March Madness in the Reading Department

It's time for the monthly round-up of recommended reading material.

• Gan, L. and J. Jiang, 1999. A test for global maximum. Journal of the American Statistical Association, 94, 847-854.
• Nowak-Lehmann, F., D. Herzer, S. Vollmer, and I. Martinez-Zarzosa, 2006. Problems in applying dynamic panel data models: Theoretical and empirical findings. Discussion Paper Nr. 140, IAI, Georg-August-Universität Göttingen.
• Olive, D. J., 2004. Does the MLE maximize the likelihood? Mimeo., Department of Mathematics, Southern Illinois University. 
• Pollock, D. S. G., 2014. Econometrics: An historical guide for the uninitiated. Working Paper No. 14/05, Department of Economics, University of Leicester.
• Terrell, G. R., 2002. The gradient statistic. Interface 2002: Computing Science and Statistics, Vol. 34.
• Wald, A., 1940. The fitting of straight lines if both variables are subject to error. Annals of Mathematical Statistics, 11, 284-300.

© 2014, David E. Giles

Forecasting from a Regression Model

There are several reasons why we estimate regression models, one of them being to generate forecasts of the dependent variable. I'm certainly not saying that this is the most important or the most interesting use of such models. Personally, I don't think this is the case.

So, why is this post about forecasting? Well, a few comments and questions that I've had from readers of this blog suggest to me that not all students of econometrics are completely clear about certain issues when it comes to using regression models for forecasting.

Let's see if we can clarify some terms that are used in this context, and in the process clear up any misunderstandings.

So Much Good Reading........

Here are my latest reading suggestions:

• Choi, I., 2013. Panel Cointegration. Working Paper, Department of Economics, Sogang University, Korea.
• Davidson, R. and J. G. MacKinnon, 2013. Bootstrap tests for overdentification in linear regression models. Economics Department Working Paper No. 1318, Queen's University.
• Deng, A., 2013. Understanding spurious regression in financial econometrics. Journal of Financial Econometrics, in press.
• Feng, C., H. Wang, Y. Han, and Y. Xia, 2013. The mean value theorem and Taylor's expansion in statistics. The American Statistician, in press.
• Kiviet, J. F. and G. D. A. Phillips, 2013. Improved variance estimation of maximum likelihood estimation in stable first-order dynamic regression models. EGC Report No. 2012/06, Division of Economics, Nanyang Technical University.
• Lanne, M., M. Meitz, and P. Saikkonen, 2013. Testing for linear and nonlinear predicatability of stock returns. Journal of Financial Econometrics, 11, 682-705.

© 2013, David E. Giles

Some More Papers for Your "To Read" List

For better, or worse, here are some of the papers I've been reading lately:

• Chambers, M. J., J. S. Ercolani, and A. M. R. Taylor, 2013. Testing for seasonal unit roots by frequency domain regression. Journal of Econometrics, in press. 
• Chicu, M. and M. A. Masten, 2013. A specification test for discrete choice models. Economics Letters, in press. 
• Hansen, P. R. and A. Lunde, 2013. Estimating the persistence and the autocorrelation function of a time series that is measured with error. Econometric Theory, in press.
• Liu, Y., J. Liu, and F. Zhang, 2013. Bias analysis for misclassificaiton in a multicategorical exposure in a logistic regression model. Statistics and Probability Letters, in press.
• Thornton, M., 2013, The aggregation of dynamic relationships caused by incomplete information. Journal of Econometrics, in press.
• Wang, H. and S. Z. F. Zhou, 2013. Interval estimation by frequentist model averaging. Communications in Statistics - Theory and Methods, in press.   

© 2013, David E. Giles

ARDL Models - Part II - Bounds Tests

[Note: For an important update of this post, relating to EViews 9, see my 2015 post, here.]

Well, I finally got it done! Some of these posts take more time to prepare than you might think.

The first part of this discussion was covered in a (sort of!) recent post, in which I gave a brief description of Autoregressive Distributed Lag (ARDL) models, together with some historical perspective. Now it's time for us to get down to business and see how these models have come to play a very important role recently in the modelling of non-stationary time-series data.

In particular, we'll see how they're used to implement the so-called "Bounds Tests", to see if long-run relationships are present when we have a group of time-series, some of which may be stationary, while others are not. A detailed worked example, using EViews, is included.

When is an Autoregressive Model Dynamically Stable?

Autoregressive processes arise frequently in econometrics. For example, we might have a simple dynamic model of the form:

            y_t = β₀ + β₁y_t-1 + ε_t   ;   ε_t ~ i.i.d.[0 , σ²]       .           (1)

Or, we might have a regression model in which everything is "standard", except that the errors follow an autoregressive process:

            y_t = β₀ + β₁x_t + u_t               (2)

             u_t = ρ u_t-1 + ε_t    ;  ε_t ~ i.i.d.[0 , σ²] .

In each of these examples a first-order autoregressive, or AR(1), process is involved.

Higher-order AR processes are also commonly used. Although most undergrad. econometrics students are familiar with the notion of "stationarity" in the context of an AR(1) process, often they're not aware of the conditions needed to ensure the stationarity of more general AR models. Let's take a look at this issue.

ARDL Models - Part I

I've been promising, for far too long, to provide a post on ARDL models and bounds testing. Well, I've finally got around to it!

"ARDL" stands for "Autoregressive-Distributed Lag". Regression models of this type have been in use for decades, but in more recent times they have been shown to provide a very valuable vehicle for testing for the presence of long-run relationships between economic time-series.

I'm going to break my discussion of ARDL models into two parts. Here, I'm going to describe, very briefly, what we mean by an ARDL model. This will then provide the background for a second post that will discuss and illustrate how such models can be used to test for cointegration, and estimate long-run and short-run dynamics, even when the variables in question may include a mixture of stationary and non-stationary time-series.

Is the Cochrane-Orcutt Estimator Unique?

One of the work-horses of econometric modelling is the Cochrane-Orcutt (1949) estimator, or some variant of it such as the Beach-MacKinnon (1978) full ML estimator. The C-O estimator was proposed by Cochrane and Orcutt as a modification to OLS estimation when the errors are autocorrelated. Those authors had in mind errors that follow an AR(1) process, but it is easily adapted for any AR process.

I've blogged elsewhere about the the historical setting for the work by Cochrane and Orcutt.

Given the limited computing power available at the time, the C-O estimator was a pragmatic solution to the problem of obtaining the GLS estimator of the regression coefficients, and approximating the full ML estimator. Students of econometrics will be familiar with the iterative process associated with the C-O estimator, as outlined below.

The use of this estimator leads to some interesting questions. Is this iterative scheme guaranteed to converge in a finite number of iterations? Is there a unique solution to this convergence problem, or can multiple local solutions (minima) occur?