SME Extensions and Applications

Summary

Section 6 extends the SME to let the observation function depend on $\beta$ ($g^{\beta }$), which broadens applicability to many asset-pricing problems. Two practically important consequences: (1) mixing calculated and simulated moments — using a known analytic moment $h_j(\beta )$ for any coordinate where it is available — strictly increases efficiency over simulating all moments; (2) measurement error in observed states is accommodated by adding a mean-zero error to the observation. A leading application is option pricing, where the European option price is a conditional expectation that may be infeasible to simulate directly but feasible to compute via its analytic/conditional form.

Overview

The baseline SME assumes the observation function $f$ does not depend on $\beta$ for the actual data. Section 6 relaxes this, replacing $f_t^{\ast }$ with $g_t^{{\beta }_0}=g(Z_t,{\beta }_0)$ and matching it to a simulated $g_s^{\beta }$. This single generalization yields several useful special cases and a corresponding adjustment to the asymptotic covariance.

Main Content

$\beta$-dependent observation function

Extended SME with $g^{\beta }$ (D&S §6, Eqs. 6.1–6.4)

Add a measurable observation function $g:{\mathbb{R}}^{NL}\times \Theta \to {\mathbb{R}}^M$ (with $L$ states entering, WLOG $L=l$). Replace $f_t^{\ast }$ by $g_t^{{\beta }_0}=g[(Y_t,\dots ,Y_{t-L+1}),{\beta }_0]$, assume $\mathbb{E}[g^{{\beta }_0}-f^{{\beta }_0}]=0$, and consider the moment difference

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

The extended SME minimizes $C_T(\beta )=G_T(\beta {)}^{\prime }{W}_T{G}_T(\beta )$ as in (3.5). The relevant long-run covariance becomes the weighted matrix

$${\Sigma }_{f,g,\tau }=\tau {\Sigma }_0+{\Sigma }_1,{\Sigma }_1=\sum _{j=-\infty }^{\infty }\mathbb{E}\left(\left[g_t^{{\beta }_0}-\mathbb{E}(g_t^{{\beta }_0})\right]{\left[g_{t-j}^{{\beta }_0}-\mathbb{E}(g_t^{{\beta }_0})\right]}^{\prime }\right),$$

and (with $W_T\to {\Sigma }_{f,g,\tau }^{-1}$ and full-rank $D_0=\mathbb{E}[\partial g_t^{{\beta }_0}/\partial \beta -\partial f_{\infty }^{{\beta }_0}/\partial \beta ]$) the SME satisfies

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

The new rank condition on $D_0$ rules out trivial underidentification (e.g. multiplicative representations $g(z,{\beta }^1)\psi (z,{\beta }^1)$ vs. $f(z,{\beta }^2)\psi (z,{\beta }^2)$ with ${\beta }^1\ne {\beta }^2$). Consistent estimation of ${\Lambda }_{f,g,\tau }$ typically requires two steps, using both simulated and observed data.

Three uses of the extension

Mixing calculated and simulated moments (efficiency gain) (D&S §6)

If a coordinate function $g_j$ has a known analytic form $h_j(\beta )=\mathbb{E}[g_j(Z_{\infty },\beta )]$, set $f_j(z,\beta )=h_j(\beta )$ for all $z$ — i.e. use the calculated moment for that coordinate and simulated moments for the rest. Substituting calculated for simulated moments improves precision: the covariance ${\Lambda }_{f,g,\tau }$ is smaller than the all-simulated covariance $\Lambda$, because simulation noise is removed from those coordinates.

Measurement error (D&S §6)

Errors in measuring the observed state are accommodated by $g_t^{{\beta }_0}=f(Z_t,{\beta }_0)+u_t$, where $\{u_t\}$ is an ergodic, mean-zero ${\mathbb{R}}^M$-valued measurement error. Asymptotic efficiency is increased by ignoring the measurement error in simulation and comparing sample moments of the simulated $\{f(Z_t^{\beta },\beta )\}$ with the noisily-measured $\{g_t^{\beta }\}$.

Option pricing via conditional expectations (D&S §6)

A coordinate may take the conditional-expectation form

$$g_j[(Y_t,\dots ,Y_{t-l+1}),\beta ]=\mathbb{E}\left[h_j(Y_{t+1},\dots ,Y_{t+l+2},\beta )\mid Y_t,\dots ,Y_{t-l+1}\right].$$

Directly simulating $g_j(Z_t^{\beta },\beta )$ may be infeasible, but by the law of iterated expectations the feasible observation $f_j(Z_t^{\beta },\beta )=h_j(Z_t^{\beta },\beta )$ has the same mean as $g_j(Z_t^{\beta },\beta )$, so $h_j$ can be used instead. The leading illustration is the European option price: the price $g_j(Z_t^{\beta },\beta )$ is the conditional expectation of the option’s discounted payoff at maturity, which can be matched using the realized discounted payoff $h_j$.

Connections

• Generalizes the covariance $\Lambda$ of SME Asymptotic Distribution to ${\Lambda }_{f,g,\tau }$; the calculated-moment efficiency gain is the analytic-vs-simulated trade-off seen in Method of Simulated Moments.
• The option-pricing/conditional-expectation device connects the SME to derivative-pricing applications and to auxiliary-model ideas where intractable objects are matched in mean.
• Measurement-error handling parallels errors-in-variables treatments in SMM inference.

See Also

• SME Asymptotic Distribution — the base covariance $\Lambda$ this generalizes
• Simulated Moments Estimator Definition — the estimator being extended
• Simulated Moments Estimation - Overview — paper summary and section map