More on Regression Coefficient Interpretation

I get a lot of direct email requests from people wanting help/guidance/advice of various sorts about some aspect of econometrics or other. I like being able to help when I can, but these requests can lead to some pitfalls -  for both of us.

More on that in a moment. Meantime, today I got a question from a Ph.D student, "J", which was essentially the following:

" Suppose I have the following regression model

             log(y_i) = α + βX_i + ε_i    ;  i = 1, 2, ...., n .

How do interpret the (estimated) value of β?"

I think most of you will know that the answer is:

"If X changes by one unit, then y changes by (100*β)%".

If you didn't know this, then some trivial partial differentiation will confirm it. And after all, isn't partial differentiation something that grad. students in ECON should be good at?

Specifically,

      β = [∂log(y_i) / ∂X_i] = [∂logy_i / ∂y_i][∂y_i / ∂X_i] = [∂y_i  / ∂X_i] / y_i,

which is the proportional change in y for a unit change in X. Multiplying by 100 puts the answer into percentage terms.

So, I responded to "J" accordingly.

So far, so good.

But then I got a response:

"Actually, my model includes an interaction term, and really it looks like this:

    log(y_i) = α + βX_i + γ [X_iΔlog(Z_i)] + ε_i    ;  i = 1, 2, ...., n.

How do I interpret β?"

Whoa! That's not the question that was first asked - and now my previous answer (given in good faith) is totally wrong! 

Let's do some partial differentiation again, with this full model. We still have:

[∂log(y_i) / ∂X_i] = [∂logy_i / ∂y_i][∂y_i / ∂X_i] = [∂y_i  / ∂X_i] / y_i.

However, this expression now equals [β + γ Δlog(Z_i)].

So, a one unit change in X leads to a percentage change in y that's equal to 100*[β + γ Δlog(Z_i)]%.

This percentage change is no longer constant - it varies as Z takes on different sample values. If you wanted to report a single value you could evaluate the expression using the estimates for β and γ, and either the sample average, or sample median, value for Δlog(Z).

This illustrates one of the difficulties that I face sometimes. I try to respond to a question, but I really don't know if the question being asked is the appropriate one; or if it's been taken out of context; or if the information I'm given is complete or not.

If you're a grad. student, then discussing your question in person with your supervisor should be your first step!

© 2018, David E. Giles

Interpreting Dummy Variable Coefficients After Non-Linear Transformations

Dummy variables - ones that take only the values zero and one - are commonly used as regressors in regression models. I've devoted several posts to discussing various aspects of such variables, notably here, but also here, here, and here.

When the regression model in question is linear, in both the variables and the parameters, the interpretation of coefficient of such a dummy variable is simple. Suppose that the model takes the form:

    y_i = α + β D_i + Σ_j γ_j X_ji + ε_i     ;     E(ε_i ) = 0   ;   i = 1, ...., n.                          (1)

The range of summation in the term on the right-hand side of (1) is from 1 to k, if there are k regressors in addition to the dummy variable, D. (There is no loss of generality in assuming a single dummy regressor in what follows, and no further distributional assumptions about the error term will be needed or used.)

As you'll know, if D_i = 0, then the intercept coefficient in (1) is just α; and it shifts to (α + β) if D_i = 1. It changes by an amount equal to β, and so does the predicted mean value of y. Conversely, this amount changes by -β  if D_i changes from 1 to 0. Estimating (1) by OLS will give us an estimate of the effect on y of D_i sw from 0 to 1 in value, or vice versa.

But a bit more on estimation issues below!


Another way of interpreting what is going on is to think about the growth rate in the expected value of y that is implied when D changes its value. Setting D_i = 0, and then D_i = 1, this growth rate is:

   g_01i = [ (α + β + Σ_j γ_j X_ji) - (α + Σ_j γ_j X_ji)] / (α + Σ_j γ_j X_ji) = [β /  (α + Σ_j γ_j X_ji)] ,

which you can multiply by 100 to convert it into a percentage rate of growth, if you wish. 

Note that this growth rate depends on the other parameters in the model, and also on the sample values for the other regressors. 

Conversely, when D changes in value from 1 to 0, this growth rate is different, namely:

   g_10i = - [β / (α + β + Σ_j γ_j X_ji)]                            (i = 1, ...., n).

In this fully linear model these growth rates offer a somewhat less appealing way of summarizing what is going on than does the amount of change in the expected value of y. The latter doesn't depend on the other parameters of the model, or on the sample values of the regressors.

However, this situation can change very quickly once we move to a regression model that is non-linear, either in the variables or in the parameters (or both). 

That's what I want to focus on in this post. 

Let's consider some interesting examples that involve common transformations of the dependent variable in a regression model. Apart from anything else, such transformations are often undertaken to make the assumption of a normally distributed error term more reasonable.

Some Reading Suggestions for July

Some summertime reading:

• Chen, T., DeJuan, J., & R. Tian, 2018. Distributions of GDP across versions of  the Penn World Tables: A functional data analysis approach. Economics Letters, in press. 
• Clements, K.W., H. Liu, & Y. Tarverdi, 2018. Alcohol consumption, censorship and misjudgment. Applied Economics, online
• Jin, H., S. Zhang, J. Zhang,& H. Hao, 2018. Modified tests for change points in variance in the possible presence of mean breaks. Journal of Statistical Computation and Simulation, online
• Pata, U.K., 2018. The Feldstein Horioka puzzle in E7 countries: Evidence from panel cointegration and asymmetric causality analysis. Journal of International Trade and Economic Development, online.
• Sen, A., 2018. A simple unit root testing methodology that does not require knowledge regarding the presence of a break. Communications in Statistics - Simulation and Computation, 47, 871-889.
• Wright, T., M. Klein, &K. Wieczorek, 2018. A primer on visualizations for comparing populations, including the issue of overlapping confidence intervals. American Statistician, online.

© 2018, David E. Giles

Dummy Variables in a Semilogarithmic Regression: Exact Distributional Results

For better or worse, semilogarithmic regression models are used a lot in empirical economics. 

It would be nice to think that this is because the researcher found that a logarithmic transformation of the model's dependent variable led to residuals that were more "normally" distributed than without the transformation. Unfortunately, however, it's often just "for convenience". With this transformation, the estimates of the regression coefficients have a simple interpretation, as explained below

I hate it when the latter situation arises. I've long since lost track of the number of times I've been at a seminar where the speaker has used this "simple interpretation" as an excuse for their choice of a semilogarithmic regression specification. For goodness sake, the choice of the model's functional form should be based on more than "convenience"!

For some of my previous comments about this point, see this post.

Most of you will know that when our semilogarithmic model includes a dummy (zero-one) regressor, we have to be careful about how we interpret that regressor's estimated coefficient. Suppose that we have the following regression model, where D is a dummy variable, and the X's are regresssors that are measured "continuously"

   ln(y_i) = α + β D_i + Σ_j γ_j X_ji + ε_i     ;     E(ε_i ) = 0   ;   i = 1, ...., n.                         

Note that there's no loss of generality here in having just one dummy variable in the model.

Then, the interpretation of the regression coefficients is:

1. A one-unit change in X_j leads to a proportional change of  γ_j (or a percentage change of 100γ_j) in y.
2. When the dummy variable changes from D = 0 to D = 1, the proportional change in y is [exp(β) -1]. Conversely, going from D = 1 to D = 0 implies a proportional change in y of  [exp(-β) -1]. Again, multiply by 100 to get a percentage change.

See Halvorsen and Palmquist (1980) for an explanation of the second of these results, and my comments in this earlier post.

Kennedy (1981) and Giles (1982) discuss the issue of estimating this proportional change in the case of the dummy variable. Their results relate to point estimation - with a focus on unbiased estimation of the proportional change, when the model's errors are normally distributed..

But what about interval estimation of this effect? 

Shout-Out for Marc Bellemare

If you don't follow Marc Bellemare's blog (shame on you - you should!), then you may not have caught up with his recent posts relating to his series of lectures on "Advanced Econometrics - Causal Inference With Observational Data" at the University of Copenhagen in May of this year.

Marc kept us all on tenterhooks by "releasing" the slides for these lectures progressively - smart move!

Update, 23 July 2018 -

However, all of the eight lectures in the series are now available for downloading:

• Lecture 1: Introduction
• Lecture 2: Causality
• Lecture 3: Instrumental Variables
• Lecture 4: Panel Data & Differences-in-Differences
• Lecture 5: Regression Discontinuity Designs
• Lecture 6: Nonstandard Dependent Variables
• Lecture 7: Tricks f the Trade, I
• Lecture 8: Tricks of the Trade, II

Thanks for sharing them, Marc.

© 2018, David E. Giles

The Series of Unsurprising Results in Economics (SURE)

Andrea Menclover of the University of Canterbury (New Zealand) has recently founded the SURE Journal, whose aims and scope are as follows:

'The Series of Unsurprising Results in Economics (SURE) is an e-journal of high-quality research with “unsurprising” findings. We publish scientifically important and carefully-executed studies with statistically insignificant or otherwise unsurprising results. Studies from all fields of Economics will be considered. SURE is an open-access journal and there are no submission charges. (My emphasis, DG.)

SURE benefits readers by:

• Mitigating the publication bias and thus complementing other journals in an effort to provide a complete account of the state of affairs;
• Serving as a repository of potential (and tentative) “dead ends” in Economics research.

SURE benefits writers by:

• Providing an outlet for interesting, high-quality, but “risky” (in terms of uncertain results) research projects;
• Decreasing incentives to data-mine, change theories and hypotheses ex post or exclusively focus on provocative topics.'

To find out more or to submit a manuscript, visit: http://surejournal.org.

This is a novel venture that has a lot to offer at a time when research replicability and publication bias are (rightly) receiving so much attention.

I'm delighted to be associated with the new journal as a member of its Editorial Board.

© 2018, David E. Giles

Suggested Reading for June

• Colussi, T., 2018. Social ties in academia: A friend is a treasure. Review of Economics and Statistics, 100, 45-50.
• Dette, H., K. Möllenhoff, S. Volgushev, & F. Bretz, 2018. Equivalence of regression curves. Journal of the American Statistical Association, online.
• Kumbhaker, S. C., C. F. Parmeter, & V. Zelenyuk, 2018. Stochastic frontier analysis: Foundations and advances. Working paper No. WP02/2018, Centre for Efficiency and productivity Analysis, School of Economics, University of Queensland.
• Morshed, A. A., E. Andreou, & O. Boldea, 2018. Structural break tests robust to regression misspecification. Econometrics, 6, 27.
• Zhang, L., 2018. Spurious regressions with high-order models: A reconsideration. Economics Letters, 168, 70-72.
• Zhao, S., A. Ullah, & X. Zhang, 2018. A class of model averaging estimators. Working Paper Series 18-11, Rimini Centre for Economic Analysis.

© 2018, David E. Giles

The Uniqueness of the Cointegrating Vector

Suppose that we have (only) two non-stationary time-series, X_1t and X_2t (t = 1, 2, 3, .....). More specifically, suppose that both of these series are integrated of order one (i.e., I(1)). Then there are two possibilities - either X₁ and X₂ are cointegrated, or they aren't.

You'll recall that if they are cointegrated, then there is a linear combination of X₁ and X₂ that is stationary. Let's write this linear combination as Z_t = (X_1t + αX_2t). (We can normalize the first "weight" to the value "one" without any loss of generality.) The vector whose elements are 1 and α is the so-called "cointegrating vector".

You may be aware that if such a vector exists, then it is unique.

Recently, I was asked for a simple proof of this uniqueness. Here goes.........

Results of the Econometric Game, 2018

In a recent post I mentioned the 2018 "edition" of The Econometric Game, which was held in Amsterdam earlier this month.

In random order, the finalists, after the first two days' of competition, were the teams representing:

Aarhus University
Erasmus Universiteit Rotterdam
Harvard University
Lund University
McGill University
Universiteit van Tilburg
Universiteit van Amsterdam
University Carlos III Madrid
University of Bristol
University of Toronto

These teams then competed in a further one-day event..

The team from University Carlos III Madrid emerged the winner; with those from Harvard University and Aarhus University taking second and third places respectively.

The organizers of The Game have provided a gallery of photos. here   

Congratulations to all involved for another impressive event!

© 2018, David E. Giles

April Reading

Very belatedly, here is my list of suggested reading for April:

• Biørn, E., 2017. Identification, instruments, omitted variables, and rudimentary models: Fallacies in the "experimental approach" to econometrics. Memorandum No. 13/2017, Department of Economics, Oslo University.
• Chambers, M. J., and M. Kyriacou, 2018. Jackknife bias reduction in the presence of a near-unit root. Econometrics, 6, 11.
• Derryberry, D., K. Aho, J. Edwards, and T. Peterson, 2018. Model selection and regression t-statistics. American Statistician, in press.
• Mitchell, J., D. Robertson, and S. Wright, 2018. R² bounds for predictive models: What univariate properties tell us about multivariate predictability. Journal of Business and Economic Statistics, in press. (Free download here.)
• Parker, T., 2017. Finite-sample distributions of the Wald, likelihood ratio, and Lagrange multiplier test statistics in the classical linear model. Communications in Statistics - Theory and Methods, 46, 5195-5202.
• Troster, V., 2018. Testing Granger-causality in quantiles. Econometric Reviews, 37, 850-866.

© 2018, David E. Giles