Stationary Stochastic Process o A stochastic process is stationary if for every collection of time indices 1<t<.< m the joint distribution of (tl,., x m)is the same as that of (tl+h, tm+h ) for h2 1 e Thus, stationarity implies that the x,'s are identically distributed and that the nature of any correlation between adjacent terms is the same across all periods Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Stationary Stochastic Process A stochastic process is stationary if for every collection of time indices 1 ≤ t1 < …< tm the joint distribution of (xt1, …, xtm) is the same as that of (xt1+h, … xtm+h) for h ≥ 1 Thus, stationarity implies that the xt ’s are identically distributed and that the nature of any correlation between adjacent terms is the same across all periods
Covariance Stationary Process o A stochastic process is covariance stationary ifE(x,) is constant, Var(x,)is constant and for any t, h>l, Cov(, x,+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 Economics 20- Prof anderson
Economics 20 - Prof. Anderson 2 Covariance Stationary Process A stochastic process is covariance stationary if E(xt ) is constant, Var(xt ) is constant and for any t, h ≥ 1, Cov(xt , xt+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
Weakly dependent Time series a stationary time series is weakly dependent if x, and x h are almost independent'' as h increases o If for a covariance stationary process Cor(x2x+b)→0ash→, we'll say this covariance stationary process is weakly dependent o Want to still use law of large numbers Economics 20- Prof anderson
Economics 20 - Prof. Anderson 3 Weakly Dependent Time Series A stationary time series is weakly dependent if xt and xt+h are “almost independent” as h increases If for a covariance stationary process Corr(xt , xt+h) → 0 as h → ∞, we’ll say this covariance stationary process is weakly dependent Want to still use law of large numbers
An Ma()Process o A moving average process of order one MA(I can be characterized as one where x e,t l, t=1, 2,... with e, being an lid sequence with mean 0 and variance o This is a stationary, weakly dependent/ sequence as variables 1 period apart are correlated, but 2 periods apart they are not Economics 20- Prof anderson 4
Economics 20 - Prof. Anderson 4 An MA(1) Process A moving average process of order one [MA(1)] can be characterized as one where xt = et + a1 et-1 , t = 1, 2, … with et being an iid sequence with mean 0 and variance s 2 e This is a stationary, weakly dependent sequence as variables 1 period apart are correlated, but 2 periods apart they are not
An ar(l) Process o An autoregressive process of order one AR(DI can be characterized as one where y-v+e,, t=l, 2, .with e, being an iid sequence with mean 0 and variance o2 o For this process to be weakly dependent, it must be the case that p< 1 o Corr(,, ]i+h)=Cov(,,y+h(o,ov=pI Which becomes small as h Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 An AR(1) Process An autoregressive process of order one [AR(1)] can be characterized as one where yt = ryt-1 + et , t = 1, 2,… with et being an iid sequence with mean 0 and variance se 2 For this process to be weakly dependent, it must be the case that |r| < 1 Corr(yt ,yt+h) = Cov(yt ,yt+h)/(sysy ) = r1 h which becomes small as h increases