A simulation-based explanation of consistency and asymptotic normality

Overview

In the frequentist approach to statistics, estimators are random variables because they are functions of random data. The finite-sample distributions of most of the estimators used in applied work are not known, because the estimators are complicated nonlinear functions of random data. These estimators have large-sample convergence properties that we use to approximate their behavior in finite samples.

Two key convergence properties are consistency and asymptotic normality. A consistent estimator gets arbitrarily close in probability to the true value. The distribution of an asymptotically normal estimator gets arbitrarily close to a normal distribution as the sample size increases. We use a recentered and rescaled version of this normal distribution to approximate the finite-sample distribution of our estimators.

I illustrate the meaning of consistency and asymptotic normality by Monte Carlo simulation (MCS). I use some of the Stata mechanics I discussed in Monte Carlo simulations using Stata.

Consistent estimator

A consistent estimator gets arbitrarily close in Read more…

Programming an estimation command in Stata: Certifying your command

\(\newcommand{\xb}{{\bf x}}
\newcommand{\betab}{\boldsymbol{\beta}}\)Before you use or distribute your estimation command, you should verify that it produces correct results and write a do-file that certifies that it does so. I discuss the processes of verifying and certifying an estimation command, and I present some techniques for writing a do-file that certifies mypoisson5, which I discussed in previous posts.

This is the twenty-fifth 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.

Verification versus certification

Verification is the process of establishing Read more…

Fitting distributions using bayesmh

This post was written jointly with Yulia Marchenko, Executive Director of Statistics, StataCorp.

As of update 03 Mar 2016, bayesmh provides a more convenient way of fitting distributions to the outcome variable. By design, bayesmh is a regression command, which models the mean of the outcome distribution as a function of predictors. There are cases when we do not have any predictors and want to model the outcome distribution directly. For example, we may want to fit a Poisson distribution or a binomial distribution to our outcome. This can now be done by specifying one of the four new distributions supported by bayesmh in the likelihood() option: dexponential(), dbernoulli(), dbinomial(), or dpoisson(). Previously, the suboption noglmtransform of bayesmh‘s option likelihood() was used to fit the exponential, binomial, and Poisson distributions to the outcome variable. This suboption continues to work but is now undocumented.

For examples, see Beta-binomial model, Bayesian analysis of change-point problem, and Item response theory under Remarks and examples in [BAYES] bayesmh.

We have also updated our earlier “Bayesian binary item response theory models using bayesmh” blog entry to use the new dbernoulli() specification when fitting 3PL, 4PL, and 5PL IRT models.

Programming an estimation command in Stata: Making predict work

I make predict work after mypoisson5 by writing an ado-command that computes the predictions and by having mypoisson5 store the name of this new ado-command in e(predict). The ado-command that computes predictions using the parameter estimates computed by ado-command mytest should be named mytest_p, by convention. In the next section, I discuss mypoisson5_p, which computes predictions after mypoisson5. In section Storing the name of the prediction command in e(predict), I show that storing the name mypoisson5_p in e(predict) requires only a one-line change to mypoisson4.ado, which I discussed in Programming an estimation command in Stata: Adding analytical derivatives to a poisson command using Mata.

This is the twenty-fourth post in the Read more…

How to generate random numbers in Stata

Overview

I describe how to generate random numbers and discuss some features added in Stata 14. In particular, Stata 14 includes a new default random-number generator (RNG) called the Mersenne Twister (Matsumoto and Nishimura 1998), a new function that generates random integers, the ability to generate random numbers from an interval, and several new functions that generate random variates from nonuniform distributions.

Random numbers from the uniform distribution

In the example below, we use runiform() to create Read more…

Programming an estimation command in Stata: Adding analytical derivatives to a poisson command using Mata

\(\newcommand{\xb}{{\bf x}}
\newcommand{\betab}{\boldsymbol{\beta}}\)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…

Programming an estimation command in Stata: Allowing for robust or cluster–robust standard errors in a poisson command using Mata

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(robust | cluster clustervar) noconstant]

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

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

Vector autoregression—simulation, estimation, and inference in Stata

\(\newcommand{\epsb}{{\boldsymbol{\epsilon}}}
\newcommand{\mub}{{\boldsymbol{\mu}}}
\newcommand{\thetab}{{\boldsymbol{\theta}}}
\newcommand{\Thetab}{{\boldsymbol{\Theta}}}
\newcommand{\etab}{{\boldsymbol{\eta}}}
\newcommand{\Sigmab}{{\boldsymbol{\Sigma}}}
\newcommand{\Phib}{{\boldsymbol{\Phi}}}
\newcommand{\Phat}{\hat{{\bf P}}}\)Vector autoregression (VAR) is a useful tool for analyzing the dynamics of multiple time series. VAR expresses a vector of observed variables as a function of its own lags.

Simulation

Let’s begin by simulating a bivariate VAR(2) process using the following specification,

\[
\begin{bmatrix} y_{1,t}\\ y_{2,t}
\end{bmatrix}
= \mub + {\bf A}_1 \begin{bmatrix} y_{1,t-1}\\ y_{2,t-1}
\end{bmatrix} + {\bf A}_2 \begin{bmatrix} y_{1,t-2}\\ y_{2,t-2}
\end{bmatrix} + \epsb_t
\]

where \(y_{1,t}\) and \(y_{2,t}\) are the observed series at time \(t\), \(\mub\) is a \(2 \times 1\) vector of intercepts, \({\bf A}_1\) and \({\bf A}_2\) are \(2\times 2\) parameter matrices, and \(\epsb_t\) is a \(2\times 1\) vector of innovations that is uncorrelated over time. I assume a \(N({\bf 0},\Sigmab)\) distribution for the innovations \(\epsb_t\), where \(\Sigmab\) is a \(2\times 2\) covariance matrix.

I set my sample size to 1,100 and Read more…

Programming an estimation command in Stata: Handling factor variables in a poisson command using Mata

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] [, noconstant]

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

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

Testing model specification and using the program version of gmm

This post was written jointly with Joerg Luedicke, Senior Social Scientist and Statistician, StataCorp.

The command gmm is used to estimate the parameters of a model using the generalized method of moments (GMM). GMM can be used to estimate the parameters of models that have more identification conditions than parameters, overidentified models. The specification of these models can be evaluated using Hansen’s J statistic (Hansen, 1982).

We use gmm to estimate the parameters of a Poisson model with an endogenous regressor. More instruments than regressors are available, so the model is overidentified. We then use estat overid to calculate Hansen’s J statistic and test the validity of the overidentification restrictions.

In previous posts Read more…