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Algorithms for estsimp

Recall that the estsimp command performs two functions: it estimates the main and ancillary parameters ($\gamma$) of the statistical model, and it draws simulations of those parameters from their asymptotic sampling distribution.

Typically, the sampling distribution is multivariate normal with mean equal to the point-estimates of the parameters ( $\hat\gamma$) and variance equal to the variance-covariance matrix of estimates $\hat{\text{V}}(\hat\gamma)$. The current version of ${\mathfrak{C}}$larify contains two exceptions to this rule.

In the case of linear regression, the effect coefficients ($\beta$s) are drawn from a multivariate normal, but simulations of the homoskedastic variance $\sigma^2$ are obtained in a separate step from a scaled inverse $\chi^2$ distribution with $\nu = n-k$ degrees of freedom, where $n$ is the number of observations in the dataset and $k$ is the number of explanatory variables, including the constant term (Gelman, et al., 1995, p. 237). The two-step procedure is legitimate because the effect coefficients and the variance parameter are orthogonal in a linear regression; the procedure is desirable because $\sigma^2$ is strictly positive, and therefore more appropriately drawn from its exact posterior than from a normal distribution. To obtain simulations of $\sigma^2$, the program draws $c$ from a $\chi^2$ with $\nu$ degrees of freedom, and then calculates $\tilde\sigma^2 = \nu\sigma^2 / c$. The resulting draws have an expected value of $\left( \frac{\nu}{\nu-2} \right) \hat\sigma^2$, which approaches $\hat\sigma^2$ as $\nu \rightarrow \infty$.

Likewise, the effect coefficients ($\beta$s) of a seemingly unrelated regression are drawn from the multivariate normal, but simulations of the variance matrix $\Sigma$ are obtained in a separate step. Here, the appropriate posterior distribution is the inverse Wishart (Gelman, et al., 1995, p. 481) with $\nu$ degrees of freedom and dimension $p$, where $p$ is the number of equations in the seemingly unrelated regression model. In cases where the number of explanatory varables varies from one equation to the next, ${\mathfrak{C}}$larify calculates $n-k$ for each equation and sets $\nu$ equal to the mean of those values. To obtain simulations of $\Sigma$, the program draws from a Wishart with scale factor $(\nu\hat{\Sigma})^{-1}$ and inverts the draws. The algorithm for drawing from the Wishart relies on Bartlett's decomposition, which is concisely summarized in Johnson (1987, p. 204) and Ripley (1987, pp. 99-100). estsimp produces draws that have an expected value of $\frac{\nu}{\nu-p-1}\hat\Sigma$, which approaches $\hat\Sigma$ as $\nu$ goes to infinity. In small samples this procedure is conservative, since $\nu > \nu-p-1$, implying that $E(\tilde\Sigma) > \hat\Sigma$.

For all models, simulations of the main and ancillary parameters are random. This means that, in any given run of estsimp, the average value of $\tilde\gamma$ may be slightly smaller or larger than the point estimate $\hat\gamma$, though the approximation becomes more precise with a higher number of simulations. Users can force the mean of the simulated parameters to equal the vector of point estimates by requesting antithetical simulations (Stern 1997, pp. 2028-29). The antisim option instructs the program to draw random numbers in pairs from the uniform[0,1] distribution, with the second draw being the complement of the first. For instance, if the first draw is 0.3 then the complementary draw is 0.7. The draws are, therefore, exactly balanced around the mean of the uniform distribution. These anthithetical simulations are then used to obtain antithetical or balanced draws from the multivariate normal.

When users are analyzing a single dataset, ${\mathfrak{C}}$larify estimates a single vector $\hat\gamma$ with variance $\hat{V}(\hat\gamma)$ and draws all $M$ simulations based on those estimates. The table that appears on the screen gives the exact point estimates and standard errors, instead of reporting the means and standard deviations of the simulations.

The procedure is somewhat more complicated when the researcher employs the mi option to analyze several imputed datasets. In this case, estsimp repeats the following algorithm $I$ times, where $I$ is the number of completed datasets: estimate the parameters and their variance-covariance matrix conditional on the information in dataset $i$ $(i=1,2, \ldots ,I)$, and then draw $M/I$ sets of parameters from their sampling distribution. By repeating this algorithm $I$ times, the program generates $M$ sets of simulated parameters. The output table gives the analytical point-estimate, standard error, and $t$-statistic for each parameter, instead of reporting the means and standard deviations of the simulations. Specifically, the multiple-imputation point estimate for parameter $q$ is $\bar{q} = \frac{1}{I} \sum_{i=1}^{I}\hat{q}_i$ and the variance associated with $\bar{q}$ is a weighted combination of the within-imputation and between-imputation variances: $V(\bar{q}) = \bar{w} + (1 + I^{-1})b$, where $\bar{w} = \frac{1}{I} \sum_{i=1}^{I}V(\hat{q}_i)$ and $b = \frac{1}{I-1} \sum_{i=1}^{I}(\hat{q}_i - \bar{q})^2$. The ratio of $\bar{q}$ (the parameter estimate) to $V(\bar{q})^{1/2}$ (its standard error) forms a $t$-statistic with degrees of freedom $\nu = (I-1)[1+\frac{\bar{u}}{(1+I^{-1})b}]^2$. For more information about these procedures, see King, et al. (2001) and Schafer (1997, pp. 109-110).