2STOCHASTIC TIME SERIES MODELSIn the general model] oe 0Ð Ñ ttt%when the effect of time is embodied in we have a "stochastic process" (SP) with%tÊa probabilistic structure.What is a SP ? It is a sequence of random variables (RV's) indexed by time.In analyzing stochastic TS, we have a long sequence of random variables:á]]]]]. , , , , , , . . . .21012Êan infinite series going indefinitely into the future and past.The sequence of n observations , , , is a sample from this long series.]]á ]12nGoallonger: to make inference about the probability structure of the series (on thebasis of sample).1
3LINEAR STOCHASTIC PROCESSConsider an , that is,autoregressive process of order one] œ]tt1t9%Ö] ×t- observed values of a random variable.Ö×%t- white-noise error terms; unobserved random variables.9- unknown constant.We can write from the above that]œ]] œ]œ]t-1t-2t-1tt-1tt-2t-1t9%9%9 9%%, ()and if we continue this waywe obtain] œttt-1t-2t-323%9%9%9 %. . . . .Ê]the effect of time is embodied in , that is, 's are correlated as a result of sharing%ttcommon error terms.A stochastic process generated by taking a linear combination of (moving average of)the white noise terms is known as a .linear process2
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4STATIONARY PROCESSESWold's theorem: any stationary process can be represented as a decomposition of alinear process and a deterministic part. We will consider some special cases of linearprocesses.We need some simplifying assumptions about the probability structure of the longerseries (about the underlying process):Stationarity Assumption.Ê• What is a Stationary TS ?The behavior of a set of random variables at one point in time is theprobabilisticallysame as the behavior of a set at another point in timethe probabilistic structure of the TS is not time-dependent.Ê• Why do we need this ?You can take your from any portion of the long series and the probabilityfinitesamplestructure will be the same. Note that what we have (the sample) is one realization fromthe long-series.3