To develop quadratic discriminant functions for each of the ARMA models considered and define the iterative steps (algorithm) to be followed in application of the functions. To apply the method to both simulated and real time series data. To briefly compare the proposed method with existing methods.

 

To develop quadratic discriminant functions for each of the ARMA models considered and define the iterative steps (algorithm) to be followed in application of the functions.

To apply the method to both simulated and real time series data.

To briefly compare the proposed method with existing methods.

CHAPTER TWO

LITERATUR REVIEW

 INTRODUCTION

Though, no work has ever been done on application of Multivariate analytical methods in Time Series ARMA model identification, a lot has been done on model identification. In this section, we are going to briefly look at the extent of work that has been done on model identification, from use of ACF, PACF, EACF and Canonical correlation to test of hypothesis and later, use of information criteria.

REVIEW OF LITERATURE

Box and Jenkins (1976) have the credit for the most popular judgmental model identification method. Their approach adopts theoretical models to be fitted into Time Series based on careful examination of the sample ACF and PACF. Behaviour of sample ACF and PACF calculated from the Series to be fitted are compared with that of available theoretical model and the model with similar behavior is suspected to have generated the Series to be fitted and therefore considered. Theoretically, the ACF and PACF of an AR(p) tail off and cut off at lag p respectively. ACF and PACF of MA(q) cut off at lag q and tails off respectively while both ACF and PACF of ARMA(p,q) tail off. This approach has come under serious criticism based on a couple of arguments. First is that the method is highly judgmental because the behaviour of the sample ACF and PACF cannot be perfectly matched with the theoretical behaviour in most cases, therefore deciding whether ACF or PACF cut off at certain lag depends on individual Judgment. Box and Jenkins approach is merely tentative, the model identified at this stage is subject to series of modifications and the models selected at the end are usually different from the initial model. This will obviously lead to fitting and re-fitting of several models and series of goodness-of-fit tests. It is based on the arguments above and other similar arguments that Anderson (1975) concluded that Box and Jenkins approach, though attractive but in practice both time consuming, difficult and relatively expensive computationally.

One of the shortfalls of the Box and Jenkins approach is specifying the value of p and q in ARMA(p,q) model where  since both ACF and PACF tail off in this case. Tsay and Tiao (1984) proposed the extended autocorrelation approach to address this problem. The EACF method uses the fact that if the AR part of a mixed ARMA model is known, “filtering out” the autoregression from the observed series results in a pure MA process which now enjoy the cut off property. Chan (1999) in a comparative simulation study concluded that the EACF approach has a good sampling property for moderately large sample size. In formalizing the procedure, EACF is usually calculated and compared with the theoretical behaviour of EACF for available models and the values of p and q are consequently considered. Cryer and Chan (2008) specifically pointed out that there will never be a clear cut off in the sample EACF as we have in the theoretical EACF. This makes the method even more difficult than the Box and Jenkins approach.

Bhansali (1993) worked on a hypothesis testing approach to model identification. His approach involve pre-fitting all the suspected models and testing hypothesis concerning the order until the right model is established. Consider an AR(p), the procedure is to test the hypothesis that the model is an AR(p) against an alternative that it is an AR(p+1). We continue increasing the value of value of p by 1 and repeating the hypothesis until the null hypothesis is accepted.

Whittle (1963) introduced an order selection procedure which is based on residual variance plot. Take an AR(p) for example, the lowest possible value of p is chosen and the model fitted, the error variance is estimated from the fitted model as , if the order fitted is lower than the actual order then  is greater than . p is increased and the model fitted with calculated for each p. p’s are plotted against ’s and the value of p that corresponds to the point where  stopped reducing is adopted as the order of the process. Jenkins and Watts (1968) also used this method and further suggested that the method can be used for order selection of MA and ARMA models.

Whittle’s (1963) procedure motivated Akaike into research on order selection for which he is so popular today.

Akaike (1969) improved on Whittle (1963) procedure for order selection of AR processes. He suggested that AR(p) be fitted for p=0,1,

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