**On Equal Predictive Ability and Parallelism of Self-Exciting Threshold Autoregressive Model**

**ABSTRACT**

Several authors have developed statistical procedures for testing whether two models are similar. In this work, we not only present the notion of equivalence but also extend this to a measure of predictive ability of a time series following a stationary self-exciting threshold autoregressive (SETAR) process. A proposition and a lemma were used to join the structure of the predictability measure to the coefficients and sample autocorrelation of the SETAR process. Illustrative examples are given to show how to conduct the test which can help practitioners avoid mistakes in decision making.

TABLE OF CONTENTS

Certification …………………………………………………………………………………………..ii

Dedication ……………………………………………………………………………………………..iii

Acknowledgements …………………………………………………………………………………iv

Abstract …………………………………………………………………………………………………vi

Chapter One: INTRODUCTION

1.1 Introduction ………………………………………………………………………………………1

1.2 Statement of Problem………………………………………………………………………….4

1.3 Research Objectives………………………………………………………………………..4

1.4 Significance of the Study……………………………………………………………………..5

1.5 Scope of the Study………………………………………………………………………………5

Chapter Two: LITERATURE REVIEW

2.1 Review of Related Literatures………………………………………………………………7

Chapter Three: METHODOLOGY

3.1 Method…………………………………………………………………………………………….14

3.2 Definition of Basic Concepts……………………………………………………………….15

3.3 R2 Defined for SETAR Time Series Models…………………………………………..17

3.3.1 Parallelism and Equal Predictive Ability……………………………………………..20

3.3.2 Testing for Equal Predictive Ability……………………………………………………23

Chapter Four: SOME APPLICATIONS

4.1 Numerical Examples…………………………………………………………………………..25

4.1.1 Example 1………………………………………………………………………………………25

4.1.2 Example 2………………………………………………………………………………………34

4.1.3 Example 3………………………………………………………………………………………42

Chapter Five: SUMMARY, CONCLUSION AND RECOMMENDATIONS

5.1 Summary…………………………………………………………………………………………..51

5.2 Conclusion………………………………………………………………………………………..52

5.3 Recommendations……………………………………………………………………………..52

REFERENCES

**CHAPTER ONE**

**INTRODUCTION**

Popularization and extensive research for linear time series modeling began in 1927 with Yule’s Autoregressive models, used in studying sunspot numbers. In the decades that followed, these models have been successfully applied in different fields, this is because as far as one-step ahead prediction is concerned, linear time series models are often adequate.

However, this is not always so as can be seen from the Sunspot numbers (which will be discussed later). The causes of this are mentioned later herein.

Nonlinear time series analysis gained attention in the 1970’s. The interest grew due to the need to model nonlinear changes in everyday time series data exhibiting nonlinearity.

Autoregressive Integrated Moving Average (ARIMA) models cannot describe adequately limit cycles, time-irreversibility, amplitude-frequency dependency and jump phenomena (the Sunspot numbers mentioned earlier is a good example). As a result, Tong (1978) came up with a procedure for modelling nonlinear changes in time series data in which different Autoregressive (AR) processes are functioning, and the switch between these AR models depends on the delay parameter and threshold value(s), which are certain time lag values from the given time series. Tong and Lim (1980) and Tong (1983) followed up the work with an extensive description of the procedure. Tsay (1989) proposed a much simpler procedure. Tsay (1989) noted that the Tong’s (1983) procedure is not statistically adequate for formally determining if a given data can be described using a threshold model (see Tsay 1989).

Several nonlinear time series models (Nonlinear Autoregressive model (AR) and Closed loop Threshold Autoregressive model (TARSC)) have been proposed over the years and the Threshold Autoregressive (TAR) models, which is the piece-wise linearization of nonlinear models over the state space by the introduction of the thresholds fro; :::; rig, has

been of significant interest because of its ability to model nonlinear data adequately. Common notion were employed by Priestly (1965), and Ozaki and Tong (1975), in the analysis of non-stationary time series and time dependent systems, in which local stationarity was the counterpart of our present local linearity. The overall process is nonlinear when there are at least two regimes with different parameters and/or process order. Tong and Lim (1980) proposed the following requirements for the modeling of nonlinear time series, in order of preference: statistical identification of an appropriate model should not entail excessive computation; they should be general enough to capture some of the nonlinear phenomena mentioned previously; one-step-ahead prediction should be easily obtained from the fitted model and, if the adopted model is nonlinear, its overall prediction performance should be an improvement upon the model; the fitted model should preferably reflect, to some extent the structure of the mechanism generating the data based on theories outside statistics; and they should preferably possess some degree of generality and be capable of generalization to the multivariate case, not just in theory but also in practice.

Predictive ability in time series informs on the degree to which the past can be used in ascertaining the future. Predictive ability is fundamental in time series analysis. Assessing whether there is predictability among macroeconomic variables has always been a central issue for applied researchers. For example, much effort has been devoted to analyzing whether money has predictive content for output. This question has been addressed by

using both simple linear Granger Causality (GC) tests (e.g. Stock and Watson (1989)) as well as tests that allow for non-linear predictive relationships (e.g. Amato and Swanson (2001) and Stock and Watson (1999), among others). Several authors have studied predictive ability and used it in several fields; for instance tourism, finance etc. However, not much has been done to investigate whether more than one series have equal predictive ability (Otranto and Traccia (2007)). Testing whether the models provide similar forecast performance represents a test of equal predictive ability. Testing equal predictive ability is essential in risk management; where, it could be interesting to establish if time series which have the same variables (economic, climate, etc), recorded in different spatial areas or calculated with different methodologies, have equal predictive abilities.

This work presents a test of equal predictive ability in relation to parallelsim of the Self-Exciting Threshold Autoregressive (SETAR) model. We use the Wald test used by Steece and Wood (1985) and Otranto and Triacca (2007) to investigate the similarity of SETAR processes.

**1.2 Statement of Problem**

Previous research works on parallelism and equal predictive ability centered on Autoregressive Integrated Moving Average models (ARIMA) and the Generalised Autoregressive Conditional Heteroscedastic (GARCH) models. Here we consider parallelism and equal predictive ability for Self-Exciting Threshold Autoregressive model. We link a

measure of equal predictive ability and the structure of the model using the autocorrelation and coefficients of the model. It will be necessary to also consider whether transformations are parallel to the original data this is because in building time series models, Box and Jenkins (1970) have devised an iterative strategy of model identification, estimation and diagnostic checking. The identification stage of their model building cycle relies on the recognition of typical patterns of behaviour or structure in the sample autocorrelation function and the partial autocorrelation. We investigate these in this work.

**1.3 Research Objectives**

This work deals with the predictive ability in time series exhibiting nonlinearity. The study aims to achieve the following objectives:

1. to apply a test of equal predictive ability to suit nonlinear time series,

2. to establish the condition necessary for parallelism and equal predictive ability of a nonlinear time series,

3. to validate the test with real life data.

**1.4 Significance of the Study**

When testing for equal predictive ability, the question that is of interest is whether one forecast model is better than another. This question can be addressed by testing the null hypothesis that the two series have the same structure. This testing problem is important for applied analysts, because several ideas and specifications are often used before a model is selected. This test can be narrowed down to testing if the different series are parallel which is a way of checking similarities in the structure of different series. Instead of testing for predictive equality we can test for similarity in the structure of the series (parallelism). There are several instances where it is important to check if two or more time series are equivalent. For instance, the task of predicting the demand for common items in different markets may be possible if it can be shown that the models characterizing demand are equivalent in various markets. If the hypothesis of parallelism between two time series is accepted, one can obtain better estimates of the model parameters by pooling the data sets, also by using series with more similar structure one can forecast the volatility of one series from the other(s) and it can be used to choose among several procedures of seasonal adjustment.

**1.5 Scope of the Study**

We consider Self-Exciting Threshold Autoregressive models in relation to parallelism and equal predictive ability. Since the R2 index can be used to test for the predictive ability we show that it can be expressed as a function of the parameters of the time series model and autocorrelation of the given time series. These helps in describing the structure of the series. We use this index to test equal predictive ability and parallelism between different models. We test the hypothesis by considering a test proposed by Steece andWood (1985) where they presented a simple method for assessing the equivalence of k time series, we then relate this to the predictive ability of different time series.