Overview

In this post, I show how to use mlexp to estimate the degree of freedom parameter of a chi-squared distribution by maximum likelihood (ML). One example is unconditional, and another example models the parameter as a function of covariates. I also show how to generate data from chi-squared distributions and I illustrate how to use simulation methods to understand an estimation technique. Read more…

Overview

A Monte Carlo simulation (MCS) of an estimator approximates the sampling distribution of an estimator by simulation methods for a particular data-generating process (DGP) and sample size. I use an MCS to learn how well estimation techniques perform for specific DGPs. In this post, I show how to perform an MCS study of an estimator in Stata and how to interpret the results.

Large-sample theory tells us that the sample average is a good estimator for the mean when the true DGP is a random sample from a \(\chi^2\) distribution with 1 degree of freedom, denoted by \(\chi^2(1)\). But a friend of mine claims this estimator will not work well for this DGP because the \(\chi^2(1)\) distribution will produce outliers. In this post, I use an MCS to see if the large-sample theory works well for this DGP in a sample of 500 observations. Read more…

Two-step estimation problems can be solved using the gmm command.

When a two-step estimator produces consistent point estimates but inconsistent standard errors, it is known as the two-step-estimation problem. For instance, inverse-probability weighted (IPW) estimators are a weighted average in which the weights are estimated in the first step. Two-step estimators use first-step estimates to estimate the parameters of interest in a second step. The two-step-estimation problem arises because the second step ignores the estimation error in the first step.

One solution is to convert the two-step estimator into a one-step estimator. My favorite way to do this conversion is to stack the equations solved by each of the two estimators and solve them jointly. This one-step approach produces consistent point estimates and consistent standard errors. There is no two-step problem because all the computations are performed jointly. Newey (1984) derives and justifies this approach. Read more…

Last time, I showed you a way to graph and to think about matrices. This time, I want to apply the technique to eigenvalues and eigenvectors. The point is to give you a picture that will guide your intuition, just as it was previously.

Before I go on, several people asked after reading part 1 for the code I used to generate the graphs. Here it is, both for part 1 and part 2: matrixcode.zip. Read more…

I want to show you a way of picturing and thinking about matrices. The topic for today is the square matrix, which we will call A. I’m going to show you a way of graphing square matrices, although we will have to limit ourselves to the 2 x 2 case. That will be, as they say, without loss of generality. The technique I’m about to show you could be used with 3 x 3 matrices if you had a better 3-dimensional monitor, and as will be revealed, it could be used on 3 x 2 and 2 x 3 matrices, too. If you had more imagination, we could use the technique on 4 x 4, 5 x 5, and even higher-dimensional matrices. Read more…

The manual entry for xtmixed documents all the official features in the command, and several applications. However, it would be impossible to address all the models that can be fitted with this command in a manual entry. I want to show you how to include covariates in a crossed-effects model.

Let me start by reviewing the crossed-effects notation for xtmixed. I will use the homework dataset from Kreft and de Leeuw (1998) (a subsample from the National Education Longitudinal Study of 1988). You can download the dataset from the webpage for Rabe-Hesketh & Skrondal (2008) (http://www.stata-press.com/data/mlmus2.html), and run all the examples in this entry. Read more…

The fourth quarter Stata News came out today. Among other things, it contains an article by Bobby Gutierrez, StataCorp’s Director of Statistics, about competing risks survival analysis. If any of you are like me, conversant in survival analysis but not an expert, I think you will enjoy Bobby’s article. In a mere page and a half, I learned the primary differences between competing risks analysis and the Cox proportional hazards model and why I will sometimes prefer competing risks. Bobby’s article can be read at http://www.stata.com/news/statanews.25.4.pdf.