Question

Q1. (10 points total) Consider an ARMA(1,1) model X4 = 0.9X-1 + Z+ +0.527-1, {Z4}~ IID N(0,1). 1. (2 points) (i) Generate n = 200 observations from this ARMA model. (ii) Find the maximum likelihood (ML) estimates of the three parameters o, e, and o2. 2. (8 points) Repeat (i) and (ii) in part 1 nine more times using all different seed numbers. Compare the estimates to their true values. Are the average of 10 estimates for each parameter close to the corresponding true parameter? Comment on the results.

Answer 1

Solution for the above question was solved in R language. The solution with commented code is given in the following images. arima.sim() command is used to simmulate time series of given parameters.

Solution w = arima.sim (n=200, list(order=c(1,0,1), ar=0.9, ma=0.5), innov=rnorm (200)) s = arima (w, order=c(1,0,1)) phi = s$coef[1] #ML estimate of phi phi ## ar1 ## 0.8931091 theta = s$coef[2] theta #ML estimate of theta ## ma1 ## 0.5773168 sigma.sq = s$sigma2 #ML estimate of sigma.sq sigma.sa ## [1] 0.9616727 The parameter values and the ML estimates are nearly equal. 2) Here we simulate 10 time series with phi=0.9 and theta=0.5, sigmasq=1 phi.est=c() theta.est=c() sigma.sq.est=c() phi.est[1]=phi theta.est[1]=theta sigma.sq.est[1]=sigma.sq for(i in 2:10) #The remaining 9 estimates w = arima. sim(n=200, list (order=c(1,0,1), ar=0.9, ma=0.5), innov=rnorm (200)) s = arima (w, order=c(1,0,1)) phi.est[i] = s$coef[1] theta.est[i] = s$coef[2] sigma.sq.est[i] = $sigma2 D1<-data.frame("phi(true-0.9) "=phi.est, "theta(true-0.5) "=theta.est, "sig 2(true-1)"=sigma.sq.est) D1 ## 1 ## 2 ## 3 ## 4 ## 5 ## 6 ## 7 ## 8 ## 9 phi.true.0.9. theta. true.0.5. sig. 2.true. 1. 0.8931091 0.5773168 0.9616727 0.8789105 0.4918848 0.8762433 0.9063024 0.4829612 0.9325800 0.9205943 0.5864775 1.0648856 0.8894522 0.4994760 1.0859504 0.9394616 0.5724518 0.9285419 0.8704324 0.4647876 0.9678046 0.9231398 0.3414607 0.9246606 0.8794089 0.5605245 0.9002414

## 10 0.8739027 0.6356538 1.0566945 The ML estimate of phi, theta and sigma are near to the original values. The following graph shows fluctuation from original value. The black line is the original one and the blue line is the estimated value. par (mfrow=c(3,1)) plot(phi.est, type="o", col="blue", main=expression (phi)) abline (h=0.9) plot(theta.est, type="o", col="blue", main=expression(theta)) abline (h=0.5) plot (sigma.sq.est, type="o", col="blue", main=expression(sigma 2)) abline (h=1) phi.est 0.87 ME 1100 Index theta.est ol 0.35 E FO + Index Q sigma.sq.est 0.90 6 Index p = mean(phi.est) t = mean(theta.est) sig = mean(sigma.sq.est) D <-data.frame("parameter"=c("phi","theta","sigma2"), "estimated"=C(p,t,sig), "original"=c(0.9,0.5,1)) ## ## 1 ## 2 ## 3 parameter estimated original phi 0.8974714 0.9 theta 0.5212995 0.5 sigma 2 0.9699275 1.0 The mean values are also nearly same to the orignal values of parameters.