10. GMM in Explicit Discount Factor Models
10. GMM in Explicit Discount Factor Models
本章导读 Part II 转向估计与评价：先估自由参数（如 \(m=\beta(c_{t+1}/c_t)^{-\gamma}\) 中的 \(\beta,\gamma\)，或 \(m=b'f\) 中的 \(b\)），再判断模型好坏。本章（Cochrane 第 10 章，Part II 开篇）讲 GMM（广义矩估计）——它与贴现因子表述天然契合，因为只需用样本矩替代总体矩。基本思路极简单：模型预测 \(E(p_t)=E(m_{t+1}x_{t+1})\)，那就让两个样本均值尽量接近，由此估参数；再为估计量与定价误差给出分布理论，从而检验"定价误差是否大到不像运气"。§10.1 配方（Hansen-Singleton 1982）；§10.2 解读（矩 = 定价误差 = Jensen's alpha；一阶/二阶估计如 OLS/GLS；\(S^{-1}\) 为何最优；\(J_T\) 检验）；§10.3 应用（工具变量、平稳性、单位选择）。
10. GMM in Explicit Discount Factor Models
Overview Part II turns to estimation and evaluation: first estimate the free parameters (e.g. \(\beta,\gamma\) in \(m=\beta(c_{t+1}/c_t)^{-\gamma}\), or \(b\) in \(m=b'f\)), then judge the model. This chapter (Cochrane Ch 10, the opener of Part II) covers GMM (Generalized Method of Moments) — a natural fit for the discount-factor formulation, since we just replace population moments with sample moments. The basic idea is very simple: the model predicts \(E(p_t)=E(m_{t+1}x_{t+1})\), so make the two sample averages as close as possible to estimate parameters; then derive a distribution theory for the estimates and the pricing errors, to test whether the errors are "too big to be luck." §10.1 the recipe (Hansen-Singleton 1982); §10.2 interpretation (moments = pricing errors = Jensen's alpha; first/second stage like OLS/GLS; why \(S^{-1}\) is optimal; the \(J_T\) test); §10.3 applying it (instruments, stationarity, choice of units).
10.1 配方 / The Recipe
任何资产定价模型都蕴含 \(E(p_t)=E(m_{t+1}(b)x_{t+1})\)，写成 \(E(\cdot)=0\) 形式即矩条件：
10.1 The Recipe
Any asset pricing model implies \(E(p_t)=E(m_{t+1}(b)x_{t+1})\); written in \(E(\cdot)=0\) form these are the moment conditions:
$$E\bigl[m_{t+1}(b)x_{t+1}-p_t\bigr]=0.\tag{10.4}$$
GMM 配方（Hansen-Singleton 1982） / The GMM recipe
定义：误差 \(u_{t+1}(b)\equiv m_{t+1}(b)x_{t+1}-p_t\)；样本均值 \(g_T(b)\equiv E_T[u_t(b)]\)（\(E_T(\cdot)=\tfrac1T\sum_{t=1}^T(\cdot)\)）；谱密度矩阵 \(S\equiv\sum_{j=-\infty}^{\infty}E[u_t(b)u_{t-j}(b)']\)。
估计：\(\hat b_2=\arg\min_b\ g_T(b)'S^{-1}g_T(b)\)。
标准误：\(\operatorname{var}(\hat b_2)=\tfrac1T(d'S^{-1}d)^{-1}\)，\(d\equiv\partial g_T(b)/\partial b\)。
模型检验（过度识别约束）：\(TJ_T=T\min_b[g_T(b)'S^{-1}g_T(b)]\sim\chi^2(\#\text{moments}-\#\text{params})\)。Definitions: errors \(u_{t+1}(b)\equiv m_{t+1}(b)x_{t+1}-p_t\); sample mean \(g_T(b)\equiv E_T[u_t(b)]\) (with \(E_T(\cdot)=\tfrac1T\sum_{t=1}^T(\cdot)\)); spectral density matrix \(S\equiv\sum_{j=-\infty}^{\infty}E[u_t(b)u_{t-j}(b)']\).
Estimate: \(\hat b_2=\arg\min_b\ g_T(b)'S^{-1}g_T(b)\).
Standard errors: \(\operatorname{var}(\hat b_2)=\tfrac1T(d'S^{-1}d)^{-1}\), \(d\equiv\partial g_T(b)/\partial b\).
Test of the model (overidentifying restrictions): \(TJ_T=T\min_b[g_T(b)'S^{-1}g_T(b)]\sim\chi^2(\#\text{moments}-\#\text{parameters})\).
具体流程：一阶估计 \(\hat b_1=\arg\min_b g_T(b)'Wg_T(b)\)，\(W\) 任取（常取 \(W=I\)），此估计已是一致且渐近正态的——你常常可以、也应该在此止步。用 \(\hat b_1\) 构造 \(S\) 的估计，再做二阶估计 \(\hat b_2=\arg\min_b g_T(b)'S^{-1}g_T(b)\)，它一致、渐近正态、且渐近有效（在所有"把 \(g_T\) 的线性组合置零"的估计量中方差最小）。标准误可用于检验参数是否为零（\(\hat b_i/\sqrt{\operatorname{var}(\hat b)_{ii}}\sim N(0,1)\)）。
10.2 解读 GMM / Interpreting the GMM Procedure
The flow: a first-stage estimate \(\hat b_1=\arg\min_b g_T(b)'Wg_T(b)\) for an arbitrary \(W\) (often \(W=I\)), already consistent and asymptotically normal — you can and often should stop here. Use \(\hat b_1\) to construct an estimate of \(S\), then a second-stage estimate \(\hat b_2=\arg\min_b g_T(b)'S^{-1}g_T(b)\), which is consistent, asymptotically normal, and asymptotically efficient (smallest variance among all estimators that set linear combinations of \(g_T\) to zero). The standard errors test whether parameters are zero (\(\hat b_i/\sqrt{\operatorname{var}(\hat b)_{ii}}\sim N(0,1)\)).
10.2 Interpreting the GMM Procedure
矩就是定价误差 = Jensen's alpha / The moments are pricing errors = Jensen's alpha 每个矩 \(g_T(b)=E_T[m_{t+1}(b)x_{t+1}]-E_T[p_t]\) 就是实际价格与预测价格之差，即定价误差。在期望收益语言里它正比于实际与预测收益之差——即 Jensen's alpha：\(g(b)=E(mR^e)=\tfrac1{R^f}(\text{actual}-\text{predicted})=\tfrac1{R^f}\alpha_i\)。还有什么比"选参数使预测价格尽量贴近实际、再看残余定价误差有多大"更自然呢？Each moment \(g_T(b)=E_T[m_{t+1}(b)x_{t+1}]-E_T[p_t]\) is the difference between actual and predicted price — the pricing error. In expected-return language it is proportional to the actual-minus-predicted return, i.e. Jensen's alpha: \(g(b)=E(mR^e)=\tfrac1{R^f}(\text{actual}-\text{predicted mean return})=\tfrac1{R^f}\alpha_i\). What could be more natural than picking parameters so predicted prices are as close as possible to actual, then judging the model by how big the leftover errors are?
一阶估计。 矩条件（收益乘工具）通常多于参数（否则理论与事实一样多就空洞了），无法令每个 \(g_T=0\)，故最小化二次型 \(\min_b g_T(b)'Wg_T(b)\)。\(W=I\) 时对称对待所有资产、即最小化平方定价误差之和。\(g_T(b)\) 可能非线性，需数值搜索，但局部二次故通常顺利。
二阶估计：为何 \(S^{-1}\)？ 不同资产的 \(m_tR_t-1\) 方差不同，高方差资产的样本均值测得更不准，理应少给关注。由于 \(g_T=E_T(u_{t+1})\) 是样本均值，其方差为
First-stage estimates. There are usually more moment conditions (returns times instruments) than parameters (else a theory with as many parameters as facts is vacuous), so we cannot set every \(g_T=0\) and instead minimize the quadratic form \(\min_b g_T(b)'Wg_T(b)\). With \(W=I\) all assets are treated symmetrically — minimize the sum of squared pricing errors. \(g_T(b)\) may be nonlinear, needing a numerical search, but it is locally quadratic so the search is usually easy.
Second-stage estimates: why \(S^{-1}\)? Different assets' \(m_tR_t-1\) have different variances; high-variance assets' sample means are measured less accurately and deserve less attention. Since \(g_T=E_T(u_{t+1})\) is a sample mean, its variance is
$$\operatorname{var}(g_T)\to\frac1T\sum_{j=-\infty}^{\infty}E(u_tu'_{t-j})=\frac1T\,S.$$
这里 \(S\) 即 \(u_t\) 在零频率的谱密度矩阵。Hansen (1982) 正式证明：取 \(W=S^{-1}\) 是统计最优的加权矩阵（产生渐近方差最小的估计）。它把最多关注放在"当前数据信息量最大"的矩的线性组合上——这与线性回归中由 OLS 到 GLS 的异方差/相关性校正完全同理。事实上当 \(u_t\) 时间上不相关时，\(\operatorname{var}(g_T)=\operatorname{var}(u)/T\)——这是你最早学过的统计公式（样本均值的方差）；在 GMM 里它也是你最后会用到的公式：GMM 不过是把"样本均值的分布"这一简单思想推广到参数估计与一般统计情形。
标准误。 \(\operatorname{var}(\hat b_2)=\tfrac1T(d'S^{-1}d)^{-1}\) 最简单地理解为 delta 方法（\(f(x)\) 的渐近方差为 \(f'(x)^2\operatorname{var}(x)\)）：单参数单矩时 \(S/T\) 是 \(g_T\) 的方差，\(d^{-1}=\partial b/\partial g_T\)，故 \(\operatorname{var}(\hat b_2)=\tfrac{\partial b}{\partial g_T}\operatorname{var}(g_T)\tfrac{\partial b}{\partial g_T}\)，向量化即得。
Here \(S\) is the spectral density matrix of \(u_t\) at frequency zero. Hansen (1982) proves \(W=S^{-1}\) is the statistically optimal weighting matrix (lowest asymptotic variance). It pays most attention to the linear combinations of moments about which the data have the most information — exactly the heteroskedasticity/correlation correction that takes you from OLS to GLS. Indeed when \(u_t\) is uncorrelated over time, \(\operatorname{var}(g_T)=\operatorname{var}(u)/T\) — the very first statistical formula you ever saw (the variance of a sample mean); in GMM it is the last one too: GMM just generalizes the distribution of the sample mean to parameter estimation and general statistical contexts.
Standard errors. \(\operatorname{var}(\hat b_2)=\tfrac1T(d'S^{-1}d)^{-1}\) is most simply an instance of the delta method (the asymptotic variance of \(f(x)\) is \(f'(x)^2\operatorname{var}(x)\)): with one parameter and moment, \(S/T\) is the variance of \(g_T\), \(d^{-1}=\partial b/\partial g_T\), so \(\operatorname{var}(\hat b_2)=\tfrac{\partial b}{\partial g_T}\operatorname{var}(g_T)\tfrac{\partial b}{\partial g_T}\), generalized to vectors.
\(J_T\) 检验：定价误差是否"太大" / The \(J_T\) test: are the pricing errors "too big"? 估出最优参数后，问"拟合多好"？看定价误差是否大到不像运气：\(TJ_T=T\,g_T(\hat b)'S^{-1}g_T(\hat b)\sim\chi^2(\#\text{moments}-\#\text{params})\)。因 \(S\) 是 \(g_T\) 的协方差阵，此统计量就是"定价误差除以其协方差阵"；样本均值趋正态，故均值平方除方差趋 \(\chi^2\)。自由度减去参数个数，是因为我们在每个样本里把 \(g_T\) 的某些线性组合置了零（真实 \(g_T\) 协方差阵奇异，秩为 \(\#\text{moments}-\#\text{params}\)）。Having estimated the best-fitting parameters, ask "how well does it fit?" — are the pricing errors too big to be luck? \(TJ_T=T\,g_T(\hat b)'S^{-1}g_T(\hat b)\sim\chi^2(\#\text{moments}-\#\text{parameters})\). Since \(S\) is the covariance matrix of \(g_T\), this is the minimized pricing errors divided by their covariance; sample means go to normal, so means squared over variance go to \(\chi^2\). Degrees of freedom drop by the number of parameters because we set linear combinations of \(g_T\) to zero in each sample (the true covariance of \(g_T\) is singular, rank \(\#\text{moments}-\#\text{parameters}\)).
10.3 应用 GMM / Applying GMM
记号、工具与收益。 大部分功夫只是把问题映射到这套通用记号。用收益时矩条件为 \(E[m_{t+1}(b)R_{t+1}-1]=0\)。常加工具变量：把 \(1=E_t[m_{t+1}R_{t+1}]\) 两边乘以 \(t\) 时可观测的 \(z_t\) 再取非条件期望，得 \(0=E[(m_{t+1}R_{t+1}-1)z_t]\)。对收益与工具的整个向量，用 Kronecker 积紧凑写成
10.3 Applying GMM
Notation, instruments, returns. Most of the effort is just mapping a problem into this general notation. With returns the moments are \(E[m_{t+1}(b)R_{t+1}-1]=0\). One commonly adds instruments: multiply \(1=E_t[m_{t+1}R_{t+1}]\) by a \(t\)-observed \(z_t\) and take unconditional expectations, giving \(0=E[(m_{t+1}R_{t+1}-1)z_t]\). For a whole vector of returns and instruments, the Kronecker product writes it compactly:
$$E\bigl[(m_{t+1}(b)R_{t+1}-1)\otimes z_t\bigr]=0.\tag{10.9}$$
预测误差与工具。 模型说：尽管期望收益随时间与资产变化，期望贴现收益总应等于 1。误差 \(u_{t+1}=m_{t+1}R_{t+1}-1\) 是事后贴现收益，像任何预测误差一样应条件与非条件均值为零。在计量语境里 \(z\) 是"工具"，因它应与 \(u_{t+1}\) 不相关——\(E(z_tu_{t+1})\) 是 \(u_{t+1}\) 对 \(z_t\) 回归系数的分子；加工具本质上是检验"事后贴现收益不能被线性回归预测"。如 §8.1，加工具 = 加管理组合，原则上能捕捉模型全部预测。
Forecast errors and instruments. The model says: although expected returns vary across time and assets, the expected discounted return should always be 1. The error \(u_{t+1}=m_{t+1}R_{t+1}-1\) is the ex-post discounted return and, like any forecast error, should be conditionally and unconditionally mean zero. In econometric terms \(z\) is an "instrument" because it should be uncorrelated with \(u_{t+1}\) — \(E(z_tu_{t+1})\) is the numerator of a regression coefficient of \(u_{t+1}\) on \(z_t\); adding instruments checks that the ex-post discounted return is unforecastable by linear regressions. As in §8.1, adding instruments = adding managed portfolios, and in principle captures all the model's predictions.
平稳性与单位选择 / Stationarity and choice of units GMM 分布理论最重要的假设是 \(m,p,x\) 须为平稳随机变量（"平稳"指 \(x_t,x_{t-j}\) 的联合分布只依赖 \(j\) 而非 \(t\)，不是指恒定或 i.i.d.）；样本均值须收敛到总体均值。确保平稳通常归结为选对单位：股价 \(p\) 与股利 \(d\) 随时间增长、非平稳，故不要写 \(p_t=E_t[m_{t+1}(d_{t+1}+p_{t+1})]\)，而应除以 \(p_t\) 写成 \(1=E_t[m_{t+1}R_{t+1}]\)（收益大体平稳），或除以股利写成价格股利比形式。债券是对 1 美元的索取权，价格/收益率不随时间增长，故 \(p^b_t=E(m_{t+1}\cdot1)\) 可不变换。注意：变量"越不平稳"（样本中摆动越久，如名义利率虽根本上平稳却可二十年单向漂移），渐近分布越不可靠——届时应用模拟/自助法求有限样本分布。也要让检验资产平稳（故常把股票按特征排成组合，组合的统计特征比个股稳定得多）。GMM 的方差公式不需要 i.i.d./正态/同方差假设（可加以简化公式与改善小样本表现，但非必需）。The most important assumption of the GMM distribution theory is that \(m,p,x\) be stationary random variables ("stationary" means the joint distribution of \(x_t,x_{t-j}\) depends only on \(j\) not \(t\) — not constant or i.i.d.); sample averages must converge to population means. Ensuring stationarity usually amounts to a choice of sensible units: stock prices \(p\) and dividends \(d\) grow over time and are nonstationary, so don't write \(p_t=E_t[m_{t+1}(d_{t+1}+p_{t+1})]\); divide by \(p_t\) to get \(1=E_t[m_{t+1}R_{t+1}]\) (returns are plausibly stationary), or by dividends for a price/dividend form. Bonds are a claim to a dollar, so \(p^b_t=E(m_{t+1}\cdot1)\) needs no transform. Note: the "less stationary" a variable (the longer its swings — e.g. nominal interest rates are fundamentally stationary yet drift one way for 20 years), the less reliable the asymptotic distribution — then use simulation/bootstrap for a finite-sample distribution. Choose test assets to be stationary too (hence sorting stocks into portfolios by characteristics, whose statistics are far more constant than individual stocks'). GMM's variance formulas need no i.i.d./normal/homoskedastic assumptions (you may add them to simplify formulas and improve small-sample performance, but you need not).
小结 / Summary
GMM 把"样本均值的分布"这一最基本统计思想推广为完整的估计与检验框架：把矩条件（= 定价误差 = Jensen's alpha）的加权平方和最小化以估参数（一阶 \(W=I\) 如 OLS，二阶 \(W=S^{-1}\) 如 GLS、渐近有效），用 delta 方法得标准误，用 \(J_T=T\cdot(\text{min objective})\sim\chi^2\) 检验过度识别约束。它与 \(p=\mathbb E(mx)\) 天然契合、不需分布假设，唯一关键前提是平稳性（靠选对单位与检验资产保证）。下一章给出 GMM 的一般公式与更多应用。
Summary
GMM generalizes the most basic statistical idea — the distribution of a sample mean — into a full estimation and testing framework: minimize a weighted sum of squared moment conditions (= pricing errors = Jensen's alpha) to estimate parameters (first stage \(W=I\) like OLS, second stage \(W=S^{-1}\) like GLS and asymptotically efficient), get standard errors via the delta method, and test the overidentifying restrictions with \(J_T=T\cdot(\text{minimized objective})\sim\chi^2\). It fits \(p=\mathbb E(mx)\) naturally and needs no distributional assumptions; its one key prerequisite is stationarity (ensured by choosing sensible units and test assets). The next chapter gives the general GMM formulas and more applications.