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 AR2SIM.OMS Script File: ```# Description: # Generates realizations of an AR(2) process. # # The AR(2) process has specified covariance at lags 0, 1, and 2. # We compare the specified covariance at each lag with the average of # x(n) * x(n + lag). # clear # number of points in each simulated realization N = 50 # number of realizations M = 9 # covariance of process: x(n) = sqrt(.75) * w(n) + .5 * x(n - 1) r = {1., .5, .25} # Simulate M realizations of length N. x = arcov(N, M, r) # set format for printing format real "f10.5" # number of covariances specified m = rowdim(r) # initialize sum as zero for each lag total_sum = fill(0., 1, m) # loop over realizations print "Sample covariances for each realization" for j =1 to M begin # dimension the sample covariance for this realization s = fill(0., 1, m) # loop over lag value for lag = 0 to m - 1 begin # copy of this realization from index 1 to N - lag xb = x.blk(1, j, N - lag, 1) # copy of this realization from index 1 + lag to N xf = x.blk(1 + lag, j, N - lag, 1) # sample covariance for this lag is average of xb(i) * xf(i) s(lag + 1) = xb' * xf / (N - lag) end # print sample covariance for this realization write("screen", s) # include in the total sum total_sum = total_sum + s end print "Average of sample covariances" write("screen", total_sum / M) print "True covariance for each lag for the AR process" write("screen", r') ``` Output: ```Sample covariances for each realization 0.85329 0.34820 0.14347 0.99553 0.49420 0.17096 1.20703 0.55181 0.58899 0.81702 0.17073 0.15266 1.09309 0.68059 0.40324 0.90018 0.26048 0.10037 0.75273 0.31750 0.02205 1.26920 0.79067 0.32335 1.08788 0.40272 0.07164 Average of sample covariances 0.99733 0.44632 0.21964 True covariance for each lag for the AR process 1.00000 0.50000 0.25000 ```
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