Hajivassiliou-McFadden (1998) The Method of Simulated Scores for the Estimation of LDV Models

Summary

This paper develops the Method of Simulated Scores (MSS) for estimating limited-dependent-variable (LDV) models — multiperiod (panel) probit and Tobit, multinomial choice — that have a flexible correlation structure in the unobservables and therefore require high-dimensional numerical integration under classical maximum likelihood. Rather than simulate moment conditions (as in the Method of Simulated Moments) or the likelihood itself (Maximum Simulated Likelihood), MSS simulates the score — the logarithmic derivative of the likelihood — directly. The authors propose three score simulators, establish consistency and asymptotic normality (CAN) for each, derive the rate at which the number of simulations must grow when a biased simulator is used, and apply the method to the incidence and extent of developing-country external-debt-repayment problems under a credit-rationing model. (First circulated as Cowles Foundation Discussion Paper No. 967, December 1990; published in Econometrica 66(4): 863–896, 1998.)

Key Claims

• The problem. Classical ML for LDV models with flexible error correlation (panel probit/Tobit, MNP with general substitution) demands high-dimensional integration and is intractable. Simulation estimators solve this, but the Method of Simulated Moments (MSM; McFadden 1989; Pakes–Pollard 1989) yields criterion functions that are discontinuous in the parameters (requiring empirical-process theory and defeating standard gradient optimizers).
• The MSS idea (after Ruud 1986). For the linear-exponential family the score can be written as a conditional expectation, which can be simulated directly. This makes MSS applicable to any LDV model expressible as a set of linear inequality constraints on latent variables whose distribution is linear-exponential — no need to invent an ad-hoc simulator per model.
• Three simulators, three CAN results.
  1. Acceptance–rejection (unbiased) simulator — generalizes acceptance–rejection sampling to give an unbiased simulation of the score; discontinuous in the parameters, but the resulting MSS estimator is CAN for a finite number of simulations $R$.
  2. Recursive-triangularization (GHK-type) simulator — smooth and continuous in the parameters via a recursive triangular factorization of the multivariate-normal density (unbiased for the likelihood contribution, asymptotically unbiased for the score); the MSS estimator is CAN provided the number of simulations grows faster than $\sqrt{N}$.
  3. Gibbs-resampling simulator — uses the conditionals of the multivariate normal with Gibbs resampling (Geman–Geman 1984); smooth, and CAN provided the number of resamplings per simulation grows only at rate $\log N$ (a much milder requirement than $\sqrt N$).
• Efficiency. Because MSS simulates the score directly, it corresponds to MSM using the optimal (asymptotically efficient) instruments; hence MSS attains the efficiency of maximum likelihood within the class of simulation estimators, while (with smooth simulators) avoiding MSM's computational discontinuities.
• Bias–rate tradeoff. When smooth but biased simulators are used, consistency is preserved only if $R$ (or the resampling count) rises with $N$ at the derived rate; the smoother simulator (Gibbs) buys a much slower required rate ($\log N$) than the triangularization simulator ($\sqrt N$).
• Application — external debt crises. The incidence and extent of LDC debt-repayment problems are modeled as optimized choices of national authorities under credit rationing, implemented as a multi-period probit and Tobit system with persistent unobserved heterogeneity and state dependence. MSS accommodates a flexible temporal correlation structure that traditional ML could not, and the results indicate that the restrictive error structures imposed by earlier studies may have produced unreliable estimates.

Concepts Introduced or Extended

• Method of Simulated Scores
• Simulation-Based Estimation
• Multinomial Probit
• Tobit Model

Entities Mentioned

• Vassilis A. Hajivassiliou
• Daniel L. McFadden
• Cowles Foundation

Quotes

"The method of simulated scores (MSS) is presented for estimating LDV models with flexible correlation structure in the unobservables. We propose simulators that are continuous in the unknown parameter vectors, and hence standard optimization methods can be used to compute the MSS estimators."

"Our findings show that the restrictive error structures imposed by past studies may have led to unreliable econometric results."

My Take

MSS is the theoretical capstone of the simulation-estimation trio (MSM / MSL / MSS): by simulating the score it inherits MSM's finite-$R$ consistency and ML's efficiency, and the paper's main practical gift is the pair of smooth simulators (recursive triangularization — the GHK device — and Gibbs resampling) that make gradient-based optimization feasible. The elegant unifying observation is Ruud's: any LDV model that is a set of linear inequalities on latent Gaussian variables has a score expressible as a conditional expectation, so one machine handles panel probit, Tobit, and MNP alike. The catch, well captured in Stern's survey, is that MSS remained the least-used of the three in practice — the required simulation rates and bookkeeping are heavier than MSL with GHK — even as its components (GHK, Gibbs) became workhorses elsewhere. The debt-crisis application is also an early, serious use of flexible panel-error LDV models in international finance.