Final Exam Cheat Sheet - A stationary time series is one...

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A stationary time series is one whose properties do not depend on the time at which the series is observed o time series with trends, or with seasonality, are not stationary; cyclic behavior (but not trend or seasonality) is stationary o Stationary = no predictable patterns in the long-term; Time plots will be roughly horizontal, some cyclic behavior, with constant variance. Differencing : compute the differences between consecutive observations; help stabilize the mean of a time series by removing changes in the level of a time series, and so eliminating trend and seasonality o For a stationary time series, the ACF will drop to zero relatively quickly, while the ACF of non-stationary data decreases slowly. Also, for non-stationary data, the value of r 1 is often large and positive. o The differenced series: y t = yt yt −1. When it is white noise: yt yt −1= et or yt = yt −1+ et Random walks typically have long periods of apparent trends up or down, and sudden and unpredictable changes in direction. allows the differences to have a non-zero mean: yt yt −1= c + et or yt = c + yt −1+ et o Occasionally the differenced data will not appear stationary and it may be necessary to difference the data a second time to obtain a stationary series: y ′′ t = y t y t −1 = ( yt yt −1)−( yt −1− yt −2) = yt −2 yt −1+ yt −2. o A seasonal difference is the difference between an observation and the corresponding observation from the previous year. So y t = yt yt m where m = number of seasons. These are also called “lag- m differences” as we subtract the observation after a lag of m periods. If seasonally differenced data appear to be white noise, then an appropriate model for the original data is yt = yt m + et . If y t = yt yt m denotes a seasonally differenced series, then the twice-differenced series is y ′′ t = y t y t −1 = ( yt yt m )− ( yt −1− yt m −1) = yt yt −1− yt m + yt m −1. When both seasonal and first differences are applied, it makes no difference which is done first—the result is same. One of the most popular unit root tests is the Augmented Dickey-Fuller (ADF) test . For this test, the following regression model is estimated: y t = yt ϕ −1+ β 1 y t −1+ β 2 y t −2+ + βky t k , where y t denotes the first-differenced series, y t = yt yt −1 and k is the number of lags to include in the regression (often set to be about 3). If the original series, yt , needs differencing, then the coefficient ϕ ^ should be approximately zero. If yt is already stationary, then ϕ ^<0. R Code: adf.test(x, alternative = "stationary") So large p-values are indicative of non-stationarity, and small p-values suggest stationarity. Using the usual 5% threshold, differencing is required if the p-value is greater than 0.05.
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