26. Bayesian Learning with Subjective Beliefs

The economy has an asset paying dividend \(D_t\) at \(t\), price \(P_t\) (interpretable as the whole equity market or an individual stock; an individual stock is more complex due to cross-sectional learning across the same industry, so assume the aggregate market). Denote \(d_t\equiv\ln D_t\), \(\Delta d_t=d_t-d_{t-1}\), dividend growth (26.1): \(\Delta d_t=\mu+\varepsilon_t\), \(\varepsilon_t\overset{iid}\sim\mathcal N(0,\sigma^2)\), \(\mu\) the objective (true) mean. \(p_t\equiv\ln P_t\). The Campbell-Shiller decomposition (7.5) (\(\kappa_0\to k\), \(\kappa_1\to\rho\)) (26.2):

\(r_t\) is the log gross return, the risk-free rate \(r_f\) constant. \(\tilde{\mathbb E}_t[\cdot]\) is the time-\(t\) subjective conditional expectation (under the subjective measure); separately the econometrician uses the objective measure \(\mathbb E_t[\cdot]\) and knows the true \(\mu\).

Crucial assumption: the agent has constant subjective risk premium \(\tilde\theta\) (26.3): \(\tilde{\mathbb E}_t[r_{t+j}]=\tilde\theta+r_f\); and a time-varying subjective estimate of the (constant) unconditional dividend growth rate (for all future periods) (26.4): \(\tilde{\mathbb E}_t[\Delta d_{t+j}]=\tilde\mu_t\), \(\forall t\ge1\). The true unconditional mean of \(\Delta d\) is \(\mu\) in (26.1); the time subscript on \(\tilde\mu_t\) just indicates the estimate varies with information. (One could also let \(\tilde\theta\) vary; here we focus on a single source of subjective beliefs.)

26.2 Return Predictability

At \(t\), \(p_t,d_t\) are observed by both. Taking the subjective expectation of (26.2) using (26.3)/(26.4) gives (26.5):

Today's price \(p_t\) varies with the subjective dividend-growth expectation \(\tilde\mu_t\). Econometrician's view: (26.2)≡(7.4) → \(r_{t+1}=d_t-p_t+k+\rho(p_{t+1}-d_{t+1})+\Delta d_{t+1}\) (26.6). Plugging (26.5) into (26.6), then taking the objective expectation of both sides, gives return predictability (26.8):

The predictability in (26.8) comes from two parts: Part 1 \(\mu-\tilde\mu_t\) (predictable misperception of next period's dividend growth — if the agent over-predicts, \(\mu-\tilde\mu_t<0\), the econometrician knows the objective risk premium is lower next period); Part 2 \(\mathbb E_t[\tilde\mu_{t+1}]-\tilde\mu_t\) (predictable dynamics of the agent's subjective beliefs — if the econometrician knows the agent will be more optimistic next period, \(\mathbb E_t[\tilde\mu_{t+1}]-\tilde\mu_t>0\), the objective risk premium is higher next period).

26.3 Bayesian Learning

26.3.1 Assumptions

The parameter the agent wants to know is \(\mu\) in (26.1), \(\varepsilon_t\overset{iid}\sim\mathcal N(0,\sigma^2)\), and the p.d.f. \(f(\Delta d_t\mid\mu)\) is an exponential family (26.9). The agent knows \(\sigma^2\) but not \(\mu\) (reasonable: large-sample/high-frequency data improves \(\sigma^2\) but not \(\mu\), see Chapter 39). Accumulated information at \(t\): \(\mathcal H_t=\{\Delta d_t,\Delta d_{t-1},\dots,\Delta d_1\}\). Bayesian view: the agent treats \(\mu\) as a random variable (subjective uncertainty about it).

26.3.2 Conjugate Prior

The agent has a conjugate prior \(\pi_0(\mu)\) before observing any data (He 2019a §7.3) (26.10). Rewriting the numerator (26.11) shows the conjugate prior is a normal distribution with mean \(\tilde\mu_0\), variance \(\nu_0\sigma^2\), \(\tilde\mu_0,\nu_0>0\) arbitrary (26.12):

If \(\nu_0=+\infty\), the agent has no prior information (flat prior, equal zero weights on any \(\mu\)).

26.3.3 Bayesian Updating

Application: after observing \(\mathcal H_1\), the posterior \(\mu\sim\mathcal N(\tilde\mu_1,\sigma_1^2)\). By Proposition 26.1 and induction, the posterior at \(t\) (26.13):

• If \(\nu_0=+\infty\) (no prior information): \(\nu_t=\frac1t\) (simple induction), so the agent effectively takes the sample average of \(\mathcal H_t\) to estimate \(\mu\) (intuitive: with no prior, weight each new piece equally).
• For any \(\nu_0\), \(\nu_t\) is decreasing in time (since \(\frac1{1+\nu_{t-1}}\in(0,1)\)), so the information gain from observing \(\Delta d_t\) decreases over time — decreasing-gain updating.

26.3.4 Dynamic Subjective Distribution of Dividend Growth

\(\Delta d_{t+1}=\mu_t+\varepsilon_{t+1}\), \(\varepsilon\overset{iid}\sim\mathcal N(0,\sigma^2)\), with mean \(\mu_t\) following the subjective distribution \(\mu_t\sim\mathcal N(\tilde\mu_t,\nu_t\sigma^2)\). \(\varepsilon_{t+1}\) (time \(t+1\) info) \(\perp\mu_t\) (time \(t\) info) → \(\tilde{\mathbb E}_t[\Delta d_{t+1}]=\tilde\mu_t\), \(\widetilde{\text{Var}}_t(\Delta d_{t+1})=(1+\nu_t)\sigma^2\), i.e. (26.17):

26.3.5 Martingale Posterior Beliefs

Rewrite (26.14) for \(t\to t+1\): \(\tilde\mu_{t+1}=\tilde\mu_t+\nu_{t+1}(\Delta d_{t+1}-\tilde\mu_t)\) (26.18), the only random part being \(\Delta d_{t+1}-\tilde\mu_t\) (posterior at \(t\), prior at \(t+1\)), \(\nu_{t+1}\) deterministic by (26.15). By (26.17), \(\Delta d_{t+1}-\tilde\mu_t\sim\mathcal N(0,(1+\nu_t)\sigma^2)\), so the time-\(t\) distribution of \(\tilde\mu_{t+1}\) is \(\mathcal N(\tilde\mu_t,\nu_{t+1}^2(1+\nu_t)\sigma^2)\) (26.19). Equivalently (26.20)/(26.21):

\(\tilde\epsilon_{t+1}\) is the i.i.d. shock in the agent's perspective. (26.20) shows the agent's belief is a martingale: \(\tilde\mu_t=\tilde{\mathbb E}_t[\tilde\mu_{t+1}]\) (intuitive: the agent uses up all time-\(t\) information, with no expected nonzero difference remaining, else they would immediately adjust \(\tilde\mu_t\) to make martingality hold).

26.3.6 Revisit Return Predictability

(26.8) under Bayesian learning: first compute \(\mathbb E_t[\tilde\mu_{t+1}]-\tilde\mu_t=\nu_{t+1}(\mathbb E_t[\Delta d_{t+1}]-\tilde\mu_t)=\nu_{t+1}(\mu-\tilde\mu_t)\) (26.23) (\(\nu_{t+1}\) deterministic, \(\mathbb E_t[\Delta d_{t+1}]=\mu\) objective). Substituting into (26.22) gives (26.24):

Even when the agent is fully rational, dynamically Bayesian-learning, and uses every piece of information to update beliefs, returns (the risk premium) are still predictable, with predictability from \(\mu-\tilde\mu_t\). The crucial assumption is that the econometrician knows the objective true \(\mu\).

26.4 Asset Pricing with Subjective Learning

26.4.1 Environment

Endowment economy, representative agent, consumption \(C_t\) whose only source is the asset dividend. \(c_t\equiv\ln C_t\). Market clearing \(C_{t+1}=D_{t+1}\), so log consumption growth = log dividend growth \(\Delta c_{t+1}=\Delta d_{t+1}\). By (26.1) (26.25): \(\Delta c_{t+1}=\mu+\varepsilon_{t+1}\), \(\varepsilon\overset{iid}\sim\mathcal N(0,\sigma^2)\).

26.4.2 Epstein-Zin Preference

The representative agent has Epstein-Zin preference (9.1), i.e. (26.26): \(V_t=[(1-\delta)C_t^{1-1/\psi}+\delta(\tilde{\mathbb E}_t[V_{t+1}^{1-\gamma}])^{\frac{1-1/\psi}{1-\gamma}}]^{\frac1{1-1/\psi}}\). \(V_t,\delta,\psi\) map to \(U_t,\beta,1/\rho\) in (9.1); the subjective measure evaluates the expectation (agent has no objective-measure info); \(\gamma\) relative risk aversion, \(\psi\) IES. Wealth portfolio \(W_{t+1}=R_{w,t+1}(W_t-C_t)\) (26.27), by (9.15) \(W_t=\frac1{1-\delta}V_t^{1-1/\psi}C_t^{1/\psi}\) (26.28). EZ subjective SDF (9.12) (26.29): \(\tilde M_{t+1}=\delta(\frac{C_{t+1}}{C_t})^{-1/\psi}(\frac{V_{t+1}}{(\tilde{\mathbb E}_t[V_{t+1}^{1-\gamma}])^{1/(1-\gamma)}})^{1/\psi-\gamma}\).

Below we focus on \(\psi=1\), \(\gamma\neq1\). (26.28) → \(\frac{W_t}{C_t}=\frac1{1-\delta}\) constant (26.30). By (9.3), (26.26) → \(V_t=(C_t)^{1-\delta}(R_t)^\delta\) (26.31), \(R_t=\tilde{\mathbb E}_t[V_{t+1}^{1-\gamma}]^{\frac1{1-\gamma}}\). \(V_{t+1}\) is h.o.d. 1 in all future consumption → h.o.d. 1 in \(W_{t+1}\), \(V_{t+1}=\alpha W_{t+1}=\alpha C_{t+1}\frac1{1-\delta}\) (26.32). Substituting into (26.29) and taking logs gives the log subjective SDF (26.34):

The agent plugs the time-\(t\) posterior \(\tilde\mu_t\) into (26.25), completely ignoring future uncertainty and updating of \(\mu\): \(\Delta c_{t+1}=\tilde\mu_t+\varepsilon_{t+1}\), \(\varepsilon\overset{iid}\sim\mathcal N(0,\sigma^2)\) (26.35) (\(\Delta c\sim\mathcal N(\tilde\mu_t,\sigma^2)\), the agent fixes \(\tilde\mu_t\) as a permanent truth). Call this anticipated utility. By log-normality \(\tilde{\mathbb E}_t[e^{(1-\gamma)\Delta c_{t+1}}]=e^{(1-\gamma)\tilde\mu_t+\frac12(1-\gamma)^2\sigma^2}\) (26.36), substituting into (26.34):

Risk-free rate \(\frac1{R_{f,t}}=\tilde{\mathbb E}_t[\tilde M_{t+1}]\) (26.38) → \(r_{f,t}=-\tilde\mu_m+\tilde\mu_t-\frac12\gamma^2\sigma^2\) (26.39). Wealth portfolio: in an endowment economy the wealth portfolio is the price of the security generating the aggregate dividends, \(R_{w,t+1}=\frac1\delta\frac{C_{t+1}}{C_t}\) (26.40), \(r_{w,t+1}=-\ln\delta+\Delta c_{t+1}\) (26.41), subjective variance \(\widetilde{\text{Var}}(r_{w,t+1})=\sigma^2\) (ignoring \(\mu\) uncertainty). Subjective log risk-premium \(\tilde{rp}_{t+1}\equiv\ln\tilde{\mathbb E}_t[R_{w,t+1}]-\ln R_{f,t}\) (26.42), and objective \(rp_{t+1}\equiv\ln\mathbb E_t[R_{w,t+1}]-\ln R_{f,t}\) (26.44):

The difference \(rp-\tilde{rp}=\mu-\tilde\mu_t\): when the agent is over-optimistic (\(\mu-\tilde\mu_t<0\)), the objective risk premium is smaller than the subjective.

The agent plugs \(\tilde\mu_t\) into (26.25) but is fully aware that \(\mu\) is uncertain and views on its distribution will change in the future. \(\Delta c_{t+1}=\tilde\mu_t+\varepsilon_{t+1}\) (26.46) with (26.17) \(\Delta c_{t+1}\sim\mathcal N(\tilde\mu_t,(1+\nu_t)\sigma^2)\). The uncertainty about \(\mu\) is priced through the dynamics of \(\nu_t\) — priced subjective growth uncertainty. Log SDF: \(\tilde{\mathbb E}_t[e^{m(\Delta c_{t+1})}]=e^{m\tilde\mu_t+\frac12m^2(1+\nu_t)\sigma^2}\) (26.47). By Remark 26.2, the SDF (26.29) still holds, but \(R_t\) must take into account the time-\(t\) expectation of \(V_{t+1}\) under all possible subjective distributions at \(t+1\). Solving the value function (see the collapsible derivation) gives the log subjective SDF (26.58):

\(\tilde\epsilon_{t+1}=\frac{\Delta c_{t+1}-\tilde\mu_t}{\sqrt{1+\nu_t}\sigma}\sim\mathcal N(0,1)\).

Risk-free rate \(r_{f,t}=-\tilde\mu_{m,t}+\tilde\mu_t-\frac12\xi_t^2\sigma^2\) (26.61). The wealth-consumption ratio is still constant ((26.30) holds, since (9.15) only depends on the SDF form (26.29)), \(R_{w,t+1}=\frac1\delta\frac{C_{t+1}}{C_t}\) (26.62), \(r_{w,t+1}=-\ln\delta+\Delta c_{t+1}\) (26.63), subjective variance \(\widetilde{\text{Var}}(r_{w,t+1})=(1+\nu_t)\sigma^2\) (taking \(\mu\) uncertainty into account). Subjective and objective log risk-premia (26.65)/(26.67):

By (26.59)(26.15), as \(t\to\infty\), \(\nu_t\to0\), \(\xi_t\to\gamma\), so (26.65) approaches the anticipated-utility (26.43) \(\gamma\sigma^2\). Return predictability comes from \(\mu-\tilde\mu_t\). Subjective-objective difference \(rp-\tilde{rp}=(\mu-\tilde\mu_t)-\frac{\nu_t}2\sigma^2\) (26.68).

26.5 Learning From Fading Memory: Nagel and Xu (2019)

26.5.1 Key Points

The representative agent learns the unconditional mean of endowment growth with fading memory. Assumptions: fundamentals i.i.d. across periods; agent has Epstein-Zin utility; risk aversion constant; memory decays over time. Model predicts: objective equity premium high and strongly counter-cyclical; subjective equity premium constant. Consistent evidence: the experienced payout growth rate (subjective growth expectations as a weighted average of past growth rates) is negatively related to future stock-market excess returns; negatively related to subjective expectation errors (the objective-subjective wedge) in surveys; positively related to analyst forecasts of long-run earnings growth (a proxy for subjective growth expectations).

26.5.2 Constant-Gain Updating

Log stock-market payout \(\Delta d_{t+1}=\mu+\varepsilon_{t+1}\), \(\varepsilon\) i.i.d. Same Bayesian learning as §26.3.3 but with fading memory. Time-\(t\) posterior (26.69): \(\pi_t(\mu)\propto\pi_0(\mu)\prod_{j=0}^\infty[e^{-\frac12((\Delta d_{t-j}-\mu)/\sigma)^2}]^{(1-\nu)^j}\), \(\nu>0\) the fading-memory parameter. The agent starts with no information (\(\pi_t(0)\) flat), (26.69)→(26.70) giving the posterior mean (26.71): \(\tilde\mu_t=\nu\sum_{j=0}^\infty\Delta d_{t-j}(1-\nu)^j\), posterior variance \(\nu\sigma^2\) constant. Recursing (26.71) gives constant-gain updating (26.72):

The gain from new information \(\Delta d_{t+1}\) when updating \(\tilde\mu\) is the constant \(\nu\), hence constant-gain updating; \(\frac1\nu\) is the effective sample size for estimating \(\mu\) (Nagel-Xu use \(\nu=0.018\)). Two motivations: (1) aggregate argument (Malmendier-Nagel 2016, §27.2: despite heterogeneous gain parameters across age cohorts, the aggregate gain parameter = equally weighted average is constant; the aggregate parameter matters for asset pricing → reasonable for a representative agent to do constant-gain learning); (2) psychology argument for individual portfolio choice (recency bias: more emphasis on recent events → decreasing weights for further history is a good assumption).

26.5.3 Empirical Analysis

Data: \(\Delta d\) (CRSP quarterly since 1926; earlier: Piketty et al. 2018 household dividend from tax 1913–1926, Wright 2004 corporate non-farm non-financial dividends 1900–1913, Barro-Ursua 2008 per-capita real GDP growth 1871–1900 as proxy); market index returns (CRSP value-weighted since 1926; Shiller 2005 S&P Composite 1871–1926, CPI-deflated); surveys (quantitative: UBS/Gallup 1998–2007, Vanguard/Ameriks 2019, Lease et al. 1974; qualitative: Michigan Survey 1987–2016, Roper 1974–1977); analyst forecasts (I/B/E/S long-term median EPS growth since 1981; SPF inflation forecast).

Testable hypotheses: from Campbell-Shiller (26.2) derive (26.73); substituting the constant subjective risk premium (26.3) and simplifying gives three regressions (26.76)/(26.77)/(26.78):

In (26.76)/(26.77), \(\tilde\mu_t\) can be constructed from both aggregate dividend payout (\(\tilde\mu_{d,t}\)) and aggregate return (\(\tilde\mu_{r,t}\)); \(\tilde\mu_{r,t}\) contains more noise than \(\tilde\mu_{d,t}\) (may contain investor sentiment orthogonal to market fundamentals). Empirical results (Figures 26.1–26.3 are regression-coefficient tables):

• Figure 26.1 (regression 26.76; Panel A uses \(\tilde\mu_{d,t}\), Panel B \(\tilde\mu_{r,t}\); dependent variable the log quarterly CRSP value-weighted return minus 3-month T-bill): the coefficient of \(\tilde\mu_t\) (both panels) is significantly negative, confirming (26.76).
• Figure 26.2 (Panel A regression 26.78, Panel B regression 26.77; dependent variable the average subjective expected stock return of survey respondents in quarter \(t\) minus the one-year treasury yield at the end of quarter \(t-1\)): Panel A's \(\tilde\mu_t\) coefficient is non-significantly different from zero (confirming 26.78); Panel B's \(\tilde\mu_{d,t}\) coefficient is significantly negative (confirming 26.77).
• Figure 26.3 (regression 26.72/26.79 \(\tilde\mu_{d,t+1}=(1-\nu)\tilde\mu_{d,t}+\nu\Delta d_{t+1}\); dependent variable the aggregate long-term EPS growth analyst forecast from I/B/E/S deflated by SPF inflation forecast): the \(\tilde\mu_{d,t}\) coefficient is significantly positive, confirming (26.72). Analyst forecasts are used because no survey asks expectations of dividend payout.

26.5.4 Asset Pricing Model with Epstein-Zin Preference

The agent prices subjective growth uncertainty → a special case of §26.4.4 (\(\nu\) constant here), so use the results directly. Log subjective SDF (26.80): \(\tilde m_{t+1}=\tilde\mu_m-\tilde\mu_t-\xi\sigma\tilde\epsilon_{t+1}\), \(\tilde\mu_m=\ln\delta-\frac12(1-\gamma)^2(1+U_v\nu)^2(1+\nu)\sigma^2\), \(\xi=[1-(1-\gamma)(U_v\nu+1)]\sqrt{1+\nu}\), \(U_v=\frac\delta{1-\delta}\). Risk-free rate \(r_{f,t}=-\tilde\mu_m+\tilde\mu_t-\frac12\xi^2\sigma^2\); wealth return \(r_{w,t+1}=-\ln\delta+\Delta c_{t+1}\) (26.63); subjective log risk-premium \(\tilde{rp}_{t+1}=\xi\sqrt{1+\nu}\sigma^2\) (26.81); objective \(rp_{t+1}=(\mu-\tilde\mu_t)-\frac\nu2\sigma^2+\xi\sqrt{1+\nu}\sigma^2\) (26.82); difference \(rp-\tilde{rp}=(\mu-\tilde\mu_t)-\frac\nu2\sigma^2\) (26.68).

Further assume \(\Delta d_{t+1}\) is a function of \(\Delta c_{t+1}\) (26.83): \(\Delta d_{t+1}=\lambda\Delta c_{t+1}-\alpha(d_t-c_t-\mu_{dc})+\sigma_d\eta_{t+1}\) (\(\alpha>0\)), deriving the log risk-premium of a dividend strip in the infinite future: subjective (26.84) constant; objective (26.85) time-varying and counter-cyclical (high current payout → high \(\tilde\mu_t\) → low risk premium).

26.5.5 / 26.5.6 Contribution & Discussion

Contribution: the fading-memory learning framework is simple yet realistic; the analysis is thorough (including simulation); the data are comprehensive (merging multiple sources for the longest sample). Discussion (motivation): the authors motivate fading memory via Malmendier-Nagel (2016), but M-N reach constant-gain by aggregating gain parameters across age cohorts (as long as each cohort's share is constant) — this is fine for aggregate-level asset pricing (representative agent), but the fading-memory framework is better suited to modeling specific investor types, so it should motivate constant-gain with a non-aggregate argument (recency bias) for specific investor types, supported by empirical evidence. The earlier (pre-1926) and survey/analyst data may not be of comparable standard and quality, so the simulation analysis should perhaps use a cleaner subset rather than the full sample.