Stern (1997) Simulation-Based Estimation

Summary

Journal of Economic Literature (JEL) survey (34 pages) covering the theory and practice of simulation-based estimation for econometric models where moment conditions or likelihood functions require evaluating intractable high-dimensional integrals. Stern organizes the survey around four estimation families (method of simulated moments (MSM), method of simulated likelihood (MSL), method of simulated scores (MSS), Markov chain Monte Carlo (MCMC)) and five simulator types, walking through stylized versions of five canonical model types: probit, multinomial probit, dynamic programming, market entry games, and unobserved heterogeneity.

Key Claims

• The generic problem is evaluating $Eh(U)=\int h(u)f(u)\,du$; the sample-mean simulator $(1/R)\sum_r h(u^r)$ is unbiased with variance $\to 0$ as $R\to\infty$.
• The frequency simulator (step function in $\theta$) is discontinuous and unsuitable for derivative-based optimization; better simulators — GHK, decomposition, importance sampling — are smooth, bounded in $(0,1)$, and differentiable.
• MSM: consistent at any fixed $R$ as $N\to\infty$; simulation inflates the asymptotic covariance matrix by $[1+1/R]$; inflation vanishes as $R\to\infty$.
• MSL: technically requires $R\to\infty$ as $N\to\infty$ for strict consistency (Jensen's inequality bias in $\log\hat{P}$); Börsch-Supan–Hajivassiliou (1993) show this is negligible in practice for MNP with good smooth simulators at small $R$.
• MSL vs MSM: MSL preferred for MNP-type problems — (a) maximum likelihood (ML) is more efficient than method of moments (MOM), and (b) MSL simulates only the chosen alternative's probability while MSM must simulate all $J$.
• GHK algorithm (Geweke-Hajivassiliou-Keane): best-performing MNP simulator in Monte Carlo studies; alternates between analytical truncated-normal cumulative distribution functions (CDFs) and draws from truncated normals; strictly bounded $(0,1)$; continuous and differentiable in $\theta$; importance-sampling interpretation.
• Antithetic acceleration: for symmetric $F$ and monotone $h$, reduces variance to order $N^{-1}$ of simulation loss, making standard-error correction unnecessary.
• MCMC (Gibbs sampling): described as one of four estimation families; characterized in 1997 as "very expensive relative to MSM and MSL" — a historically significant assessment that dated quickly with Kim-Shephard-Chib (KSC, 1998) and fast forward-filtering backward-sampling (FFBS) algorithms.
• Multinomial probit application (Börsch-Supan et al. 1992): 243-choice elderly living arrangement model; flexible error structures (AR(1) + random effects) dominate simple $\Omega=I$; small $R=3$ sufficient with GHK.
• Pakes (1986) patent renewal: stochastic DP, T-dimensional integral for reservation return sequence, feasible only via simulation.
• Berry (1992) entry game: $EN_i$ easy to simulate even though $N_i$ likelihood is not; MSM required because likelihood hard to simulate.
• Berkovec-Stern (1991) retirement: T=32 planning horizon, three-component error (person/job-type/time-specific); unobserved heterogeneity standard deviations dominate the independent time-specific error — would be excluded without simulation.

Concepts Introduced or Extended

• Simulation-Based Estimation
• Multinomial Probit
• Dynamic Discrete Choice Model
• Markov Chain Monte Carlo

Entities Mentioned

• Steven Stern
• Michael Keane

Quotes

"Bayesian econometrics has always been hampered by the large computation costs associated with evaluating posterior distributions... Simulation (and rapidly improving computer technology) knock down this computation hurdle."

"My experience (Stern forthcoming) suggests that Gibbs sampling methods are very expensive relative to MSM and MSL."

My Take

A well-organized methodological survey that situates simulation-based estimation within a unified $Eh(U)$ framework. Its primary value in this wiki is as a reference for GHK, the MSM/MSL taxonomy, and as a historical marker showing where MCMC stood in 1997 relative to classical simulation methods. The "MCMC is too expensive" remark is notable — by 2000, with KSC (1998) and FFBS, Bayesian MCMC had become the standard approach for the same class of problems. The conceptual clarity on the MSM/MSL distinction and simulator requirements (boundedness, smoothness, differentiability) remains useful framing.