Description

Q1 (5 points): Find the autocorrelation function (ACF) of the following ARMA models 1. AR(1) model: zt = 0.7zt−1 + wt , where wt wn(0, σ2 w). 2. AR(2) model: zt = 0.1zt−1 + 0.3zt−2 + wt , where wt wn(0, σ2
w). 3. MA(1) model: zt = wt − 0.5wt−1, where wt wn(0, σ2 w). 4. MA(2) model: zt = wt − 1.1wt−1 + 0.28wt−2, where wt wn(0, σ2 w). 5. ARMA(1, 1) model: zt = 0.3zt−1 + wt + 0.7wt−1, where wt wn(0, σ2 w).well (do the residuals look white)?

Q2 (2 points): Consider the model (1 − 1.1B + 0.8B 2 )xt = (1 − 1.7B + 0.72B 2 )wt 1. Verify whether it is stationary, or invertible, or both. Hint: |a + bi| = √ a 2 + b 2 . 2. Express the model in an MA representation if it exists.

Q3 (2 points): An AR model has AR characteristic polynomial (1 − 1.6z + .7z 2 )(1 − .8z 12) 1. Is the model stationary? Explain! 2. Identify the model as a certain seasonal ARIMA model.

Q4 (2 points): Identify the following as certain multiplicative seasonal ARIMA models: 1. xt = .5xt−1 + xt−4 − .5xt−5 + wt − .3wt−1. 2. xt = xt−1 + xt−12 − xt−13 + wt − .5wt−1 + −.5wt−12 + .25wt−13.

Q5 (2 points): 1. Consider the ARMA(1, 1): zt = ϕ1zt−1 + wt + θ1wt−1, where ϕ1 = −θ1 and wt wn(0, σ2 w). Show that this model is not really an ARMA(1, 1), but it is a white noise model ARMA(0, 0). 2. Consider the ARMA(2, 1): zt = −0.3zt−1 + 0.18zt−2 + wt + 0.6wt−1, where wt wn(0, σ2 w). Show that this mode is not really an ARMA(2, 1), but it is an AR(1) ≡ ARMA(1, 0).

Q6 (5 points): Fit an ARIMA(p, d, q) model to the sulfur dioxide series, so2, (in the package astsa) performing all of the necessary diagnostics. After deciding on an appropriate model, forecast the data into the future four time periods ahead (about one month) and calculate 95% prediction intervals for each of the four forecasts. Comment.

Q7 (9 points): Quarterly earnings per share for the Johnson & Johnson Company are given in the data file named jj. The data cover the years from 1960 through 1980. 1. Plot the time series and also the logarithm of the series. Argue that we should transform by logs to model this series. 2. Display a time series plot of the differences of the transformed values. Does this plot suggest that a stationary model might be appropriate for the differences? 3. Display a quantile-quantile plot of the transformed data. Are they looks like normal? 4. Calculate and graph the sample ACF of the first differences. Interpret the results. 5. Display the plot of seasonal differences and the first differences. Interpret the plot. 6. Graph and interpret the sample ACF of seasonal differences with the first differences. 7. Fit the model ARIMA(0, 1, 1)×(0, 1, 1)4, and assess the significance of the estimated coefficients. 8. Perform all of the diagnostic tests on the residuals. 9. Calculate and plot forecasts for the next two years of the series. Also include the 95% forecast limits.

Q8 (8 points): Let xt be a time series of 11 values 1, , 1, 3, −1, , 0, 4, , 2, 0, −2, , 1, , 2 1. Use the Yule-
Walker equations to estimate the parameters of AR(2) model. 2. Construct the 95% confidence interval around the parameter ϕ1. 3. Calculate the first three sample partial autocorrelations. 4. Calculate the residuals of the fitted model in part (1). 5. Use Durbin-Watson test to check for first-order autoregressive errors. 6. Test whether there is a unit root versus the alternative that the process is stationary. 7. Apply the Ljung-Box portmanteau test on the residuals of the fitted AR(2) model at lag m = 3. What you can conclude? 8. Redo parts (1)-(7) using R and confirm you get similar results!

Q9 (2 points): Generate n = 100 observations from the three models ARMA(0, 1), ARMA(1, 0), and ARMA(1, 1). Use the same parameters that are given in Question 1, parts 3, 1, and 5 respectively. 1. Compute the sample ACF for each model and compare it to the theoretical values. 2. Compute the sample PACF for each of the generated series and compare it to the theoretical values.

Q10 (2 points): Fit a seasonal ARIMA model of your choice to the U.S. Live Birth Series (birth) available from the package astsa. Use the estimated model to forecast the next 12 months.

Q11 (7 points): Let St represent the monthly sales data in sales (n = 150), and let Lt be the leading indicator in lead. 1. Fit an ARIMA model to St , the monthly sales data. Discuss your model fitting in a step-by-step fashion, presenting your A. initial examination of the data, B. transformations, if necessary, C. initial identification of the dependence orders and degree of differencing, D. parameter estimation, E.
residual diagnostics and model choice. 2. Use the cross-correlation function (CCF) and lag plots between
St and Lt to argue that a regression of St on Lt−3 is reasonable. [Note that in lag2.plot(), the first named series is the one that gets lagged.] 3. Fit the regression model St = β0 +β1 Lt−3 +xt , where xt is an ARMA process (explain how you decided on your model for xt). Discuss your results.