Constructing and Weighting Simulated Moments

Why Match Moments?

A model can imply features of the data even when its likelihood is difficult to calculate. For example, its parameters may determine an outcome's mean, variance, or relationship with a covariate. The method of simulated moments (MSM) estimates parameters by comparing selected features of observed data with the corresponding features implied by the model. When those expectations are unavailable in closed form, simulation can approximate them.

The key choices are deliberate: which features to match, how to calculate their simulated counterparts, and how to weight discrepancies measured on different scales or with different levels of noise. MSM does not automatically make a model fit every feature of the data. It chooses parameters to make the selected, weighted discrepancies small.

Moment Conditions and Their Simulated Versions

Definition: Let \(Y\) denote an observation generated by a model with parameter vector \(\theta\), and let \(g(Y)\) be a \(k\)-dimensional vector of functions of that observation. The model-implied moment vector is \(m(\theta)=E_\theta[g(Y)]\). For observed data \(Y_1,\ldots,Y_n\), the corresponding sample moment vector is \(\bar{g}_n=n^{-1}\sum_{i=1}^n g(Y_i)\). A moment condition compares \(\bar{g}_n\) with \(m(\theta)\).

Each element of \(g(Y)\) describes a feature to match. For a scalar outcome, \(g(Y)\) might include \(Y\) and \(Y^2\), so the moments describe its mean and second raw moment. Other choices can describe event frequencies or relationships between outcomes and covariates. The chosen moments need to be defined for the observed data and for the model at candidate parameter values.

If an expectation is not available analytically, draw \(R\) simulated observations \(Y^{(1)}(\theta),\ldots,Y^{(R)}(\theta)\) from the model at \(\theta\). The simulated moment vector is

\widehat{m}_R(\theta)=\frac{1}{R}\sum_{r=1}^{R}g\!\left(Y^{(r)}(\theta)\right).

This is a Monte Carlo approximation to \(m(\theta)\), not the observed sample moment. The observed moment vector summarizes the data; the simulated moment vector summarizes model-generated data at a proposed parameter value. In practice, simulation draws may be held fixed across candidate values of \(\theta\) during optimization, so changes in the simulated criterion more clearly reflect changes in the parameters rather than fresh simulation noise. The number and design of draws matter because \(\widehat{m}_R(\theta)\) can differ from its expectation.

Worked Example: Building Moments for a Binary-Choice Model

Suppose a hypothetical model has \(P(Y=1\mid x)=\Lambda(a+bx)\), where \(Y\) is binary, \(x\) is a covariate, \(\Lambda(t)=1/(1+\exp(-t))\), and \(\theta=(a,b)\). Consider four hypothetical observations with \(x=(0,1,1,2)\) and \(Y=(1,0,1,1)\). Use \(g(Y,x)=(Y,xY)'\), where the prime denotes a transpose.

The observed moments are

\bar{g}_n= \begin{pmatrix} (1+0+1+1)/4\\ (0\cdot1+1\cdot0+1\cdot1+2\cdot1)/4 \end{pmatrix} = \begin{pmatrix} 0.75\\ 0.75 \end{pmatrix}.

To approximate the model moments, take two simulation draws for each covariate value. At the candidate \(a=b=0\), the choice probability is \(0.5\) for every observation. Use fixed uniforms \(u^{(1)}=(0.2,0.7,0.3,0.8)\) and \(u^{(2)}=(0.9,0.4,0.8,0.1)\), and set \(Y_i^{(r)}=1\{u_i^{(r)}\leq\Lambda(a+bx_i)\}\). The simulated outcomes are \((1,0,1,0)\) and \((0,1,0,1)\).

For the first draw, the two moments are \(2/4=0.5\) and \((0+0+1+0)/4=0.25\). For the second, they are \(2/4=0.5\) and \((0+1+0+2)/4=0.75\). Averaging the two draws gives \(\widehat{m}_2(0,0)=(0.5,0.5)'\). Thus the observed-minus-simulated discrepancy at this candidate is \((0.25,0.25)'\). With only two draws, this is an illustrative, noisy approximation, not an exact calculation of the population moments.

The MSM Estimator

Let \(q\) be the number of parameters and \(k\) the number of selected moments. Typically \(k\geq q\), although the number of moments alone does not ensure that the parameters are identified. Before using the estimator, specify a parameter space \(\Theta\), the data moments, the model moments (or their simulation approximation), and a weighting matrix.

Definition: For a symmetric positive definite \(k\times k\) weighting matrix \(W\), the simulated method of moments estimator is any parameter value minimizing

\widehat{\theta}_{\mathrm{MSM}} \in \underset{\theta\in\Theta}{\arg\min}\; \left[\bar{g}_n-\widehat{m}_R(\theta)\right]' W \left[\bar{g}_n-\widehat{m}_R(\theta)\right].

The discrepancy is a \(k\)-vector, so \(W\) must be \(k\times k\). Positive definiteness ensures that every nonzero discrepancy has a positive quadratic distance. A positive semidefinite matrix can be used in some formulations, but it may assign zero weight to some directions; the positive definite case is a straightforward default. If the simulated moments depend on fixed draws, the optimization is over the criterion computed using those draws.

The parameter value is determined by the selected moments and their weights. For a unique, meaningful fit, different parameter values must produce distinguishable moment vectors in the directions being matched. This is an identification requirement, not something guaranteed by minimizing the criterion. If two values of \(\theta\) give the same selected moments, the criterion cannot distinguish them.

Worked Example: Matching a Mean and a Second Moment

Consider the hypothetical location-scale model \(Y=\theta+\sigma U\), where \(U\sim N(0,1)\), \(\sigma>0\), and \(\theta\) is a location parameter. Use \(g(Y)=(Y,Y^2)'\). Suppose the observed sample moments are \(\bar{Y}=4\) and \(n^{-1}\sum_iY_i^2=25\). For illustration, use the fixed simulation draws \(u=(-1,-1,1,1)\) at each candidate \((\theta,\sigma)\).

The simulation average of \(u\) is \(0\), and the average of \(u^2\) is \(1\). Therefore the simulated moments for these draws are

\widehat{m}_R(\theta,\sigma) = \begin{pmatrix} \theta+\sigma(0)\\ \theta^2+2\theta\sigma(0)+\sigma^2(1) \end{pmatrix} = \begin{pmatrix} \theta\\ \theta^2+\sigma^2 \end{pmatrix}.

The observed-minus-simulated discrepancy is \((4-\theta,\;25-\theta^2-\sigma^2)'\). With identity weighting, the criterion is

Q(\theta,\sigma)=(4-\theta)^2+(25-\theta^2-\sigma^2)^2.

Both squared terms can equal zero: the first requires \(\theta=4\), and the second then requires \(\sigma^2=25-4^2=9\). Since \(\sigma>0\), the minimizing value is \((\widehat{\theta},\widehat{\sigma})=(4,3)\). The simulated outcomes at that value are \(1,1,7,7\); their mean is \(4\), and their mean square is \((1+1+49+49)/4=25\), confirming the match. The example shows exact matching for the chosen moments and draws; it does not imply that these few draws generally reproduce the model's entire distribution.

Choosing the Weighting Matrix

The weighting matrix determines which discrepancies matter most. The identity matrix gives equal numerical weight to all moment discrepancies, but that need not mean equal substantive importance: a discrepancy of one unit in dollars is not directly comparable with one unit in a probability. Rescaling moments before using identity weights can make the criterion more interpretable.

A diagonal matrix allows separate weights for each moment. A general positive definite matrix also accounts for correlation between discrepancies: its off-diagonal elements affect how combinations of moment errors are evaluated. A common covariance-based choice is to weight by the inverse of an estimated covariance matrix for the moment vector. This gives relatively less weight to directions with greater estimated variability and accounts for correlation. In practice, covariance estimates can be unstable, especially when moments are strongly correlated, so scaling or regularization may be needed. Weighting does not fix poorly chosen or uninformative moments.

Worked Example: How Moment Weights Change the Estimate

Suppose a hypothetical model has one parameter \(\theta\) and two simulated moments \(m(\theta)=(\theta,2\theta)'\). The observed moments are \((2,5)'\). The discrepancies are \(d(\theta)=(2-\theta,\;5-2\theta)'\). Compare identity weighting with \(W=\operatorname{diag}(1,1/4)\). For illustration, the latter could reflect estimated moment variances of 1 and 4, respectively, with no estimated covariance.

With identity weighting,

Q_I(\theta)=(2-\theta)^2+(5-2\theta)^2.

Differentiating gives \(Q_I'(\theta)=2(\theta-2)-4(5-2\theta)=10\theta-24\). Setting the derivative to zero gives \(\widehat{\theta}_I=2.4\). The two discrepancies at this value are \(-0.4\) and \(0.2\), so \(Q_I(2.4)=0.16+0.04=0.20\).

With the second moment downweighted,

Q_W(\theta)=(2-\theta)^2+\frac{1}{4}(5-2\theta)^2.

Now \(Q_W'(\theta)=2(\theta-2)-(5-2\theta)=4\theta-9\), so \(\widehat{\theta}_W=2.25\). The discrepancies are \(-0.25\) and \(0.5\), and the weighted criterion is \(Q_W(2.25)=0.25^2+(1/4)(0.5)^2=0.125\). The estimate moves toward matching the first moment more closely because the second discrepancy receives less weight. Neither estimate matches both moments exactly: the observed pair \((2,5)'\) does not lie on the model's line \((\theta,2\theta)'\).

A Short Justification for Scaling Moments

Suppose a moment vector is rescaled by an invertible matrix \(A\): the new discrepancy is \(Ad\). If the original criterion uses \(W\), then the same numerical distance is preserved by using the transformed weight \(W_{\mathrm{new}}=A^{-T}WA^{-1}\), since

(Ad)'A^{-T}WA^{-1}(Ad)=d'Wd.

Thus rescaling moments and changing weights are linked. Using identity weights after rescaling generally defines a different criterion from using identity weights on the original scale. This is why a weighting choice should be reported together with the definition and scale of the moments, rather than treated as a cosmetic detail.

Practical Checks and Common Pitfalls

Before interpreting an MSM estimate, verify that the simulated moments correspond to the same functions and covariate structure as the observed moments. Check that the simulation scheme represents the model at each candidate parameter value, that the weighting matrix has the intended dimension and is symmetric, and that parameters obey their model constraints. If a simulated indicator moment changes abruptly with parameters, the objective may also be irregular; simulation design and optimization then require care.

Using incompatible moments. Comparing an observed mean with a simulated median does not implement the stated moment condition. Define the same function \(g\) for both.
Assuming equal weights are automatically neutral. Identity weighting depends on the units and scales of the moments. Consider standardization or an explicitly justified weighting matrix.
Calling the closest fit an exact fit. With more moments than parameters, the observed moment vector may not be attainable under the model. Report the criterion and discrepancies rather than implying every moment was matched.
Ignoring simulation error. A simulated moment is an approximation that depends on the draw count and design. A small discrepancy with a particular finite set of draws need not mean the exact model moments fit equally well.

Key takeaway: MSM compares observed sample moments with model-implied moments, approximating the latter by simulation when necessary. The parameter estimate minimizes a weighted quadratic discrepancy, so moment construction, simulation, identification, and the weighting matrix all shape the result.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

In a model with \(g(Y)=(Y,Y^2)'\), what two features of the outcome distribution do the selected moments describe?
Explain the difference between \(\bar{g}_n\) and \(\widehat{m}_R(\theta)\).
Why must the weighting matrix have the same number of rows and columns as the moment vector?
In the final worked example, why does downweighting the second moment move the estimate from 2.4 to 2.25?
What is one reason identity weighting can be difficult to interpret when moments use different units?