STOR 556 Time Series Data Analysis Spring 2022 Lecture date January 25 Scribe Younghoon Kim Lecture 5 Stationary time series models We focus on random irregular fluctuations left after the classical decomposition which can still exhibit a dependence over time Let s look at stationary time series models whose statistical properties do not change over time For example A typical example of non stationary model is a random walk Xt Zt Zt 1 Zt is iid noise Xt Zt Zt 1 Z1 Xt 1 Zt because its variability increases over time Stationarity means that the mean does not depend on time t and the covariance of Xt and Xt h does not depend on t but possibly depends on lag h We name these covariances of Xt and Xt h autocovariance function ACVF of X Auto refers to both variables Xt and Xt h being for X Note that Cov Xs Xt X t s Cov Xt Xt Var Xt X 0 For autocorrelation function ACF of X obviously X 0 1 at h 0 And it is symmetric X h X h ACVF has the same property since by assuming EXt 0 for simplicity X h E XtXt h E Xt hXt X h Lastly the non negative definiteness here means that if you think of a vector X1 XT and its correlation as a matrix it is a non negative matrix Sample quantities For stationary models because their properties do not change over time we replace any expected values EXt or EXtXt h and so on by averages of the data over time For the sample ACVF described in the class notes h xt h x xt x 1 1 T T h cid 88 t 1 5 1 Figure 1 Illustration of the example The example of correlogram is depicted in the left panel of Figure 1 Note that summation in 1 is up to T h which is decreasing in h This is because we make T h pairs of data One can illustrate this through the middle panel of Figure 1 So as the lag h gets larger it is more difficult to estimate because we lack data Each point of correlogram represents the correlation between xt and xt h at the right panel of Figure 1 as we have seen before Finally note that the denominator in the sample ACVF 1 is T not T h Here the sample ACF can be computed for any time series For example suppose the model has a trend Then the corresponding correlogram looks like below You are asked to think in homework about the case when the model has a seasonal compo nent White noise 5 2 Two things are mentioned One the blue dashed lines represent 95 confidence intervals for each calculated sample ACF This is used for choosing a model Two for the time being we don t have a distinction between i i d noise and white noise technically i i d model is also a white noise model but converse does not hold in general Autoregressive series A convenient tool for writing an AR model is a backward shift operator B For AR 1 Xt 1Xt 1 cid 124 cid 123 cid 122 cid 125 1BXt Zt I 1B Xt Zt B Xt Zt z 1 1z I is used for identity IXt Xt Consider AR 1 model and set 1 First let s discuss the existence of a stationary model solution that satisfies the AR 1 equation Note that 2 3 This candidate solution makes sense when 1 It turns out that i it is well defined and stationary Also ii it satisfies AR 1 equation since Xt Xt 1 Zt Xt 2 Zt 1 Zt 2Xt 2 Zt 1 Zt 3Xt 3 2Zt 2 Zt 1 Zt Zt Zt 1 2Zt 2 3Zt 3 jZt j cid 88 j 0 Xt Zt Zt 1 2Zt 2 Zt Zt 1 Zt 2 Zt Xt 1 5 3 There is also a stationary solution when 1 By rearranging 2 Xt 1 1Xt 1Zt Xt 1Xt 1 1Zt 1 1 1Xt 2 1Zt 2 1Zt 1 2Xt 2 2Zt 2 1Zt 1 3Xt 3 3Zt 3 2Zt 2 1Zt 1 1Zt 1 2Zt 2 3Zt 3 jZt j cid 88 j 1 4 Indeed this candidate solution also satisfies i and ii The solution 3 is called causal 4 is called non causal When 1 there is no stationary solution Homwork prob lem Finally in modeling we choose the causal model because it is more natural in the sense that we regress on past observed values and not on the future values 5 4