The Forecasting Performance of Models for Cointegrated Data

Here's an interesting practical question that arises when you're considering different forms of econometric models for forecasting time-series data:

"Which type of model will perform best when the data are non-stationary, and perhaps cointegrated?"

To answer this question we have to think about the alternative models that are available to us; and we also have to decide on what we mean by 'best'. In other words, we have to agree on some sort of loss function or performance criterion for measuring forecast quality.

Notice that the question I've posed above allows for the possibility that the data that we're using are integrated, and the various series we're working with may or may not be cointegrated. This scenario covers a wide range of commonly encountered situations in econometrics.

In an earlier post I discussed some of the basic "mechanics" of forecasting from an Error Correction Model. This type of model is used in the case where our data are non-stationary and cointegrated, and we want to focus on the short-run dynamics of the relationship that we're modelling. However, in that post I deliberately didn't take up the issue of whether or not such a model will out-perform other competing models when it comes to forecasting.

Let's look at that issue here.

Beyond MSE - "Optimal" Linear Regression Estimation

In a recent post I discussed the fact that there is no linear minimum MSE estimator for the coefficients of a linear regression model. Specifically, if you try to find one, you end up with an "estimator" that is non-operational, because it is itself a function of the unknown parameters of the model. It's note really an estimator at all, because it can't be computed.

However, by changing the objective of the exercise slightly, a computable "optimal estimator" can be obtained. Let's take a look at this.

A Regression "Estimator" that Minimizes MSE

Let's talk about estimating the coefficients in a linear multiple regression model. 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?”

More Interesting Papers to Read

Here's my latest list of suggested reading:

• Bayer, C. and C. Hanck, 2012. Combining non-cointegration tests. Journal of Time Series Analysis, DOI: 10.1111/j.1467-9892.2012.814.x 
• Cipollina, M., L. De Benedictis, L. Salvatici, and C. Vicarelli, 2013.  A note on dummies for policies in gravity models: A Monte Carlo experiment. Working Paper no. 180, Dipartimento di Economia, Università degli studi Roma Tre.
• Fair, R. C., 2013. Reflections on macroeconometric modelling. Cowles Foundation Discussion Paper No. 1908, Yale University.
• Kourouklis, S., 2012. A new estimator of the variance based on minimizing mean squared error. The American Statistician, 66, 234-236.
• Kulish, M. and A. R. Pagan, 2013. Issues in estimating new-Keynesian Phillips curves in the presence of unknown structural change. Research Discussion Paper, RDP 2012-11, Reserve Bank of Australia.
• Little, R. J., 2013. In praise of simplicity, not mathematistry! Ten simple powerful ideas for the statistical scientist. Journal of the American Statistical Association, 108, 359-369.
• Zhang, L., X. Xu, and G. Chen, 2012. The exact likelihood ratio test for equality of two normal populations. The American Statistician, 66, 180-184.

© 2013, David E. Giles

What Have You Been Reading?

Here are some of the papers that I was reading last week:

• Arel-Bundock, V., 2013. A solution to the weak instrument bias in 2SLS estimation: Indirect inference with stochastic approximation, Economics Letters, in press.
• Behar, R., P. Grima, and L. Marco-Almagro, 2013. Twenty-five analogies for explaining statistical concepts. American Statistician, 67(1), 44-48.
• Chang, C-L., P. H. Frances, and M. McAleer, 2013, Are forecast updates progressive? MPRA Paper No. 46387.
• Chortareas, G., and G. Kapetanios, 2013. How puzzling is the PPP puzzle? An alternative half-life measure of convergence to PPP. Journal of Applied Econometrics, 28, 435-457.
• Davidson, R. and J. B. MacKinnon, 1998. Graphical methods for investigating the size and power of hypothesis tests. Manchester School, 66, 1-26.
• Hood, W. C. and T. C. Koopmans, 1953. Studies in Econometric Method. Cowles Commission Monograph for Research in Economics, Monograph No. 14. Wiley, New York.
• Kourouklis, S., 2012. A new estimator of the variance based on minimizing mean squared error. American Statistician, 66(4), 234-236.
• Lanne, M. and P. Saikkonen, 2013. Noncausal vector autoregression. Econometric Theory, 29, 447-482.

© 2013, David E. Giles

Minimum MSE Estimation of a Regression Model

Students of econometrics encounter the Gauss-Markhov Theorem (GMT) at a fairly early stage - even if they don't see a formal proof to begin with. This theorem deals with a particular property of the OLS estimator of the coefficient vector, β, in the following linear regression model:

                        y = Xβ + ε  ;  ε ~ [0 , σ² I_n] ,

where X is (n x k), non-random, and of rank k.

The GMT states that among all linear estimators of β that are also unbiased estimators, the OLS estimator of β is most efficient. That is, OLS is the BLU estimator for β.

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!