eThesis - Kauppakorkeakoulun sähköiset gradut

Linear time series models are useful tools for analyzing the dynamic structure of time series. Usual assumption when using these models is that innovations are normally distributed. As a consequence modeled variable should be normally distributed as well. However, in practice for example logarithmic equity returns and index returns are not normally distributed. In order to enlarge the ability of linear time series models to model non-normally distributed variables, one could let innovations to be non-normally distributed. When the assumption of normality is relaxed, the relationship between the lag length, innovation distribution and values of model parameters becomes interesting. The aim of this study is to examine how these three factors affect the ability of linear time series models to model variables with various distributions. Main emphasizes is on finite MA models.

This study uses limit theorems and simulations to discuss how the lag length, innovation distribution and values of model parameters affect the distributional properties of variable generated by the finite MA model. Characteristic functions are used to provide factorization condition distribution of modeled variable has to satisfy so that it is possible to model variable by the finite MA model. It is shown that distributions having this kind of factorization property contain the class of infinitely divisible distributions as a subgroup. It is also shown that indecomposable distributions cannot be factorized in this manner and as a consequence variables following this kind of distribution cannot be modeled by the finite MA model if one or more lagged terms are involved. This result is extended also for the AR and ARMA model types. At the end of the study, some foundational properties of different classes of distributions are discussed to encompass model builders to choose between different innovation distributions.