● Please describe in precise terms stationarity of time series. Provide a simple example of a stationary time series, and verify stationarity. Then, answer the question that follows which gives you more insight of the concept of stationarity.
● Let w0, w1, w2,… be a white noise process with variance w2 , and let | ϕ | < 1 be a constant. Consider the process x0=w0, and
xt=φxt-1+wt t=0, 1, 2, 3…
a. Show that xt=j=1tjwt-j
b. Show that varxi=w21-21-2t+1
c. Show that covxt+h,xt=h varxi
d. Comment if the series is stationary
e. Argue that if t→∞ the process becomes stationary, so in a sense the process is asymptotically stationary.
f. Simulate n observations of the process based on your choice for and w. Discuss how these results can be used to generate a non-trivial stationary Gaussian AR(1).
Question 2(6 pages, 6 references)
Let ct be the cardiovascular mortality series (cmort) discussed in Example 3.5 of the textbook, and
xt=ct
be the differenced data.
(a) Plot xt and compare it to the actual data plotted in Figure 3.2 in the textbook. Why does differencing seem reasonable in this case?
(b) Calculate and plot the sample ACF and PACF of xt and using Table 4.1 in the textbook argue that an AR(1) is appropriate for xt.
(c) Fit an AR(1) to xt using maximum likelihood (basically unconditional least squares) as in Section 4.3 in the textbook. The easiest way to do this is to use sarima from astsa. Comment on the significance of the regression parameter estimates of the model. What is the estimate of the white noise variance?
(d) Examine the residuals and comment on whether or not you think the residuals are white.
(e) Assuming the fitted model is the true model, find the forecasts over a four-week horizon, xn+m for m = 1,2,3,4, and the corresponding 95% prediction intervals; n = 508 here. The easiest way to do this is to use sarima.for from astsa.