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
Trends revisited o A trending series cannot be stationary, since the mean is changing over time o A trending series can be weakly dependent o If a series is weakly dependent and is stationary about its trend, we will call it a trend-stationary process o As long as a trend is included, all is well Economics 20- Prof anderson 6
Economics 20 - Prof. Anderson 6 Trends Revisited A trending series cannot be stationary, since the mean is changing over time A trending series can be weakly dependent If a series is weakly dependent and is stationary about its trend, we will call it a trend-stationary process As long as a trend is included, all is well
Assumptions for consistency o Linearity and Weak Dependence A weaker zero conditional mean assumption: E(ux =0, for each t ◆ No Perfect t Co collinearity Thus, for asymptotic unbiasedness (consistency ), we can weaken the exogeneity assumptions somewhat relative to those for unbiasedness Economics 20- Prof anderson 7
Economics 20 - Prof. Anderson 7 Assumptions for Consistency Linearity and Weak Dependence A weaker zero conditional mean assumption: E(ut |xt ) = 0, for each t No Perfect Collinearity Thus, for asymptotic unbiasedness (consistency), we can weaken the exogeneity assumptions somewhat relative to those for unbiasedness
Large-Sample Inference o Weaker assumption of homoskedasticity Var(u,lx) =o, for each t o Weaker assumption of no serial correlation E(ullx,x)=0fort≠S o With these assumptions, we have asymptotic normality and the usual standard errors t statistics. F statistics and lM statistics are valid Economics 20- Prof anderson 8
Economics 20 - Prof. Anderson 8 Large-Sample Inference Weaker assumption of homoskedasticity: Var (ut |xt ) = s 2 , for each t Weaker assumption of no serial correlation: E(utus | xt , xs ) = 0 for t s With these assumptions, we have asymptotic normality and the usual standard errors, t statistics, F statistics and LM statistics are valid
Random walks e Arandom walk is an ar(1)model where Pr 1, meaning the series is not weakly dependent o With a random walk, the expected value of y is always yo- it doesnt depend on t o Var(=oet, so it increases with t o We say a random walk is highly persistent since Evi+hv=y, for all h2 1 Economics 20- Prof anderson 9
Economics 20 - Prof. Anderson 9 Random Walks A random walk is an AR(1) model where r1 = 1, meaning the series is not weakly dependent With a random walk, the expected value of yt is always y0 – it doesn’t depend on t Var(yt ) = se 2 t, so it increases with t We say a random walk is highly persistent since E(yt+h|yt ) = yt for all h ≥ 1
Random Walks(continued) A random walk is a special case of what's known as a unit root process e Note that trending and persistence are different things -a series can be trending but weakly dependent. or a series can be highly persistent without any trend A random walk with drift is an example of a highly persistent series that is trending Economics 20- Prof anderson 10
Economics 20 - Prof. Anderson 10 Random Walks (continued) A random walk is a special case of what’s known as a unit root process Note that trending and persistence are different things – a series can be trending but weakly dependent, or a series can be highly persistent without any trend A random walk with drift is an example of a highly persistent series that is trending