Extend the first order case of the AR(1) model to the AR(2) case. Detail how the variance of the predictions is computed. Simulate this with an empirical example where Ptt+1 is computed for one-step ahead under a Gaussian error. AR(1) Example: # Number of observations N <- 500 # Simulation x <- arima.sim(list(ar=0.9), n=N) # Durbin-Levinson algorithm DLalgo <- function(x) { T <- length(x) # MSE matrix P <- matrix(0,T,1) # Phi matrix phi <- matrix(0,T,1) # Get the rhos rhos <- acf(x, lag.max=length(x), plot=FALSE)$acf # Intialize for an AR(1) phi[1,1] <- rhos[1,1,1] P[1,1] <- var(x)/(1-phi[1,1]^2) # Now loop over the computation for(i in 2:length(x)) { phi[i-1,1] <- (rhos[i-1,1,1] sum(phi[(T-i-1)]*rhos[T-i,1,1])) / (1-sum(rhos[-i,1,1])) } # print out the results print(DLalgo(x)[1:10]) > print(DLalgo(x)[1:10]) [1] 0.7162427 0.6905250 0.6612465 0.6315899 0.5995472 0.5495968 0.5181615 [8] 0.4748376 0.4370178 0.3894195