Exam 14: Time Series: Descriptive Analyses, Models, and Forecasting Available on CD

Retail sales for a home improvement store in quarters 1-4 over a five-year period are shown (in millions of dollars) in the table below. Quarter Year 1 2 3 4 1 1.2 1.4 1.5 1.1 2 1.3 1.6 1.5 1.2 3 1.4 1.8 1.8 1.6 4 1.4 1.7 1.9 1.6 5 1.6 2.0 2.1 1.9 a. Write a regression model that contains trend and seasonal components to describe the sales data. b. Use least squares regression to fit the model. c. Use the regression model to forecast the quarterly sales during Year 6. Give 95% prediction intervals for the forecasts.

Using just the wool prices and a smoothing constant w = 0.8, find the exponentially smoothed value for May.

The exponential smoothing constant can be any number between 0 and 100.

Smoothing techniques are used to remove rapid fluctuations in a time series so the general trend can be seen.

(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991 The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is shown in the table. Year Time Wheat Harvested by Coop. Member (y, in thousands of bushels) 1991 1 75 1992 2 78 1993 3 82 1994 4 82 1995 5 84 1996 6 85 1997 7 87 1998 8 91 1999 9 92 2000 10 92 2001 11 93 2002 12 96 2003 13 101 2004 14 102 -Suppose the least squares regression equation is $\hat{ y } _ { \mathrm { t } } = 75 + 2 \mathrm { t }$ . Use the regression model to forecast the harvest in 2005.

Consider the table below which displays the price of a commodity for six consecutive years. Year Price (dollars) 1 250 2 255 3 253 4 255 5 259 6 261 a. Calculate the values in the exponentially smoothed series using $\mathrm { w } = 0.6$ . b. Calculate the forecast errors for Years 7-10 if the actual values in those years are 262, 264, 263, 266 respectively. c. Calculate MAD, MAPE, and RMSE, using the forecast errors for Years 7-10.

The Holt forecasting model consists of both an exponentially smoothed component and a trend component.

Since the theoretical distributional properties of the forecast errors with smoothing methods are unknown, many analysts regard smoothing methods as descriptive procedures rather than inferential procedures.

(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, $y _ { t }$ as a function of total labor income, $\mathrm { x } _ { 1 } \mathrm { t }$ , and total property income, $\mathrm { x } _ { 2 } \mathrm { t }$ , with the following results. Assume data for $\mathrm { n } = 40 \mathrm { years }$ were used in the analysis. $\hat { y } _ { t } = 7.81 + 0.91 x _ { 1 t } + 0.57 x _ { 2 t } \quad s = 1.29 \quad \text { Durbin-Watson } d = 2.09$ -For the situation above, set up the null and alternative hypotheses for testing for the presence of autocorrelation of residuals.

(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below. Year G.M. Ford Chrysler 1986 8993 5810 1796 1987 7101 4328 1225 1988 6762 4313 1283 1989 6244 4255 1182 1990 7769 4934 1494 1991 8256 5585 2034 1992 9305 5551 2157 -Using 1986 as the base year, and only using the Chrysler sales data, find the simple index for 1992.

The value of the Durbin-Watson d-statistic always falls in the interval from 0 to 1.

(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below. Year G.M. Ford Chrysler 1986 8993 5810 1796 1987 7101 4328 1225 1988 6762 4313 1283 1989 6244 4255 1182 1990 7769 4934 1494 1991 8256 5585 2034 1992 9305 5551 2157 -Using 1986 as the base year, find the simple composite index for 1992.

A composite index number represents combinations of the prices or quantities of several commodities.

It is common to use dummy variables to describe seasonal differences in a time series.

With N time periods in your data, a good rule of thumb is to forecast ahead no more than 2N time periods.

A major advantage of forecasting with smoothing techniques is that the standard deviation of the forecast errors is known prior to observing the future values.

(Situation H) The prices of coffee, gasoline, and sugar for each month of 1983 are shown below in the table. Month Price of Coffee (per pound) Price of Gasoline (per gallon) Price of Sugar (per pound) January \ 1.47 \ 1.15 \ 0.32 February \ 1.47 \ 1.10 \ 0.33 March \ 1.47 \ 1.06 \ 0.32 April \ 1.39 \ 1.13 \ 0.33 May \ 1.36 \ 1.18 \ 0.33 June \ 1.36 \ 1.20 \ 0.34 July \ 1.36 \ 1.21 \ 0.34 August \ 1.36 \ 1.20 \ 0.34 September \ 1.36 \ 1.19 \ 0.34 October \ 1.36 \ 1.17 \ 0.34 November \ 1.36 \ 1.16 \ 0.33 December \ 1.36 \ 1.15 \ 0.33 -Using just the price of gasoline and a smoothing constant of $\mathrm { w } = 0.4$ lculate the exponentially smoothed value for March.

(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years. -To test for first-order autocorrelation, we use the _______ test.

(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table. Year Commercial Manufacturing Retail Trade Wholesale Trade 1985 1637 2262 1645 4799 1089 1986 1331 1770 1360 4139 1028 1987 1041 1463 1122 3406 887 1988 773 1204 1013 2889 740 1989 930 1378 1165 3183 908 1990 1594 2355 1599 4910 1284 1991 2366 3614 2223 6882 1709 1992 3840 4872 3683 9730 2783 1993 8627 5247 4433 11,429 3598 1994 12,787 6936 5759 13,787 4882 1995 16,647 7004 5662 13,501 4835 1996 20,911 7035 5641 13,509 4808 -Using just the wholesale trade failures and a smoothing constant $\mathrm { w } = 0.7$ , calculate the exponentially smoothed value for 1988.

(Situation K) Foreign Exchange rates, the values of foreign currency in U.S. dollars, are important to investors and international travelers. The table lists the monthly foreign exchange rates of the British pound (in U.S. dollars per pound) for a certain year. January 1.13 February 1.10 March 1.13 April 1.23 May 1.25 June 1.28 July 1.38 August 1.39 September 1.36 October 1.42 November 1.44 December 1.44 -Calculate the value of the exponentially smoothed series for April using a smoothing constant of $w = 0.7$