Simulated Moments Estimation - Overview

Summary

Duffie & Singleton (1993) develop the Simulated Moments Estimator (SME): an extension of GMM to dynamic asset-pricing models whose moment restrictions have no analytic representation in terms of observable variables and the parameter vector. The estimator matches sample moments of the data, $f(Y_t,{\beta }_0)$, to sample moments of a simulated equilibrium series $f(Y_s^{\beta },\beta )$, choosing $\beta$ to minimize the distance. The paper’s contribution is the rigorous large-sample theory — consistency (weak and strong) and asymptotic normality — under conditions on the Markov state process that overcome two problems unique to simulation: the nonstationarity of the simulated series and the feedback dependence of the entire simulated path on $\beta$.

Overview

Many dynamic structural/asset-pricing models are characterized by a time-homogeneous Markov state vector $Y_t$ whose transition depends on an unknown parameter ${\beta }_0$. Asset prices are observed as $f(Y_t,{\beta }_0)$ for some function $f$. When the mapping $\beta \mapsto \mathbb{E}[f(Z_t,\beta )]$ is known analytically, Hansen’s (1982) GMM applies. But for a large class of asset-pricing models (e.g. with unobserved taste shocks, where Euler-equation GMM is infeasible) this expectation has no closed form. The SME replaces the population moment $\mathbb{E}[f(Z_t,\beta )]$ with a Monte-Carlo sample counterpart computed from a simulated equilibrium series.

This note is the entry point for the Duffie–Singleton (1993) paper; the formal estimator, its assumptions, and its asymptotics are developed in the linked notes.

Main Content

Research question

How can one consistently estimate, and characterize the asymptotic distribution of, the parameter ${\beta }_0$ of a time-series asset-pricing model when the model’s moment restrictions cannot be written analytically as a function of observables and $\beta$ — and must instead be approximated by simulating the model?

The estimator in one line

Match data moments to simulated moments:

$$b_T=\arg \underset{\beta \in \Theta }{\min }{G}_T(\beta {)}^{\prime }{W}_T{G}_T(\beta ),G_T(\beta )=\frac{1}{T}\sum _{t=1}^T{f}_t^{\ast }-\frac{1}{\mathcal{T}(T)}\sum _{s=1}^{\mathcal{T}(T)}{f}_s^{\beta },$$

where $f_t^{\ast }$ are observed-data moments and $f_s^{\beta }$ are moments of a simulated series of length $\mathcal{T}(T)$. See Simulated Moments Estimator Definition.

Two problems that simulation creates (and the paper’s answers)

Standard GMM/SME regularity conditions (Hansen 1982; Andrews 1987; Lee & Ingram 1991; McFadden 1989; Pakes & Pollard 1989) do not directly apply to estimation of time-series models by simulation, for two reasons:

1. Nonstationarity of the simulated series. Pre-sample values are needed to simulate a time series, but the model’s stationary distribution (as a function of $\beta$) is unknown, so the simulation is generally started off its ergodic distribution and is therefore nonstationary. → Answer: assume geometric ergodicity of the state process, so simulated processes are asymptotically stationary with an ergodic distribution independent of starting values. See Geometric Ergodicity and Uniform LLN.

2. Parameter feedback through the law of motion. Functions of the current simulated state depend on $\beta$ both through the model (as in any GMM problem) and indirectly through the entire simulated history, because $\beta$ enters the transition law. This compounding feedback breaks the first-moment-continuity conditions used by Hansen (1982)/Andrews (1987). → Answer: impose a damping condition on the feedback effect — the “Asymptotic Unit-Circle (AUC) condition” — so past shocks decay geometrically. See SME Consistency.

Structure of the paper / note map

Section	Content	Note
§2	Illustrative stochastic-growth asset-pricing model	Duffie-Singleton Asset-Pricing Model
§3	Formal definition of the SME	Simulated Moments Estimator Definition
§4.1–4.2	Geometric ergodicity + uniform weak LLN	Geometric Ergodicity and Uniform LLN
§4.3–4.5	Weak & strong consistency (Theorems 1–3)	SME Consistency
§5	Asymptotic normality (Theorem 4, Cor. 3.1)	SME Asymptotic Distribution
§6	Extensions: $\beta$-dependent moments, calculated moments, measurement error, option pricing	SME Extensions and Applications

Connections

• Extends Method of Simulated Moments / Hansen (1982) GMM from analytic to simulated moments, and from i.i.d. (McFadden 1989; Pakes & Pollard 1989) to time-series Markov settings (cf. Lee & Ingram 1991).
• The applied MSM/SMM workflow in Method of Simulated Moments and SMM Weighting Matrix and Inference is the practitioner’s version of this estimator; this paper supplies its time-series asymptotic theory.
• Contrast with Indirect Inference / Efficient Method of Moments, which match an auxiliary model rather than raw moments.

See Also

• Simulation-Based Estimation - Overview — the broader sub-topic overview (MSM vs. indirect inference vs. EMM)
• Duffie-Singleton Asset-Pricing Model — the motivating example
• Simulated Moments Estimator Definition — the formal estimator