\(\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…

I show how to use the undocumented command _vce_parse to parse the options for robust or cluster-robust estimators of the variance-covariance 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 cluster-robust methods.

**myregress12.ado** performs ordinary least-squares (**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 ado-subroutines 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 estimation-postestimation 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 ado-command
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 do-file that implements them. I subsequently introduce Read more…

I discuss a command that computes ordinary least-squares (**OLS**) results in Mata, paying special attention to the structure of Stata programs that use Mata work functions.

This command builds on several previous posts; at a minimum, you should be familiar with Programming an estimation command in Stata: A first ado-command using Mata and Programming an estimation command in Stata: Computing OLS objects in Mata.

This is the fifteenth 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.

**An OLS command with Mata computations**

The Stata command **myregress11** computes the results in Mata. The syntax of the **myregress11** command is

**myregress11** *depvar* [*indepvars*] [if] [in] [, __nocons__tant]

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

In the remainder of this post, I discuss the code for **myregress11.ado**. I recommend that you click on the file name to download the code. To avoid scrolling, view the code in the do-file editor, or your favorite text editor, to see the line numbers.

I do not discuss Read more…

We often use probit and logit models to analyze binary outcomes. A case can be made that the logit model is easier to interpret than the probit model, but Stata’s **margins** command makes any estimator easy to interpret. Ultimately, estimates from both models produce similar results, and using one or the other is a matter of habit or preference.

I show that the estimates from a probit and logit model are similar for the computation of a set of effects that are of interest to researchers. I focus on the effects of changes in the covariates on the probability of a positive outcome for continuous and discrete covariates. I evaluate these effects on average and at the mean value of the covariates. In other words, I study the average marginal effects (AME), the average treatment effects (ATE), the marginal effects at the mean values of the covariates (MEM), and the treatment effects at the mean values of the covariates (TEM).

First, I present the results. Second, I discuss the code used for the simulations.

**Results**

In Table 1, I present the results of a simulation with 4,000 replications when the true data generating process (DGP) satisfies the assumptions of a probit model. I show the Read more…

\(\newcommand{\epsilonb}{\boldsymbol{\epsilon}}

\newcommand{\ebi}{\boldsymbol{\epsilon}_i}

\newcommand{\Sigmab}{\boldsymbol{\Sigma}}

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

\newcommand{\eb}{{\bf e}}

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

\newcommand{\xbit}{{\bf x}_{it}}

\newcommand{\xbi}{{\bf x}_{i}}

\newcommand{\zb}{{\bf z}}

\newcommand{\zbi}{{\bf z}_i}

\newcommand{\wb}{{\bf w}}

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

\newcommand{\ub}{{\bf u}}

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

\newcommand{\Mb}{{\bf M}}

\newcommand{\Xtb}{\tilde{\bf X}}

\newcommand{\Wb}{{\bf W}}

\newcommand{\Vb}{{\bf V}}\)I present the formulas for computing the ordinary least-squares (**OLS**) estimator and show how to compute them in Mata. This post is a Mata version of Programming an estimation command in Stata: Using Stata matrix commands and functions to compute OLS objects. I discuss the formulas and the computation of independence-based standard errors, robust standard errors, and cluster-robust standard errors.

This is the fourteenth 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.

**OLS formulas**

Recall that the **OLS** point estimates are given by

\[

\widehat{\betab} =

\left( \sum_{i=1}^N \xb_i’\xb_i \right)^{-1}

\left(

\sum_{i=1}^N \xb_i’y_i

\right)

\]

where \(\xb_i\) is the \(1\times k\) vector of independent variables, \(y_i\) is the dependent variable for each of the \(N\) sample observations, and the model for \(y_i\) is

\[

y_i = \xb_i\betab’ + \epsilon_i

\]

If the \(\epsilon_i\) are independently and identically distributed (**IID**), we estimate Read more…

I discuss a sequence of ado-commands that use Mata to estimate the mean of a variable. The commands illustrate a general structure for Stata/Mata programs. This post builds on Programming an estimation command in Stata: Mata 101, Programming an estimation command in Stata: Mata functions, and Programming an estimation command in Stata: A first ado-command.

This is the thirteenth 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 Mata in ado-programs**

I begin by reviewing the structure in **mymean5.ado**, which I discussed Read more…

I show how to write a function in Mata, the matrix programming language that is part of Stata. This post uses concepts introduced in Programming an estimation command in Stata: Mata 101.

This is the twelfth 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.

**Mata functions**

Commands do work in Stata. Functions do work in Mata. Commands operate on Stata objects, like variables, and users specify options to alter the behavior. Mata functions accept arguments, operate on the arguments, and may return a result or alter the value of an argument to contain a result.

Consider **myadd()** defined below.

**Code block 1: myadd()**
mata:
function myadd(X, Y)
{
A = X + Y
return(A)
}
end

**myadd()** accepts two arguments, **X** and **Y**, puts the sum of **X** and **Y** into **A**, and returns **A**. For example, Read more…

I introduce Mata, the matrix programming language that is part of Stata.

This is the eleventh 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. Read more…

I make two improvements to the command that implements the ordinary least-squares (**OLS**) estimator that I discussed in Programming an estimation command in Stata: Allowing for options. First, I add an option for a cluster-robust estimator of the variance-covariance of the estimator (**VCE**). Second, I make the command accept the modern syntax for either a robust or a cluster-robust estimator of the **VCE**. In the process, I use subroutines in my ado-program to facilitate the parsing, and I discuss some advanced parsing tricks.

This is the tenth 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. Read more…