Autoregressive (AR) and movingaverage (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 quasimaximum 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 simulationbased 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 twentythird 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 loglikelihood function for the Poisson maximumlikelihood 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 twentysecond 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 Poissonregression results in Mata. The syntax of the mypoisson3 command is
mypoisson3 depvar indepvars [if] [in] [, vce(robust  cluster clustervar) noconstant]
where indepvars can contain factor variables or timeseries 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 twentyfirst 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] [, noconstant]
where indepvars can contain factor variables or timeseries 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 Poissonregression results in Mata. The code in mypoisson1.ado is remarkably similar to the code in myregress11.ado, which computes ordinary leastsquares (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 adocommand 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 loglikelihood 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 realvalued 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 secondorder 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…
I show how to use the undocumented command _vce_parse to parse the options for robust or clusterrobust estimators of the variancecovariance of the estimator (VCE). I then discuss myregress12.ado, which performs its computations in Mata and computes VCE estimators based on independently and identically distributed (IID) observations, robust methods, or clusterrobust methods.
myregress12.ado performs ordinary leastsquares (OLS) regression, and it extends myregress11.ado, which I discussed in Programming an estimation command in Stata: An OLS command using Mata. To get the most out of this post, you should be familiar with Programming an estimation command in Stata: Using a subroutine to parse a complex option and Programming an estimation command in Stata: Computing OLS objects in Mata.
This is the sixteenth 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.
Parsing the vce() option
I used adosubroutines to simplify the parsing of the options vce(robust) and vce(cluster cvarname) in myregress10.ado; see Programming an estimation command in Stata: Using a subroutine to parse a complex option. Part of the point was to Read more…
I have posted a series of entries about programming an estimation command in Stata. They are best read in order. The comprehensive list below allows you to read them from first to last, at your own pace.

Programming estimators in Stata: Why you should
To help you write Stata commands that people want to use, I illustrate how Stata syntax is predictable and give an overview of the estimationpostestimation structure that you will want to emulate in your programs.

Programming an estimation command in Stata: Where to store your stuff
I discuss the difference between scripts and commands, and I introduce some essential programming concepts and constructions that I use to write the scripts and commands.

Programming an estimation command in Stata: Global macros versus local macros
I discuss a pair of examples that illustrate the differences between global macros and local macros.

Programming an estimation command in Stata: A first adocommand
I discuss the code for a simple estimation command to focus on the details of how to implement an estimation command. The command that I discuss estimates the mean by the sample average. I begin by reviewing the formulas and a dofile that implements them. I subsequently introduce Read more…