Biased Estimation of Marginal Effects

I began a recent post with the comment:

"One thing that a lot of practitioners seem to be unaware of (or they choose to ignore it) is that in many of the common situations where we use regression analysis to estimate elasticities, these estimators are biased.

And that's true even if all of the conditions needed for the coefficient estimator (e.g., OLS) to be unbiased are fully satisfied."

Exactly the same point can be made in respect of estimated marginal effects, and that's what this post is about.

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.

The Bias of Certain Elasticity Estimators

In a recent post I discussed some aspects of estimating elasticities from regression models, and the interpretation of these values. That discussion should be kept in mind in reading what follows.

One thing that a lot of practitioners seem to be unaware of (or they choose to ignore it) is that in many of the common situations where we use regression analysis to estimate elasticities, these estimators are biased.

And that's true even if all of the conditions needed for the coefficient estimator (e.g., OLS) to be unbiased are fully satisfied.

Let's look at some common situations leading to the estimation of elasticities and marginal effects, and see if we can summarize what's going on.

Econometric Society World Congress

The Econometric Society holds a World Congress every five years. Right now, the 2015 Congress is taking place in Montréal, Canada.

Here's the full program. Enjoy!

© 2015, David E. Giles

Classic Data Visualizations

My thanks to Veronica Johnson at Investech.com for drawing my attention a recent piece of theirs relating to Classic Data Visualizations.

As they say:

"A single data visualization graphic can be priceless. It can save you hours of research. They’re easy to read, interpret, and, if based on the right sources, accurate, as well.  And with the highly social nature of the web, the data can be lighthearted, fun and presented in so many different ways. 

What’s most striking about data visualizations though is that they aren’t as modern a concept as we tend to think they are. 

In fact, they go back to more than 2,500 years—before computers and tools for easy visual representation of data even existed."

Here are the eleven graphics that they highlight:

Symmetry and Skewness

After taking your first introductory course in statistics you probably agreed wholeheartedly with the following statement:

"A statistical distribution is symmetric if and only if it is not skewed."

After all, isn't that how we define "skewness"?

In fact, that statement is incorrect. There are distributions which have a skewness coefficient of zero, but are asymmetric.

Before considering some examples of this phenomenon, let's take a closer look at the meaning of "skewness" in the statistical context.

The H-P Filter and Unit Roots

The Hodrick-Prescott (H-P) filter is widely used for trend removal in economic time-series, and as a basis for business cycle analysis, etc. I've posted about the H-P filter before (e.g., here).

There's a widespread belief that application of the H-P filter will not only isolate the deterministic trend in a series, but it will also remove stochastic trends - i.e., unit roots. For instance, you'll often hear that if the H-P filter is applied to quarterly data, the filtered series will be stationary, even if the original series is integrated of order up to 4.

Is this really the case?

Let's take a look at two classic papers relating to this topic, and a very recent one that provides a bit of an upset.

Estimating Elasticities, All Over Again

I had some interesting email from Andrew a while back to do with computing elasticities from log-log regression models, and some related issues.

In his first email, Andrew commented:

"I am interested in the elasticity of H with respect to W, e.g., hours with respect to wages. For simplicity, assume that W is randomly assigned, and that the elasticity is identical for everyone.

Standard practice would be to regress log(H) on a constant and log(W). The coefficient on log(W) then seems to be the elasticity, as it estimates d log(H) / d log(W).

But changes in log( ) are only equal to changes in percent in the limit as the changes go to zero. In practice, one typically uses discrete data. Because the changes in W may be large, the resulting coefficient is just a first order approximation of the elasticity, and is not identical to the true elasticity."

Let's focus on the third paragraph. Keep in mind that log( ), here, refers to "natural" (base 'e') logarithms.

Andrew is quite correct, and this is something that we often overlook when teaching econometrics, or when interpreting someone's regression results. I sometimes refer students to this useful piece by Kenneth Benoit. Here's a key extract from p.4:

August reading

Here's my (slightly delayed) August reading list:

• Ahelegbey, A. F., 2015. The econometrics of networks: A review. Working Paper  2015/13, Department of Economics, University of Venice.
• Clemens, M. A., 2015. The meaning of failed replications: A review and proposal. IZA Discussion Paper No.9000.
• Fair, R. C., 2015. Information limits of aggregate data. Discussion Paper No. 2011, Cowles Foundation, Yale University.
• Phillips, P. C. B., 2015. Inference in near singular regression. Discussion Paper No. 2009, Cowles Foundation, Yale University.
• Stock, J. H. and M. W. Watson, 2015. Core inflation and trend inflation. NBER Working Paper 21282.
• Ullah, A. and X. Zhang, 2015. Grouped model averaging for finite sample size. Working paper, Department of Economics, University of California, Riverside.

© 2015, David E. Giles