace prior, MAP with lasso penalty giving sparsity) in-sample \(R^2\) also good (Fig 30.11) and rejection probability also always rejected for large \(J\) (Fig 30.12); with excess shrinkage, only the excess-shrinkage model generates positive out-of-sample portfolio return under both normal and Laplace priors.

Empirical (CRSP 1971–2019, excluding small stocks below the NYSE 20th percentile of market cap or price below 1 USD; using \(t-2\) to \(t-120\) month returns and squared returns, \(J=238\) predictors): run a panel regression with ridge penalty (leave-one-year-out cross-validation for the ridge tuning). Results: the regression coefficients of 119 simple return predictors show short-term momentum, long-term reversal, and momentum seasonality (Fig 30.13), with squared-return regressions showing similar patterns; the in-sample vs out-of-sample comparison (12-month moving average \(R^2\) from a 20-year rolling-window ridge regression on 238 predictors, Fig 30.14) shows in-sample \(R^2\) clearly higher than out-of-sample; the predicted-return-weighted portfolio return (Fig 30.15) is also higher in-sample than out-of-sample. Summary (Fig 30.16):

So in-sample predictability (high \(R^2\) and portfolio return) is higher than both forward and backward out-of-sample predictability.

30.2.7 / 30.2.8 Contribution & Discussion

Contribution: focuses on in-model decision-making with ML tools under high-dimensional factors, contributing to a literature that mainly did out-of-model econometric analysis; points out that in-sample predictability factors are not that interesting (given the ever-increasing dimension of predictors), and the focus should be on out-of-sample performance — a conclusive and meaningful implication for future research. Discussion: comparing in-sample vs out-of-sample predictability is itself not novel in the literature; the framework used in model/simulation is not closely related to the empirical analysis (though the basic point is the same); firm characteristics assumed time-constant significantly reduces the difficulty of investor learning, so it would be interesting to consider at least one of \(\mathbf X\) and \(\mathbf g\) being time-varying; the assumption that investors know \(\boldsymbol\Sigma_e\) is too strong (investors don't even know \(\mathbf g\) for sure, so there's no way to estimate \(\boldsymbol\Sigma_e\) independently of observed data); the homogeneous Bayesian agents assumption is too strong, leaving out heterogeneity — a potentially crucial driving force of out-of-sample return predictability; it would be interesting to consider a model with agents heterogeneous in beliefs and information-processing abilities.