All About Spherically Distributed Regression Errors

This post is based on a handout that I use for one of my courses, and it relates to the usual linear regression model,

                                  y = Xβ + ε

In our list of standard assumptions about the error term in this linear multiple regression model, we include one that incorporates both homoskedasticity and the absence of autocorrelation. That is, the individual values of the errors are assumed to be generated by a random process whose variance (σ²) is constant, and all possible distinct pairs of these values are uncorrelated. This implies that the full error vector, ε, has a scalar covariance matrix, σ²I_n. 

We refer to this overall situation as one in which the values of the error term follow a “Spherical Distribution”. Let's take a look at the origin of this terminology.

Good Old R-Squared!

My students are often horrified when I tell them, truthfully, that one of the last pieces of information that I look at when evaluating the results of an OLS regression, is the coefficient of determination (R²), or its "adjusted" counterpart. Fortunately, it doesn't take long to change their perspective!

After all, we all know that with time-series data, it's really easy to get a "high" R² value, because of the trend components in the data. With cross-section data, really low R² values are really common. For most of us, the signs, magnitudes, and significance of the estimated parameters are of primary interest. Then we worry about testing the assumptions underlying our analysis. R² is at the bottom of the list of priorities.

Finite Sample Properties of GMM

In a comment on a post earlier today,  Stephen Gordon quite rightly questioned the use of GMM estimation with relatively small sample sizes. The GMM estimator is weakly consistent, the "t-test" statistics associated with the estimated parameters are asymptotically standard normal, and the J-test statistic is asymptotically chi-square distributed under the null. But what can be said in finite samples?

Of course, this question applies to almost all of the estimators that we use in practice - IV, MLE, GMM, etc. Indeed, lots of work has been done to explore the finite-sample properties of such estimators. For instance, consider my own work on bias corrections for MLEs (see here, here, and here). So, I'm more than sympathetic to the general point that Stephen made.

Estimating an Euler Equation Using GMM

In one of my grad. econometrics courses we cover Generalized Method of Moments (GMM) estimation. I thought that some readers might be interested in the material that I use for one of the associated lab. classes.

The lab. exercise involves estimating the Euler equation associated with the "Consumption-Based Asset-Pricing Model" (e.g., Campbell, 1993, 1996.) This is a great example for illustrating GMM estimation, because the Euler equation is a natural "moment equation".

The basic statement of the problem is given below, taken from the handout that accompanies the lab. class exercises:

Some Official Data Come With Standard Errors!

Without intending to, I seem to have been on a bit of a rant about data quality and reliability recently! For example, see here, here, and here.

This post is about a related topic that's bugged me for a long time. It's to do with the measures of uncertainty that some statistical agencies (e.g., Statistics Canada) correctly report with some of their survey-based statistics.

A good example of what I have in mind is the Labour Force Survey (LFS) from Statistics Canada.

What (Some of) My Colleagues Are Up To

There's plenty of empirical research going on in the Department of Economics at the University of Victoria, where I work. Readers of the blog get to read plenty about what I've been doing, but what about other empirical work by some of my colleagues?

The following is a small cross-section of some of the quantitative papers that have been produced recently in this department. I've limited myself to very papers that are readily available for downloading, so not all of my empirically oriented colleagues are represented here - sorry!.

Confidence Intervals for Impulse Response Functions

An impulse response function gives the time-path for a variable explained in a VAR model, when one of the variables in the model is "shocked". We get a "picture" of how the variable in question responds to the shock over several periods of time.

An impulse response function (IRF) is essentially a type of conditional forecast. It's a messy function of the estimated coefficients in the VAR model, and the data. So, it's really just a point estimate, period by period. There's some uncertainty associated with the IRF, of course - this comes from the uncertainty associated with the estimated coefficients in the model. So, we really need to report a confidence band, period by period, to go with the IRF.

More on the Quality of Economic Data

Yesterday I posted two pieces relating to the quality of economic data, in general terms, and with reference to China.

I'm firmly of the view that we need to be paying more attention to data quality than we currently do as economists. We also need to keep in mind that data are frequently revised, and this has implications for policy conclusions based on preliminary figures.

To help you with your reading on this topic, here's a small selection of papers that touch on different aspects of this topic:

Now That the Semester is Over....

Another teaching term is done, and the exams are all graded!

HT to my colleague, Emma Hutchinson, for this timely item:

Bias Reduction Paper Published

Another of our papers on bias reduction for Maximum Likelihood estimators has now been published. This one is titled, "On the Bias of the Maximum Likelihood Estimator for the Two-Parameter Lomax Distribution", and is co-authored with Ryan Godwin and Helen Feng. It's in Vol. 42 (11) of Communications in Statistics - Theory and Methods, and is available here.

This paper stems from an ongoing research program with Helen, Ryan, and others. Other posts relating to this program can be found here and here. There's more of this on the way!

© 2013, David E. Giles