Topics in autoregression

An overview of the thesis is given in Chapter 1. Chapter 2 discusses a symbolic form for the exact maximum likelihood estimator in the stationary normal AR(1) process. We derive the finite sample inference properties of the exact maximum likelihood estimator. We establish its consistency and its empirical cumulative distribution for a random walk case. The power of our one-tail unit root test overall outperforms that of previous proposals in the unknown mean AR(1) model. Chapter 3 provides a general technique to describe the shape of the admissible region of AR(p). As applications, we have visualized the admissible regions for AR(3) and AR(4). For the AR(4) process, all possible subset admissible regions for the model re-parametrized in terms of partial autocorrelations are obtained and it is demonstrated that these regions are quite complex and hence this re-parameterization is not so useful in the subset case. Chapter 4 develops an algorithm for computing the expectations of time series products given the autocovariance function. Using it as our tool, we evaluate the bias and variance of the Burg estimate to order n-1 in the first order autoregressive model and find that Burg estimate and the least-squares estimate have the same bias and variance to order n-1 in that case. We also obtain explicit formulae for the large sample bias of Burg estimates in the second order cases. Both simulations and theory indicates that Burg estimates have biases similar to the least-squares estimates in the second order cases. The advantages of the Burg estimates over the least-squares estimates are briefly indicated. Chapter 5 is an extension of Chapter 3. A new more computationally efficient general purpose algorithm for computing the exact maximum likelihood estimates in an AR(p) model is developed. Then this algorithm is used to develop a new approach to subset autoregression modelling in which the subsets are obtained by containing some of the zeta parameters to zero. After the exact maximum likelihood estimation algorithm for the subset models is presented, it is shown how a tentative identification of possible subset AR models can be accomplished using the AIC or BIC criterion and the partial autocorrelation function. The distribution of the residual autocorrelations for subset AR models is also derived and appropriate diagnostic checks for model adequacy are discussed. Several illustrative examples are presented.