Question

2.6 Consider a process consisting of a linear trend with an additive noise term consisting of independent random variables wt with zero means and variances o that is where Bo, B1 are fixed constants (a) Prove t is nonstationary. (b) Prove that the first difference series Vxt finding its mean and autocovariance function Xt t-s stationary by

Answer 1

calienray Mean mix) = E(X) =ECB +++2) - Po + Bt t = tz and mit, mit Hence Xt is non station cory (b) Difference series 7lt = +- *-1 = ($) +- (t-1)} +Wt-Wes = B + Wt – Wt-1 m (7x+) = B The mean does not depend on timne t Now By Auto correlation function R(4.8) = E( *773) = E[[ + W5 -ws-1) (P + Ws -W5-1)] -E [B² + BWs - F, We-1 + But + We ws - Wews- - We-1B - WEWS - Wt. Ws-1] = 8 + E (WWE) + (W+WS-1) +E (WT-1W) + E (WH-1 Ws-1) Now, E (WEWs)=0 forts and t=0 for t=s. Hence auto correlation function is ROHS) = 720 Which is independent of time.