Simulation is implementing the story encoded in the model, step by step, to produce something which should look like the data. Stochastic models have random components and so require some random steps. Stochastic models specified through conditional distributions are simulated by chaining together random variables. Methods for generating random variables with specified distributions: the transformation or inverse-quantile method; the rejection method; Markov chain Monte Carlo (Metropolis or Metropolis-Hastings method). Simulation shows us what a model predicts (expectations, higher moments, correlations, regression functions, sampling distributions); analytical probability calculations are short-cuts for exhaustive simulation. Simulation lets us check aspects of the model: does the data look like typical simulation output? if we repeat our exploratory analysis on the simulation output, do we get the same results? Simulation-based estimation: the method of simulated moments.
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