Stat 510 Homework 3 Due Tuesday Sept
Stat 510 Homework 3 Due Tuesday Sept. 21
For all three questions, use the data in the Minitab worksheet assign1.MTW that can be linked at www.stat.psu.edu/~rho/stat510fa04/. On the Penn State network, clicking on the link will allow you to open Minitab with the data in place. On other machines, you might have to download the file, open Minitab, and then read the worksheet from the download location.
Several plots are asked for in these questions. It’s not necessary to turn in the plots as part of your answer.
The Sept. 14 handout gives some Minitab guidance.
1. The first column (LogOil) gives a measure of the price of oil in the United States for 100 consecutive months. For month t, this measure (in C3) is where x = actual price.
A. Do a time series plot of the data in C1 (LogOil). Briefly describe the noteworthy features.
B. Create a column of first differences. Use Stat>Time Series>Differences to do this. Then, do a time series plot of the column of differences. Describe the noteworthy features.
C. Determine the sample ACF of the column of first differences.
(Stat>Time Series>Autocorrelations).
· List the autocorrelations for the first seven lags.
· Discuss whether an AR(1) is a feasible model for the differenced data.
· Discuss whether an MA(1) is a feasible model for the differenced data.
2. The third column (Mortality) gives weekly mortality rates due to cardiovascular causes in Los Angeles County of California.
A. Do a time series plot of the data in C3 (Mortality). Briefly describe the noteworthy features.
B. Create a column of first differences. Use Stat>Time Series>Differences to do this. Then, do a time series plot of the column of differences. Describe the noteworthy features.
C. Determine the sample ACF of the column of first differences.
(Stat>Time Series>Autocorrelations).
· List the autocorrelations for the first seven lags.
· Discuss whether an AR(1) is a feasible model for the differenced data.
· Discuss whether some form of a moving average might be a feasible model for the differenced data. Identify a possible order for this MA model.
3. The fifth column (quakes) of the Minitab worksheet gives worldwide annual numbers of earthquakes with seismic magnitude >7, from the year 1900 until 1998
A. Do a time series plot of the variable quakes. Write a brief interpretation of the plot. Focus on issues like – do you think there is a trend, are there outliers, does the variance seem to be relatively constant?
B. Determine the sample ACF of the variable quakes. Explain whether the pattern (if any) is consistent (or not) with the ACF of an AR(1) model.
C. Regardless of how you answered the previous part, fit an AR(1) model to quakes. Use Stat>Time Series as described on the second sheet of the Sept. 1 handout. As part of the procedure, graph the ACF of the residuals and graph residuals versus fits.
Give the ARIMA output, and write out the estimated model for .
D. Based on the ACF of the residuals, do you think the model is suitable? Explain.
E. Write a brief interpretation of the plot of residuals versus fitted values. What is indicated about the data and/or model?
F. Your answer to part B could be “maybe not .” Try fitting an AR(2) model to these data. The form of this model is where wt is iid(0, ).
In Stat>Time>Series>ARIMA, enter 2 in the box for the nonseasonal autoregressive part of the model.
· What is the MS error for this model?
· Explain whether the sample estimates and are statistically significant or not (look at the p-values associated with the t-statistics for these estimated coefficients).
F. Try fitting a model that contains a 1st order AR term and a 1st order MA term. This model might be denoted as ARIMA(1,0,1) and its form is where wt is iid(0, ). In Stat>Time>Series>ARIMA, enter 1 in the box for the nonseasonal autoregressive part of the model and 1 in the box for the nonseasonal moving average part of the model.
· What is the MS error for this model?
· Explain whether the sample estimates and are statistically significant or not.
G. Which of the three models tried in this problem is the most suitable? Briefly explain.