Markov chain Monte Carlo Methods Are Important in many Areas of Applied Econometrics

Using Bayesian methods, it is often important to be able to sample from a posterior distribution. Markov chain Monte Carlo (MCMC) methods are of particular importance. Popular random walk MCMC methods include the Metropolis-Hastings algorithm and Gibbs sampling (which is a special case of Metropolis-Hastings).

Why would we need MCMC methods? The most common application is the numerical calculation of mutli-dimensional integrals. Numerical integration is key in many areas of applied econometrics. A simple example is the random effects binary model. The dependent variable takes only values 0 and 1 and the individual effects are normally distributed. This model is estimated with maximum likelihood estimation. The log-likelihood function consists of an integral for which no analytical solution exists. Thus, numerical methods are needed. In this case, integration is one-dimensional, so methods other than MCMC can be used. But it illustrates that even in basic econometric applications (that are not necessarily based on Bayesian methods) it is possible that numerical integration plays a key role.