Tarrows, Pearson, Foster and Zuligar (TPF&Z) is one of the largest actuarial consulting firms in the United States. In addition to providing its clients with expert advice on executive compensation programs and employee benefits programs, TPF&Z also helps its clients determine the amounts of money they must contribute annually to defined benefit retirement programs.

Most companies offer two different types of retirement programs to their employees: defined contribution plans and defined benefit plans. Under a defined contribution plan, the company contributes a fixed percentage of an employee's earning to fund the employee's retirement. Individual employees covered by this type of plan determine how their money is to be invested (e.g., stocks, bonds, or fixed-income securities), and whatever the employees are able to accumulate over the years constitutes their retirement fund. In a defined benefit plan, the company provides covered employees with retirement benefits that are usually calculated as a percentage of the employee's final salary (or sometimes an average of the employee's highest five years of earnings). Thus, under a defined benefit plan, the company is obligated to make payments to retired employees, but the company must determine how much of its earnings to set aside each year to cover these future obligations. Actuarial firms such as TPF&Z assist companies in making this determination.

Several of TPF&Z's clients offer employees defined benefit retirement plans that allow for cost of living adjustments (COLAs). Here, an employee's retirement benefit is still based on some measure of his or her final earnings, but these benefits are increased over time as the cost of living rises. These COLAs are often tied to the national consumer price index (CPI), which tracks the cost of a fixed-market basket of items over time. Each month, the Federal government calculates and publishes the CPI. Monthly CPI data from January 1991 through December 2015 is given in the file CPIData.xlsx. To assist their clients in determining the amount of money to accrue during a year for their annual contribution to their defined benefit programs, TPF&Z must forecast the value of the CPI one year into the future. Pension assets represent the largest single source of investment funds in the world. As a result, small changes or differences in TPF&Z's CPI forecast translate into hundreds of millions of dollars in corporate earnings being diverted from the bottom line into pension reserves. Needless to say, the partners of TPF&Z want their CPI forecasts to be as accurate as possible.

1. Prepare an XY scatter plot of the data. (Set the minimum value on the Y-axis on this chart to 110.) Discuss any concerns you might have about fitting a time series model to this data.

2. Prepare an XY scatter plot of the data starting from January 1, 2010. (Set the minimum value on the Y-axis on this chart to 210.) You should use this data set for the remaining questions.

3. Fit the data with a linear trend model (as discussed in section 11.16 of SMDA).
a. What is the equation of the model you estimated?

b. For each value in the sample data, plot the estimated CPI values produced by this model against the actual CPI values. How well does this model seem to fit the data?

c. What is the value of the R2 statistic for this model? Interpret this value.

d. Forecast the next 12 CPI values using this model.

4. Fit the data with a second order polynomial (or quadratic) trend model (as discussed in section 11.17 of SMDA).
a. What is the equation of the model you estimated?

b. For each value in the sample data, plot the estimated CPI values produced by this model against the actual CPI values. How well does this model seem to fit the data?

c. What is the value of the R2 statistic for this model? Interpret this value.

d. Forecast the next 12 CPI values using this model.

5. Fit the data with an additive seasonal regression model with second order polynomial trend component. (This process is described in section 11.20 of SMDA for quarterly data where p=4. In this case, the data is monthly so you'd need to use p=12.)
a. What is the equation of the model you estimated?

b. For each value in the sample data, plot the estimated CPI values produced by this model against the actual CPI values. How well does this model fit the data?

c. What is the value of the R2 statistic for this model? Interpret this value.

d. Forecast the next 12 CPI values using this model.

6. Which of the three previous models would you have greatest confidence in using? Why?