In this blog post, I’d like to give you a relatively nontechnical introduction to Markov chain Monte Carlo, often shortened to “MCMC”. MCMC is frequently used for fitting Bayesian statistical models. There are different variations of MCMC, and I’m going to focus on the Metropolis–Hastings (M–H) algorithm. In the interest of brevity, I’m going to omit some details, and I strongly encourage you to read the [BAYES] manual before using MCMC in practice.

Let’s continue with the coin toss example from my previous post Introduction to Bayesian statistics, part 1: The basic concepts. We are interested in the posterior distribution of the parameter \(\theta\), which is the probability that a coin toss results in “heads”. Our prior distribution is a flat, uninformative beta distribution with parameters 1 and 1. And we will use a binomial likelihood function to quantify the data from our experiment, which resulted in 4 heads out of 10 tosses. Read more…

In this blog post, I’d like to give you a relatively nontechnical introduction to Bayesian statistics. The Bayesian approach to statistics has become increasingly popular, and you can fit Bayesian models using the **bayesmh** command in Stata. This blog entry will provide a brief introduction to the concepts and jargon of Bayesian statistics and the **bayesmh** syntax. In my next post, I will introduce the basics of Markov chain Monte Carlo (MCMC) using the Metropolis–Hastings algorithm. Read more…

Quantile regression models a quantile of the outcome as a function of covariates. Applied researchers use quantile regressions because they allow the effect of a covariate to differ across conditional quantiles. For example, another year of education may have a large effect on a low conditional quantile of income but a much smaller effect on a high conditional quantile of income. Also, another pack-year of cigarettes may have a larger effect on a low conditional quantile of bronchial effectiveness than on a high conditional quantile of bronchial effectiveness.

I use simulated data to illustrate what the conditional quantile functions estimated by quantile regression are and what the estimable covariate effects are. Read more…

**teffects ipw** uses multinomial logit to estimate the weights needed to estimate the potential-outcome means (POMs) from a multivalued treatment. I show how to estimate the POMs when the weights come from an ordered probit model. Moment conditions define the ordered probit estimator and the subsequent weighted average used to estimate the POMs. I use **gmm** to obtain consistent standard errors by stacking the ordered-probit moment conditions and the weighted mean moment conditions. Read more…

For a nonlinear model with heteroskedasticity, a maximum likelihood estimator gives misleading inference and inconsistent marginal effect estimates unless I model the variance. Using a robust estimate of the variance–covariance matrix will not help me obtain correct inference.

This differs from the intuition we gain from linear regression. The estimates of the marginal effects in linear regression are consistent under heteroskedasticity and using robust standard errors yields correct inference.

If robust standard errors do not solve the problems associated with heteroskedasticity for a nonlinear model estimated using maximum likelihood, what does it mean to use robust standard errors in this context? I answer this question using simulations and illustrate the effect of heteroskedasticity in nonlinear models estimated using maximum likelihood. Read more…

I illustrate that exact matching on discrete covariates and regression adjustment (RA) with fully interacted discrete covariates perform the same nonparametric estimation. Read more…

We estimate the average treatment effect (ATE) for an exponential mean model with an endogenous treatment. We have a two-step estimation problem where the first step corresponds to the treatment model and the second to the outcome model. As shown in *Using gmm to solve two-step estimation problems*, this can be solved with the generalized method of moments using **gmm**.

This continues the series of posts where we illustrate how to obtain correct standard errors and marginal effects for models with multiple steps. In the previous posts, we used **gsem** and **mlexp** to estimate the parameters of models with separable likelihoods. In the current model, because the treatment is endogenous, the likelihood for the model is no longer separable. We demonstrate how we can use **gmm** to estimate the parameters in these situations. Read more…

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\newcommand{\betab}{\boldsymbol{\beta}}\)Differences in conditional probabilities and ratios of odds are two common measures of the effect of a covariate in binary-outcome models. I show how these measures differ in terms of conditional-on-covariate effects versus population-parameter effects. Read more…

\(\newcommand{\Eb}{{\bf E}}\)The change in a regression function that results from an everything-else-held-equal change in a covariate defines an effect of a covariate. I am interested in estimating and interpreting effects that are conditional on the covariates and averages of effects that vary over the individuals. I illustrate that these two types of effects answer different questions. Doctors, parents, and consultants frequently ask individuals for their covariate values to make individual-specific recommendations. Policy analysts use a population-averaged effect that accounts for the variation of the effects over the individuals. Read more…

I want to estimate, graph, and interpret the effects of nonlinear models with interactions of continuous and discrete variables. The results I am after are not trivial, but obtaining what I want using **margins**, **marginsplot**, and factor-variable notation is straightforward. Read more…