Multivariate approach to time series model identification

Project Abstract

<p> </p><p>This work suggests an exact and systematic model identification approach which is<br>entirely new and addresses most of the challenges of existing methods. We<br>developed quadratic discriminant functions for various orders of autoregressive<br>moving average (ARMA) models. An Algorithm that is to be used alongside our<br>functions was also developed. In achieving this, three hundred sets of time series<br>data were simulated for the development of our functions. Another twenty five sets<br>of simulated time series data were used in testing out the classifiers which correctly<br>classified twenty three out of the twenty five sets. The two cases of<br>misclassification merely imply that our Algorithm will require a second iteration to<br>correctly identify the model in question. The Algorithm was also applied to some<br>real life time series data and it correctly classified it in two iterations.</p><p>&nbsp;</p> <br><p></p>

Project Overview

<p> INTRODUCTION<br>1.1 INTRODUCTION<br>Model identification is a crucial part of Time Series model development. The main<br>task of Time Series Modeling is to first examine the series at hand so as to<br>establish the theoretical model that generates the Series. This task seems to be the<br>most challenging and most ambiguous in Time Series Modeling. It has been<br>approached from different perspectives over time. One of the most popular<br>approach is the Box and Jenkins approach presented in Box and Jenkins (1976).<br>Their method involves going through some iterative steps before a final model is<br>selected. The initial step involves calculating the sample autocorrelation function<br>(ACF) and partial autocorrelation function (PACF) of the series at various lags and<br>comparing their behaviour with known behaviour of some theoretical model and<br>the model that best approximates the sample behavior is tentatively selected. There<br>are two serious problems with their method. First is the fact that one will need to fit<br>several models or do several adjustments to arrive at the final model. This makes<br>the method computationally expensive. Another serious problem is the inability of<br>the method to accurately differentiate between some classes of models. For<br>example, it is not easy to determine the values of p and q in fitting ARMA (p, q)<br>x<br>when p,q ¹ 0 since both the ACF and PACF tail off. However, Tsay and Tiao<br>(1984, 1985) addressed this problem by proposing the use of the extended<br>autocorrelation (EACF) and smallest canonical correlation SCAN respectively.<br>Tsay and Tiao methods are mere extension of the Box and Jenkins approach as<br>they involve comparing behaviour of sample EACF and smallest canonical<br>correlation with the theoretical behaviour. The difficulties in matching these<br>behaviour is even more in these approach because the clear cut off in theoretical<br>EACF table for example is hardly observed in sample EACF, Cryer and Chan<br>(2008).<br>Far away from the Box and Jenkins approach, Akaike (1969) and lots of other<br>scientists have done several works on various forms of information criteria. Their<br>approach is based on values calculated from residual of already fitted models. The<br>statistic calculated from residual of these fitted models is perceived as information<br>loss as a result of fitting the model. The order of the model that minimizes the<br>information loss is finally adopted. Model identification stage of time series<br>modeling has suffered severe deficiency over time. All the available methods are<br>deficient in terms of accurately selecting the model fit. There is no well defined<br>procedure that gives the exact model or at least with a known error margin before<br>going ahead to fit the model. The approach presented here is well spelt out rules<br>that guides model development. Some of the model identification approaches<br>xi<br>especially the Box and Jenkins approach is more of art than science as it is highly<br>judgmental.<br>The information criteria approach is not an exemption in this deficiency. Fitting<br>models with virtually all order before selecting a particular order is almost the<br>same as fitting all the possible models and selecting the one that passes the<br>goodness of fit test. Model selection is a stage that should be put to rest before<br>considering estimation, if one must come back to this stage then the initial<br>approach is supposed to have pre-specified the next model/order to consider. One<br>other important shortfall of the information criteria approach is that its results are<br>comparative meaning that only selected tentative models will have their<br>information criteria calculated compared and then the model with minimum<br>information criteria chosen as the best model.<br>In this work, we are proposing an exact model identification method which will<br>address most of the issues with available model identification methods. Our<br>method does not require fitting any model into series at hand before selecting the<br>right model. It will be capable of selecting the exact model with very high level of<br>certainty and in event where the selected model is not the appropriate model (since<br>there may be a small error margin); the method also predetermines the next model<br>(from all models under consideration) to be considered. With further<br>transformation as we have in this work, The behaviour of sample ACF and PACF<br>xii<br>is actually enough information for model identification. The choice of the lag is<br>informed by the fact that ACF and PACF of stationary ARMA models usually cut<br>off with higher lags. The ACF at lag 1 to 4 and PACF at lag 2 to 5 are the<br>information used in this proposed method. Our method utilized the variations<br>inherent in selected ACF and PACF in the classifiers and used them to classify<br>models as ARMA (p, q) with p + q £ 2 (with ARMA(22) added to demonstrate the<br>usefulness of our methods in cases of mixed models)<br>1.2 STATEMENT OF PROBLEMS<br>In Time Series analysis, there is need to establish the theoretical model to be fitted<br>into a particular time Series at hand before proceeding with subsequent steps, this<br>work particularly, seeks to develop a new method of achieving this.<br>In this work, we want to build discriminant functions for each of the classes of<br>theoretical ARMA models considered (i.e. AR(1), AR(2), MA(1), MA(2),<br>ARMA(1,1) and ARMA(2,2)) and also develop an algorithm which is a well<br>organized iterative steps to be followed in applying the functions. We equally hope<br>to test our new method out using both simulated and real life time series and<br>comment on its performance.<br>1.3 SIGNIFICANCE OF THE STUDY<br>Series of work have been done on time series model identification but all existing<br>methods are characterized with lack of precise problem formulation, presence of<br>xiii<br>heavy individual judgments, and associated high computational cost due to several<br>adjustments needed to arrive at the final model. Existing model identification<br>methods only give merely tentative and inexact perception of the appropriate<br>theoretical model before fitting hence several models are usually fitted at the initial<br>stage. This work is aimed at developing an exact model identification approach<br>which will address the challenges indicated above.<br>Our method is hoped to provide a systematic and well organized problem<br>formulation which will be capable of reducing computational cost and rigour<br>associated with time series modeling.<br>1.4 OBJECTIVES OF THE STUDY<br>This work is aimed at developing an entirely new model identification method<br>which addresses most of the challenges of existing methods. In achieving this, our<br>specific objectives are outlined below.<br>· To develop quadratic discriminant functions for each of the ARMA models<br>considered and define the iterative steps (algorithm) to be followed in<br>application of the functions.<br>· To apply the method to both simulated and real time series data.<br>· To briefly compare the proposed method with existing methods<br>1.5 SCOPE AND LIMITATION OF THE STUDY<br>xiv<br>This work is limited to six classes of ARMA models which are AR(1), AR(2),<br>MA(1), MA(2), ARMA(1,1) and ARMA(2,2). This method can only be applied to<br>those classes of ARMA model meaning that other classes of models like<br>Autoregressive integrated moving average model (ARIMA), seasonal ARIMA,<br>Autoregressive conditional heteroscedasticity (ARCH) models, Generalized<br>Conditional Heteroscedasticity Model (GARCH) etc. are not covered by our<br>method. <br></p>