Estimation and setting starting values in ARMA algorithms

For the given observations set of the ARMA (autoregressive moving average) process, the likelihood function depends, not only on model parameters, but on the starting values of the input and output. Therefore, it is called theconditional likelihood function. Theunconditional likelihood function can be obtained in two ways. The first is to set the starting values to zero, as is often done, and the second is to set them to the properly estimated values. The difference between these two types of likelihood functions is significant when the given data sequence is short, and any of the zeros of the moving average part is close to the boundary of the unit circle.In this paper the direct method of starting value estimation and its application to two off-line ARMA estimation algorithms, the maximum likelihood (ML) algorithm and the iterative inverse filtering (ITIF) algorithm, is proposed. Experimental results prove both increased efficiency and stability of these algorithms.The importance of setting the starting values properly is also significant when the recursive algorithm, with previously estimated parameters, has to be restarted. The advantage of the proposed reinitialization method is shown on the recursive lattice algorithm working in the block mode.