Method of Simulated Scores

Definition

The Method of Simulated Scores (MSS) is a simulation-based estimator for limited-dependent-variable (LDV) models — multiperiod (panel) probit and Tobit, multinomial probit — in which the likelihood requires high-dimensional integration over correlated latent errors. Instead of simulating moment conditions (Method of Simulated Moments) or the likelihood itself (Maximum Simulated Likelihood), MSS simulates the score, $\partial\log L_i/\partial\theta$ , directly and solves the simulated first-order conditions. Developed by Hajivassiliou and McFadden (1998) from a suggestion of Ruud (1986), it attains the asymptotic efficiency of maximum likelihood while keeping the finite-simulation consistency of moment-based simulation estimators.

Key Ideas

Simulate the derivative of the log-likelihood. For the linear-exponential family the score can be written as a conditional expectation of the latent variables given the observed inequality pattern, which is directly simulatable — so a single machinery covers any LDV model expressible as linear inequality constraints on latent Gaussian variables.
Efficiency = MSM with optimal instruments. Because the score is the efficient set of moment conditions, MSS corresponds to the Method of Simulated Moments using the asymptotically optimal (GLS) instruments; it therefore reaches ML efficiency within the class of simulation estimators.
Smoothness matters. Simulators that are continuous and differentiable in $\theta$ let one use standard gradient-based optimizers — the key practical advantage over MSM, whose frequency simulators give discontinuous criterion functions.
Bias–rate tradeoff. Unbiased score simulators give consistency at fixed $R$ ; smooth but biased simulators are consistent only if the number of draws grows with the sample size $N$ , at a rate that depends on the simulator.

How It Works

Hajivassiliou–McFadden propose three score simulators, each with its own consistency–asymptotic-normality (CAN) condition:

Acceptance–rejection simulator — generalizes acceptance–rejection sampling to give an unbiased simulation of the score. It is discontinuous in $\theta$ , but the MSS estimator is CAN for a finite number of simulations $R$ .
Recursive-triangularization simulator (the GHK device) — factors the multivariate-normal density recursively so that draws come from a sequence of univariate truncated normals; smooth and continuous in $\theta$ , unbiased for the likelihood contribution and asymptotically unbiased for the score. The MSS estimator is CAN if the number of simulations grows faster than $\sqrt{N}$ .
Gibbs-resampling simulator — draws the latent variables from their multivariate-normal conditionals via Gibbs resampling (Geman–Geman 1984); smooth, with the MSS estimator CAN if the resampling count per simulation grows only at rate $\log N$ — a far milder requirement than $\sqrt{N}$ .

Estimation then solves the simulated score equations for $\theta$ , using a gradient optimizer when a smooth simulator is used.

Why It Matters

Unlocks flexible-error LDV panels. MSS makes multiperiod probit/Tobit models with rich temporal correlation and unobserved heterogeneity estimable without high-dimensional quadrature — models that were previously intractable under classical ML.
Completes the MSM/MSL/MSS trio. It is the efficiency benchmark of simulation-based estimation: MSM is robust and simple but inefficient; MSL is efficient but Jensen-biased at fixed $R$ ; MSS targets ML efficiency directly.
Component reuse. Its two smooth simulators are foundational elsewhere — the recursive-triangularization simulator is the GHK simulator (Geweke–Hajivassiliou–Keane) used throughout Bayesian and classical MNP estimation, and the Gibbs-resampling device is the latent-variable data augmentation logic of MCMC.

Open Questions