Stationarity and Autocorrelation

EXt = μ Cov(Xt, Xt−k) = γk (lag-k autocovariance). The lag zero autocovariance γ0 is just the variance of the time series. γk’s are of central importance in time series analysis as they characterize the serial dependence over observations (i.e. over time). We usually do not have iid data in time series contexts. A white noise series is stationary. A white noise (WN) series is defined as an uncorrelated series with EXt = 0 and EX2 t = σ 2. A trend model is not stationary. Let Xt = α+ βt+ εt, where εt is white noise. A random walk is not stationary either. Let Xt be such that Xt = Xt−1 + εt (where εt is WN). Assuming X0 = 0 (the initial value), Xt = εt + εt−1 + · · ·+ ε1. Although EXt = 0, Var(Xt) = tσ2 depends on t.