SMM Weighting Matrix and Inference

Summary

The choice of weighting matrix $W$ in the SMM criterion function controls both the efficiency of the estimator and the standard errors of ${\hat{\theta }}_{SMM}$. Four strategies exist in increasing order of optimality: identity matrix, two-step variance-covariance estimator, iterated estimator, and Newey-West HAC estimator. Separately, the parameter variance-covariance matrix ${\hat{\Sigma }}_{SMM}$ is computed from the Jacobian of the moment error vector — an R×K matrix of numerical derivatives evaluated at the SMM estimate.

Overview

The Method of Simulated Moments estimator minimizes:

$${\hat{\theta }}_{SMM}=\theta :\underset{\theta }{\min }e(\tilde{x},x|\theta {)}^{T}We(\tilde{x},x|\theta )$$

where $e(\tilde{x},x|\theta )$ is the $R\times 1$ vector of moment errors (simulated minus data moments, typically as percent deviations). The $R\times R$ weighting matrix $W$ controls how each moment is weighted in the minimization. Different choices of $W$ produce estimators with different asymptotic variances.

Percent-Deviation Scaling

Scale the moment error vector as percent deviations $e_r=({\hat{m}}_r-m_r)/m_r$ (not raw differences). This puts all moments in the same units and prevents moments with larger absolute values from receiving unintended extra weight. Exception: percent deviations are not valid when data moments can be zero or change sign.

Weighting Matrix Strategies

1. Identity Matrix ($W=I$)

$${\hat{\theta }}_{SMM}=\theta :\underset{\theta }{\min }e(\tilde{x},x|\theta {)}^{T}e(\tilde{x},x|\theta )$$

Gives each moment equal weight. Simple and sufficient when:

• The problem is well-conditioned
• Moments are in the same units (or after percent-deviation scaling)
• A quick estimate is needed before computing the optimal $W$

2. Two-Step Variance-Covariance Estimator

Definition: Two-Step SMM Estimator

Step 1. Estimate with identity matrix:

$${\hat{\theta }}_{1,SMM}=\theta :\underset{\theta }{\min }e(\tilde{x},x|\theta {)}^{T}Ie(\tilde{x},x|\theta )$$

Step 2. Compute the $R\times S$ error matrix at the Step 1 estimates, where each column is the moment error vector from one simulation:

$$E(\tilde{x},x|{\hat{\theta }}_{1,SMM})=\left[\begin{array}{ccc}{m}_1({\tilde{x}}_1|\hat{\theta })-m_1(x)& \cdots & m_1({\tilde{x}}_S|\hat{\theta })-m_1(x)\\ ⋮& \ddots & ⋮\\ m_R({\tilde{x}}_1|\hat{\theta })-m_R(x)& \cdots & m_R({\tilde{x}}_S|\hat{\theta })-m_R(x)\end{array}\right]$$

(Or in percent-deviation form: divide each row by the corresponding data moment $m_r(x)$.)

Step 3. Estimate the variance-covariance matrix of the moment errors:

$${\hat{\Omega }}_2=\frac{1}{S}E(\tilde{x},x|{\hat{\theta }}_{1,SMM})E(\tilde{x},x|{\hat{\theta }}_{1,SMM}{)}^T$$

The $(r,s)$ element is ${\hat{\Omega }}_{2,r,s}=\frac{1}{S}{\sum }_{i=1}^S[m_r({\tilde{x}}_i|\hat{\theta })-m_r(x)][m_s({\tilde{x}}_i|\hat{\theta })-m_s(x)]$.

Step 4. Set the optimal weighting matrix ${\hat{W}}^{two-step}={\hat{\Omega }}_2^{-1}$ and re-estimate:

$${\hat{\theta }}_{2,SMM}=\theta :\underset{\theta }{\min }e(\tilde{x},x|\theta {)}^T{\hat{W}}^{two-step}e(\tilde{x},x|\theta )$$

Intuition: Downweight moments with high simulation variance; upweight moments that are estimated precisely across simulations.

Practical result: The two-step optimal $W$ may not change point estimates much compared to $W=I$, but it reduces standard errors by efficiently weighting moments.

3. Iterated Variance-Covariance Estimator

The truly optimal $W^{opt}$ is the fixed point of the two-step procedure. Iterate:

$${\hat{\theta }}_{i,SMM}=\theta :\underset{\theta }{\min }{e}^T{\hat{W}}_{i}e$$ $${\hat{W}}_{i+1}={\hat{\Omega }}_{i+1}^{-1},{\hat{\Omega }}_{i+1}=\frac{1}{S}E(\tilde{x},x|{\hat{\theta }}_{i,SMM})E(\tilde{x},x|{\hat{\theta }}_{i,SMM}{)}^T$$

The iterated SMM estimator ${\hat{\theta }}_{it,SMM}$ is the ${\hat{\theta }}_{i,SMM}$ such that:

$$\Vert {\hat{W}}_{i+1}-{\hat{W}}_i\Vert <ϵ$$

In practice, the two-step estimator usually suffices — the gain from additional iterations is typically small.

4. Newey-West HAC Estimator

When simulated data are autocorrelated (time series models), the variance-covariance matrix of the moments requires a HAC (heteroskedasticity and autocorrelation consistent) estimator.

Definition: Newey-West Weighting Matrix

The asymptotically optimal weighting matrix in the presence of autocorrelation:

$${\hat{W}}_{nw}={\Gamma }_{0,S}+\sum _{v=1}^q\left(1-\frac{v}{q+1}\right)({\Gamma }_{v,S}+{\Gamma }_{v,S}^T)$$

where:

$${\Gamma }_{v,S}=\frac{1}{S}\sum _{i=v+1}^{S}E({\tilde{x}}_i,x|\theta )E({\tilde{x}}_{i-v},x|\theta {)}^T$$

The bandwidth parameter $q$ controls how many lags are included.

The optimal weighting matrix ${\hat{W}}^{opt}$ in the autocorrelated case:

$${\hat{W}}^{opt}=\underset{S\to \infty }{\lim }\frac{1}{S}\sum _{i=1}^S\sum _{l=-\infty }^{\infty }E({\tilde{x}}_i,x|\theta )E({\tilde{x}}_{i-l},x|\theta {)}^T$$

The Newey-West estimator approximates this via the weighted sum of autocovariance matrices.

When to use: Essential for structural models where simulated time series exhibit serial correlation (e.g., DSGE models, asset pricing models with persistence).

Variance-Covariance of ${\hat{\theta }}_{SMM}$

The parameter estimates ${\hat{\theta }}_{SMM}$ are asymptotically normal. The estimated $K\times K$ variance-covariance matrix ${\hat{\Sigma }}_{SMM}$ is computed from the Jacobian of the moment error vector.

Theorem: SMM Parameter Variance-Covariance

Define the $R\times K$ Jacobian matrix $d(\tilde{x},x|\theta )$ of derivatives of the moment error vector with respect to each parameter:

$$d(\tilde{x},x|\theta )\equiv \left[\begin{array}{ccc}\frac{\partial e_1}{\partial {\theta }_1}& \cdots & \frac{\partial e_1}{\partial {\theta }_K}\\ ⋮& \ddots & ⋮\\ \frac{\partial e_R}{\partial {\theta }_1}& \cdots & \frac{\partial e_R}{\partial {\theta }_K}\end{array}\right]$$

The SMM estimates are asymptotically normal as $S\to \infty$:

$${\text{plim}}_{S\to \infty }\sqrt{S}({\hat{\theta }}_{SMM}-{\theta }_0)\sim N(0,[d^{T}Wd{]}^{-1})$$

The estimated variance-covariance matrix is:

$${\hat{\Sigma }}_{SMM}=\frac{1}{S}[d(\tilde{x},x|{\hat{\theta }}_{SMM}{)}^{T}Wd(\tilde{x},x|{\hat{\theta }}_{SMM}){]}^{-1}$$

Computing the Jacobian numerically: Use centered finite differences:

$$\frac{\partial e_r}{\partial {\theta }_k}\approx \frac{e_r(\theta +h{e}_k)-e_r(\theta -h{e}_k)}{2h}$$

where $e_k$ is the $k$-th unit vector and $h$ is the step size. The step size $h$ must be large enough that the criterion function changes detectably — see Practical Issues in Simulation Estimation for step-size guidelines.

Identification

A model is:

• Exactly identified if $K=R$ (parameters = moments): unique solution, no overidentifying restrictions to test
• Overidentified if $K<R$: more moments than needed, enables specification testing (J-test — see SMM Copula Specification Testing)
• Underidentified if $K>R$: cannot estimate $\theta$ consistently

Practical Moment Selection

Not all moments are orthogonal — some moments convey the same information and don’t separately identify additional parameters. Good practice:

1. Choose moments with a clear theoretical connection to specific parameters
2. Overidentify the model ($R>K$) to gain efficiency and enable specification testing
3. Out-of-sample moment check: after estimation, verify that data moments not used in estimation are matched by the estimated model — a powerful diagnostic

Connections

• Method of Simulated Moments — the estimator these weighting strategies apply to
• SMM Copula Specification Testing — J-test for overidentified copula SMM (uses efficient $W$)
• SMM Copula Asymptotic Theory — Propositions 2-3 cover variance estimation for the copula case
• Practical Issues in Simulation Estimation — step-size guidelines for numerical Jacobians
• Standard Errors and Clustering — analogous HAC issues in GMM/OLS
• Indirect Inference — uses auxiliary model parameters as moments; the same W strategies apply

See Also

• SMM Python Implementation — Python code implementing two-step W and Jacobian-based Σ̂
• Method of Simulated Moments — consistency and asymptotic normality theorems
• Practical Issues in Simulation Estimation — common random numbers, step-size selection

Sources

• Computational Methods for Economists — Ch. 19 — Evans (2024), Sections 19.2–19.5
• Newey, W.K. and K.D. West (1987), “A Simple, Positive, Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica 55(3), 703–708
• Adda, J. and R. Cooper (2003), Dynamic Economics, MIT Press, pp. 82–100