May Reading List

Here's a selection of suggested reading for this month:

• Athey, S. & G. W. Imbens, 2019. Machine learning methods economists should know about. Mimeo.
• Bhagwat, P. & E. Marchand, 2019. On a proper Bayes but inadmissible estimator. American Statistician, online.
• Canals, C. & A. Canals, 2019. When is n large enough? Looking for the right sample size to estimate proportions. Journal of Statistical Computation and Simulation, 89, 1887-1898.
• Cavaliere, G. & A. Rahbek, 2019. A primer on bootstrap testing of hypotheses in time series models: With an application to double autoregressive models. Discussion Paper 19-03, Department of Economics, University of Copenhagen.
• Chudik, A. & G. Geogiardis, 2019. Estimation of impulse response functions when shocks are observed at a higher frequency than outcome variables. Globalization Institute Working Paper 356, Federal Reserve Bank of Dallas.
• Reschenhofer, E., 2019. Heteroscedasticity-robust estimation of autocorrelation. Communications in Statistics - Simulation and Computation, 48, 1251-1263.

© 2019, David E. Giles

Some April Reading for Econometricians

Here are my suggestions for this month:

• Hyndman, R. J., 2019. A brief history of forecasting competitions. Working Paper 03/19, Department of Econometrics and Business Statistics, Monash University.
• Kuffner, T. A. & S. G. Walker, 2019. Why are p-values controversial?. American Statistician, 73, 1-3.
• Sargan, J. D.,, 1958. The estimation of economic relationships using instrumental variables. Econometrica, 26, 393-415. (Read for free online.)  
• Sokal, A. D., 1996. Transgressing the boundaries: Towards a trasnformative hermeneutics of quantum gravity. Social Text, 46/47, 217-252.
• Zeng, G. & Zeng, E., 2019. On the relationship between multicollinearity and separation in logistic regression. Communications in Statistics - Simulation and Computation, published online.
• Zhang, X., S. Paul, & Y-G. Yang, 2019. Small sample bias correction or bias reduction? Communications in Statistics - Simulation and Computation, published online.

© 2019, David E. Giles

New Year Reading Suggestions for 2019

With a new year upon us, it's time to keep up with new developments -

• Basu, D., 2018. Can we determine the direction of omitted variable bias of OLS estimators? Working Paper 2018-16, Department of Economics, University of Massachusetts, Amherst.
• Jiang, B., Y. Lu, & J. Y. Park, 2018. Testing for stationarity at high frequency. Working Paper 2018-9, Department of Economics, University of Sydney. 
• Psaradakis, Z. & M. Vavra, 2018. Normality tests for dependent data: Large-sample and bootstrap approaches. Communications in Statistics - Simulation and Computation, online.
• Spanos, A., 2018. Near-collinearity in linear regression revisited: The numerical vs. the statistical perspective. Communications in Statistics - Theory and Methods, online.
• Thorsrud, L. A., 2018. Words are the new numbers: A newsy coincident index of the business cycle. Journal of Business Economics and Statistics, online. (Working Paper version.)
• Zhang, J., 2018. The mean relative entropy: An invariant measure of estimation error. American Statistician, online.

© 2019, David E. Giles

Monte Carlo Simulation Basics, III: Regression Model Estimators

This post is the third in a series of posts that I'm writing about Monte Carlo (MC) simulation, especially as it applies to econometrics. If you've already seen the first two posts in the series (here and here) then you'll know that my intention is to provide a very elementary introduction to this topic. There are lots of details that I've been avoiding, deliberately.

In this post we're going to pick up from where the previous post about estimator properties based on the sampling distribution left off. Specifically, I'll be applying the ideas that were introduced in that post in the context of regression analysis. We'll take a look at the properties of the Least Squares estimator in three different situations. In doing so, I'll be able to illustrate, through simulation, some "text book" results that you'll know about already.

If you haven't read the immediately preceding post in this series already, I urge you to do so before continuing. The material and terminology that follow will assume that you have.

Monte Carlo Simulation Basics, II: Estimator Properties

In the early part of my recent post on this series of posts about Monte Carlo (MC) simulation, I made the following comments regarding its postential usefulness in econometrics:

".....we usually avoid using estimators that are are "inconsistent". This implies that our estimators are (among other things) asymptotically unbiased. ......however, this is no guarantee that they are unbiased, or even have acceptably small bias, if we're working with a relatively small sample of data. If we want to determine the bias (or variance) of an estimator for a particular finite sample size (n), then once again we need to know about the estimator's sampling distribution. Specifically, we need to determine the mean and the variance of that sampling distribution. 

If we can't figure the details of the sampling distribution for an estimator or a test statistic by analytical means - and sometimes that can be very, very, difficult - then one way to go forward is to conduct some sort of MC simulation experiment."

Before proceeding further, let's recall just what we mean by a "sampling distribution". It's a very specific concept, and not all statisticians agree that it's even an interesting one.

March Reading List

Now is a good time to catch up on some Econometrics reading. Here are my suggestions for this month:

• Carrasco, M. and R. Kotchoni, 2016. Efficient estimation using the characteristic function. Econometric Theory, in press.
• Chambers, M. J., 2016. The estimation of continuous time models with mixed frequency data. Discussion Paper No. 777, Department of Economics, University of Essex.
• Cuaresma, J. C., M. Feldkircher, and F. Huber, 2016. Forecasting with global vector autoregressive models: A Bayesian approach. Journal of Applied Econometrics, in press.
• Hendry, D., 2016. Deciding between alternative approaches in macroeconomics. Discussion Paper No. 778, Department of Economics, University of Oxford.
• Reed, W. R., 2016. Univariate unit root tests perform poorly when data are cointegrated. Working Paper No. 1/2016, Department of Economics and Finance, University of Canterbury.

© 2016, David E. Giles

The Distribution of a Ratio of Correlated Normals

Suppose that the random variables X₁ and X₂ are jointly distributed as bivariate Normal, with means of θ₁ and θ₂, variances of σ₁² and σ₂² respectively, and a correlation coefficient of ρ.

In this post we're going to be looking at the distribution of the ratio, W = (X₁ / X₂).

You probably know that if X₁ and X₂ are independent standard normal variables, then W follows a Cauchy distribution. This will emerge as a special case in what follows.

The more general case that we're concerned with is of interest to econometricians for several reasons.

Maximum Likelihood Estimation & Inequality Constraints

This post is prompted by a question raised by Irfan, one of this blog's readers, in some email correspondence with me a while back.

The question was to do with imposing inequality constraints on the parameter estimates when applying maximum likelihood estimation (MLE). This is something that I always discuss briefly in my graduate econometrics course, and I thought that it might be of interest to a wider audience.

Here's the issue.

Regression Coefficients & Units of Measurement

A linear regression equation is just that - an equation. This means that when any of the variables - dependent or explanatory - have units of measurement, we also have to keep track of the units of measurement for the estimated regression coefficients.

All too often this seems to be something that students of econometrics tend to overlook.

Consider the following regression model:

               y_i = β₀ + β₁X_1i + β₂x_2i + β₃x_3i + ε_i    ;    i = 1, 2, ...., n                   (1)

where y and x₂ are measured in dollars; x₁ is measured in Kg; and x₃ is a unitless index.

Because the term on the left side of (1) has units of dollars, every term on the right side of that equation must also be expressed in terms of dollars. These terms are β₀, (β₁x_1i), (β₂x_2i), (β₃x_3i), and ε_i.

In turn, this implies that β₀ and β₃ have units which are dollars; the units of β₁ are ($ / Kg); and β₂ is unitless. In addition, the error term, ε, has units that are dollars, and so does its standard deviation, σ.

What are some of the implications of this?

Recent Reading

From my "Recently Read" list:

• Born, B. and J. Breitung, 2014. Testing for serial correlation in fixed-effects panel data models. Econometric Reviews, in press.
• Enders, W. and Lee. J., 2011. A unit root test using a Fourier series to approximate smooth breaks, Oxford Bulletin of Economics and Statistics, 74, 574-599.
• Götz, T. B. and A. W. Hecq, 2014. Testing for Granger causality in large mixed-frequency VARs. RM/14/028, Maastricht University, SBE, Department of Quantitative Economics.
• Kass, R. E., 2011. Statistical inference: The big picture. Statistical Science, 26, 1-9.
• Qian, J. and L. Su, 2014. Structural change estimation in time series regressions with endogenous variables. Economics Letters, in press.
• Wickens, M., 2014. How did we get to where we are now? Reflections on 50 years of macroeconomic and financial econometrics. Discussion Paper No. 14/17, Department of Economics and Related Studies, University of York.

© 2014, David E. Giles

Estimation & Accuracy After Model Selection

This was the title of Brad Efron's invited paper at the 2014 Joint Statistical Meetings in Boston this morning. It was a great presentation, with excellent discussants - Lan Wang, Lawrence Brown, and Soumendra Lahiri.

The paper and discussion are scheduled to appear in the September 2014 issue of JASA.

A lot of what Brad and his discussants had to say related to one of the main points in one of my recent posts. Namely, if you search for a model specification, then this affects all of your subsequent inferences - and usually in a rather complicated way. Typically, even after searching for a preferred model, we tend to "pretend" that we haven't done this, and that the model's form was known from the outset. Naughty! Naughty!

What Brad has done is to address the "pre-test" issue in a rather nice way. You won't be surprised to learn that bootstrapping features heavily in the methodology that he's developed. Using two examples - one non-parametric, and one parametric - he showed how to take account of model selection via Mallows' C_p statistic, and the lasso (respectively), when constructing regression confidence intervals.

One important feature of his analysis involves "smoothing" the results to take account of the discontinuities that inherent in model selection. Although it wasn't mentioned in Brad's talk, these discontinuities are the source of some of the most important problems associated with pre-testing in general. For example, traditional pre-test estimators of regression coefficients (based, say, on a prior test of linear restrictions on those coefficients) are inadmissible under a range of standard loss functions. This inadmissibility is entirely due to the fact that these pre-test estimators are discontinuous functions of the random sample data.

All in all it was a great session, with some nice take-away quotes:

• "The discussants actually discussed my paper."
• "Simulations are hard to do."
• "Model averaging is perfectly easy to do, but model selection is not."

I took some comfort from the last two of these comments!

© 2014, David E. Giles

Unbiased Estimation of a Standard Deviation

Frequently, we're interested in using sample data to obtain an unbiased estimator of a population variance. We do this by using the sample variance, with the appropriate correction for the degrees of freedom. Similarly, in the context of a linear regression model, we use the sum of the squared OLS residuals, divided by the degrees of freedom, to get an unbiased estimator of the variance of the model's error term.

But what if we want an unbiased estimator of the population standard deviation, rather than the variance?

Variance Estimators That Minimize MSE

In this post I'm going to look at alternative estimators for the variance of a population. The following discussion builds on a recent post, and once again it's really directed at students. Well, for the most part.

Actually, some of the results relating to populations that are non-Normal probably won't be familiar to a lot of readers. In fact, I can't think of a reference for where these results have been assembled in this way previously. So, I think there's some novelty here. But we'll get to that in due course.

I can just imagine you smacking your lips in anticipation!

What's the Variance of a Sample Variance?

This post is really pitched at students who are taking a course or two in introductory economic statistics. It relates to a couple of estimators of the variance of a population that we all meet in such courses - plus another one that you might not have met. In addition, I'll be emphasising the fact that some "standard" results depend crucially on certain assumptions. Not surprisingly - but  not always made clear by instructors and text books.

What's Your Favourite Estimator?

It's interesting to dwell on the popularity of different estimators that econometricians use. Some estimators are "in vogue" for a period, and then give way to others as new developments come along. Different topics have captured the attention of theoreticians and practitioners alike at different times in history.

Here's a Google Ngram showing the extent to which some familiar estimators for simultaneous equations models have been mentioned in books since 1960:

Not too surprisingly, good old OLS just goes on and on:

I was going to include the GMM estimator in these plots, but this acronym has meanings other than the obvious one that comes to mind. So, the results would have been misleading. To be safe, let's use the full phrase Generalized Method of Moments and allow for case sensitivity:

Interestingly, the phrase appeared in some books before the publication of Hansen's classic 1982 paper.

© 2013, David E. Giles

Being Unbiased Isn't Everything!

When we first learn about estimation, we encounter various properties that estimators might possess. Unless your first course in statistics or econometrics takes a fully Bayesian stance, then these properties will be ones based on the sampling distribution of the statistic that is being used as the estimator.

There are plenty of unsettling things that can be raised against the notion of the sampling distribution, but let's put those to one side here. In elementary courses, attention usually focuses on just the mean and variance of an estimator's sampling distribution. I'm not endorsing this - it's just a fact of life.

Seminar on Pre-test Estimation & Testing

Last Friday I gave a seminar in the Department of Mathematics and Statistics, here at UVic. The Statistics seminar series is always very enjoyable, and I really enjoy interacting with this friendly and capable group.

My talk was titled, "The Effects of Prior Hypothesis Testing on the Sampling Properties of Estimators and Tests: An Overview". Preliminary test ( or pre-test) estimation (& testing) was a research topic that I was heavily involved in for about a decade, from the mid 1980's to the mid 1990's. A lot of that work was done with Judith Clarke. I've been looking at some related problems again recently.

If you're interested in this topic, you'll find the slides from my talk here.

© 2013, David E. Giles

Measuring the Quality of an Estimator

In which, with almost no symbols, I encourage students and practitioners to question what they've been taught............

When it comes to introducing our students to the notion of the "quality" of an estimator, most of us begin by observing that estimators are functions of the random sample data, and hence they are "statistics" in the literal sense. As such, estimators have a probability distribution. We give this distribution a special name - the "sampling distribution" of the estimator in question.

It's understandable that students sometimes find the concept of the sampling distribution a little tricky when they first encounter it. After all, it's based on a "thought game" of sorts. We have to consider the idea of repeatedly drawing samples of a fixed size, for ever, constructing the statistic in question, and then keeping track of all of the possible values that the statistic can take, together with the relative frequency of occurrence for each value. A Monte Carlo experiment is the obvious way to introduce students to this concept.

Assessing Heckman's Two-Step Estimator

Good survey papers are worth their weight in gold. Reading and digesting a thoughtful, constructive, and well-researched survey can save you a lot of work. It can also save you from making poor choices in your own research, or even from "re-inventing the wheel".

For these reasons, The Journal of Economic Surveys is a great resource. Over the years it has published some really fine peer-reviewed survey articles, many of which I've benefited from personally.

Another piece of good news is that Wiley (the journal's publisher) makes a number of the most highly-cited articles available for free.

Let's be Consistent

One of the standard, large-sample, properties that we hope our estimators will possess is "consistency". Indeed, most of us take the position that if an estimator isn't consistent, then we should probably throw it away and look for one that is!

When you're talking about the consistency of an estimator, it's a really good idea to be quite clear regarding the precise type of consistency you have in mind - especially if you're talking to a statistician! For example, there's "weak consistency", "strong consistency", "mean square consistency", and "Fisher consistency", at least some of which you'll undoubtedly encounter from time to time as an econometrician.