Simulated Moments Estimator Definition

Summary

The formal Section-3 definition of the SME. From primitives — a transition function $H$ and observation function $f$ — one generates the actual state process $\{Y_t\}$ (driven by shocks ${\epsilon }_t$ with true parameter ${\beta }_0$) and, using an independent shock sequence $\{{\hat{\epsilon }}_t\}$ identical in distribution, a simulated state process $\{Y_t^{\beta }\}$ for any candidate $\beta$. The SME $b_T$ minimizes a quadratic form in the difference $G_T(\beta )$ between the sample mean of the observed moments $f_t^{\ast }$ and the sample mean of the simulated moments $f_s^{\beta }$. It generalizes GMM by replacing the population moment $\mathbb{E}[f(Z_t,\beta )]$ — needed in closed form for GMM — with its simulation-based sample counterpart.

Overview

This note states the estimator precisely. It is the formal core that the consistency and asymptotic-distribution notes build on. The crucial conceptual point is that the simulated series is built from a fresh shock sequence independent of the data’s shocks, so the simulated and observed moments are independent — a fact that drives the $(1+\tau )$ variance inflation in SME Asymptotic Distribution.

Main Content

Primitives (D&S §3)

• A measurable transition function $H:{\mathbb{R}}^N\times {\mathbb{R}}^p\times \Theta \to {\mathbb{R}}^N$, with compact parameter set $\Theta \subset {\mathbb{R}}^Q$, for positive integers $N,p,Q$.
• A measurable observation function $f:{\mathbb{R}}^{Nl}\times \Theta \to {\mathbb{R}}^M$, for positive integers $l$ and $M$, with $M\ge Q$ (at least as many moments as parameters).

Actual state and observation processes (D&S §3, Eq. 3.1)

The ${\mathbb{R}}^N$-valued state process $\{Y_t{\}}_{t=1}^{\infty }$ is generated by

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

where ${\beta }_0$ is to be estimated and $\{{\epsilon }_t\}$ is an i.i.d. ${\mathbb{R}}^p$-valued sequence. With $Z_t=(Y_t,Y_{t-1},\dots ,Y_{t-l+1})$, the observed moments are $f_t^{\ast }\equiv f(Z_t,{\beta }_0)$.

Simulated state and observation processes (D&S §3, Eq. 3.3)

The econometrician has access to an ${\mathbb{R}}^p$-valued sequence $\{{\hat{\epsilon }}_t\}$ that is identical in distribution to, and independent of, $\{{\epsilon }_t\}$. For any initial point ${\hat{Y}}_1$ and any $\beta \in \Theta$, the simulated state process is built inductively with $Y_1^{\beta }={\hat{Y}}_1$ and

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

The simulated observation process is $f_t^{\beta }=f(Z_t^{\beta },\beta )$ with $Z_t^{\beta }=(Y_t^{\beta },\dots ,Y_{t-l+1}^{\beta })$.

Moment difference and the SME (D&S §3, Eqs. 3.4–3.5)

Let $\mathcal{T}(T)$ be the simulation sample size for actual sample size $T$, with $\mathcal{T}(T)\to \infty$ as $T\to \infty$. Define the difference in sample moments

$$G_T(\beta )=\frac{1}{T}\sum _{t=1}^T{f}_t^{\ast }-\frac{1}{\mathcal{T}(T)}\sum _{s=1}^{\mathcal{T}(T)}{f}_s^{\beta }.$$

If $\{f_t^{\ast }\}$ and $\{f_s^{\beta }\}$ obey a law of large numbers, then ${\lim }_T{G}_T(\beta )=0$ iff $\beta ={\beta }_0$ (with identification). Introducing a sequence $W=\{W_T\}$ of $M\times M$ positive semi-definite distance matrices (rank $\ge Q$), the SME for ${\beta }_0$ given $(H,\epsilon ,\mathcal{T},{\hat{Y}}_1,W)$ is

$$b_T=\arg \underset{\beta \in \Theta }{\min }{G}_T(\beta {)}^{\prime }{W}_T{G}_T(\beta )\equiv \arg \underset{\beta \in \Theta }{\min }{C}_T(\beta ).$$

Relation to GMM

GMM as the analytic-moment special case (D&S §3, Eq. 3.2)

When $\beta \mapsto \mathbb{E}[f(Z_t,\beta )]$ is known and independent of $t$, the GMM estimator (Hansen 1982) is

$$b_T=\arg \underset{\beta \in \Theta }{\min }{\left[\frac{1}{T}\sum _{t=1}^T{f}_t^{\ast }-\mathbb{E}[f(Z_t,\beta )]\right]}^{\prime }{W}_T\left[\frac{1}{T}\sum _{t=1}^T{f}_t^{\ast }-\mathbb{E}[f(Z_t,\beta )]\right].$$

The SME replaces the population moment $\mathbb{E}[f(Z_t,\beta )]$ with its simulated sample counterpart $\frac{1}{\mathcal{T}}{\sum }_s{f}_s^{\beta }$, which can be computed for a large class of asset-pricing models where the analytic form is unavailable.

Connections

• Instantiated by the Duffie-Singleton Asset-Pricing Model (its $H$ is the equilibrium transition; its $f$ extracts prices/consumption moments).
• The applied counterparts of $G_T$, $W_T$, and $C_T$ appear in Method of Simulated Moments and SMM Weighting Matrix and Inference; the optimal weight is $W_0={\Sigma }_0^{-1}$ (see SME Asymptotic Distribution).
• The independence of $\{{\hat{\epsilon }}_t\}$ from $\{{\epsilon }_t\}$ is exactly what makes the two terms of $G_T$ asymptotically independent in SME Asymptotic Distribution.

See Also

• Simulated Moments Estimation - Overview — narrative motivation
• SME Consistency — when $b_T\to {\beta }_0$
• SME Asymptotic Distribution — the limiting distribution of $\sqrt{T}(b_T-{\beta }_0)$
• SME Extensions and Applications — letting $f_t^{\ast }$ itself depend on $\beta$