STAT 112 --- Quiz 7 --- Solutions

This is a closed book quiz; you may use a calculator. Show your work to receive full credit.

Problem 1.The SAS printout for the model is reproduced on the next page. The data are quarterly sales indexes for one brand of graphing calculator at a campus store over 1996-2000.The quarters are based on an academic year, so the first quarter represents fall, the second winter; the third, spring and the fourth, summer.

We defined the time variable as t=1 for the first quarter of 1996, t=2 for the second quarter of 1996, etc. The seasonal dummy variables are as follows: Q1=1 if quarter 1, 0 otherwise, Q2=1 if quarter 2, 0 otherwise, and Q3=1 if quarter 3, 0 otherwise.

  1. (4 pts) What are the value and the meaning of the least squares estimate of β4? Of β0?

The estimate of β4 is 105.5125: The average sales in the spring are up by 105.5125 compared to the sales in the summer.

The estimate of β0 is 119.85: The average sales in the summer are 119.85.

  1. (4 pts) Which of the assumptions about the random error component is in doubt when a regression model is fitted to time series data?

The assumption of independence.

  1. (4 pts) Find the forecasts and the 95% prediction intervals for the 2001 quarterly sales. Interpret the result.

Q1: 729.0, (662.8, 795.1)

Q2: 706.0, (639.8, 772.1)

Q3: 605.1, (539.0, 671.3)

Q4: 516.1, (450.0, 582.3)

Meaning: For the first quarter of 2001, for example, we expect the range of the sales to fluctuate between 662.8 and 795.1 95% of the time.

Problem 2. (8 pts) The decrease in the value of the dollar,Yt, from 1960 to 1997 is illustrated by the data in the table. The buying power of the dollar (compared with 1982) is listed for each year. The first order model was fit to the data using the method of least squares. The STATISTIX printout and a plot of the regression residuals are attached.

  1. (3 pts) Examine the plot of the regression residuals against t. Is there a tendency for the residuals to have long positive and negative runs? To what do you attribute this phenomenon?

Positive auto-correlation: As expected, high values of the dollar are followed by high values and low values by low.

  1. (5 pts) Locate the Durbin-Watson d statistic on the printout and test the null hypothesis that the time series residuals are uncorrelated. Use α=.10.

The computer output provides the value of the DW test statistic, d=0.0692, and also the p-value of positive autocorrelation (0). Hence, the p-value for testing presence of autocorrelation is also 0.