using SparseArrays, LinearAlgebra, Optim, StatsBase,Plots,DelimitedFiles function loglike_ARMA11(psi,X,y) T = length(y) H = sparse(I,T,T) + psi*sparse(2:T,1:T-1,ones(T-1),T,T) HH = H*H' rhohat = (X'*(HH\X))\(X'*(HH\y)) uhat = y-X*rhohat sigma2hat = uhat'*(HH\uhat)/T l = -T/2*log(2*pi*sigma2hat) - .5/sigma2hat*uhat'*(HH\uhat); return l, rhohat, sigma2hat end USCPI= readdlm("USCPI.csv") y0 = USCPI[1] y = USCPI[2:end] T = length(y) X = [ones(T,1) [y0; y[1:end-1]]] f = psi -> -loglike_ARMA11(psi,X,y)[1] res = optimize(f,-1,1) psihat = res.minimizer l, rhohat, sigma2hat = loglike_ARMA11(psihat,X,y)