Linear Time Series Analysis

Abstract

From a purely statistical point of view, one major difference between time series and data sets as discussed in the previous chapters is that temporally consecutive measurements are usually highly dependent, thus violating the assumption of identically and independently distributed observations on which most of conventional statistical inference relies. Before we dive deeper into this topic, we note that the independency assumption is not only violated in time series but also in a number of other common test situations. Hence, beyond the area of time series, statistical models and methods have been developed to deal with such scenarios. Most importantly, the assumption of independent observations is given up in the class of mixed models which combine fixed and random effects, and which are suited for both nested and longitudinal (i.e., time series) data (see, e.g., Khuri et al. 1998; West et al. 2006, for more details). Aarts et al. (2014) discuss these models specifically in the context of neuroscience, where dependent and nested data other than time series frequently occur, e.g., when we have recordings from multiple neurons, nested within animals, nested within treatment groups, thus introducing dependencies. Besides including random effects, mixed models can account for dependency by allowing for much more flexible (parameterized) forms for the involved covariance matrices. For instance, in a regression model like Eq. (2.6) we may assume a full covariance matrix for the error terms [instead of the scalar form assumed in Eq. (2.6)] that captures some of the correlations among observations. Taking such a full covariance structure for Σ into account, under the multivariate normal model the ML estimator for parameters β becomes (West et al. 2006)