Univariate time series models
Andrew C. Harvey · Cambridge University Press eBooks · 1990
A univariate time series consists of a set of observations on a single variable, y . If there are T observations, they may be denoted by y t , t = 1,…, T . A univariate time series model for y t is formulated in terms of past values of y t and/or its position with respect to time. Forecasts from such a model are therefore nothing more than extrapolations of the observed series made at time T . These forecasts may be denoted by ŷ T+l|T , where l is a positive integer denoting the lead time . No univariate statistical model can be taken seriously as a mechanism describing the way in which the observations are generated. If we are to start building workable models from first principles, therefore, it is necessary to begin by asking the question of what we expect our models to do. The ad hoc forecasting procedures described in section 2.2 provide the starting point. These procedures make forecasts by fitting functions of time to the observations but do so by placing relatively more weight on the more recent observations. This discounting of past observations is intuitively sensible but lacks any explicit statistical foundation. The first part of section 2.3 introduces the idea of a class of statistical models known as stochastic processes. Structural time series models are then built up by formulating stochastic components which, when combined, give forecasts of the required form. It turns out that these models provide a statistical rationale for the ad hoc procedures introduced earlier.