Structural Analysis for the First Order Autoregressive Stochastic Models

Abstract

The response variables of a first order autoregressive stochastic process with autocorrelation $\rho$ can be constructed as a location-scale transform of a set of error variables whose distribution depends on the autocorrelation parameter, and the model can be treated as a composite response model with an error quantity $\rho$ (Fraser 1968, page 192). For known value of $\rho$ the model is a conditional structural model (Fraser 1968, page 188). Inference concerning $\rho$ is naturally based on the marginal likelihood function of $\rho$ obtained from the marginal probability distribution of the orbit of the response. The results are specialized to cover the normal error distribution. For the same model with error variables having a periodic structure a suitable transformation reduces the error variable into uncorrelated variables and the resultant transformed model can be treated as a location-scale structural model. The orbit of the transformed response depends on $\rho$ and so inference concerning $\rho$ can be made from the marginal likelihood of $\rho$ obtained from the marginal probability distribution of the orbit. It has been found that without any approximation being used the marginal likelihood function of $\rho$ thus obtained depends on the first order circular serial correlation coefficient in general, and for normal distribution in particular. For both the cases the general approximate distribution of the serial correlation coefficient has been derived by likelihood modulation. Using Anderson's (1942) result the general exact distribution of the circular serial correlation coefficient has also been obtained.