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
vii
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
1.1 Introduction
Popularisation and extensive research for linear time series modelling 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)
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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 Closedloop
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 modelling of nonlinear time series, in
order of preference:
statistical identification of an appropriate model should not entail excessive compu
tation;
they should be general enough to capture some of the nonlinear phenomena men
tioned previously;
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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 improve
ment upon the model;
the fitted model should preferably reflect, to some extent the structure of the mecha
nism generating the data based on theories outside statistics;
and they should preferably possess some degree of generality and be capable of gen
eralization 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.
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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:
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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.
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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.
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