Instructions:

• Please hand in a pdf (or word) file of your homework which includes all the graphs and your answers to the questions via HuskyCT. And please hand in a R script which includes all your R codes via HuskyCT. Note that in this homework, only Q2 needs R programming.

• This homework is due at the end of the day (i.e., 11:59 pm) on 11/7/2022.

• No late submission is accepted unless there is some unexpected or emergency situations.

Question 1: Calculating Forecasts from Trend Models.

You work for the International Monetary Fund in Washington DC, monitoring Singa- pore’s real consumption expenditures. Using a sample of real consumption data (mea- sured in billions of 2005 Singapore dollars), yt, where t = 1990 : Q1, 1990 : Q2, · · · , 2021 : Q4, you estimate the linear consumption trend model,

yt = β0 + β1TIMEt + et,

where et  ∼ N(0, σ2 ) for all t, obtaining the estimates 0  = 0.37, 1  =  1.78, and s  = 6.  Based upon your estimated trend model, construct feasible point, interval(90%) and density forecasts for 2023: Q1.

Question 2: Mechanics of Trend Estimation and Forecasting.

Use the ”CPI.csv”file I uploaded on HuskyCT to answer the following questions. Choose your series such that it spans at least ten years, and such that it ends at the end of a year (i.e., in December).

a. Produce a time series plot of it. Discuss.

b. Fit linear, quadratic and exponential trend models to your series. Discuss the asso- ciated diagnostic statistics and residual plots.

c. Select a trend model using the AIC and using the BIC. Do the selected models agree? If not, which do you prefer?

d. Use your preferred model to forecast each of the twelve months of the next year. Discuss.

e. The residuals from your fitted model are effectively a detrended version of your original series. Why? Plot them and discuss.

Question 3: Constructing Seasonal Models.

Describe how you would construct a purely seasonal model for the following monthly series. In particular, what dummy variable(s) would you use to capture the relevant ef- fects?

a. A sporting goods store finds that detrended monthly sales are roughly the same for each month in a given three-month season.  For example, sales are similar in the winter months of January, February and March, in the spring months of April, May and June, and so on.

b. A campus bookstore finds that detrended sales are roughly the same for all first, all second, all third, and all fourth months of each trimester.  For example, sales are similar in January, May, and September, the first months of the first, second, and third trimesters, respectively.

c. A Christmas ornament store is only open in November and December, so sales are zero in all other months.

Question 4: Calendar Effects.

You run a large catering firm, specializing in Sunday brunches and weddings. You model the firm’s monthly income as

yt = β0 + 6SSt+ 6WWt + et,

where y is monthly income, and S and W are calendar effect variables indicating the number of Sundays and weddings in a month.

a. What are the units of β0, 6S and 6W?

b. How could you estimate the average income the firm receives per wedding?

c. Over the past thirty years, you have regularly increased your prices to keep pace with inflation. How would you modify the model to account for the effects of such increases?