STAT/MATH 463 Homework #11, due Wednesday, December 7, 2016 Show your work to the extent possible. Also, you must label the answers to the questions - if you paste plots and computer code into your document without labeling things clearly, you will lose points. 1. Suppose that Yt = Xtd + Zt where Xt and Zt are uncorrelated white noise both with variance 2 and d is an integer. Find the cross-correlation function (CCF) between X and Y at lag k, k (X, Y ). 2. Consider again the following two State College temperature data sets: (a) http://sites. stat.psu.edu/~mharan/463/hwdir/StateCollege_tmax.dat, (b) http://sites.stat.psu. edu/~mharan/463/hwdir/StateCollege_tavg.dat (a) Plot a periodogram for each data set. Summarize briefly what it tells you about the data. (b) Suppose you have to do an analysis of this data in the spectral/frequency domain. Based on the periodogram describe mathematically, and in as much detail as possible, the model you would try to fit to this data. 3. Simulate an AR(1) model with = 0.6 of length n = 100. Plot a periodogram of the data set. Now, plot the theoretical spectral density using: library(TSA) ARMAspec(model=list(ar= - 0.6)) Comment briefly on how well the periodogram estimates the theoretical spectal density. Repeat for n = 1000. 4. Simulate a signal + noise time series from the model in Equation (13.2.4) on page 323 of Cryer and Chan. Use the R code below: win.graph(width=4.875,height=2.5,pointsize=8) set.seed(123); t=1:96; integer=sample(48,2) freq1=integer[1]/96; freq2=integer[2]/96 A1=rnorm(1,0,2); B1=rnorm(1,0,2) A2=rnorm(1,0,3); B2=rnorm(1,0,3); w=2*pi*t y=A1*cos(w*freq1)+B1*sin(w*freq1)+A2*cos(w*freq2)+ B2*sin(w*freq2)+rnorm(96,0,1) plot(t,y,type='o',ylab=expression(y[t])) (a) Plot the time series and look for the periodicities. Can you see them? (b) Plot the periodogram for the simulated series. Are the periodicities clear now? 5. The number of overseas visitors to New Zealand is recorded for each month over the period 1977 to 1995 in the file osvisit.dat. Let xt be the number of overseas visitors in time period t (in months) and zt = ln(xt ). (a) Comment on the main features in the correlogram for zt . (b) Fit an ARIMA (1, 1, 0) model to zt giving the estimated AR parameter and the standard deviation of the residuals. Comment on the correlogram of the residuals of this fitted ARIMA model. (c) Fit a multiplicative seasonal ARIMA(1, 1, 0) (0, 1, 0)12 model to zt and plot the correlogram of the residuals of this model. Has seasonal differencing removed the seasonal effect? Comment. (d) Choose the best-fitting model from the following: ARIMA (1, 1, 0) (1, 1, 0)12 , ARIMA (0, 1, 1) (0, 1, 1)12 , ARIMA (1, 1, 0) (0, 1, 1)12 , ARIMA (0, 1, 1) (1, 1, 0)12 , ARIMA (1, 1, 1) (1, 1, 1)12 , ARIMA (1, 1, 1) (1, 1, 0)12 , ARIMA (1, 1, 1) (0, 1, 1)12 . Base your choice on the AIC, and comment on the correlogram of the residuals of the best-fitting model. (e) Express the best-fitting model in part (d) above in terms of zt , white noise components, and the backward shift operator. (f) Forecast the number of overseas visitors for each month in the next year (1996), and give the total number of visitors expected in 1996 under the fitted model