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Ch 11 - Stationary Stochastic Process Time Series Data yt =...

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1 Econometrics--Q. He 1 Time Series Data y t = 0 + 1 x t1 + . . .+ k x tk + u t 2. Further Issues Econometrics--Q. He 2 Stationary Stochastic Process A stochastic process is stationary if for every collection of time indices 1 t 1 < …< t m the joint distribution of ( x t1 , … , x tm ) is the same as that of ( x t1+h , … x tm+h ) for h 1 Thus, stationarity implies that the x t ’s are identically distributed and that the nature of any correlation between adjacent terms is the same across all periods Econometrics--Q. He 3 Covariance Stationary Process A stochastic process is covariance stationary if E( x t ) is constant, Var( x t ) is constant and for any t , h 1, Cov( x t , x t+h ) depends only on h and not on t. Thus, this weaker form of stationarity requires only that the mean and variance are constant across time, and the covariance just depends on the distance across time. Econometrics--Q. He 4 Weakly Dependent Time Series A stationary time series is weakly dependent if x t and x t+h are “almost independent”as h increases If for a covariance stationary process Corr( x t , x t+h ) 0 as h → ∞ , we’ ll say this covariance stationary process is weakly dependent Want to still use law of large numbers Econometrics--Q. He 5 An MA(1) Process A moving average process of order one [MA(1)] can be characterized as one where x t = e t + 1 e t-1 , t = 1, 2, … with e t being an iid sequence with mean 0 and variance 2 e This is a stationary, weakly dependent sequence as variables 1 period apart are correlated, but 2 periods apart they are not Econometrics--Q. He 6 An AR(1) Process An autoregressive process of order one [AR(1)] can be characterized as one where y t = y t-1 + e t , t = 1, 2,… with e t being an iid sequence with mean 0 and variance e 2 For this process to be weakly dependent, it
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