Indirect Inference

Summary

Indirect inference estimates structural parameters $\theta$ by matching properties of an auxiliary model — a possibly misspecified but tractable model — between observed and simulated data. The structural model is too complex for direct estimation, but the auxiliary model can be estimated by quasi-MLE. The key idea: find $\theta$ such that simulated data from the structural model “looks like” real data through the lens of the auxiliary model. Two formulations exist: minimum distance (match auxiliary parameter estimates) and score-based (match auxiliary model scores). Both are consistent and asymptotically equivalent, but differ computationally.

Overview

The MSM approach requires the researcher to specify moment conditions, which may not fully capture the model’s features. Indirect inference offers an alternative: instead of choosing moments directly, the researcher chooses an auxiliary model whose parameters implicitly define a rich set of moment conditions.

The approach was proposed by Gouriéroux, Monfort, and Renault (1993) for the minimum distance formulation, and by Gallant and Tauchen (1996a) for the score-based formulation (which they termed “EMM” when combined with the SNP auxiliary model — see Efficient Method of Moments).

Setup

Let ${\mathcal{M}}^{\ast }=\{h^{\ast }(y_t|z_t;\lambda ),\lambda \in \Lambda \}$ denote the auxiliary model, where $\lambda$ is a $q$-dimensional parameter vector with $q\ge p$ (at least as many auxiliary parameters as structural parameters).

The auxiliary model is generally misspecified: there is no ${\lambda }^{\ast }$ such that $h^{\ast }(y_t|z_t;{\lambda }^{\ast })=h(y_t|z_t;{\theta }_0)$ exactly. However, the quasi-MLE of $\lambda$ is well-defined:

$${\tilde{\lambda }}_T=\arg \underset{\lambda }{\max }Q(Y,X;\lambda )=\arg \underset{\lambda }{\max }{T}^{-1}\sum _{t=1}^T\log h^{\ast }(y_t|z_t;\lambda )$$

Definition: Binding Function

The binding function $b(\theta )$ links the structural parameters $\theta$ to the auxiliary parameters $\lambda$. It is defined as the solution to:

$$E_{\theta }[g(Y,X;b(\theta ))]=0$$

where $g(Y,X;\lambda )=\frac{\partial Q(Y,X;\lambda )}{\partial \lambda }$ is the score vector of the auxiliary model and the expectation is taken with respect to the joint distribution $h(Y,X|\theta )$ implied by the structural model.

The QML estimate ${\tilde{\lambda }}_T$ converges in probability to the pseudo-true value ${\lambda }_0=b({\theta }_0)$.

Minimum Distance Formulation

Since the binding function $b(\theta )$ is generally unknown, it must be approximated by simulation.

Definition: Simulated Binding Function

Generate $R$ simulated paths $y_1^{(r)}(\theta ),\dots ,y_T^{(r)}(\theta )$ from the structural model. For each path, estimate the auxiliary model parameters:

$${\tilde{\lambda }}_T^{(r)}(\theta )=\arg \underset{\lambda }{\max }{T}^{-1}\sum _{t=1}^T\log h^{\ast }[y_t^{(r)}(\theta )|z_t^{(r)}(\theta );\lambda ]$$

The simulated binding function is the average:

$${\hat{b}}_R(\theta )=\frac{1}{R}\sum _{r=1}^R{\tilde{\lambda }}_T^{(r)}(\theta )$$

Definition: Minimum Distance Indirect Inference Estimator

The minimum distance indirect inference estimator is:

$${\hat{\theta }}_{MD}^R=\arg \underset{\theta }{\min }{\left[{\tilde{\lambda }}_T-{\hat{b}}_R(\theta )\right]}^{\prime }A\left[{\tilde{\lambda }}_T-{\hat{b}}_R(\theta )\right]$$

where $A$ is a positive definite weight matrix. The estimator finds the structural parameter $\theta$ for which the simulated auxiliary estimates ${\hat{b}}_R(\theta )$ are as close as possible to the real-data estimates ${\tilde{\lambda }}_T$.

Score-Based Formulation

Definition: Score-Based Indirect Inference Estimator

The score-based formulation, suggested by Gallant and Tauchen (1996a), uses the moment conditions implied by the scores of the auxiliary model:

$$Eg[Y,X;b({\theta }_0)]=0$$

Using path simulations to approximate $E_{\theta }g$, the estimator is:

$${\hat{\theta }}_{GT}^R=\arg \underset{\theta }{\min }{\hat{g}}_R(\theta ,{\tilde{\lambda }}_T{)}^{\prime }A{\hat{g}}_R(\theta ,{\tilde{\lambda }}_T)$$

where:

$${\hat{g}}_R(\theta ,{\tilde{\lambda }}_T)=\frac{1}{R}\sum _{r=1}^R\frac{1}{T}\sum _{t=1}^T\frac{\partial \log h^{\ast }[y_t^{(r)}(\theta )|z_t^{(r)}(\theta );{\tilde{\lambda }}_T]}{\partial \lambda }$$

The estimator searches for $\theta$ such that the simulated scores, evaluated at the real-data QML estimate ${\tilde{\lambda }}_T$, are close to zero.

Asymptotic Theory

Theorem: Consistency and Asymptotic Normality of Indirect Inference

Both the minimum distance and score-based indirect inference estimators are consistent as $T\to \infty$ for any fixed $R\ge 1$, and asymptotically normal:

$$T^{1/2}({\hat{\theta }}^R-{\theta }_0)\stackrel{d}{\to }N(0,\text{avar}({\hat{\theta }}^R))$$

Both approaches yield asymptotically equivalent estimators (Gouriéroux, Monfort, and Renault, 1993).

Theorem: Optimal Weight Matrix for Minimum Distance

For the minimum distance estimator ${\hat{\theta }}_{MD}^R$, the asymptotic optimal weight matrix is:

$$A_0=J_0{I}_0^{-1}{J}_0$$

where:

$$J_0=\underset{T\to \infty }{\lim }E\left\{\frac{{\partial }^{2}Q(Y,X;{\lambda }_0)}{\partial \lambda \partial {\lambda }^{\prime }}\right\},I_0=\underset{T\to \infty }{\lim }\text{var}\left\{\sqrt{T}g(Y,X;{\lambda }_0)-E[\sqrt{T}g(Y,X;{\lambda }_0)|X]\right\}$$

The asymptotic variance is:

$$\text{avar}({\hat{\theta }}_{MD}^R)=\left(1+\frac{1}{R}\right)[B^{\prime }{A}_{0}B{]}^{-1}$$

where $B=\partial b({\theta }_0)/\partial {\theta }^{\prime }$.

Theorem: Optimal Weight Matrix for Score-Based Estimator

For the score-based estimator ${\hat{\theta }}_{GT}^R$, the asymptotic optimal weight matrix is:

$$A_0=I_0^{-1}$$

The asymptotic variance is:

$$\text{avar}({\hat{\theta }}_{GT}^R)=\left(1+\frac{1}{R}\right)[B^{\prime }{A}_{0}B{]}^{-1}$$

Computational Comparison

Feature	Minimum Distance (${\hat{\theta }}_{MD}^R$)	Score-Based (${\hat{\theta }}_{GT}^R$)
Each iteration of $\theta$	Requires $R$ nested optimizations to estimate ${\tilde{\lambda }}_T^{(r)}(\theta )$	Requires only one optimization for ${\tilde{\lambda }}_T$ (done once)
Optimal $A_0$	Requires $J_0$ (Hessian estimate)	Only requires $I_0$ (variance estimate)
Score vector availability	Not required	Must have analytical score $g(Y,X;\lambda )$
Overall cost	Higher (nested optimization)	Lower

Efficiency and the Choice of Auxiliary Model

The efficiency of indirect inference depends critically on the auxiliary model’s ability to approximate the structural model:

Smoothly Embedded Condition

If $h(y_t|z_t;{\theta }_0)=h^{\ast }(y_t|z_t;b({\theta }_0))$ in some neighborhood of ${\theta }_0$ — i.e., the structural model is smoothly embedded within the auxiliary model — then the indirect inference estimator is asymptotically efficient (equal to MLE).

Two strategies for choosing the auxiliary model:

1. Simple, close auxiliary model: Choose a tractable model that captures the salient features of the structural model. For example, use a GARCH model as auxiliary for an SV structural model, since both capture volatility clustering.

2. Data-dependent flexible model (SNP): Use the semi-nonparametric density of Gallant and Nychka (1987), increasing its flexibility with sample size. This is the Efficient Method of Moments approach.

Connections

• Builds on Method of Simulated Moments — both are simulation-based, but indirect inference uses an auxiliary model rather than direct moment conditions
• Leads to Efficient Method of Moments when combined with SNP auxiliary models
• The binding function concept relates to simulation-based estimation more broadly
• Practical Issues in Simulation Estimation covers variance reduction and auxiliary model selection

See Also

• Simulation-Based Estimation - Overview — comparison of all methods
• Method of Simulated Moments — direct moment-matching approach
• Efficient Method of Moments — data-driven auxiliary model (SNP)
• Practical Issues in Simulation Estimation — implementation guidance

Sources

• tdb136.pdf — Liesenfeld & Breitung (1998), Section 4
• Gouriéroux, C., A. Monfort, and E. Renault (1993), “Indirect Inference,” Journal of Applied Econometrics 8, S85-S118
• Gallant, A.R. and G.E. Tauchen (1996a), “Which Moments to Match?,” Econometric Theory 12, 657-681
• Smith, A.A. (1993), “Estimating Nonlinear Time-Series Models Using Simulated Vector Autoregressions,” Journal of Applied Econometrics 8, S63-S84