IV. Suppose that {u: t = ...,-2,-1,0,1,2,...} is an independent time series with mean zero, variance σ²= 4.0. Suppose that the time series {x: t = ...,-2,-1,0,1,2,...} satisfies the equation.: X₁ = .5 Xt-1 + 1.0 + 1 - .4 µ4-1 · a) Determine the autocovariance function and autocorrelation function of the time series Xt. b) Find the random shock form of the time series. c) Suppose that the first four observations of the time series are x₁ = 2.75, x2 = 1.10, X3 = 1.38 and x4 =0.94 i) Use these observations to compute prediction intervals for the next 2 observations. (Compute both 95% and 66.7% prediction limits) ii) If the fifth observation turns out to be x5 = 4.22 use this information to re-compute prediction intervals for the next observation. (Compute both 95% and 66.7% prediction limits)