22. Overview
本章是 Part V 简约实证分析 (reduced-form empirical analysis) 的开篇路线图。实证资产定价从基本方程 \(1=\mathbb E[m_{t+1}\mathbf R_{t+1}]\) (22.1) 出发，有两条路：(1) 先给定结构模型的 SDF \(m_{t+1}\)（如 C-CAPM），再用收益率数据检验 (22.1) 是否成立；(2) 不设任何结构模型，直接令 (22.1) 成立并从数据中反解出 SDF。本章聚焦第二条路，并引入主观信念：计量经济学家用客观概率 \(\mathbf P\) 与客观 SDF \(m\)，而投资者用主观概率 \(\tilde{\mathbf P}\) 与主观 SDF \(\tilde m\)。市场完全时二者的 Arrow-Debreu 状态价格必须一致 (22.4)，由此把客观 SDF 分解为三部分 (22.6)：主观 SDF（Part A）、乐观/悲观带来的偏差（Part B，\(\tilde{\mathbf P}/\mathbf P\) 与收益的协方差）、风险误判带来的偏差（Part C，\(\tilde{\mathbf P}/\mathbf P\) 与 \(\tilde m R\) 的协方差）。理性预期 (RE) 是 \(\tilde{\mathbf P}=\mathbf P\) 的特例，比"投资者理性"更强。
This chapter is the opening road map for Part V, reduced-form empirical analysis. Empirical asset pricing starts from the fundamental equation \(1=\mathbb E[m_{t+1}\mathbf R_{t+1}]\) (22.1), and offers two routes: (1) take the SDF \(m_{t+1}\) from a structural model (e.g. C-CAPM), then use return data to test whether (22.1) holds; (2) impose no structural model, just require (22.1) and extract the SDF from data. This chapter focuses on the second route and introduces subjective beliefs: the econometrician uses objective probabilities \(\mathbf P\) and objective SDF \(m\), while investors use subjective probabilities \(\tilde{\mathbf P}\) and subjective SDF \(\tilde m\). Under market completeness their Arrow-Debreu state prices must agree (22.4), which decomposes the objective SDF into three parts (22.6): the subjective SDF (Part A), the optimism/pessimism distortion (Part B, covariance of \(\tilde{\mathbf P}/\mathbf P\) with return), and the misperception-of-risk distortion (Part C, covariance of \(\tilde{\mathbf P}/\mathbf P\) with \(\tilde m R\)). Rational expectation (RE) is the special case \(\tilde{\mathbf P}=\mathbf P\), which is stronger than mere "investor rationality".
22.1 Two Approaches in Empirical Asset Pricing
资产定价的基本方程（毛收益率版本）(22.1)：
The fundamental equation of asset pricing (gross return version) (22.1):
$$\mathbf 1=\mathbb E_t\!\left[m_{t+1}\mathbf R_{t+1}\right]\quad\Rightarrow\quad \mathbf 1=\mathbb E\!\left[m_{t+1}\mathbf R_{t+1}\right]\tag{22.1}$$
其中 \(\mathbf R_{t+1}\) 是 \(N\times1\) 风险毛收益率向量，\(m_{t+1}\) 是 SDF，\(\mathbf 1\) 是 \(N\times1\) 全 1 向量。(22.1) 引出两条实证路径：
- 从 \(m_{t+1}\) 出发，检验 (22.1)：从结构模型（如 C-CAPM）取出 SDF \(m_{t+1}\)；用 \(\mathbf R_{t+1}\) 数据检验 (22.1) 是否成立，以判断该结构模型是否合理。
- 从 (22.1) 出发，反解 \(m_{t+1}\)：不设任何结构模型，只要求 (22.1) 成立；用 \(\mathbf R_{t+1}\) 数据与概率分布反推出 SDF \(m_{t+1}\)。
where \(\mathbf R_{t+1}\) is an \(N\times1\) vector of risky gross returns, \(m_{t+1}\) the SDF, and \(\mathbf 1\) an \(N\times1\) vector of ones. (22.1) leads to two empirical approaches:
- Start from \(m_{t+1}\) and test (22.1): take the SDF \(m_{t+1}\) from a structural model (e.g. C-CAPM); use data on \(\mathbf R_{t+1}\) to test whether (22.1) holds, determining whether the structural model makes sense.
- Start from (22.1) and extract \(m_{t+1}\): impose no structural model, just require (22.1) to hold; use data on \(\mathbf R_{t+1}\) and the probability distribution to back out the SDF \(m_{t+1}\).
22.2 Subjective Belief and Stochastic Discount Factor
聚焦第二条路。设市场完全（可放松），存在 Arrow-Debreu 证券，对每个状态 \(s\in\mathcal S=\{s_1,\dots,s_K\}\) 给出状态价格 \(\Pi(s)\)。由 (3.2)：
Focus on the second route. Suppose the market is complete (can be relaxed), with Arrow-Debreu securities giving a state price \(\Pi(s)\) for each state \(s\in\mathcal S=\{s_1,\dots,s_K\}\). By (3.2):
$$1=\sum_{s\in\mathcal S}\mathbf P\{s\}\,m(s)\,R(s)\tag{22.2}$$
其中 \(\mathbf P\{s\}\) 是状态 \(s\) 的客观概率；\(m(s)\equiv\frac{\Pi(s)}{\mathbf P\{s\}}\) 是状态价格密度（对应 (3.2) 中的 \(\pi(s)\)），即完全市场下状态 \(s\) 实现的 SDF；\(R(s)\) 是状态 \(s\) 实现的毛收益。可放松完全性、但仍假设状态数有限，此时 (22.2) 因 (22.1) 仍成立，只是 \(\mathbf P\{s\}m(s)\) 不再能解释为 Arrow-Debreu 状态价格。下文为简化仍保留完全市场假设。
设计量经济学家能得到客观概率分布 \(\mathbf P\) 和客观 SDF \(m\)。再考虑投资者用主观概率分布 \(\tilde{\mathbf P}\) 与主观 SDF \(\tilde m\) 给资产定价，于是 (22.2) 变为 (22.3)：
where \(\mathbf P\{s\}\) is the objective probability of state \(s\); \(m(s)\equiv\frac{\Pi(s)}{\mathbf P\{s\}}\) is the state-price density (corresponding to \(\pi(s)\) in (3.2)), i.e. the SDF realized in state \(s\) under complete markets; \(R(s)\) is the realized gross return in state \(s\). Completeness can be relaxed (still with finitely many states), in which case (22.2) still holds by (22.1) but \(\mathbf P\{s\}m(s)\) can no longer be interpreted as an Arrow-Debreu state price. For simplicity we keep the completeness assumption below.
Suppose the econometrician can obtain the objective distribution \(\mathbf P\) and objective SDF \(m\). Now investors use a subjective distribution \(\tilde{\mathbf P}\) and subjective SDF \(\tilde m\) to price assets, so (22.2) becomes (22.3):
$$1=\sum_{s\in\mathcal S}\tilde{\mathbf P}\{s\}\,\tilde m(s)\,R(s)\tag{22.3}$$
设无风险收益 \(R_f\) 存在，则 (22.2) 与 (22.3) 一起意味着 \(\sum_s\tilde{\mathbf P}\{s\}\tilde m(s)=\sum_s\mathbf P\{s\}m(s)=\frac1{R_f}\)。在完全市场下，Arrow-Debreu 状态价格必须被计量经济学家与投资者共同认同，故 (22.4)：
If a risk-free gross return \(R_f\) exists, then (22.2) and (22.3) together imply \(\sum_s\tilde{\mathbf P}\{s\}\tilde m(s)=\sum_s\mathbf P\{s\}m(s)=\frac1{R_f}\). Under complete markets the Arrow-Debreu state price must be agreed on by both the econometrician and investors, which implies (22.4):
$$\tilde{\mathbf P}\{s\}\,\tilde m(s)=\mathbf P\{s\}\,m(s)\quad\text{for }\forall s\in\mathcal S\tag{22.4}$$
Remark 22.1（理性预期 RE） 当投资者的主观信念恰好等于客观信念（真实数据生成过程）时，称其有理性预期 (rational expectation, RE)，即 \(\tilde{\mathbf P}\{s\}=\mathbf P\{s\}\) 对 \(\forall s\) 成立，且 \(\tilde{\mathbb E}[\cdot]=\mathbb E[\cdot]\)。但 RE 远强于"投资者是理性的"：后者只要求投资者正确学习参数（如第 26 章的贝叶斯学习），而前者要求投资者精确地知道参数。
Remark 22.1 (Rational Expectation, RE) When investors' subjective beliefs are exactly the objective beliefs (the true data-generating process), they are said to have rational expectation (RE): \(\tilde{\mathbf P}\{s\}=\mathbf P\{s\}\) for \(\forall s\), and \(\tilde{\mathbb E}[\cdot]=\mathbb E[\cdot]\). But RE is much stronger than "investors are rational": the latter only requires investors to learn the parameters correctly (e.g. Bayesian learning in Chapter 26), while the former requires investors to know the parameters exactly.
22.3 Subjective Stochastic Discount Factor Decomposition
分解客观 SDF \(m\)。由 (22.4) 改写并展开 (22.5)：
Decompose the objective SDF \(m\). Rewrite (22.4) and expand (22.5):
$$m(s)=\frac{\tilde{\mathbf P}\{s\}}{\mathbf P\{s\}}\tilde m(s)=\tilde m(s)+\left(\frac{\tilde{\mathbf P}\{s\}}{\mathbf P\{s\}}-1\right)\frac1{R_f}+\left(\frac{\tilde{\mathbf P}\{s\}}{\mathbf P\{s\}}-1\right)\left(\tilde m(s)-\frac1{R_f}\right)\tag{22.5}$$
代入 (22.2) 得三部分分解 (22.6)：
Plug into (22.2) to get the three-part decomposition (22.6):
$$1=\underbrace{\sum_{s}\mathbf P\{s\}\tilde m(s)R(s)}_{\text{Part A}}+\underbrace{\sum_{s}\mathbf P\{s\}\!\left(\frac{\tilde{\mathbf P}\{s\}}{\mathbf P\{s\}}-1\right)\!\frac{R(s)}{R_f}}_{\text{Part B}}+\underbrace{\sum_{s}\mathbf P\{s\}\!\left(\frac{\tilde{\mathbf P}\{s\}}{\mathbf P\{s\}}-1\right)\!\tilde m(s)R(s)}_{\text{Part C}}\tag{22.6}$$
- Part A 是反映投资者对状态条件支付偏好的主观 SDF，被投资者的乐观/悲观与风险误判所扭曲。
- Part B 是 \(\frac{\tilde{\mathbf P}}{\mathbf P}\) 与 \(\frac{R}{R_f}\) 的协方差（因 \(\sum_s\mathbf P\{s\}(\frac{\tilde{\mathbf P}\{s\}}{\mathbf P\{s\}}-1)=0\)）。Part B $>0$ ⟹ 投资者对高收益状态赋予更高概率权重 = 乐观；Part B $<0$ = 悲观。故 Part B 度量乐观/悲观调整。
- Part C 是 \(\frac{\tilde{\mathbf P}}{\mathbf P}\) 与 \(\tilde m(s)R(s)\) 的协方差。Part C $>0$ ⟹ 投资者对高收益且高边际效用 \(\tilde m(s)\) 状态（灾难状态，\(\tilde m\) 高像保险）赋予更高权重，即相信灾难很可能发生；Part C $<0$ ⟹ 投资者低估灾难状态概率。故 Part C 度量风险误判调整。
(22.6) 的四种解读：
- 理性预期：\(\tilde{\mathbf P}\{s\}=\mathbf P\{s\}\) 且 \(\tilde m(s)=m(s)\)。Part B、Part C 消失，计量经济学家提取的 SDF 即投资者真用的 SDF，问题易解。但 RE 即便在完全理性下也未必合理。
- 投资者学习：\(\tilde{\mathbf P}\{s\}\xrightarrow{p}\mathbf P\{s\}\) 且 \(\tilde m(s)\to m(s)\)（收敛到客观）。
- 数据挖掘 (data mining)：\(\tilde{\mathbf P}\{s\}\neq\mathbf P\{s\}\)、\(\tilde m(s)\neq m(s)\)，甚至计量经济学家得到的 \(\mathbf P\{s\}\) 因数据挖掘/操纵而非客观分布。
- 部分投资者非完全理性：对某些投资者 \(\tilde{\mathbf P}\neq\mathbf P\)、\(\tilde m\neq m\)，而对另一些（套利者）\(\tilde{\mathbf P}=\mathbf P\)、\(\tilde m=m\)。
Remark 22.2（路线图） 本节为实证资产定价提供路线图。两条路：第一，检验结构模型的有效性——取决于有什么结构模型；第二，不设结构模型从数据提取 SDF——取决于做什么假设：若为简化假设理性预期，可直接用数据提取 SDF，难点在于如何对 SDF 降维（详见第 23 章）；若不假设理性预期，则需研究投资者的主观信念。
- Part A is the subjective SDF reflecting investors' preferences over state-contingent payoffs, distorted by their optimism/pessimism and misperception of risk.
- Part B is the covariance between \(\frac{\tilde{\mathbf P}}{\mathbf P}\) and \(\frac{R}{R_f}\) (since \(\sum_s\mathbf P\{s\}(\frac{\tilde{\mathbf P}\{s\}}{\mathbf P\{s\}}-1)=0\)). Part B $>0$ ⟹ investors place higher probability weights on high-return states = optimism; Part B $<0$ = pessimism. So Part B measures the optimism/pessimism adjustment.
- Part C is the covariance between \(\frac{\tilde{\mathbf P}}{\mathbf P}\) and \(\tilde m(s)R(s)\). Part C $>0$ ⟹ investors place higher weights on states with high return and high marginal utility \(\tilde m(s)\) (disaster states, where high \(\tilde m\) acts like insurance), i.e. they believe disaster is very likely; Part C $<0$ ⟹ investors underestimate the likelihood of disaster states. So Part C measures the misperception-of-risk adjustment.
Four interpretations of (22.6):
- Rational expectation: \(\tilde{\mathbf P}\{s\}=\mathbf P\{s\}\) and \(\tilde m(s)=m(s)\). Parts B and C disappear, and the econometrician's extracted SDF is the real SDF used by investors — problem easily solved. But RE is not always reasonable even under perfect rationality.
- Investor learning: \(\tilde{\mathbf P}\{s\}\xrightarrow{p}\mathbf P\{s\}\) and \(\tilde m(s)\to m(s)\) (converging to the objective).
- Data mining: \(\tilde{\mathbf P}\{s\}\neq\mathbf P\{s\}\), \(\tilde m(s)\neq m(s)\), and even the \(\mathbf P\{s\}\) obtained by the econometrician may not be the objective distribution due to data mining/manipulation.
- Imperfect rationality of some investors: \(\tilde{\mathbf P}\neq\mathbf P\), \(\tilde m\neq m\) for some investors, while \(\tilde{\mathbf P}=\mathbf P\), \(\tilde m=m\) for others (arbitrageurs).
Remark 22.2 (Road map) This section provides a road map for empirical asset pricing. Two ways to go: first, test the validity of a structural model — depends on what structural model we have; second, extract the SDF from data without a structural model — depends on assumptions: if we assume rational expectation for simplicity, we can directly extract the SDF from data, the difficulty being how to reduce the dimension of the SDF (discussed in detail in Chapter 23); alternatively, without assuming rational expectation, we must study the subjective beliefs of investors.