12. Suggested Explanation: Habit Persistence

Jun He May 30, 2026

资产定价Asset Pricing 习惯形成Habit Persistence Campbell-Cochrane 外部习惯External Habit 内部习惯Internal Habit 学习笔记Study Note

Note

习惯形成 (habit persistence) 思路：让效用不再直接定义在消费 $c_t$ 上，而是定义在消费与一个参考点 (habit) 之差 $s_t=c_t-x_t$ 上。当消费接近习惯水平时，边际效用急剧上升，投资者变得极度厌恶风险——于是 SDF 在衰退期波动放大，用合理的风险厌恶也能产生高股权溢价、低无风险利率（让 SDF 进入图 8.2 的可行区）。本章给出两类模型：(i) Campbell–Cochrane (1999) 外部习惯——习惯是经济整体的滞后消费（"攀比效应"），盈余消费比 $y_t$ 服从均值回复过程，敏感度函数 $\lambda(\cdot)$ 制造逆周期风险厌恶，几乎拟合了所有时序矩；(ii) 内部习惯与 Ferson–Constantinides (1991) 的"耐用 vs 习惯"框架——把消费的时间不可分拆成耐用品效应（$b_s>0$）与习惯效应（$b_s<0$）的净效应。批评：外部习惯校准出的风险厌恶高得离谱、敏感度函数特设、内部习惯模型仍拟合不了无风险利率与股权溢价。

Note

The habit persistence idea: define utility not directly on consumption $c_t$ but on the gap $s_t=c_t-x_t$ between consumption and a reference point (habit). As consumption nears the habit level, marginal utility rises sharply and the investor becomes extremely risk-averse — so the SDF becomes volatile in recessions, and a reasonable risk aversion can generate a high equity premium and low risk-free rate (pushing the SDF into the feasible region of Figure 8.2). This chapter covers two model classes: (i) Campbell–Cochrane (1999) external habit — the habit is the economy's lagged consumption ("keeping up with the Joneses"), the surplus consumption ratio $y_t$ follows a mean-reverting process, and the sensitivity function $\lambda(\cdot)$ creates countercyclical risk aversion, fitting almost all time-series moments; (ii) internal habit and the Ferson–Constantinides (1991) "durability vs. habit" framework — decomposing time non-separability into a durability effect ($b_s>0$) and a habit effect ($b_s<0$) as a net effect. Critiques: external habit calibrates an implausibly high risk aversion, the sensitivity function is ad hoc, and internal-habit models still fail to fit the risk-free rate and equity premium.

消费基础 CRRA 模型的 SDF 是 (1.12) $m_{t+1}=\beta(\frac{C_{t+1}}{C_t})^{-\gamma}$。我们想提高其波动率，即写成

The consumption-based CRRA SDF is (1.12) $m_{t+1}=\beta(\frac{C_{t+1}}{C_t})^{-\gamma}$. We want to raise its volatility by writing

$$m_{t+1}=\beta\Big(\frac{C_{t+1}}{C_t}\Big)^{-\gamma}f_{t+1},$$

使该 SDF：(i) 落入可行区（图 8.2 的红线）；(ii) 用合理 $\gamma$ 产生高股权溢价；(iii) 用合理 $\gamma$ 产生低无风险利率。习惯形成正是给出这样一个调整因子 $f_{t+1}$。

so that the SDF: (i) sits inside the feasible region (the red line of Figure 8.2); (ii) generates a high equity premium with a reasonable $\gamma$; (iii) generates a low risk-free rate with a reasonable $\gamma$. Habit persistence provides exactly such an adjustment factor $f_{t+1}$.

12.1 External Habit Model: Campbell and Cochrane (1999)

效用定义在消费与外部习惯 $x_t^a$ 之差上：盈余 $s_t^a=c_t-x_t^a$，其中 $x_t^a=\psi c_{t-1}^a$ 是经济整体的滞后消费（参考点，"攀比效应"）。外部指个体把 $x_t^a$ 当作给定，不内部化自己消费对它的影响。盈余上的 CRRA 周期效用与终身效用：

Utility is defined on the gap between consumption and an external habit $x_t^a$: the surplus $s_t^a=c_t-x_t^a$, where $x_t^a=\psi c_{t-1}^a$ is the economy-wide lagged consumption (the reference point, "keeping up with the Joneses"). External means the agent takes $x_t^a$ as given and does not internalize the effect of their own consumption on it. CRRA period and lifetime utility on the surplus:

$$u(s_t)=\frac{s_t^{1-\gamma}-1}{1-\gamma},\qquad \sum_{t=0}^\infty\beta^t u(s_t).$$

证明 / Proof：外部习惯 SDF $m_t^*=m_t^c\cdot f_{t+1}$

SDF 为 $m_t^*=\beta\frac{MU_{t+1}}{MU_t}$，$MU_t$ 为消费的边际效用。由链式法则，$MU_t=u'(s_t)\frac{\partial s_t}{\partial c_t}$。外部习惯下 $\frac{\partial s_t}{\partial c_t}=1$（$x_t^a$ 外生），故

The SDF is $m_t^*=\beta\frac{MU_{t+1}}{MU_t}$ with $MU_t$ the marginal utility of consumption. By the chain rule $MU_t=u'(s_t)\frac{\partial s_t}{\partial c_t}$. Under an external habit $\frac{\partial s_t}{\partial c_t}=1$ ($x_t^a$ exogenous), so

$$m_t^*=\beta\frac{s_{t+1}^{-\gamma}}{s_t^{-\gamma}}=\beta\Big(\frac{c_{t+1}-x_{t+1}^a}{c_t-x_t^a}\Big)^{-\gamma}=\beta\Big(\frac{c_{t+1}}{c_t}\Big)^{-\gamma}\Big(\frac{1-x_{t+1}^a/c_{t+1}}{1-x_t^a/c_t}\Big)^{-\gamma}.$$

即 $m_t^*=m_t^c\cdot f_{t+1}$。$\blacksquare$

i.e. $m_t^*=m_t^c\cdot f_{t+1}$. $\blacksquare$

$$ m_t^* = \cssId{hb1}{\beta\Big(\frac{c_{t+1}}{c_t}\Big)^{-\gamma}}\ \cssId{hb2}{\Big(\frac{1-x_{t+1}^a/c_{t+1}}{1-x_t^a/c_t}\Big)^{-\gamma}} $$

定义盈余消费比 (surplus consumption ratio) $y_{t+1}=1-\frac{x_{t+1}^a}{c_{t+1}}=\frac{c_{t+1}-x_{t+1}^a}{c_{t+1}}$，则欧拉方程为 $\mathbb E_t[\beta(\frac{c_{t+1}}{c_t})^{-\gamma}(\frac{y_{t+1}}{y_t})^{-\gamma}R^i_{t+1}]=1$。设 $y_{t+1}$ 服从均值回复过程 (12.1)，敏感度函数 (12.2)：

Define the surplus consumption ratio $y_{t+1}=1-\frac{x_{t+1}^a}{c_{t+1}}=\frac{c_{t+1}-x_{t+1}^a}{c_{t+1}}$; the Euler equation is $\mathbb E_t[\beta(\frac{c_{t+1}}{c_t})^{-\gamma}(\frac{y_{t+1}}{y_t})^{-\gamma}R^i_{t+1}]=1$. Let $y_{t+1}$ follow a mean-reverting process (12.1) with sensitivity function (12.2):

$$y_{t+1}=(1-\psi)\bar y+\psi\,y_t+\lambda(y_t)\,\varepsilon_{c,t+1},\tag{12.1}$$

$$\lambda(y_t)=\begin{cases}\dfrac{1}{\bar Y}\sqrt{1-2(y_t-\bar y)} & \text{if }s_t\le s_{\max},\\[6pt]0 & \text{if }s_t>s_{\max},\end{cases}\tag{12.2}$$

其中 $\bar y,\psi$ 为常数，$\varepsilon_{c,t+1}\sim\mathcal N(0,\sigma_c^2)$ 与所有其它随机变量独立。

(12.1) 是均值回复的（$\psi\in(0,1)$）：盈余高于 $\bar y$ 时漂移为负，低于时为正。
(12.2) 给出逆周期风险厌恶：衰退（盈余低）时 $\lambda$ 大 $\Rightarrow$ SDF 波动大、风险溢价高；繁荣（盈余高）时 $\lambda$ 小 $\Rightarrow$ 溢价低。

实证结果。 该模型几乎拟合了所有时序矩：无条件股权溢价与无风险利率、收益的时变与可预测性、消费增长与股票收益的低相关、价格-红利比与收益的均值回复、股价/收益高波动但红利平滑、收益波动率的持久movements、长期股权溢价等。

批评：

模型唯一风险源是消费增长，故应预期消费增长与价格-红利比有高相关（模型约 0.5），但数据只有约 0.03——说明聚合消费并非主要风险驱动。
外部习惯假设使校准的相对风险厌恶高得离谱（60 到数百）。
(12.2) 的敏感度函数 $\lambda(\cdot)$ 是特设、反向工程出来的，削弱了模型的说服力。

where $\bar y,\psi$ are constants and $\varepsilon_{c,t+1}\sim\mathcal N(0,\sigma_c^2)$ is independent of all other random variables.

(12.1) is mean-reverting ($\psi\in(0,1)$): surplus above $\bar y$ has negative drift, below it positive.
(12.2) gives countercyclical risk aversion: in a recession (low surplus) $\lambda$ is large $\Rightarrow$ a volatile SDF and high risk premium; in a boom (high surplus) $\lambda$ is small $\Rightarrow$ low premium.

Empirical results. The model fits almost all time-series moments: the unconditional equity premium and risk-free rate, the time variation and predictability of returns, the low correlation of consumption growth with stock returns, the price-dividend ratio and mean reversion in returns, high stock-price/return volatility with smooth dividends, persistent movements in return volatility, the long-run equity premium, etc.

Critiques:

The model's only source of risk is consumption growth, so one would expect a high correlation between consumption growth and the price-dividend ratio (about 0.5 in the model), but the data show only about 0.03 — aggregate consumption is not the main driver of risk.
The external habit assumption makes the calibrated relative risk aversion implausibly high (60 to hundreds).
The sensitivity function $\lambda(\cdot)$ in (12.2) is ad hoc and reverse-engineered, weakening the model's appeal.

12.2 Internal Habit Model

12.2.1 One-Lag Internal Habit

效用定义在 $s_t=c_t+\theta c_{t-1}$ (12.3) 上：

Utility is defined on $s_t=c_t+\theta c_{t-1}$ (12.3):

$$s_t=c_t+\theta\,c_{t-1}.\tag{12.3}$$

$\theta<0$ 时 $s_t$ 是当期消费与加权前期消费之差（习惯）；极端情形参考点 $-\theta c_{t-1}$ 退化为常数 $b$，$s_t=c_t-b$ 即生存水平。内部习惯指个体理性地认识到当期 $c_t$ 会影响明天的 $u(s_{t+1})$。盈余上的 CRRA $u(s_t)=\frac{s_t^{1-\gamma}-1}{1-\gamma}$。SDF $m_t^*=\beta\frac{MU_{t+1}}{MU_t}$，其中 $MU_t=\frac{\partial\mathbb E[\sum_{\tau=0}^\infty\beta^\tau u(s_{t+\tau})\mid\mathcal F_t]}{\partial c_t}$ 同时含当期与未来效应。整理得 (12.4)：

When $\theta<0$, $s_t$ is the gap between current and weighted past consumption (habit); in the extreme the reference $-\theta c_{t-1}$ degenerates to a constant $b$ and $s_t=c_t-b$ is the subsistence level. Internal habit means the agent rationally recognizes that current $c_t$ affects tomorrow's $u(s_{t+1})$. CRRA on the surplus $u(s_t)=\frac{s_t^{1-\gamma}-1}{1-\gamma}$. The SDF $m_t^*=\beta\frac{MU_{t+1}}{MU_t}$, where $MU_t=\frac{\partial\mathbb E[\sum_{\tau=0}^\infty\beta^\tau u(s_{t+\tau})\mid\mathcal F_t]}{\partial c_t}$ contains both current and future effects. This gives (12.4):

$$m_t^*=m_t^c\cdot\frac{\big(1+\theta\frac{c_{t+1}}{c_t}\big)^{-\gamma}+\beta\theta\,\mathbb E_{t+1}\big[\big(\frac{c_{t+2}}{c_t}+\theta\big)^{-\gamma}\big]}{\big(1+\theta\frac{c_t}{c_{t-1}}\big)^{-\gamma}+\beta\theta\,\mathbb E_t\big[\big(\frac{c_{t+1}}{c_{t-1}}+\theta\big)^{-\gamma}\big]}.\tag{12.4}$$

Note

Remark 12.1。 (12.3) 中 $\theta<0$ 是习惯 (habit) 模型；$\theta>0$ 则是耐用品 (durability) 模型。

Remark 12.2。 内部习惯（$\theta<0$）能提高 SDF 方差：习惯条件 $u'(0)=\infty$ 使 $s_t$ 始终远离 0，即 $c_t$ 永远不接近 $-\theta c_{t-1}$；对耐用品（$\theta>0$），$c_t$ 远离 $-\theta c_{t-1}$ 的限制使 $c_t$ 更波动。

Note

Remark 12.1. In (12.3), $\theta<0$ gives a habit model; $\theta>0$ gives a durability model.

Remark 12.2. Internal habit ($\theta<0$) raises the SDF variance: the habit condition $u'(0)=\infty$ keeps $s_t$ away from 0, i.e. $c_t$ never close to $-\theta c_{t-1}$; for durability ($\theta>0$), the restriction that $c_t$ stays away from $-\theta c_{t-1}$ makes $c_t$ more volatile.

12.2.2 Durability vs. Habit: Ferson and Constantinides (1991)

耐用品与习惯是时间不可分 (time non-separability) 的两种相反建模。记代表性代理人的消费流 $c_t^F=\sum_{s=0}^\infty\delta_s c_{t-s}$（$\delta_s\ge0$，$\sum_s\delta_s=1$）——过去消费支出的加权平均，刻画耐用品效应。有效消费再减去习惯参考点：

Durability and habit are two opposite ways to model time non-separability. Let the representative agent's consumption flow be $c_t^F=\sum_{s=0}^\infty\delta_s c_{t-s}$ ($\delta_s\ge0$, $\sum_s\delta_s=1$) — a weighted average of past consumption expenditures, capturing the durability effect. Effective consumption then subtracts the habit reference point:

$$C_t=c_t^F-h\sum_{s=1}^\infty\alpha_s c_{t-s},\qquad h\ge0,\ \alpha_s\ge0.$$

合并得有效消费为过去支出的加权和 (12.5)，其权重 $b_s$ 是耐用与习惯两种相反效应的净效应 (12.6)：

Combining, effective consumption is a weighted sum of past expenditures (12.5), whose weights $b_s$ are the net effect of the two opposing durability and habit forces (12.6):

$$C_t=\delta_0\sum_{s=0}^\infty b_s\,c_{t-s},\tag{12.5}$$

$$b_s=\Big[1-\frac{h(1-a)}{\delta-a}\Big]\delta^s+\frac{h(1-a)}{\delta-a}\,a^s,\qquad \delta_s=(1-\delta)\delta^s,\ \alpha_s=(1-a)a^{s-1}.\tag{12.6}$$

纯耐用品（$h=0$）：$b_s=\delta^s>0$。
纯习惯（$\delta=0$）：$b_s=-h(1-a)a^{s-1}<0$。
故 $b_s$ 的符号反映谁占主导：$b_s>0$ 耐用品主导，$b_s<0$ 习惯主导。

由欧拉方程（对 $c_t$ 求一阶条件）得 (12.7)，对每种 $\{b_s\}$ 结构估计：

Pure durability ($h=0$): $b_s=\delta^s>0$.
Pure habit ($\delta=0$): $b_s=-h(1-a)a^{s-1}<0$.
So the sign of $b_s$ reflects which dominates: $b_s>0$ durability, $b_s<0$ habit.

The Euler equation (first-order condition in $c_t$) gives (12.7), estimated for each structure of $\{b_s\}$:

$$\mathbb E_t\!\left[\sum_{\tau=0}^\infty\beta^\tau\Big(\frac{C_{t+\tau}}{C_t}\Big)^{-\gamma}\big(R_{t+1}b_{\tau-1}-b_\tau\big)\right]=1,\qquad b_{-1}\equiv0.\tag{12.7}$$

批评（实证结果）：

一期模型（$b_s=\delta^s$，$\tau\ge2$）：对某些工具变量耐用品主导（$b_1>0$），其余则习惯主导（$b_1<0$）。
两期模型（$b_s=0$，$\tau\ge3$）：$b_1$ 与 $b_2$ 无法分别估计。
模型拟合不了历史上的低无风险利率。
模型生成不了高股权溢价。
估计出的相对风险厌恶极大且不稳健。

Critiques (empirical results):

One-lag model ($b_s=\delta^s$ for $\tau\ge2$): durability dominates ($b_1>0$) for some instruments, while habit dominates ($b_1<0$) for the rest.
Two-lag model ($b_s=0$ for $\tau\ge3$): $b_1$ and $b_2$ cannot be separately estimated.
The model cannot fit the historical low risk-free rate.
The model cannot generate the high equity premium.
The estimated relative risk aversion is extremely large and non-robust.

References

Campbell, J. Y. and J. H. Cochrane (1999). By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior. Journal of Political Economy 107(2), 205–251.
Constantinides, G. M. (1990). Habit Formation: A Resolution of the Equity Premium Puzzle. Journal of Political Economy 98(3), 519–543.
Ferson, W. E. and G. M. Constantinides (1991). Habit Persistence and Durability in Aggregate Consumption: Empirical Tests. Journal of Financial Economics 29(2), 199–240.
Sundaresan, S. M. (1989). Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth. The Review of Financial Studies 2(1), 73–89.