Specification of ARMA models with non-Gaussian and non-stationary errors
by
Kuldeep Kumar
Bond University, Gold Coast, Quuensland 4229, AUSTRALIA

Since the appearance of the book by Box and Jenkins (1970) use of Autoregressive -Moving Average (ARMA) models has become widespread in many fields for the analysis and prediction of time series data. One of the major problem in Box-Jenkins modelling is the specification of correct ARMA(p, q) model where p is the order of Autoregressive model and q is the order of moving average model. Kumar (1986, 87) has done a review of various specification methods and proposed a new method based on the theory of Pade approximation for the specification of ARMA(p, q) model.

The problem of specification of ARMA models become more complicated when the noise is non-Gaussian or non-stationary. In this paper we have proposed a new method for the specification of Autoregressive, Moving average and mixed ARMA models when the noise do not follow a normal distribution or when it is non-stationary.

The specification problem of AR, MA and ARMA models with non-normal noise can be looked in two ways. Either we can analyse the observation Zt directly or we can analyse the residuals. If the errors are normally distributed the distribution of Zt will be expected to be normal and the residuals will be also expected to be normally distributed. However, the non-normality of the distribution of Zt may be either due to non-normal noise or the model itself may be non-linear. We have shown that autocorrelation function can not distinguish between ARMA models having normal noise and those having non-normal noise. To distinguish between these two types of models we have obtained third order moments. We have also obtained fourth order moments to distinguish between MA models having non-normal but symmetric noise from moving average models having normal noise. Theoretical and simulation results show that third order moments are capable of distinguishing between ARMA models having non-normal noise and those having non-normal skewed noise.

In this paper we have also discussed the problem of specification of ARMA process when the white noise is non-stationary. This kind of behaviour is quite often observed in financial time series data. These series are found to be stationary in mean but non-stationary in variance. In this paper we have given an algorithm for the identification of such models.

Date received: October 23, 2000