What NOT To Do When Data Are Missing

Here's something that's very tempting, but it's not a good idea.

Suppose that we want to estimate a regression model by OLS. We have a full sample of size n for the regressors, but one of the values for our dependent variable, y, isn't available. Rather than estimate the model using just the (n - 1) available data-points, you might think that it would be preferable to use all of the available data, and impute the missing value for y.

Fair enough, but what imputation method are you going to use?

For simplicity, and without any loss of generality, suppose that the model has a single regressor,

             

                y_i = β x_i + ε_i ,                                                                       (1)

 and it's the n^th value of y that's missing. We have values for x₁, x₂, ...., x_n; and for y₁, y₂, ...., y_n-1.

Here's a great idea! OLS will give us the Best Linear Predictor of y, so why don't we just estimate (1) by OLS, using the available (n - 1) sample values for x and y; use this model (and x_n) to get a predicted value (y*_n) for y_n; and then re-estimate the model with all n data-points: x₁, x₂, ...., x_n; y₁, y₂, ...., y_n-1, y*_n.

Unfortunately, this is actually a waste of time. Let's see why.

Reading List for October

Some suggestions for the coming month:

• Franses, P. H., 2016. A note on the mean absolute scaled error. International Journal of Forecasting, 32, 20-22.
• Gorroochurn, P., 2015. On Galton's change from 'reversion' to 'regression'. American Statistician, in press.
• Holan, S. H., R. Lund, and G. Davis, 2010. The ARMA alphabet soup: A tour of ARMA model variants. Statistics Surveys, 4, 232-274.
• Liu, S., H. Wu, and W. Q. Meeker, 2015. Understanding and addressing the unbounded "likelihood" problem. American Statistician, 69, 191-200.
• McCracken, M. W. and S. Ng, 2015.  FRED-MD: A monthly database for macroeconomic research. Journal of Business and Economic Statistics, in press.
• Müller, U. K. and M. W. Watson, 2015. Low-frequency econometrics. NBER Working Paper No. 21564.

© 2015, David E. Giles

Thanks, Dan!

Quite out of the blue, Dan Getz kindly sent me a nice LaTeX version of my hand-written copy of "The Solution", given in my last post.

Dan used the ShareLaTeX site - https://www.sharelatex.com/ .

So, here's a nice pdf file with The Solution.

Thanks a million, Dan - that was most thoughtful of you!

© 2015, David E. Giles

The Solution

You can find a solution to the problem posed in yesterday's post here.

I hope you can read my writing!

p.s.: Dan Getz kindly supplied a LaTeX version - here's the pdf file. Thanks, Dan!

© 2015, David E. Giles

Try This Problem

Here's a little exercise for you to work on:

We know from the Gauss-Markhov Theorem that within the class of linear and unbiased estimators, the OLS estimator is most efficient. Because it is unbiased, it therefore has the smallest possible Mean Squared Error (MSE) within the linear and unbiased class of estimators. However, there are many linear estimators which, although biased, have a smaller MSE than the OLS estimator. You might then think of asking: 

“Why don’t I try and find the linear estimator that has the smallest possible MSE?”

(a) Show that attempting to do this yields an “estimator” that can’t actually be used in practice.

(You can do this using the simple linear regression model without an intercept, although the result generalizes to the usual multiple linear regression model.)

(b) Now, for the simple regression model with no intercept, 

         y_i = β x_i + ε_i       ;     ε_i ~ i.i.d. [0 , σ²] ,

find the linear estimator, β* , that minimizes the quantity:

h[Var.(β*) / σ²] + (1 - h)[Bias(β*)/ β]² , for 0 < h < 1.

Is  β* a legitimate estimator, in the sense that it can actually be applied in practice?

The answer will follow in a subsequent post.

© 2015, David E. Giles

September Reading List

• Abeln, B. and J. P. A. M. Jacobs, 2015. Seasonal adjustment with and without revisions: A comparison of X-13ARIMA-SEATS and CAMPLET. CAMA Working Paper 25/2015, Crawford School of Public Policy, Australian National University.
• Chan, J. C. C. and A. L. Grant, 2015. A Bayesian model comparison for trend-cycle decompositions of output. CAMA Working Paper 31/2015, Crawford School of Public Policy, Australian National University.
• Chen, K. and K-S. Chan, 2015. A note on rank reduction in sparse multivariate regression. Journal of Statistical Theory and Practice, in press.
• Fan, Y., S. Pastorello, and E. Renault, 2015. Maximization by parts in extremum estimation. Econometrics Journal, 18, 147-171.
• Horowitz, J., 2014. Variable selection and estimation in high-dimensional models. Cemmap Working Paper CWP35/15, Institute of Fiscal Studies, Department of Economics, University College London.
• Larson, W., 2015. Forecasting an aggregate in the presence of structural breaks in the disaggregates. RPF Working Paper No. 2015-002, Research Program on Forecasting, Center of Economic Research, George Washington University.

© 2015, David E. Giles

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