Chapter2Section2Stationarity

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Stat 4870/7870: Introduction to Time Series Chapter 2: Correlation and Stationary Time Series Section 2: Stationarity Definition 0.1 (Strictly Stationary) . A strictly stationary time series is one for which the probabilistic behavior of every collection of values and shifted values { x i 1 , x i 2 , . . . , x i k } and { x i 1 + h , x i 2 + h , . . . , x i k + h } , are identical, for all k = 1 , 2 , . . . , all time points t 1 , t 2 , . . . , t k and all time shifts h = 0 , ± 1 , ± 2 , . . . . Stationarity It is difficult to assess strict stationarity from data, however, station- ary time series data should exhibit similar behaviors over different time intervals. Instead of imposing conditions on all possible distributions of a time series, we will use a milder version that imposes conditions only on the first two moments. Definition 0.2 (Weakly Stationary) . A weakly stationary time series is a finite variance process where ( i ) the mean value function, μ t is constant and does not depend on time t , and ( ii ) the autocovariance function γ ( s, t ) depends on s and t only through their distance | s - t | . Remark. On stationary: 1
We will use the term stationary to mean weakly stationary. If a process is stationary in the strict sense, we will use the term strictly stationary. Weakly stationary is also known as covariance stationary or second- order stationary. A strictly stationary, finite variance, time series is also stationary The converse is not true in general. Specifically, in general, weak stationarity does not imply strict sta- tionarity. One important case where stationarity implies strict stationarity is if the time series is Gaussian [meaning all finite collections of the series are Gaussian]. Example 2.14: A Random Walk is Not Stationary Autocovariance function of a random walk is γ ( s, t ) = min { s, t } σ 2 w . See Example 2.9. The mean function of a random walk with drift is μ xt = δt . Thus, the regular random walk has an autcovariance function that de- pends on time. The random walk with drift has both the mean and autocovariance func- tion depending on time. Therefore, the random walk (with or without drift) is nonstationary. Stationary Time Series - Notation The mean function, E( x t ) = μ t of a stationary time series does not depend on time. Thus, for a stationary time series we write μ t = μ . The autocovariance, γ ( s, t ), of a stationary time series depends on s and t only through the time distance | s - t | . 2
To simplify notation, let s = t + h , where h represents the time shift or lag. Then, for a stationary time series, we write γ ( t + h, t ) = cov( x t + h , x t ) = cov( x h , x 0 ) = γ ( h, 0) = γ ( h ) , because the time difference between t + h and t is the same as the time difference between h and 0. Definition 0.3. The autocovariance function of a stationary time series will be written as γ ( h ) = cov( x t + h , x t ) = E[( x t + h - μ )( x t - μ )] . Definition 0.4. The autocorrelation function (ACF) of a station- ary time series will be written as ρ ( h ) = γ ( h ) γ (0) . Remark. ACF: The Cauchy-Schwarz inequality implies that - 1 ρ ( h ) 1 for all h . Thus, the relative importance of a given ACF value can be assessed by comparing with the extreme values - 1 and 1 . Example 2.17: Stationarity of White Noise The mean and autocovariance functions of the white noise series were previously discussed. Specifically, μ wt = 0 and γ w ( h ) = cov( w t + h , w t ) = ( σ 2 w if h = 0 0 if h 6 = 0 . Thus, the white noise process satisfies the conditions of Definition 0.2 and is weakly stationary. 3
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