Evaluating Direct Multi-Step Forecasts

Specialeforsvar ved Jan Owe Bauer

Titel: Evaluating Direct Multi-Step Forecasts

Abstract: Within this thesis, we will go through the theoretical results of the publication \Evaluating Direct Multi-Step Ahead Forecasts" by Clark and McCracken (2005a). They constructed four tests for equal forecast accuracy under multi-step ahead forecasting and derived the respective asymptotic null distributions. The models to be considered for forecasting are two nested regression models. There will be two approaches for considering the predictive power of a model and, hence, two approaches for constructing the tests: The Mean Squared Error (MSE) and forecast encompassing. Both approaches will be used to formulate a t-type and an F-type test, which gives us the resulting four tests overall. Note that we will further derive the asymptotic null distributions for two cases for each test: Under the case when the in-sample part is vast compared to the out-of-sample part; and under the case when it is not. It will then turn out that the limit distributions consist either of standard Gaussian distributions for the first, or of stochastic integrals of Brownian motion for the latter case, respectively.
Since it is reasonable to assume that the forecast errors are autocorrelated for multistep ahead prediction, consistency of our estimators will be a concern. We will therefore make use of Andrews' (1991) results to show that our estimators for the scaling parameters needed for the tests, and a variance estimator entering the limit distributions, are consistent indeed. This will be done by using kernel functions and by taking the link between autocovariances and spectral densities into account.
Note that we will not only show the consistency, but also give an interpretation of the construction of the scaling parameters. This will be especially interesting for the one entering the limit distributions.
Further, we will challenge Clark and McCracken's results regarding the limit distributions: While the authors propose to integrate over [_; 1], we rather state that one should integrate over [_; !T ], with !T < 1, that is to say we conclude a different interval for integration. On the other hand, our results for the Gaussian distributions will not di_er. This is quite surprising and might be due to a technical aspect. Since this thesis is based on Clark and McCracken's publication, we will refer to \Clark and McCracken (2005a)" simply as \Clark and McCracken", and to Clark and McCracken (2005b), which is the \Not-for-Publication Appendix to Evaluating Direct Multi-Step Ahead Forecasts", as \Non-Published Appendix" for convenience purposes.
The structure of the thesis will be the following: We will first recap the theoretical framework of Clark and McCracken in Chapter 1. It will also contain assumptions needed to construct the tests and to derive the limit distributions. In there, we will use slightly stronger restrictions in Assumption 3, which are needed to show the consistency of our scaling parameters; this will be done in Chapter 3. Beforehand, we will construct the four test statistics in Chapter 2. In Chapter 4, there will be the relevant theory provided, needed for deriving the asymptotic null distribution. Small changes of Clark and McCracken's results will already be made in Chapter 4; this will indicate our discrepancy later. Note that some of the proofs and calculus can be found in Appendix A and Appendix B. Finally, in Chapter 5, we will derive the limit distributions and state our deviations from Clark and McCracken's results and provide some thought-provoking extensions in Chapter 6.

Vejleder:    Rasmus Søndergaard Pedersen, Ø.I.
Censor:       Jesper Lund, CBS