SME Asymptotic Distribution

Summary

The limiting distribution of the SME. With $T/\mathcal{T}(T)\to \tau$, Theorem 4 gives $\sqrt{T}{G}_T({\beta }_0)\Rightarrow N[0,{\Sigma }_0(1+\tau )]$ — the moment gap is asymptotically normal with covariance inflated by the factor $(1+\tau )$ from independent simulation noise. Corollary 3.1 then gives $\sqrt{T}(b_T-{\beta }_0)\Rightarrow N[0,\Lambda ]$ with $\Lambda =(1+\tau )(D_0^{\prime }{\Sigma }_0^{-1}{D}_0{)}^{-1}$. As the simulation size grows relative to the data ($\tau \to 0$), the simulation penalty vanishes and the SME attains the efficiency of the analytic-moment GMM estimator $[D_0^{\prime }{\Sigma }_0^{-1}{D}_0{]}^{-1}$ — so simulation can be “almost free.”

Overview

Under the unit-circle conditions of SME Consistency, the stationary ergodic process $\{Y_t^{\infty \beta }\}$ replaces the nonstationary simulated $\{Y_t^{\beta }\}$, and asymptotic normality follows from suitably modified Hansen (1982) arguments via an intermediate-value (delta-method) expansion of $G_T$ around ${\beta }_0$. The headline is that simulation costs only a $(1+\tau )$ variance inflation, controllable by simulating a long series.

Main Content

Assumptions 6–7 (D&S §5)

• Assumption 6: (i) ${\beta }_0$ and $\{b_T\}$ are interior to $\Theta$; (ii) $f_t^{\beta }$ is continuously differentiable in $\beta$ for all $t$, $\omega$ by $\omega$; (iii) the Jacobian $D_0=\mathbb{E}[\partial f_{\infty }^{{\beta }_0}/\partial \beta ]$ exists, is finite, and has full rank (the identification/rank condition).
• Assumption 7: the family $\{D_{\beta }{f}_t^{\beta }\}$ is Lipschitz, uniformly in probability; $\mathbb{E}(|D_{\beta }{f}_{\infty }^{\beta }|)<\infty$ for all $\beta$; and $\beta \mapsto \mathbb{E}(D_{\beta }{f}_{\infty }^{\beta })$ is continuous. (Here $D_{\beta }{f}_t^{\beta }=(d/d\beta )f(Z_t^{\beta },\beta )$ is the total derivative.)

Expansion. Expanding $G_T(b_T)$ about ${\beta }_0$ (Eq. 5.1) and applying the first-order conditions $[\partial G_T(b_T)/\partial \beta {]}^{\prime }{W}_T{G}_T(b_T)=0$ yields (Eq. 5.2) a solvable system once $J_T=[\partial G_T(b_T)/\partial \beta {]}^{\prime }{W}_T\partial G^{\ast }(T)$ is invertible for large $T$; under Assumptions 5–7, ${\mathrm{plim}}_T\partial G_T(b_T)/\partial \beta =D_0$. Hence the asymptotic distribution of $\sqrt{T}(b_T-{\beta }_0)$ equals that of $(D_0^{\prime }{\Sigma }_0^{-1}{D}_0{)}^{-1}\sqrt{T}{G}_T({\beta }_0)$.

Theorem 4 (Asymptotic normality of the moment gap) (D&S §5, Eq. 5.3)

Suppose $T/\mathcal{T}(T)\to \tau$ as $T\to \infty$. Under Assumptions 1–4 and 6–7,

$$\sqrt{T}{G}_T({\beta }_0)\Rightarrow N\left[0,{\Sigma }_0(1+\tau )\right].$$

Mechanism (Eq. 5.4): $\sqrt{T}{G}_T({\beta }_0)$ splits into a data term $\frac{1}{\sqrt{T}}{\sum }_t[f_t^{\ast }-\mathbb{E}(f_{\infty }^{\ast })]$ and a simulation term scaled by $\sqrt{T/\mathcal{T}(T)}\to \sqrt{\tau }$. Each is asymptotically normal by Doob’s (1953) CLT for geometrically ergodic processes (using the $\Vert f_t^{\beta }{\Vert }_{2+\delta }$ bounds), and the two are independent because the simulation shocks $\{\hat{\epsilon }\}$ are independent of the data shocks $\{\epsilon \}$ — giving the variance ${\Sigma }_0+\tau {\Sigma }_0={\Sigma }_0(1+\tau )$.

Corollary 3.1 (Asymptotic distribution of the SME) (D&S §5, Eq. 5.5)

Under the assumptions of Theorem 4, with optimal weighting $W_0={\Sigma }_0^{-1}$,

$$\sqrt{T}(b_T-{\beta }_0)\Rightarrow N[0,\Lambda ],\Lambda =(1+\tau ){\left(D_0^{\prime }{\Sigma }_0^{-1}{D}_0\right)}^{-1}.$$

Interpretation: simulation is (almost) free

• The factor $(1+\tau )$ is the price of simulation. As $\tau \to 0$ (simulated sample size $\mathcal{T}(T)$ large relative to data size $T$), $\Lambda \to (D_0^{\prime }{\Sigma }_0^{-1}{D}_0{)}^{-1}$ — exactly the covariance of the analytic-moment GMM estimator that would require knowing $\mathbb{E}(f_{\infty }^{\beta })$ in closed form (cf. McFadden 1989; Pakes & Pollard 1989; Lee & Ingram 1991).
• Thus the SME extends the class of Markov models estimable by method-of-moments with potentially negligible loss of efficiency — one simply simulates a long series.
• ${\Sigma }_0$ depends only on the data moments $\{f_t^{\ast }\}$, so it is estimated by HAC without simulation; alternatively it can be estimated from simulated data, advantageous since $\mathcal{T}(T)$ is controllable.

Checking the rank/identification condition

Assumption 6(iii) (full-rank $D_0$) is the identification condition. When the model is solved numerically it can be hard to tell whether moment choices identify the parameters. Duffie & Singleton recommend examining sensitivity to moment choice, and note the partial-derivative matrix can be computed numerically from the simulated state:

$$D(\beta )=\frac{\partial \left[\frac{1}{\mathcal{T}}{\sum }_{t=1}^{\mathcal{T}}{f}_t^{\beta }\right]}{\partial \beta }\text{\hspace{1em}(5.6)}$$

For large $\mathcal{T}$, $D(\beta )\approx \partial \mathbb{E}(f_{\infty }^{\beta })/\partial \beta$; an orthogonalization of $D(\beta )$ reveals whether the first-order conditions form an ill-conditioned (near-underidentified) system at points in $\Theta$, including at the SME.

Connections

• Completes the consistency story with the limiting distribution; the optimal weight $W_0={\Sigma }_0^{-1}$ matches the efficient-GMM weighting in SMM Weighting Matrix and Inference and Method of Simulated Moments.
• The $(1+\tau )$ / $(1+1/R)$ variance-inflation theme recurs throughout simulation estimation — see Method of Simulated Moments and Simulation-Based Estimation - Overview.
• The covariance $\Lambda$ generalizes in SME Extensions and Applications to ${\Lambda }_{f,g,\tau }=(D_0^{\prime }{\Sigma }_{f,g,\tau }^{-1}{D}_0{)}^{-1}$ when observation functions depend on $\beta$.

See Also

• SME Consistency — assumptions and the consistency theorems this extends
• SME Extensions and Applications — efficiency gains from mixing calculated and simulated moments
• Simulated Moments Estimator Definition — definitions of $G_T$, ${\Sigma }_0$, $D_0$