Some problems in time series analysis and forecasting

Abstract

This thesis is concerned with various investigations relating to time series analysis and forecasting. Particular attention is given to fractional differencing and its applications to long memory time series models. Chapter 1 entitled "Introduction", contains the summary of the basic time series theory required for the work carried out in the remaining six chapters. In addition to the general theory, the notion of fractional differencing, time reversibility and optimal experimental designs are included as special aspects of the thesis. In Chapter 2, the general class of univariate ARMA models with time dependent coefficients is considered. Existence and uniqueness of a second-order solution to the model is established using certain AR and MA regularity conditions. A simple form of the solution~ the covariance structure and the associated model building problem are considered from the theoretical point of view. The prediction problem is solved using alternative approaches. Some recursive relations to the optimum linear predictors are obtained by the orthogonal projection in Hilbert spaces. Few examples are added in support of the general results derived in this chapter. Chapter 3 considers the multivariate generalization of the results obtained in Chapter 2. The condition for the asymptotic stationary of the associated process is obtained in terms of the spectral radii of the corresponding matrices. It is shown that the recursive relations of the predictors satisfy a matricial equation as in the univariate case. Assuming the predictive distribution to be multivariate normal, a simultaneous confidence interval for the predictors is derived. The family of ARMA models with constant coefficients and nonstationary innovations is considered in Chapter4. A particular form of the model is considered for further analysis. It is shown that the usual regularity conditions are necessary and sufficient to ensure the existence and uniqueness of second-order solutions. The covariance structure of the associated process is obtained. The prediction problem is solved using the same procedures as in the stationary innovation case. The effect of nonstationarity in noise is shown to be insignificant in the parameter estimation. In Chapter 5 the theoretical autocorrelation function of an ARIMA (0,d,q) process is obtained. An asymptotically unbiased estimator of the sample spectrum is given. The various relations of the predictors are obtained. In particular, the attention is paid to the minimum mean squared error prediction and it is shown that this predictor is not optimal from the theoretical point of view. Finally, a direct basic form of the predictors of ARIMA (0,d,q) ; d .1 model is obtained. Fractional differencing and its applications to long memory time series are discussed in Chapter 6. This new class of ARIMA models aroused the interest of many time series since it has numerous applications in several scientific A proof for the stationarity and invertibility conditions ARIMA(p,d,q) ; d . R is given. Persistence in time series the Hurst phenomenon are examined for some actual situations. For the predictors of ARIMA (p,d,q) ; are given and it is shown that the minimum mean squared optimum properties when |d| < 1/2 . Numerical examples are added. In particular the long memory characteristic of some Australian rivers are demonstrated a suitable model is fitted in each case. Chapter 7 addresses itself to some miscellaneous problems in time series analysis. The first section is devoted to the discussion of some functions of ARMA models and generalizes the existing results for the multivariate case. Some contributions to bilinear time series models are considered in Section 7.2. Various new results regarding BL(p,q,p,s) and BL(p,q,r,l) are given. Section 7.3 considers the notion of time reversibility of a stochastic process and non Gaussian ARMA models from the theoretical point of view. The last section i.e. Section 7.4 discusses the optimal experimental design problems of time series regression model, when the coefficients are stochastic. It is shown that under certain conditions the effect of randomness of the coefficients is insignificant in the optimal design problem.