Ross Bowden

This thesis explores Joint Autoregressive Moving-Average (JARMA) models for independent replicated univariate time series with common ARMA coefficients whose innovations variances are either in common, unique to each series or vary with the series mean. The constraint of a common variance is also applied to vector ARMA processes. Interleaving is shown to represent replicated series with a common variance as one series from the same process. The time and frequency domain properties of interleaved replicated stationary and invertible processes are established. As an aid to identification, hypothesis tests for comparing series are reviewed and several new tests are presented and explored along with a graphical method for identification. Unconditional maximum likelihood estimates of the parameters of various JARMA processes are derived using the methods of joint likelihood and interleaving. The properties of the estimators are examined using simulation and asymptotics. Finally JARMA models are fitted to over 60 years of daily univariate and bivariate temperature data to estimate differences in level due to location and climate change.