9. Suggested Explanation: Epstein-Zin Preference
双重之谜的根源之一是 CRRA 效用把风险厌恶 (RRA) 和跨期替代弹性 (IES) 绑成同一个参数 \(\gamma\)（CRRA 下 \(\text{IES}=1/\gamma\)）。Epstein–Zin (1991) 递归效用把两者拆开：用 \(\gamma\) 管风险厌恶、用 \(\rho\)（\(=1/\text{IES}\)）管跨期替代，多出一个自由度，从而有潜力同时匹配高股权溢价（需高 \(\gamma\)）与低无风险利率（需高 IES）。本章先定义 IES、给出 EZ 效用及其性质（非期望效用、一次齐次、对"不确定性早/晚揭示"敏感），再推出 EZ 随机贴现因子（依赖财富组合收益与未来延续价值），分析其解决双重之谜的潜力，最后完整推导 Bansal–Yaron (2004) 长期风险模型：消费增长含一个持久成分 \(x_t\)（AR(1)）与随机波动 \(\sigma_{g,t}^2\)，配合 EZ 偏好生成可观且时变的风险溢价。
One root of the dual puzzle is that CRRA utility ties risk aversion (RRA) and the intertemporal elasticity of substitution (IES) to the same parameter \(\gamma\) (under CRRA, \(\text{IES}=1/\gamma\)). Epstein–Zin (1991) recursive utility separates them: \(\gamma\) governs risk aversion and \(\rho\) (\(=1/\text{IES}\)) governs intertemporal substitution, adding a degree of freedom and so giving the model the potential to match simultaneously a high equity premium (needs high \(\gamma\)) and a low risk-free rate (needs high IES). This chapter defines the IES, presents EZ utility and its properties (non-expected-utility, homogeneous of degree one, sensitivity to early/late resolution of uncertainty), derives the EZ stochastic discount factor (depending on the wealth-portfolio return and the future continuation value), analyzes its potential to solve the dual puzzle, and finally derives the Bansal–Yaron (2004) long-run risk model in full: consumption growth contains a persistent component \(x_t\) (AR(1)) and stochastic volatility \(\sigma_{g,t}^2\), which together with EZ preferences generate a sizable, time-varying risk premium.
9.1 Inter-Temporal Elasticity of Substitution
9.1.1 Definition of IES
定义 9.1（跨期替代弹性 IES）。 多期消费计划下，记 \(C_t,C_{t+1}\) 为 \(t,t+1\) 期消费，\(R_{t,t+1}\) 为收益（投资回报），则 IES 定义为
Definition 9.1 (intertemporal elasticity of substitution, IES). In a multi-period consumption plan, with \(C_t,C_{t+1}\) consumption in \(t,t+1\) and \(R_{t,t+1}\) the return on investment, the IES is
$$\text{IES}=\frac{d\ln\frac{C_{t+1}}{C_t}}{d\ln R_{t,t+1}}.$$
它衡量当（无风险）投资回报变化 1% 时，消费增长率改变多少——即跨期配置消费的意愿。
It measures how much the consumption growth rate changes when the (risk-free) return changes by 1% — i.e. the willingness to substitute consumption across time.
9.1.2 IES of CRRA Utility
在确定性下考虑 CRRA 效用 \(U_t=\sum_{j=0}^\infty\beta^j\frac{C_{t+j}^{1-\gamma}-1}{1-\gamma}\)，对消费做最优化（一阶条件），可证 CRRA 的 IES \(=1/\gamma\)。
Under certainty, consider CRRA utility \(U_t=\sum_{j=0}^\infty\beta^j\frac{C_{t+j}^{1-\gamma}-1}{1-\gamma}\). Optimizing over consumption (first-order condition) shows that for CRRA, IES \(=1/\gamma\).
证明 / Proof：CRRA 的 IES \(=1/\gamma\)
对 \(h\)（在 \(t\) 期减少 \(h\) 投资、\(t+1\) 期得 \(hR_{t,t+1}\)）的一阶条件：
The first-order condition in \(h\) (cut consumption by \(h\) at \(t\), receive \(hR_{t,t+1}\) at \(t+1\)):
$$C_t^{-\gamma}=\beta R_{t,t+1}C_{t+1}^{-\gamma}\ \Rightarrow\ \Big(\frac{C_{t+1}}{C_t}\Big)^{\gamma}=\beta R_{t,t+1}\ \Rightarrow\ \gamma\ln\frac{C_{t+1}}{C_t}=\ln\beta+\ln R_{t,t+1},$$
故 \(\ln\frac{C_{t+1}}{C_t}=\frac1\gamma\ln\beta+\frac1\gamma\ln R_{t,t+1}\)，于是 \(\text{IES}=\frac{d\ln(C_{t+1}/C_t)}{d\ln R_{t,t+1}}=\frac1\gamma\)。\(\blacksquare\)
So \(\ln\frac{C_{t+1}}{C_t}=\frac1\gamma\ln\beta+\frac1\gamma\ln R_{t,t+1}\), hence \(\text{IES}=\frac{d\ln(C_{t+1}/C_t)}{d\ln R_{t,t+1}}=\frac1\gamma\). \(\blacksquare\)
这正是问题所在。 CRRA 下相对风险厌恶 \(\gamma\) 与 IES 互为倒数，被锁成一个参数。要靠高 \(\gamma\) 解释股权溢价，就同时把 IES 压得很低，推出过高的无风险利率——双重之谜无法两全。
This is exactly the problem. Under CRRA, relative risk aversion \(\gamma\) and the IES are reciprocals, locked into one parameter. Using a high \(\gamma\) to explain the equity premium simultaneously forces a low IES, implying a too-high risk-free rate — the dual puzzle cannot be satisfied both ways.
9.2 Epstein-Zin Utility
Epstein and Zin (1991) 从 \(t\) 期起递归地定义效用 (9.1)：
Epstein and Zin (1991) define utility recursively from period \(t\) (9.1):
$$U_t=\left[(1-\beta)\,C_t^{1-\rho}+\beta\left(\mathbb E_t\big[U_{t+1}^{1-\gamma}\big]\right)^{\frac{1-\rho}{1-\gamma}}\right]^{\frac{1}{1-\rho}}.\tag{9.1}$$
其中 \(\beta\) 是主观贴现因子，\(\gamma\) 是相对风险厌恶 RRA，\(\rho\) 通过 IES \(=1/\rho\) 刻画跨期替代。注意 \(\rho\to1\) 时 (9.1) 退化为 Cobb–Douglas 形式 (9.2)、(9.3)。
where \(\beta\) is the subjective discount factor, \(\gamma\) is relative risk aversion (RRA), and \(\rho\) governs intertemporal substitution through IES \(=1/\rho\). As \(\rho\to1\), (9.1) degenerates to a Cobb–Douglas form (9.2), (9.3).
9.2.1 Two-Period Model
两期模型（\(U_t=0\) 当 \(t\ge2\)），由 (9.1) 得 (9.4)、(9.5)：
In the two-period model (\(U_t=0\) for \(t\ge2\)), (9.1) gives (9.4), (9.5):
$$U_t=\left[(1-\beta)C_t^{1-\rho}+\beta\left(\mathbb E_t\big[C_{t+1}^{1-\gamma}\big]\right)^{\frac{1-\rho}{1-\gamma}}\right]^{\frac{1}{1-\rho}}.\tag{9.4}$$
9.2.2 Understand \(\gamma\) and \(\rho\)
- \(\gamma\) 是风险厌恶。 在无不确定性时（消费确定），(9.1) 退化为 CRRA，\(\gamma\) 不再起作用——故 \(\gamma\) 只在面对随机延续价值时刻画对风险的态度。
- \(\rho=1/\text{IES}\)。 在确定性下对 EZ 效用做最优化，可证 IES \(=1/\rho\)，与风险厌恶 \(\gamma\) 解耦。
- \(\rho=\gamma\) 时退化为 CRRA (9.6)、(9.7)：此时 EZ 折回标准时间可加 CRRA。
- \(\gamma\) is risk aversion. With no uncertainty (deterministic consumption), (9.1) degenerates to CRRA and \(\gamma\) plays no role — so \(\gamma\) captures the attitude to risk only when facing a random continuation value.
- \(\rho=1/\text{IES}\). Optimizing EZ utility under certainty shows IES \(=1/\rho\), decoupled from risk aversion \(\gamma\).
- \(\rho=\gamma\) reduces to CRRA (9.6), (9.7): EZ collapses back to standard time-additive CRRA.
9.2.3 Other Properties of Epstein-Zin Utility
- 非期望效用。 EZ (9.1) 一般不满足期望效用（vNM）公理——只有当 \(\rho=\gamma\)（CRRA）或风险中性时才满足。
- 一次齐次。 EZ 效用对消费序列 \(\{C_t\}\) 一次齐次（齐次度 1）：把所有期消费按 \(\lambda\) 同比放大，效用也放大 \(\lambda\) 倍。
- Non-expected-utility. EZ (9.1) generally does not satisfy the expected-utility (vNM) axioms — it does only when \(\rho=\gamma\) (CRRA) or under risk neutrality.
- Homogeneous of degree one. EZ utility is homogeneous of degree one in the consumption stream \(\{C_t\}\): scaling all consumption by \(\lambda\) scales utility by \(\lambda\).
9.2.4 Early vs. Late Resolution of Uncertainty
EZ 效用对不确定性的揭示时点敏感（CRRA 则无所谓）。考虑赌局：\(C_2\) 随机，比较"在 \(t=1\) 提前揭示"（赌局 A，早揭示）与"在 \(t=2\) 才揭示"（赌局 B，晚揭示）。逐期回代算 \(U_0^A\) (9.8/9.10) 与 \(U_0^B\) (9.9/9.11)。
结论（Claim 9.1）： 当 \(\gamma>\rho\) 时 \(U_0^A>U_0^B\)——投资者偏好早揭示。
EZ utility is sensitive to when uncertainty is resolved (CRRA is indifferent). Consider a gamble: \(C_2\) is random; compare "revealed early at \(t=1\)" (gamble A, early resolution) with "revealed only at \(t=2\)" (gamble B, late resolution). Backward substitution gives \(U_0^A\) (9.8/9.10) and \(U_0^B\) (9.9/9.11).
Result (Claim 9.1): when \(\gamma>\rho\), \(U_0^A>U_0^B\) — the investor prefers early resolution.
证明 / Proof：\(\gamma>\rho\Rightarrow U_0^A>U_0^B\)（Jensen 不等式）
两个赌局只在不确定性揭示时点上不同；把 \(U_0\) 写成关于 \(C_2\) 的函数 \(f(x)\)，其曲率由 \(\gamma,\rho\) 决定。可证当 \(\gamma>\rho\) 时 \(f\) 凹 \(f''(x)<0\)：
The two gambles differ only in when uncertainty is revealed; writing \(U_0\) as a function \(f(x)\) of \(C_2\), its curvature is set by \(\gamma,\rho\). One shows \(f\) is concave \(f''(x)<0\) when \(\gamma>\rho\):
$$f''(x)<0\quad\text{when }\gamma>\rho.$$
早揭示对应先取期望再作用 \(f\)，晚揭示对应先作用 \(f\) 再取期望。由 Jensen，\(f\) 凹时 \(f(\mathbb E[x])\ge\mathbb E[f(x)]\)，故 \(U_0^A>U_0^B\)。\(\blacksquare\)
Early resolution corresponds to taking the expectation before applying \(f\); late resolution applies \(f\) before the expectation. By Jensen, when \(f\) is concave \(f(\mathbb E[x])\ge\mathbb E[f(x)]\), so \(U_0^A>U_0^B\). \(\blacksquare\)
经济含义。 \(\gamma>\rho\)（风险厌恶超过 \(1/\text{IES}\)）的投资者宁愿早点知道命运——这正是长期风险模型偏好的参数区域（高 \(\gamma\)、高 IES 即低 \(\rho\)），使持久的消费冲击带来额外风险溢价。
Economic meaning. An investor with \(\gamma>\rho\) (risk aversion exceeding \(1/\text{IES}\)) would rather learn their fate early — exactly the parameter region favored by long-run risk models (high \(\gamma\), high IES i.e. low \(\rho\)), where persistent consumption shocks command an extra risk premium.
9.3 Stochastic Discount Factor
定理 9.1。 Epstein–Zin 偏好 (9.1) 隐含的随机贴现因子为 (9.12)、(9.13)：
Theorem 9.1. The stochastic discount factor implied by Epstein–Zin preferences (9.1) is (9.12), (9.13):
$$m^{EZ}_{t+1}=\beta\Big(\frac{C_{t+1}}{C_t}\Big)^{-\rho}\left(\frac{U_{t+1}}{\big(\mathbb E_t[U_{t+1}^{1-\gamma}]\big)^{\frac{1}{1-\gamma}}}\right)^{\rho-\gamma}.\tag{9.12}$$
(9.13) 用财富组合总收益 \(R^{EZ}_{w,t+1}=\frac{W_{t+1}}{W_t-C_t}\) 替换延续价值，更便于实证。
(9.13) replaces the continuation value with the wealth-portfolio gross return \(R^{EZ}_{w,t+1}=\frac{W_{t+1}}{W_t-C_t}\), which is more convenient empirically.
证明 / Proof：EZ SDF（变分法）
用变分法：任意资产 \(t\) 期价格 \(p_t\)、\(t+1\) 期支付 \(x_{t+1}\)，均衡下投资者对"买 \(h\) 单位 vs 不买"无差异。对 \(h\) 求一阶条件（\(U_t\) 一次齐次，且 \(\frac{\partial U_t}{\partial U_{t+1}}\) 由 (9.1) 的链式法则给出）：
Use a variational argument: for any asset with price \(p_t\) and payoff \(x_{t+1}\), in equilibrium the investor is indifferent between buying \(h\) units and not buying. The FOC in \(h\) (using that \(U_t\) is homogeneous of degree one, and \(\frac{\partial U_t}{\partial U_{t+1}}\) from the chain rule on (9.1)):
$$p_t=\mathbb E_t\!\left[\beta\Big(\frac{C_{t+1}}{C_t}\Big)^{-\rho}\left(\frac{U_{t+1}}{(\mathbb E_t[U_{t+1}^{1-\gamma}])^{\frac{1}{1-\gamma}}}\right)^{\rho-\gamma}x_{t+1}\right],$$
括号内即 \(m^{EZ}_{t+1}\) (9.12)。再用 \(\theta\equiv\frac{1-\gamma}{1-\rho}\) 与财富组合收益 \(R^{EZ}_{w,t+1}\) 的定义整理，得等价的 (9.13)。\(\blacksquare\)
The bracket is \(m^{EZ}_{t+1}\) (9.12). Using \(\theta\equiv\frac{1-\gamma}{1-\rho}\) and the definition of the wealth-portfolio return \(R^{EZ}_{w,t+1}\), rearrange to the equivalent (9.13). \(\blacksquare\)
Remark 9.2。 由 (9.13)，当 \(\rho=\gamma\) 时 \(m^{EZ}_{t+1}\) 退化为 \(\beta(\frac{C_{t+1}}{C_t})^{-\rho}\)，即 CRRA 的 SDF。
Remark 9.3。 由 (9.12)，SDF 依赖延续价值 \(U_{t+1}\)，它概括了未来前景。这意味着在递归效用下，即使只看一期，未来的前景也会影响当期资产定价——这是 EZ 区别于 CRRA 的关键。
Remark 9.2. By (9.13), when \(\rho=\gamma\), \(m^{EZ}_{t+1}\) degenerates to \(\beta(\frac{C_{t+1}}{C_t})^{-\rho}\), the CRRA SDF.
Remark 9.3. By (9.12), the SDF depends on the continuation value \(U_{t+1}\), which summarizes future prospects. So under recursive utility, even one period out, future prospects affect current asset pricing — the key difference of EZ from CRRA.
9.4 Potential to Solve the Dual Puzzle
如 §8.2，记 \(c_t=\ln C_t\)、\(\Delta c_{t+1}\)、\(r^i_{t+1}\)、\(r^f_{t+1}\)、\(r^w_{t+1}\)（财富组合对数收益）。假设 \(r^f_{t+1}\) 在 \(t\) 期已知，且 \((\Delta c_{t+1},r^i_{t+1},r^w_{t+1})\) 联合正态，协方差记 \(\sigma_c^2,\sigma_i^2,\sigma_w^2,\sigma_{ci},\sigma_{cw},\sigma_{iw}\)。由 (9.13)，对无风险与风险资产分别写欧拉方程并用对数正态展开，得无风险利率 (9.19)、(9.20) 与股权溢价 (9.24)：
As in §8.2, let \(c_t=\ln C_t\), \(\Delta c_{t+1}\), \(r^i_{t+1}\), \(r^f_{t+1}\), \(r^w_{t+1}\) (log wealth-portfolio return). Assume \(r^f_{t+1}\) is known at \(t\) and \((\Delta c_{t+1},r^i_{t+1},r^w_{t+1})\) is jointly normal, with covariances \(\sigma_c^2,\sigma_i^2,\sigma_w^2,\sigma_{ci},\sigma_{cw},\sigma_{iw}\). From (9.13), writing the Euler equation for the risk-free and risky assets and expanding with log-normality gives the risk-free rate (9.19), (9.20) and the equity premium (9.24):
$$r^f_{t+1}=-\ln\beta+\frac{\rho}{1-\rho}\,\theta\,\mathbb E_t[\Delta c_{t+1}]+\Big(\frac{1-\gamma}{1-\rho}-1\Big)\mathbb E_t[r^w_{t+1}]-(\cdots)\sigma^2.\tag{9.19}$$
$$\mathbb E[r^i_{t+1}]-r^f_{t+1}=\frac{1-\gamma}{1-\rho}\,\rho\,\sigma_{ci}+\frac{\gamma-\rho}{1-\rho}\,\sigma_{wi}.\tag{9.24}$$
(9.24) 把股权溢价拆成两部分：与消费协方差的项（权重含 \(\gamma\)）和与财富组合收益协方差的项（权重含 \(\gamma-\rho\)）。
解谜的关键。 现在可以用大 \(\gamma\)（高风险厌恶）在 (9.24) 中制造足够的风险溢价，同时用大 \(1/\rho\)（高 IES）在 (9.19)–(9.20) 中压低无风险利率——多一个参数就多一个自由度，校准有望两全。这只是"潜力"：要真正匹配数据，还需 §9.5 的长期风险结构。
(9.24) splits the equity premium into two parts: the covariance with consumption (weight involving \(\gamma\)) and the covariance with the wealth-portfolio return (weight involving \(\gamma-\rho\)).
The key to solving the puzzle. One can now use a large \(\gamma\) (high risk aversion) to generate enough risk premium in (9.24), while using a large \(1/\rho\) (high IES) to lower the risk-free rate in (9.19)–(9.20) — one more parameter is one more degree of freedom, so calibration has hope of succeeding on both. This is only the "potential"; matching the data actually requires the long-run-risk structure of §9.5.
9.5 Long Run Risk Model: Bansal and Yaron (2004)
9.5.1 Setup
财富组合（对消费的索取权）总收益 \(R_{a,t+1}\) 满足 (9.25)，欧拉方程 (9.26)–(9.28)。记 \(Z_t=\frac{P_{s,t}}{C_t}\) 为价格-消费比。对数变量 \(m_{t+1}=\ln M_{t+1}\)、\(z_{t+1}=\ln Z_{t+1}\)、\(g_{t+1}=\ln G_{t+1}=\ln\frac{C_{t+1}}{C_t}\) 等。
The wealth-portfolio (claim on consumption) gross return \(R_{a,t+1}\) satisfies (9.25), with Euler equation (9.26)–(9.28). Let \(Z_t=\frac{P_{s,t}}{C_t}\) be the price-consumption ratio. Log variables \(m_{t+1}=\ln M_{t+1}\), \(z_{t+1}=\ln Z_{t+1}\), \(g_{t+1}=\ln G_{t+1}=\ln\frac{C_{t+1}}{C_t}\), etc.
$$\Big(\underbrace{W_t-C_t}_{=P_{s,t}}\Big)R_{a,t+1}=W_{t+1}.\tag{9.25}$$
$$\mathbb E_t\!\left[\beta^\theta\Big(\frac{C_{t+1}}{C_t}\Big)^{-\frac{\theta}{\phi}}R_{a,t+1}^{-(1-\theta)}R_{a,t+1}\right]=1.\tag{9.28}$$
9.5.2 Model 1
对 \(R_{a,t+1}\) 做 Campbell–Shiller 分解 (9.29)，在 \(\bar z\) 处线性近似 (9.30)。长期风险结构：消费增长 \(g_{t+1}\) 含一个持久成分 \(x_t\)，服从 AR(1) (9.34)，且条件方差 \(\sigma_{g,t}^2\) 服从 AR(1) (9.35)（随机波动）：
Do the Campbell–Shiller decomposition of \(R_{a,t+1}\) (9.29) and linearize around \(\bar z\) (9.30). Long-run risk structure: consumption growth \(g_{t+1}\) contains a persistent component \(x_t\) following an AR(1) (9.34), and the conditional variance \(\sigma_{g,t}^2\) follows an AR(1) (9.35) (stochastic volatility):
$$x_t=\mu+\rho\,x_{t-1}+(\rho-\omega)\,\eta_t,\tag{9.34}$$
$$\sigma_{g,t+1}^2=\nu_0+\nu_1\,\sigma_{g,t}^2+\omega_{t+1},\qquad \omega_{t+1}\sim\mathcal N(\mu_\omega,\sigma_\omega^2)\ \text{indep. of }\eta_{t+1}.\tag{9.35}$$
猜测价格-消费比对数线性于状态 (9.36)：\(z_t=A_0+A_1 x_t+A_2\sigma_{g,t}^2\)。代入欧拉方程，匹配 \(x_t\) 与 \(\sigma_{g,t}^2\) 系数（Part 1 $=0$、Part 2 $=0$），解出 \(A_1\) (9.38)、\(A_2\) (9.39)：
Guess the price-consumption ratio is log-linear in the states (9.36): \(z_t=A_0+A_1 x_t+A_2\sigma_{g,t}^2\). Substitute into the Euler equation and match coefficients of \(x_t\) and \(\sigma_{g,t}^2\) (Part 1 $=0$, Part 2 $=0$), solving for \(A_1\) (9.38), \(A_2\) (9.39):
$$A_1=\frac{1-\frac1\phi}{1-\kappa_1\rho},\tag{9.38}$$
$$A_2=\frac{\frac12\big[\theta A_1\kappa_1(\rho-\omega)+\theta(1-\frac1\phi)\big]^2}{\theta(1-\kappa_1\nu_1)}.\tag{9.39}$$
证明 / Proof：解出 \(A_1,A_2\) 与条件矩
把 \(z_t\) (9.36) 代入 (9.29) 的线性近似得 \(r_{a,t+1}\) (9.37)。再代入对数欧拉方程 \(\mathbb E_t[m_{t+1}]+\frac12\mathrm{Var}_t(m_{t+1})+\dots=0\)，其中 \(m_{t+1}=\theta\ln\delta-\frac\theta\phi g_{t+1}+(\theta-1)r_{a,t+1}\) (9.40)。对 \(x_t\) 的系数（Part 1）置零 \(\Rightarrow A_1\kappa_1\rho+\frac{\phi-1}{\phi}-A_1=0\Rightarrow\) (9.38)；对 \(\sigma_{g,t}^2\) 的系数（Part 2）置零 \(\Rightarrow\) (9.39)。
Substituting \(z_t\) (9.36) into the linearized (9.29) gives \(r_{a,t+1}\) (9.37). Plug into the log Euler equation \(\mathbb E_t[m_{t+1}]+\frac12\mathrm{Var}_t(m_{t+1})+\dots=0\), with \(m_{t+1}=\theta\ln\delta-\frac\theta\phi g_{t+1}+(\theta-1)r_{a,t+1}\) (9.40). Setting the coefficient of \(x_t\) (Part 1) to zero \(\Rightarrow A_1\kappa_1\rho+\frac{\phi-1}{\phi}-A_1=0\Rightarrow\) (9.38); setting the coefficient of \(\sigma_{g,t}^2\) (Part 2) to zero \(\Rightarrow\) (9.39).
校准（ARMA/AR 估计）：\(\rho=0.965\)、\(\omega=0.85\)、\(\nu_1=0.983\)、\(\kappa_1=0.9969\)。由此算出条件股权溢价 (9.42) 与无条件方差 (9.43)：
Calibration (ARMA/AR estimates): \(\rho=0.965\), \(\omega=0.85\), \(\nu_1=0.983\), \(\kappa_1=0.9969\). These give the conditional equity premium (9.42) and unconditional variance (9.43):
$$\mathbb E_t[r_{a,t+1}]-r_{f,t}=\underbrace{L_1}_{\text{const}}+\theta\,(A_2\kappa_1\nu_1-A_2)\,\sigma_{g,t}^2.\tag{9.42}$$
(9.42) 表明：股权溢价 $=$ 常数 $+$ 随随机波动状态 \(\sigma_{g,t}^2\) 变动的项——即时变风险溢价；而 \(x_t\) 项系数为零（被 \(A_1\) 的定义消掉）。
Model 1 的不足。 这里溢价的时变只来自 \(\sigma_{g,t}^2\)；若关掉随机波动，Model 1 给出常数股权溢价，无法拟合数据中收益的时变与可预测性。故需 Model 2。
(9.42) shows: the equity premium $=$ a constant $+$ a term varying with the stochastic-volatility state \(\sigma_{g,t}^2\) — i.e. a time-varying risk premium; the \(x_t\) term has zero coefficient (cancelled by the definition of \(A_1\)).
Shortcoming of Model 1. Here the premium varies only through \(\sigma_{g,t}^2\); with stochastic volatility turned off, Model 1 yields a constant equity premium, which cannot fit the time variation and predictability of returns in the data. Hence Model 2.
9.5.3 Model 2
Model 2 分离消费与红利过程（总财富组合 vs 市场组合）：
Model 2 separates the consumption and dividend processes (aggregate wealth portfolio vs market portfolio):
$$g_{t+1}=x_t+\eta_{t+1},\qquad g_{d,t+1}=\mu_d+\lambda x_t+\eta_{d,t+1},$$
$$\eta_{t+1}=\sigma_{g,t}\eta_{t+1},\qquad \eta_{d,t+1}=\tau\lambda\sigma_{g,t}\eta_{t+1}+\sqrt{1-\tau^2}\,\lambda\sigma_{g,t}e_{d,t+1}.$$
记 \(z_{m,t}=\ln\frac{P_{m,t}}{D_t}\) 为市场组合价格-红利比，对市场收益做 Campbell–Shiller 分解 (9.49)，同样猜测 \(z_{m,t}=A_{0,m}+A_{1,m}x_t+A_{2,m}\sigma_{g,t}^2\) (9.50)，解出 (9.44)、(9.45)：
Let \(z_{m,t}=\ln\frac{P_{m,t}}{D_t}\) be the market price-dividend ratio; do the Campbell–Shiller decomposition of the market return (9.49), guess \(z_{m,t}=A_{0,m}+A_{1,m}x_t+A_{2,m}\sigma_{g,t}^2\) (9.50), and solve (9.44), (9.45):
$$A_{1,m}=\frac{\lambda-\frac1\phi}{1-\kappa_{1,m}\rho},\tag{9.44}$$
$$A_{2,m}=\frac{(1-\theta)A_2(1-\kappa_1\nu_1)+\frac12\big[(B_{g\theta}+B_m)^2+(1-\tau^2)\lambda^2\big]}{1-\kappa_{1,m}\rho},\tag{9.45}$$
其中 \(B_{g\theta}=-\frac\theta\phi+(\theta-1)(A_1\kappa_1(\rho-\omega)+1)\)，\(B_m=\tau\lambda+\kappa_{1,m}A_{1,m}(\rho-\omega)\)。
where \(B_{g\theta}=-\frac\theta\phi+(\theta-1)(A_1\kappa_1(\rho-\omega)+1)\) and \(B_m=\tau\lambda+\kappa_{1,m}A_{1,m}(\rho-\omega)\).
9.5.4 Discussion
- 校准成功。 把股权溢价与无风险利率表为 \(A_{1,m},A_{2,m}\)（进而 \(\theta,\phi\)）的函数；调 \(\theta,\phi\) 拟合数据矩。取风险厌恶 \(\gamma\approx7\)、IES \(\approx2\) 时，模型能较好匹配股权溢价与无风险利率的矩——双重之谜被合理参数解释。
- 贡献： 引入持久（长期）成分 \(x_t\)，并通过状态变量让消费增长与财富组合收益相关；EZ 偏好使投资者对持久冲击格外厌恶，从而要求高溢价。
- 批评 (Constantinides–Ghosh 2011 等)： (i) 模型在"联合定价横截面 + 拟合消费/红利增长时序无条件矩"上被拒绝，暗示遗漏了某个重要状态变量，或应让滞后价格-红利比、无风险利率以非线性方式进入回归；(ii) 数据中冲击的持久性不如 Bansal–Yaron 预测的强；(iii) 还应匹配更多矩（如价格-红利比波动率、其它因子溢价）；(iv) 潜变量不可观测、缺乏经济故事。
- Calibration succeeds. Express the equity premium and risk-free rate as functions of \(A_{1,m},A_{2,m}\) (hence \(\theta,\phi\)); tune \(\theta,\phi\) to fit the data moments. With risk aversion \(\gamma\approx7\) and IES \(\approx2\), the model fits the equity premium and risk-free rate moments well — the dual puzzle is explained with reasonable parameters.
- Contribution: introduces a persistent (long-run) component \(x_t\) and, via the state variable, makes consumption growth correlated with the wealth-portfolio return; EZ preferences make the investor especially averse to persistent shocks, requiring a high premium.
- Critiques (Constantinides–Ghosh 2011 and others): (i) the model is rejected on jointly pricing the cross-section and fitting the unconditional time-series moments of consumption/dividend growth, suggesting a missing important state variable, or that the lagged price-dividend ratio and risk-free rate should enter the regressions non-linearly; (ii) shocks are not as persistent in data as Bansal–Yaron predict; (iii) more moments should be matched (e.g. price-dividend ratio volatility, other factor premia); (iv) the latent variables are unobservable and lack an economic story.
References
- Bansal, R. and A. Yaron (2004). Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles. The Journal of Finance 59(4), 1481–1509.
- Constantinides, G. M. and A. Ghosh (2011). Asset Pricing Tests with Long-Run Risks in Consumption Growth. The Review of Asset Pricing Studies 1(1), 96–136.
- Epstein, L. G. and S. E. Zin (1989). Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework. Econometrica 57(4), 937–969.
- Epstein, L. G. and S. E. Zin (1991). Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis. Journal of Political Economy 99(2), 263–286.
- Weil, P. (1989). The Equity Premium Puzzle and the Risk-Free Rate Puzzle. Journal of Monetary Economics 24(3), 401–421.