EMalgorithmExample

This repository contains code examples for estimating a finite mixture model via the Expectation-Maximization (EM) algorithm for two maximum likelihood applications:

1. Multinomial logit model (Also allows for alternative-specific conditional logit model)
2. Model (1), combined with a normally-distributed continuous outcome
3. Model (2), but with censoring of the continuous outcome
4. Model (3), but with a labor supply model with stochastic employment

Folder Structure

The folder structure within each language's folder is outlined below:

• General Functions: Contains functions that can be used in any of the model:
  • MLE objective functions for optimization (used in the Maximization step of the EM algorithm)
  • Type-specific probability updating (used in the Expectation step of the EM algorithm)
  • Logit probability prediction
• mlogitOnly: Contains scripts and functions to simulate and estimate the EM algorithm on a logit-only model
• mlogitAndNormal: Contains scripts and functions to simulate and estimate the EM algorithm on a logit-and-continuous model
• mlogitAndNormalCensored: Contains scripts and functions to simulate and estimate the EM algorithm on a logit-and-continuous model that also incorporates censoring of the continuous outcome
• mlogitAndNormalCensoredAndUnemp: Contains scripts and functions to simulate and estimate the EM algorithm on a logit-and-continuous model that also incorporates censoring of the continuous outcome as well as a labor supply model with stochastic employment (and where wages are observed only in employment)

Tips

Some miscellaneous tips for estimating EM algorithms:

• Finite mixture models are not generally globally concave
• Because of this, you will get different estimates based on different starting values
• General tip is to start from a pertrubed version of the solution (if a simulation) or start from a perturbed version of the zero-types estimates (if not a simulation)
• The overall likelihood of the model should increase with each EM iteration, so long as the model is additively separable across likelihood components.
• If maximization is overly burdensome, you can loosen the convergence criteria at this step for the first few iterations of the algorithm. This may improve performance and/or be helpful for debugging.
• Convergence is typically more difficult in models where there is not a continuous outcome (i.e. the "mlogitOnly" folder). This is because there is much less variation with which to identify the unobserved type.