Duffie-Singleton Asset-Pricing Model

Summary

The illustrative dynamic asset-pricing model Duffie & Singleton (1993) use to motivate the SME: an extended stochastic-growth economy (Brock 1980; Michener 1984) with a representative consumer facing a technology shock $z_t$ and an unobserved taste shock $u_t$. Because the taste shock is unobserved to the econometrician, Euler-equation GMM is infeasible, so equilibrium prices/quantities must be simulated. The model’s state $X_t=(z_t,u_t)$ is Markov; the augmented state $Y_t=(X_t^{\prime },k_t{)}^{\prime }$ (adding the capital stock) is also Markov and is what gets simulated.

Overview

Section 2 sets up a concrete economy that exhibits the econometric problems of simulation-based estimation: nonstationarity of the simulated series and parameter-dependence of the simulated path. It is “an informal backdrop” to the formal SME, and it recurs as the running example for the regularity conditions in Geometric Ergodicity and Uniform LLN and SME Consistency.

Main Content

Production and the firm

Production and firm's problem (D&S §2, Eqs. 2.1–2.2)

A single consumption commodity is produced by

$$F(k_t,z_t)=z_t{k}_t^{\varphi },0<\varphi <1,$$

where $k_t$ is the capital stock at date $t$ and $z_t$ is a technology shock. The firm rents capital at rate $r_t^k$ and pays profits as dividends $d_t$, solving each period the static problem

$$d_t=\arg \underset{k_t}{\max }\left\{z_t{k}_t^{\varphi }-r_t^k{k}_t\right\}.$$

In equilibrium this is equivalent to maximizing share market value.

Consumer, budget, and preferences

Consumer problem with taste shock (D&S §2, Eqs. 2.3–2.4)

Given share price $p_t$, the representative consumer faces the budget constraint

$$c_t+k_{t+1}+p_t{s}_{t+1}=(d_t+p_t)s_t+(r_t^k+\mu )k_t,$$

where $c_t$ is consumption, $s_t$ are share holdings, and $(1-\mu )$ is a constant capital depreciation rate. With an unobserved (to the econometrician) additively-separable taste shock $u_t$, the consumer maximizes

$$\underset{(c_t,k_t)}{\max }\mathbb{E}\left[\sum _{t=1}^{\infty }{\delta }^t\frac{(c_t-1{)}^{1-\alpha }}{1-\alpha }{u}_t\right],\alpha <0,$$

where $\alpha$ is the constant coefficient of relative risk aversion and $\delta \in (0,1)$ is the subjective discount factor.

Markov state and the parameter vector

State process and augmented state (D&S §2, Eqs. 2.5–2.6)

The exogenous state $X_t=(z_t,u_t)$ is Markov:

$$X_t=h(X_{t-1},{\epsilon }_t,{\rho }_0),$$

with $\{{\epsilon }_t\}$ a two-dimensional i.i.d. process, $h$ a transition function, and ${\rho }_0$ an unknown parameter sub-vector. The full unknown parameter is

$${\beta }_0=(\varphi ,\alpha ,{\rho }_0,\mu ,\delta {)}^{\prime }\in \Theta .$$

Solving the system (2.1)–(2.5) analytically or numerically yields the equilibrium transition function $H$ for the augmented state $Y_t=(X_t^{\prime },k_t{)}^{\prime }$:

$$Y_{t+1}=H(Y_t,{\epsilon }_{t+1},{\beta }_0).$$

$Y_t$ is the object that is simulated; see Simulated Moments Estimator Definition.

Why simulation is needed

Three reasons (D&S §2) motivate joint solution-and-estimation by simulation:

1. Goodness-of-fit. Solving for the stochastic equilibrium lets one assess fit directly via the joint distribution of asset returns, consumption, and capital.
2. Infeasible Euler-equation GMM. With unobserved taste shocks $u_t$, Hansen–Singleton (1982) Euler-equation estimation is not feasible.
3. Temporal aggregation. GMM with temporally aggregated data can be inconsistent (Hall 1988; Hansen–Singleton 1989), but aggregation is often accommodated by the SME.

Examples

Closed-form special case (D&S §4.1, Eqs. 4.3–4.5)

Take the special case $u_t=1$ for all $t$, $\mu =0$ (100% depreciation), and $\alpha =1$ (logarithmic utility), with technology shock

$$\ln z_{t+1}={\zeta }_z+\rho \ln z_t+{\epsilon }_{t+1}.$$

Then the implied equilibrium asset-pricing function and capital law of motion are (Michener 1984):

$$p_t=\frac{\delta }{(1-\delta )}(1-\varphi )z_t{k}_t^{\varphi },d_t=(1-\varphi )z_t{k}_t^{\varphi },k_{t+1}=\delta \varphi z_t{k}_t^{\varphi }.$$

If $\{{\epsilon }_t\}$ is i.i.d. normal, the resulting $\{Y_t\}$ satisfies the irreducibility/recurrence Condition B needed for geometric ergodicity — even though the capital stock $k_{t+1}$ given $X_t$ is degenerate (so the single-period “full support” condition fails, but the weaker Condition B holds).

Conditionally heteroskedastic shock that separates ergodicity from AUC (D&S §4.5, Eq. 4.11)

Let the technology shock follow

$$z_t=\xi +\rho z_{t-1}+\sigma {\nu }_{t-1}^{\gamma }{\epsilon }_t,\gamma <1,\sigma >0,|\rho |<1,$$

with ${\nu }_t=z_t$ if $z_t\ge \eta >0$ and ${\nu }_t=\eta$ otherwise. This process is geometrically ergodic (since $|\rho |<1$), so it obeys weak/strong LLNs — yet it can violate the Asymptotic Unit-Circle condition (the Lipschitz factor $\rho +\sigma \epsilon ({\nu }^{\gamma }-{\nu }^{\prime \gamma })/(z-z^{\prime })$ can exceed unity). This is the paper’s key counterexample showing that geometric ergodicity accommodates a strictly larger class of processes than the AUC condition used for strong consistency. See SME Consistency.

Connections

• Provides the concrete $H$, $\epsilon$, $f$ primitives that the abstract Simulated Moments Estimator Definition requires.
• The closed-form case is the test bed for the geometric-ergodicity conditions; the heteroskedastic case motivates the split between weak (ergodicity-based) and strong (AUC-based) consistency.
• Structurally a stochastic growth model — compare the Brock-Mirman SMM exercise, a related production economy estimated by SMM.

See Also

• Simulated Moments Estimation - Overview — where this model fits in the paper
• Simulated Moments Estimator Definition — the estimator built on $H,f$
• Geometric Ergodicity and Uniform LLN — Condition B applied to this model