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…

Autoregressive (AR) and moving-average (MA) models are combined to obtain ARMA models. The parameters of an ARMA model are typically estimated by maximizing a likelihood function assuming independently and identically distributed Gaussian errors. This is a rather strict assumption. If the underlying distribution of the error is nonnormal, does maximum likelihood estimation still work? The short answer is yes under certain regularity conditions and the estimator is known as the quasi-maximum likelihood estimator (QMLE) (White 1982).

In this post, I use Monte Carlo Simulations (MCS) to verify that the QMLE of a stationary and invertible ARMA model is consistent and asymptotically normal. See Yao and Brockwell (2006) for a formal proof. For an overview of performing MCS in Stata, refer to Monte Carlo simulations using Stata. Also see A simulation-based explanation of consistency and asymptotic normality for a discussion of performing such an exercise in Stata.

**Simulation**

Let’s begin by Read more…

\(\newcommand{\xb}{{\bf x}}

\newcommand{\betab}{\boldsymbol{\beta}}\)Using analytically computed derivatives can greatly reduce the time required to solve a nonlinear estimation problem. I show how to use analytically computed derivatives with **optimize()**, and I discuss **mypoisson4.ado**, which uses these analytically computed derivatives. Only a few lines of **mypoisson4.ado** differ from the code for **mypoisson3.ado**, which I discussed in Programming an estimation command in Stata: Allowing for robust or cluster–robust standard errors in a poisson command using Mata.

This is the twenty-third post in the series **Programming an estimation command in Stata**. I recommend that you start at the beginning. See Programming an estimation command in Stata: A map to posted entries for a map to all the posts in this series.

**Analytically computed derivatives for Poisson**

The contribution of the *i*(th) observation to the log-likelihood function for the Poisson maximum-likelihood estimator is Read more…

**mypoisson3.ado** adds options for a robust or a cluster–robust estimator of the variance–covariance of the estimator (VCE) to **mypoisson2.ado**, which I discussed in Programming an estimation command in Stata: Handling factor variables in a poisson command using Mata. **mypoisson3.ado** parses the **vce()** option using the techniques I discussed in Programming an estimation command in Stata: Adding robust and cluster–robust VCEs to our Mata based OLS command. Below, I show how to use **optimize()** to compute the robust or cluster–robust VCE.

I only discuss what is new in the code for **mypoisson3.ado**, assuming that you are familiar with **mypoisson2.ado**.

This is the twenty-second post in the series **Programming an estimation command in Stata**. I recommend that you start at the beginning. See Programming an estimation command in Stata: A map to posted entries for a map to all the posts in this series.

**A poisson command with options for a robust or a cluster–robust VCE**

**mypoisson3** computes Poisson-regression results in Mata. The syntax of the **mypoisson3** command is

**mypoisson3** *depvar* *indepvars* [if] [in] [**,** **vce(**__r__obust | __cl__uster *clustervar***)** __nocons__tant]

where *indepvars* can contain factor variables or time-series variables.

In the remainder of this post, I discuss Read more…

**mypoisson2.ado** handles factor variables and computes its Poisson regression results in Mata. I discuss the code for **mypoisson2.ado**, which I obtained by adding the method for handling factor variables discussed in Programming an estimation command in Stata: Handling factor variables in optimize() to **mypoisson1.ado**, discussed in Programming an estimation command in Stata: A poisson command using Mata.

This is the twenty-first post in the series **Programming an estimation command in Stata**. I recommend that you start at the beginning. See Programming an estimation command in Stata: A map to posted entries for a map to all the posts in this series.

**A Poisson command with Mata computations**

**mypoisson2** computes Poisson regression results in Mata. The syntax of the **mypoisson2** command is

**mypoisson2** *depvar indepvars* [*if*] [*in*] [**,** __nocons__tant]

where *indepvars* can contain factor variables or time-series variables.

In the remainder of this post, I discuss Read more…

\(

\newcommand{\xb}{{\bf x}}

\newcommand{\betab}{\boldsymbol{\beta}}\)I discuss a method for handling factor variables when performing nonlinear optimization using **optimize()**. After illustrating the issue caused by factor variables, I present a method and apply it to an example using **optimize()**.

This is the twenty post in the series **Programming an estimation command in Stata**. I recommend that you start at the beginning. See Programming an estimation command in Stata: A map to posted entries for a map to all the posts in this series.

**How poisson handles factor variables**

Consider the Poisson regression in which I include a full set of indicator variables created from Read more…

\(

\newcommand{\xb}{{\bf x}}

\newcommand{\betab}{\boldsymbol{\beta}}\)I discuss **mypoisson1**, which computes Poisson-regression results in Mata. The code in **mypoisson1.ado** is remarkably similar to the code in **myregress11.ado**, which computes ordinary least-squares (OLS) results in Mata, as I discussed in Programming an estimation command in Stata: An OLS command using Mata.

I build on previous posts. I use the structure of Stata programs that use Mata work functions that I discussed previously in Programming an estimation command in Stata: A first ado-command using Mata and Programming an estimation command in Stata: An OLS command using Mata. You should be familiar with Read more…

\(

\newcommand{\xb}{{\bf x}}

\newcommand{\betab}{\boldsymbol{\beta}}\)I show how to use **optimize()** in Mata to maximize a Poisson log-likelihood function and to obtain estimators of the variance–covariance of the estimator (**VCE**) based on independent and identically distributed (**IID**) observations or on robust methods.

This is the eighteenth post in the series **Programming an estimation command in Stata**. I recommend that you start at the beginning. See Programming an estimation command in Stata: A map to posted entries for a map to all the posts in this series.

**Using optimize()**

There are many optional choices that one may make when solving a nonlinear optimization problem, but there are very few that one must make. The **optimize*()** functions in Mata handle this problem by making a set of default choices for you, requiring that you specify a few things, and allowing you to change any of the default choices.

When I use **optimize()** to solve a Read more…

\(\newcommand{\betab}{\boldsymbol{\beta}}

\newcommand{\xb}{{\bf x}}

\newcommand{\yb}{{\bf y}}

\newcommand{\gb}{{\bf g}}

\newcommand{\Hb}{{\bf H}}

\newcommand{\thetab}{\boldsymbol{\theta}}

\newcommand{\Xb}{{\bf X}}

\)I review the theory behind nonlinear optimization and get more practice in Mata programming by implementing an optimizer in Mata. In real problems, I recommend using the **optimize()** function or **moptimize()** function instead of the one I describe here. In subsequent posts, I will discuss **optimize()** and **moptimize()**. This post will help you develop your Mata programming skills and will improve your understanding of how **optimize()** and **moptimize()** work.

This is the seventeenth post in the series **Programming an estimation command in Stata**. I recommend that you start at the beginning. See Programming an estimation command in Stata: A map to posted entries for a map to all the posts in this series.

**A quick review of nonlinear optimization**

We want to maximize a real-valued function \(Q(\thetab)\), where \(\thetab\) is a \(p\times 1\) vector of parameters. Minimization is done by maximizing \(-Q(\thetab)\). We require that \(Q(\thetab)\) is twice, continuously differentiable, so that we can use a second-order Taylor series to approximate \(Q(\thetab)\) in a neighborhood of the point \(\thetab_s\),

\[

Q(\thetab) \approx Q(\thetab_s) + \gb_s'(\thetab -\thetab_s)

+ \frac{1}{2} (\thetab -\thetab_s)’\Hb_s (\thetab -\thetab_s)

\tag{1}

\]

where \(\gb_s\) is the \(p\times 1\) vector of first derivatives of \(Q(\thetab)\) evaluated at \(\thetab_s\) and \(\Hb_s\) is the \(p\times p\) matrix of second derivatives of \(Q(\thetab)\) evaluated at \(\thetab_s\), known as the Hessian matrix.

Nonlinear maximization algorithms start with Read more…