18. Robustness to Misspecification
本章综述对模型误设的稳健性 (robustness to misspecification) 文献核心思想（Hansen-Sargent 2001）。代理人不 100% 确信基准模型对应真实概率测度，于是做审慎调整 (cautious adjustment)：在所有"与基准模型偏离不太远"的概率测度上最小化未来连续价值，得到连续价值的保守下界。偏离的惩罚由条件相对熵 (conditional relative entropy) 度量，惩罚强度参数 \(\xi\) 代表对基准模型的信心。核心结果：求解此最小化问题得到的最优似然比是指数倾斜 (exponential tilting) 形式 (18.8)，而把它代回目标函数 (18.9) 恰好等于 Epstein-Zin 递归效用的连续价值 (18.10)，且当 \(\xi=\frac{1}{\gamma-1}\) 时二者完全相同。于是稳健性/模糊厌恶在数学上等价于更高的风险厌恶（更小的 \(\xi\) ⟺ 更大的 \(\gamma\)）。
This chapter surveys the core idea of the robustness to misspecification literature (Hansen-Sargent 2001). The agent is not 100% sure the baseline model corresponds to the true probability measure, so makes a cautious adjustment: minimizing the future continuation value over all probability measures "not too far" from the baseline, yielding a conservative lower bound on the continuation value. The penalty for deviating is measured by the conditional relative entropy, with penalty strength \(\xi\) representing confidence in the baseline. Key result: the optimal likelihood ratio solving this minimization is an exponential tilting (18.8), and plugging it back into the objective (18.9) exactly equals the continuation value of Epstein-Zin recursive utility (18.10), being identical when \(\xi=\frac{1}{\gamma-1}\). So robustness / ambiguity aversion is mathematically equivalent to higher risk aversion (smaller \(\xi\) ⟺ larger \(\gamma\)).
18.1 Setup
设代理人有递归效用（即 Epstein-Zin 偏好），等价于 (9.1) 改写的 (18.1)：
The agent has recursive utility (Epstein-Zin preference), equivalent to (9.1) rewritten as (18.1):
$$V_t=\left[(1-\beta)C_t^{1-\rho}+\beta(R_t^{1-\rho})\right]^{\frac1{1-\rho}}\tag{18.1}$$
$$R_t=\mathbb E\!\left[V_{t+1}^{1-\gamma}\mid\mathcal F_t\right]^{\frac1{1-\gamma}}\tag{18.2}$$
\(R_t\) 为连续价值，\(V_t\) 为当前价值函数，\(C_t\) 为即期消费，\(\mathcal F_t\) 为 \(t\) 时信息。对数对象 \(v_t\equiv\ln V_t\)、\(c_t\equiv\ln C_t\)、\(r_t\equiv\ln R_t\)。条件似然比 (conditional likelihood ratio) 记为 \(\frac{N_{t+1}}{N_t}\)，满足 \(\mathbb E[\frac{N_{t+1}}{N_t}\mid\mathcal F_t]=1\)（保证新概率测度积分为 1）与 \(\frac{N_{t+1}}{N_t}\geq0\)（保证非负）。故 \(\frac{N_{t+1}}{N_t}\) 是合法的改变测度密度。
18.2 The Minimization Problem
代理人求解最小化问题 (18.3)：
\(R_t\) is the continuation value, \(V_t\) the current value function, \(C_t\) instant consumption, \(\mathcal F_t\) time-\(t\) information. Log objects \(v_t\equiv\ln V_t\), \(c_t\equiv\ln C_t\), \(r_t\equiv\ln R_t\). The conditional likelihood ratio \(\frac{N_{t+1}}{N_t}\) satisfies \(\mathbb E[\frac{N_{t+1}}{N_t}\mid\mathcal F_t]=1\) (so the new measure integrates to 1) and \(\frac{N_{t+1}}{N_t}\geq0\) (non-negativity). So \(\frac{N_{t+1}}{N_t}\) is a valid change-of-measure density.
18.2 The Minimization Problem
The agent solves the minimization (18.3):
$$\min_{\frac{N_{t+1}}{N_t}}\ \mathbb E\!\left[\frac{N_{t+1}}{N_t}v_{t+1}\mid\mathcal F_t\right]+\xi\,\mathbb E\!\left[\frac{N_{t+1}}{N_t}(\ln N_{t+1}-\ln N_t)\mid\mathcal F_t\right]\tag{18.3}$$
$$\text{s.t.}\quad\mathbb E\!\left[\frac{N_{t+1}}{N_t}\mid\mathcal F_t\right]=1,\qquad\frac{N_{t+1}}{N_t}\geq0.\tag{18.4}$$
(18.5) \(\frac{N_{t+1}}{N_t}\geq0\) 自动满足（可写 \(\ln N_{t+1}\)、\(\ln N_t\)）。
解读。
- 似然比 \(\frac{N_{t+1}}{N_t}\) 把概率测度由基准模型改为某个其他测度。
- \(\mathbb E[\frac{N_{t+1}}{N_t}v_{t+1}\mid\mathcal F_t]\) 是新测度下的期望连续价值。
- \(\xi\,\mathbb E[\frac{N_{t+1}}{N_t}(\ln N_{t+1}-\ln N_t)\mid\mathcal F_t]\) 称条件相对熵 (conditional relative entropy)，度量从基准模型改到其他测度的惩罚值。其凸性与非负性保证它是合理的"距离"惩罚（定义 \(\varepsilon(n)=n\ln n\)，\(\varepsilon'(n)=1+\ln n\)、\(\varepsilon''(n)=\frac1n>0\)，故 \(\frac{N_{t+1}}{N_t}(\ln N_{t+1}-\ln N_t)\) 凸；由 Jensen \(\mathbb E[\frac{N_{t+1}}{N_t}(\ln N_{t+1}-\ln N_t)\mid\mathcal F_t]\geq\underbrace{\mathbb E[\frac{N_{t+1}}{N_t}\mid\mathcal F_t]}_{=1}\ln(\underbrace{\mathbb E[\frac{N_{t+1}}{N_t}\mid\mathcal F_t]}_{=1})=0\)，惩罚恒非负）。
- \(\xi\) 是惩罚项的拉格朗日乘子/惩罚载荷，代表对基准模型的信心：\(\xi\) 越大，偏离惩罚越大，代理人越不愿偏离基准（\(\xi\to\infty\) 时 \(\frac{N_{t+1}}{N_t}\to1\)，直接用基准模型）。
- 该审慎调整把连续价值取为所有概率测度下的保守下界（最小化），体现模糊厌恶 (ambiguity aversion)：代理人对真实概率测度本身不确定。
- \(\xi\) 可时变，意味着在宏观经济好/坏时对基准模型的信心不同。
(18.5) \(\frac{N_{t+1}}{N_t}\geq0\) is automatically satisfied (one can write \(\ln N_{t+1}\), \(\ln N_t\)).
Interpretation.
- The likelihood ratio \(\frac{N_{t+1}}{N_t}\) changes the measure from the baseline to some other measure.
- \(\mathbb E[\frac{N_{t+1}}{N_t}v_{t+1}\mid\mathcal F_t]\) is the expected continuation value under the new measure.
- \(\xi\,\mathbb E[\frac{N_{t+1}}{N_t}(\ln N_{t+1}-\ln N_t)\mid\mathcal F_t]\) is the conditional relative entropy, the penalty for moving from the baseline to another measure. Its convexity and non-negativity make it a sensible "distance" penalty (define \(\varepsilon(n)=n\ln n\), \(\varepsilon'(n)=1+\ln n\), \(\varepsilon''(n)=\frac1n>0\), so \(\frac{N_{t+1}}{N_t}(\ln N_{t+1}-\ln N_t)\) is convex; by Jensen \(\mathbb E[\frac{N_{t+1}}{N_t}(\ln N_{t+1}-\ln N_t)\mid\mathcal F_t]\geq\underbrace{\mathbb E[\frac{N_{t+1}}{N_t}\mid\mathcal F_t]}_{=1}\ln(\underbrace{\mathbb E[\frac{N_{t+1}}{N_t}\mid\mathcal F_t]}_{=1})=0\), always non-negative).
- \(\xi\) is the Lagrange multiplier / penalty loading, representing confidence in the baseline: larger \(\xi\) means a larger deviation penalty, so the agent is less willing to deviate (\(\xi\to\infty\) gives \(\frac{N_{t+1}}{N_t}\to1\), using the baseline directly).
- This cautious adjustment sets the continuation value to a conservative lower bound across all measures (minimization), reflecting ambiguity aversion: the agent is unsure of the true measure itself.
- \(\xi\) may be time-dependent, meaning confidence in the baseline differs in good/bad macroeconomic times.
18.3 Solve the Minimization Problem
对 (18.3)–(18.4) 作拉格朗日（乘子 \(\mu\)），对 \(\frac{N_{t+1}}{N_t}\) 取一阶条件 (18.6)：
Form the Lagrangian for (18.3)–(18.4) (multiplier \(\mu\)), take the f.o.c. w.r.t. \(\frac{N_{t+1}}{N_t}\) (18.6):
$$\frac{N_{t+1}}{N_t}=e^{-\frac1\xi v_{t+1}}e^{-\frac1\xi\mu-1}.\tag{18.6}$$
代入约束 (18.4) 钉住乘子 (18.7)：\(e^{-\frac1\xi\mu-1}=\frac{1}{\mathbb E[e^{-\frac1\xi v_{t+1}}\mid\mathcal F_t]}\)。结合 (18.6) 得最优似然比 (18.8)：
Substituting into the constraint (18.4) pins the multiplier (18.7): \(e^{-\frac1\xi\mu-1}=\frac{1}{\mathbb E[e^{-\frac1\xi v_{t+1}}\mid\mathcal F_t]}\). Combined with (18.6), the optimal likelihood ratio (18.8):
$$\frac{N_{t+1}}{N_t}=\frac{e^{-\frac1\xi v_{t+1}}}{\mathbb E\!\left[e^{-\frac1\xi v_{t+1}}\mid\mathcal F_t\right]}.\tag{18.8}$$
Remark 18.1. (18.8) 是指数倾斜 (exponential tilting)：给较小的 \(v_{t+1}\)（坏结果）赋更高权重，故求解出的 \(\frac{N_{t+1}}{N_t}\) 把测度倾向坏结果——保守。\(\xi\to\infty\) 时 \(\frac{N_{t+1}}{N_t}\to1\)（回到基准，无倾斜）。
理解目标函数。 把 (18.8) 代回目标函数 (18.3)，化简得 (18.9)：
Remark 18.1. (18.8) is exponential tilting: it places higher weight on smaller \(v_{t+1}\) (bad outcomes), so the solved \(\frac{N_{t+1}}{N_t}\) tilts the measure toward bad outcomes — conservative. As \(\xi\to\infty\), \(\frac{N_{t+1}}{N_t}\to1\) (back to baseline, no tilt).
Understanding the objective. Substituting (18.8) back into the objective (18.3) and simplifying gives (18.9):
$$\text{Objective (18.3)}=-\xi\ln\!\left(\mathbb E\!\left[e^{-\frac1\xi v_{t+1}}\mid\mathcal F_t\right]\right).\tag{18.9}$$
而由 (18.2)，递归效用的连续价值对数 (18.10)：\(r_t=\frac{1}{1-\gamma}\ln(\mathbb E[e^{(1-\gamma)v_{t+1}}\mid\mathcal F_t])\)。比较 (18.9) 与 (18.10)：若设
By (18.2), the log continuation value of recursive utility (18.10): \(r_t=\frac{1}{1-\gamma}\ln(\mathbb E[e^{(1-\gamma)v_{t+1}}\mid\mathcal F_t])\). Comparing (18.9) and (18.10): if we set
$$\xi=\frac{1}{\gamma-1},$$
则 (18.9) 与 (18.10) 完全相同。
解读。
- 最小化问题的目标函数 (18.3) 恰是递归（Epstein-Zin）效用下的期望连续价值——这为递归效用提供了一个辩护：对风险持保守态度的代理人本就会拥有递归效用函数。
- 递归效用因此可理解为内嵌模糊厌恶：代理人在所有可能概率测度上最小化、用最坏不确定性下的期望连续价值。
- 之所以是最小化而非 max-min 问题，是因为消费序列外生、无可选；在带生产与消费-投资权衡的经济中，递归效用的连续价值才会是 max-min 问题的解。
- \(\gamma\) 很大 ⟺ \(\xi\) 很小：偏离基准的惩罚很小、会选更保守的测度。故"选更保守测度（更小 \(\xi\)）"等价于"更高风险厌恶（更大 \(\gamma\)）"——再次印证稳健性与递归效用的联系。
偏好冲击 (preference shock)。
- 一种理解：偏好冲击通过 \(\xi\) 体现。\(\xi\) 时变会改变所选似然比 \(\frac{N_{t+1}}{N_t}\) 与对应测度，从而改变连续价值。
- 另一种理解：通过 \(C_t\)。仍用 \(C_t\) 表递归效用，但 \(C_t\) 为有效消费，显式定义 \(C_t=\tilde C_t D_t\)，\(\tilde C_t\) 为实际消费、\(D_t\) 为外生偏好移位项（如平稳一阶差分过程、AR 过程、随机波动率过程等）。
- 两种方式都可把价格的一部分归于偏好冲击；但需谨慎："偏好冲击"的故事也可能其实经由其他渠道（如改变生产能力）影响资产价格，未必真经由偏好。
then (18.9) and (18.10) are exactly the same.
Interpretation.
- The objective (18.3) of the minimization is exactly the expected continuation value under recursive (Epstein-Zin) utility — a justification for recursive utility: agents with conservative attitudes toward risk would naturally have a recursive utility function.
- Recursive utility can thus be understood as embedding ambiguity aversion: the agent minimizes over all possible measures, using the expected continuation value under the worst uncertainty.
- It is a minimization rather than a max-min problem because the consumption sequence is exogenous with no choice; in an economy with production and a consumption-investment trade-off, the recursive-utility continuation value would be the solution to a max-min problem.
- Large \(\gamma\) ⟺ small \(\xi\): the deviation penalty is small and a more conservative measure is chosen. So "choosing a more conservative measure (smaller \(\xi\))" equals "higher risk aversion (larger \(\gamma\))" — again confirming the link between robustness and recursive utility.
Preference shock.
- One view: the preference shock acts through \(\xi\). A time-varying \(\xi\) changes the selected likelihood ratio \(\frac{N_{t+1}}{N_t}\) and its measure, hence the continuation value.
- Another view: through \(C_t\). Still use \(C_t\) for recursive utility, but \(C_t\) is effective consumption, explicitly \(C_t=\tilde C_t D_t\), with \(\tilde C_t\) the actual consumption and \(D_t\) an exogenous preference shifter (e.g. a stationary first-difference, AR, or stochastic-volatility process).
- Either way one can attribute part of the price to a preference shock; but be cautious — the "preference shock" story may in fact operate through other channels (e.g. changing production capacity) rather than truly through preferences.
References
- Hansen, L. P. and T. J. Sargent (2001). Robust Control and Model Uncertainty. American Economic Review 91(2), 60–66.