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Models Zelig Can Run

This section describes the mathematical components of the models supported by Zelig, using whenever possible the classification and notation of (). Most models have a stochastic component (probability density given certain parameters) and a systematic component (deterministic functional form that specifies how one or more of the parameters varies over the observed values 102#102 as a function of the explanatory variables 69#69 ).

Let 158#158 be a random outcome variable, realized as 70#70 observations 102#102 . For the probability density 159#159 with systematic feature 160#160 varying over 4#4 and a scalar ancillary parameter 21#21 (constant over 4#4 ), the stochastic component is given by

For a functional form 162#162 , 3#3 explanatory variables 163#163 , and effect parameters 12#12 , the systematic component is:

Using the definitions of King, Tomz, and Wittenberg, 2000, Zelig generates at least two quantities of interest:

• The predicted value is a random draw from the stochastic component given random draws of 12#12 and 21#21 from their sampling (or posterior) distribution.
• The expected value is the mean of the stochastic component given random draws of 12#12 and 21#21 from their sampling (or posterior) distributions. For computational efficiency, Zelig deterministically calculates the expected values from the simulated parameters whenever possible.

Both the predicted values and expected values produced by Zelig can be displayed as histograms or density estimates (to summarize the full sampling or posterior density), or summarized with confidence intervals (by sorting the simulations and taking the 5th and 95th percentile values for a 90% confidence interval for example), standard errors (by taking the standard deviation of the simulations), or point estimates (by averaging the simulations). The point estimate of predicted and expected values are the same only in linear models. In almost all situations, simulations from predicted values have more variance than expected values. As the number of simulations increases the distribution of the expected values tends toward a constant; the distribution of the predicted values does not collapse as the number of simulations increases.

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