On equal predictive ability and parallelism of self-exciting threshold autoregressive model

Project Abstract

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

Project Overview

<p> INTRODUCTION<br>1.1 Introduction<br>Popularisation and extensive research for linear time series modelling began in 1927<br>with Yule’s Autoregressive models, used in studying sunspot numbers. In the decades that<br>followed, these models have been successfully applied in different fields, this is because as<br>far as one-step ahead prediction is concerned, linear time series models are often adequate.<br>However, this is not always so as can be seen from the Sunspot numbers (which will be<br>discussed later). The causes of this are mentioned later herein.<br>Nonlinear time series analysis gained attention in the 1970’s. The interest grew due to<br>the need to model nonlinear changes in everyday time series data exhibiting nonlinearity.<br>Autoregressive Integrated Moving Average (ARIMA) models cannot describe adequately<br>limit cycles, time-irreversibility, amplitude-frequency dependency and jump phenomena<br>(the Sunspot numbers mentioned earlier is a good example). As a result, Tong (1978)<br>1<br>came up with a procedure for modelling nonlinear changes in time series data in which<br>different Autoregressive (AR) processes are functioning, and the switch between these<br>AR models depends on the delay parameter and threshold value(s), which are certain time<br>lag values from the given time series. Tong and Lim (1980) and Tong (1983) followed up<br>the work with an extensive description of the procedure. Tsay (1989) proposed a much<br>simpler procedure. Tsay (1989) noted that the Tong’s (1983) procedure is not statistically<br>adequate for formally determining if a given data can be described using a threshold model<br>(see Tsay 1989).<br>Several nonlinear time series models (Nonlinear Autoregressive model (AR) and Closedloop<br>Threshold Autoregressive model (TARSC)) have been proposed over the years and<br>the Threshold Autoregressive (TAR) models, which is the piece-wise linearization of nonlinear<br>models over the state space by the introduction of the thresholds fro; :::; rig, has<br>been of significant interest because of its ability to model nonlinear data adequately. Common<br>notion were employed by Priestly (1965), and Ozaki and Tong (1975), in the analysis<br>of non-stationary time series and time dependent systems, in which local stationarity was<br>the counterpart of our present local linearity. The overall process is nonlinear when there<br>are at least two regimes with different parameters and/or process order. Tong and Lim<br>(1980) proposed the following requirements for the modelling of nonlinear time series, in<br>order of preference:<br>statistical identification of an appropriate model should not entail excessive compu<br>tation;<br>they should be general enough to capture some of the nonlinear phenomena men<br>tioned previously;<br>2<br>one-step-ahead prediction should be easily obtained from the fitted model and, if the<br>adopted model is nonlinear, its overall prediction performance should be an improve<br>ment upon the model;<br>the fitted model should preferably reflect, to some extent the structure of the mecha<br>nism generating the data based on theories outside statistics;<br>and they should preferably possess some degree of generality and be capable of gen<br>eralization to the multivariate case, not just in theory but also in practice.<br>Predictive ability in time series informs on the degree to which the past can be used in<br>ascertaining the future. Predictive ability is fundamental in time series analysis. Assessing<br>whether there is predictability among macroeconomic variables has always been a central<br>issue for applied researchers. For example, much effort has been devoted to analyzing<br>whether money has predictive content for output. This question has been addressed by<br>using both simple linear Granger Causality (GC) tests (e.g. Stock and Watson (1989)) as<br>well as tests that allow for non-linear predictive relationships (e.g. Amato and Swanson<br>(2001) and Stock and Watson (1999), among others). Several authors have studied predictive<br>ability and used it in several fields; for instance tourism, finance etc. However,<br>not much has been done to investigate whether more than one series have equal predictive<br>ability (Otranto and Traccia (2007)). Testing whether the models provide similar forecast<br>performance represents a test of equal predictive ability. Testing equal predictive ability<br>is essential in risk management; where, it could be interesting to establish if time series<br>which have the same variables (economic, climate, etc), recorded in different spatial areas<br>or calculated with different methodologies, have equal predictive abilities.<br>3<br>This work presents a test of equal predictive ability in relation to parallelsim of the<br>Self-Exciting Threshold Autoregressive (SETAR) model. We use the Wald test used by<br>Steece and Wood (1985) and Otranto and Triacca (2007) to investigate the similarity of<br>SETAR processes.<br>1.2 Statement of Problem<br>Previous research works on parallelism and equal predictive ability centered on Autoregressive<br>Integrated Moving Average models (ARIMA) and the Generalised Autoregressive<br>Conditional Heteroscedastic (GARCH) models. Here we consider parallelism<br>and equal predictive ability for Self-Exciting Threshold Autoregressive model. We link a<br>measure of equal predictive ability and the structure of the model using the autocorrelation<br>and coefficients of the model. It will be necessary to also consider whether transformations<br>are parallel to the original data this is because in building time series models, Box<br>and Jenkins (1970) have devised an iterative strategy of model identification, estimation<br>and diagnostic checking. The identification stage of their model building cycle relies on<br>the recognition of typical patterns of behaviour or structure in the sample autocorrelation<br>function and the partial autocorrelation. We investigate these in this work.<br>1.3 Research Objectives<br>This work deals with the predictive ability in time series exhibiting nonlinearity. The<br>study aims to achieve the following objectives:<br>4<br>1. to apply a test of equal predictive ability to suit nonlinear time series,<br>2. to establish the condition necessary for parallelism and equal predictive ability of a<br>nonlinear time series,<br>3. to validate the test with real life data.<br>1.4 Significance of the Study<br>When testing for equal predictive ability, the question that is of interest is whether one<br>forecast model is better than another. This question can be addressed by testing the null<br>hypothesis that the two series have the same structure. This testing problem is important<br>for applied analysts, because several ideas and specifications are often used before a model<br>is selected. This test can be narrowed down to testing if the different series are parallel<br>which is a way of checking similarities in the structure of different series. Instead of testing<br>for predictive equality we can test for similarity in the structure of the series (parallelism).<br>There are several instances where it is important to check if two or more time series are<br>equivalent. For instance, the task of predicting the demand for common items in different<br>markets may be possible if it can be shown that the models characterizing demand are<br>equivalent in various markets. If the hypothesis of parallelism between two time series<br>is accepted, one can obtain better estimates of the model parameters by pooling the data<br>sets, also by using series with more similar structure one can forecast the volatility of one<br>series from the other(s) and it can be used to choose among several procedures of seasonal<br>adjustment.<br>5<br>1.5 Scope of the Study<br>We consider Self-Exciting Threshold Autoregressive models in relation to parallelism<br>and equal predictive ability. Since the R2 index can be used to test for the predictive ability<br>we show that it can be expressed as a function of the parameters of the time series model<br>and autocorrelation of the given time series. These helps in describing the structure of the<br>series. We use this index to test equal predictive ability and parallelism between different<br>models. We test the hypothesis by considering a test proposed by Steece andWood (1985)<br>where they presented a simple method for assessing the equivalence of k time series, we<br>then relate this to the predictive ability of different time series.<br>6 <br></p>