TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 34

CHAPTER 67 Timeseries Analysis. A time series y with typical element y s is a (finite or infinite) sequence of random variables. Usually, the subscript s goes from 1 to ∞, ., the time series is written y 1 , y 2 , . . ., but it may have different (finite or inifinite) starting or ending values. | CHAPTER 67 Timeseries Analysis A time series y with typical element ys is a finite or infinite sequence of random variables. Usually the subscript s goes from 1 to to . the time series is written y1 y2 . but it may have different finite or infinite starting or ending values. . Covariance Stationary Timeseries A time series is covariance-stationary if and only if E ys y for all s var ys to for all s cov ys ys k Yk for all s and k 1435 1436 67. TIMESERIES ANALYSIS . the means do not depend on s and the covariances only depend on the distances and not on s. A covariance stationary time series is characterized by the expected value of each observation p the variance of each observation a2 and the autocorrelation function pk for k 1 or alternatively by p and the autocovariance function Yk for k 0. The autocovariance and autocorrelation functions are vectors containing the unique elements of the covariance and correlation matrices. The simplest time series has all yt IID p a2 . all covariances between different elements are zero. If p 0 this is called white noise. A covariance-stationary process yt t 1 . n with expected value p E yi is said to be ergodic for the mean if 1 A plim-V yt P. n œ n t i We will usually require ergodicity along with stationarity. PROBLEM 548. Ham94 pp. 46 7 Give a simple example for a stationary time series process which is not ergodic for the mean. ANSWER. White noise plus a mean which is drawn once and for all from a N 0 t2 independent of the white noise. . COVARIANCE STATIONARY TIMESERIES 1437 . Moving Average Processes. The following is based on Gra89 pp. 63-91 and on End95 . We just said that the simplest stationary process is a constant plus white noise all autocorrelations zero . The next simplest process is a moving average process of order 1 also called a MA 1 process yt h t fist-1 t IID 0 ct2 where the first y say it is y1 depends on the pre-sample q. PROBLEM 549. Compute the .

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