Index: Simulation-Based Estimation

Simulation-Based Estimation

Routing Summary

This folder covers simulation-based estimation methods — the general framework for estimating structural models when moment conditions or likelihoods are intractable. Contains 15 notes from Liesenfeld & Breitung (1998), Evans (2024) Ch. 19, and Duffie & Singleton (1993) Econometrica (the foundational time-series SME theory).

• Need the high-level overview (MSM vs. indirect inference vs. EMM)? → Simulation-Based Estimation - Overview
• Need the rigorous time-series SME theory (Duffie–Singleton 1993: consistency, asymptotic normality for Markov asset-pricing models)? → Simulated Moments Estimation - Overview
• Need the formal SME definition (primitives $H,f$; simulated state; $G_T$; estimator)? → Simulated Moments Estimator Definition
• Need geometric ergodicity / uniform LLN conditions for simulated series? → Geometric Ergodicity and Uniform LLN
• Need SME weak/strong consistency + the AUC unit-circle condition? → SME Consistency
• Need the SME limiting distribution + the $(1+\tau )$ simulation-noise inflation? → SME Asymptotic Distribution
• Need SME extensions ($\beta$-dependent moments, calculated-vs-simulated efficiency, measurement error, option pricing)? → SME Extensions and Applications
• Need MSM/SMM theory (criterion function, consistency, asymptotic normality)? → Method of Simulated Moments
• Need indirect inference (auxiliary model as moments)? → Indirect Inference
• Need asymptotically efficient simulation estimation (SNP/score-based)? → Efficient Method of Moments
• Need variance reduction, step sizes, common random numbers? → Practical Issues in Simulation Estimation
• Need weighting matrix strategies (identity, two-step, Newey-West) and Σ̂? → SMM Weighting Matrix and Inference
• Need Python code (scipy workflow, eps fix, Jacobian)? → SMM Python Implementation
• Need the Brock-Mirman (1972) structural macro exercise (latent TFP, policy function, 6 moments, 4 params)? → Brock-Mirman Model - SMM Estimation Exercise

Concept Map

Concept	Note	Type	Depends On	Key Result
MSM vs. indirect inference vs. EMM — overview	Simulation-Based Estimation - Overview	overview	Standard Errors and Clustering, Copula Estimation	Replace intractable criteria with Monte Carlo; variance inflated by $(1+1/R)$
MSM criterion function, consistency, asymptotic normality	Method of Simulated Moments	concept/theorem	Simulation-Based Estimation - Overview	Consistent for any $R\ge 1$; variance inflated by $(1+1/R)$
Auxiliary model approach: min-distance and score-based	Indirect Inference	concept/theorem	Method of Simulated Moments	Match auxiliary model estimates between real and simulated data
SNP density, EMM procedure, asymptotic efficiency	Efficient Method of Moments	concept/theorem	Indirect Inference	Achieves MLE efficiency via flexible SNP auxiliary model
Common RNGs, variance reduction, step sizes	Practical Issues in Simulation Estimation	concept	Method of Simulated Moments, Indirect Inference	Step size must be $\gg 1/\sqrt{T}$; never use software defaults
Identity W, two-step W, Newey-West W, Σ̂_SMM via Jacobian	SMM Weighting Matrix and Inference	concept/theorem	Method of Simulated Moments, Standard Errors and Clustering	Optimal W = Ω̂⁻¹; Σ̂ = (1/S)[dᵀWd]⁻¹
Python workflow: scipy, eps stepsize fix, numerical Jacobian	SMM Python Implementation	tutorial	SMM Weighting Matrix and Inference, Method of Simulated Moments, Practical Issues in Simulation Estimation	L-BFGS-B needs options={'eps': 1.0} when moments are in the 100s
Brock-Mirman (1972) model: system, policy function, 6-moment SMM exercise	Brock-Mirman Model - SMM Estimation Exercise	example	SMM Python Implementation, SMM Weighting Matrix and Inference	Latent TFP motivates SMM; policy function $k_{t+1}=\alpha \beta e^{z_t}{k}_t^{\alpha }$ enables efficient simulation
SME (Duffie–Singleton 1993): paper overview, 2 simulation challenges	Simulated Moments Estimation - Overview	overview	Method of Simulated Moments	Simulate moments when $\mathbb{E}[f(Z_t,\beta )]$ has no closed form; handles nonstationarity + parameter feedback
Stochastic-growth asset-pricing model (Brock/Michener + taste shock)	Duffie-Singleton Asset-Pricing Model	example	Simulated Moments Estimation - Overview	Unobserved taste shock makes Euler-GMM infeasible → simulate; closed-form $k_{t+1}=\delta \varphi z_t{k}_t^{\varphi }$
Formal SME definition ($H,f$; $G_T$; $b_T$)	Simulated Moments Estimator Definition	definition	Duffie-Singleton Asset-Pricing Model, Method of Simulated Moments	$b_T=\arg \min G_T(\beta {)}^{\prime }{W}_T{G}_T(\beta )$; replaces $\mathbb{E}[f]$ with simulated mean
Geometric ergodicity, Condition B, uniform weak LLN	Geometric Ergodicity and Uniform LLN	theorem	Simulated Moments Estimator Definition	$\rho$-ergodicity (Lemma 1, Mokkadem) + uniform weak LLN (Lemma 2)
SME weak/strong consistency, AUC & $L^2$-UC conditions	SME Consistency	theorem	Geometric Ergodicity and Uniform LLN	Thm 1 (weak, ergodicity); Thms 2–3 (strong, AUC damping)
SME asymptotic normality, simulation-noise inflation	SME Asymptotic Distribution	theorem	SME Consistency	$\sqrt{T}(b_T-{\beta }_0)\Rightarrow N[0,(1+\tau )(D_0^{\prime }{\Sigma }_0^{-1}{D}_0{)}^{-1}]$
SME extensions: $g^{\beta }$, calculated moments, measurement error, options	SME Extensions and Applications	concept	SME Asymptotic Distribution	Calculated moments cut variance; option price via iterated-expectations

Notes

• Simulation-Based Estimation - Overview — CONTAINS: MSM vs. indirect inference vs. EMM comparison, SV and diffusion motivating examples, $(1+1/R)$ variance structure
• Method of Simulated Moments — CONTAINS: MSM criterion function, conditional vs. unconditional moments, consistency theorem, asymptotic normality, optimal weight matrix, SV model example
• Indirect Inference — CONTAINS: Binding function, minimum distance estimator, score-based estimator, auxiliary model choice, smoothly embedded condition
• Efficient Method of Moments — CONTAINS: SNP density (location, scale, Hermite polynomial), EMM estimator, asymptotic efficiency theorem, model selection for SNP
• Practical Issues in Simulation Estimation — CONTAINS: Common random numbers, antithetic variates, control variates, auxiliary model selection strategies, step-size guidelines, simulation size trade-offs, implementation checklist
• SMM Weighting Matrix and Inference — CONTAINS: Identity W, two-step W procedure (R×S error matrix, Ω̂₂ = (1/S)EEᵀ, W̃ = Ω̂₂⁻¹), iterated W, Newey-West HAC W, Σ̂_SMM via Jacobian, identification (exact/over/under)
• SMM Python Implementation — CONTAINS: General Python SMM workflow, fixed random draws, trunc_norm_draws (inverse CDF), err_vec/criterion functions, L-BFGS-B eps stepsize fix, numerical Jacobian, two-step W code, indirect inference pattern, results comparison table
• Brock-Mirman Model - SMM Estimation Exercise — CONTAINS: BM1972 six-equation system, latent TFP AR(1), closed-form policy function, simulation algorithm, 6-moment estimation setup (mean c, mean k, mean c/y, var y, corr(c,c-1), corr(c,k)), two-part exercise (identity W + two-step W)
• Simulated Moments Estimation - Overview — CONTAINS: research question, SME-extends-GMM contribution, the two simulation challenges (nonstationarity, parameter feedback), section/note map
• Duffie-Singleton Asset-Pricing Model — CONTAINS: production/firm (Eqs. 2.1–2.2), consumer w/ taste shock (2.3–2.4), Markov state & augmented state (2.5–2.6), closed-form special case (4.4–4.5), conditionally-heteroskedastic counterexample (4.11)
• Simulated Moments Estimator Definition — CONTAINS: primitives $H,f$; actual & simulated state processes (3.1, 3.3); moment gap $G_T$ (3.4); SME (3.5); GMM comparison (3.2)
• Geometric Ergodicity and Uniform LLN — CONTAINS: $\rho$-ergodic / geometrically ergodic def (4.1), Condition B (4.2), Lemma 1 (Mokkadem), Lipschitz-uniformly-in-probability def, Lemma 2 (uniform weak LLN, 4.7)
• SME Consistency — CONTAINS: Assumptions 1–4, Theorem 1 (weak consistency), AUC condition (4.9), $S$-smoothness, Lemmas 3–4, Theorem 2 & Theorem 3 (strong consistency, $L^2$-UC), weak-vs-strong model classes
• SME Asymptotic Distribution — CONTAINS: Assumptions 6–7, Theorem 4 ($\sqrt{T}{G}_T({\beta }_0)\Rightarrow N[0,{\Sigma }_0(1+\tau )]$), Corollary 3.1 ($\Lambda =(1+\tau )(D_0^{\prime }{\Sigma }_0^{-1}{D}_0{)}^{-1}$), efficiency interpretation, $D(\beta )$ rank diagnostic (5.6)
• SME Extensions and Applications — CONTAINS: $\beta$-dependent observation $g^{\beta }$ (6.1–6.4), ${\Sigma }_{f,g,\tau }=\tau {\Sigma }_0+{\Sigma }_1$, calculated-vs-simulated efficiency gain, measurement error, option-pricing via iterated expectations

Sources

• tdb136.pdf — Liesenfeld & Breitung (1998), “Simulation Based Methods of Moments in Empirical Finance”
• 19. Simulated Method of Moments Estimation — Computational Methods for Economists using Python — Evans (2024), Computational Methods for Economists, Ch. 19: full SMM tutorial with Python code, truncated normal example, Brock-Mirman exercise
• Duffie Singleton 1993 - Simulated Moments Estimation of Markov Models of Asset Prices — Duffie & Singleton (1993), Econometrica 61(4):929–952: foundational SME theory for time-series Markov asset-pricing models (geometric ergodicity, AUC condition, consistency Thms 1–3, asymptotic normality Thm 4)

See Also

• Copula SMM — application of these methods to copula models (Oh & Patton 2011)
• Standard Errors and Clustering — analogous HAC inference in GMM/OLS
• Copula Estimation — Bayesian alternative for copula models