Statistics for Business and Economics: Global Edition

Newbold P., Carlson W.L., Thorne B.M.

Chapter 16

Forecasting with Time-Series Models - all with Video Answers

Educators

Chapter Questions

Problem 1

The data file Housing Starts shows private housing units started per thousand of population in the United States over a period of 24 years. Use a computer to prepare a time plot of this series and comment on the components of the series revealed by this plot.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 2

The data file Earnings per Share shows earnings per share of a corporation over a period of 28 years.
Use a computer to prepare a time plot of this series and comment on the components of the series revealed by this plot.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 2

The data file Quarterly Earnings shows quarterly sales of a corporation over a period of 6 years. Use the Holt-Winters seasonal method to obtain forecasts of sales up to eight quarters ahead. Employ smoothing constants $\alpha=0.4, \beta=0.5$, and $\gamma=0.6$. Graph the data and the forecasts.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 3

The data file Fargo Electronics Earnings shows quarterly sales of a corporation over a period of 6 years.
a. Draw a time plot of this series and discuss its features.
b. Use the seasonal-index method to seasonally adjust this series. Graph the seasonally adjusted series and discuss its features.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 4

The data file Fargo Electronics Sales shows quarterly sales of a corporation over a period of 6 years.
a. Draw a time plot of this series and discuss its features.
b. Use the seasonal-index method to seasonally adjust this series. Graph the seasonally adjusted series and discuss its features.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 5

The data file Gold Price shows the year-end price of gold (in dollars) over 14 consecutive years. Compute a simple, centered 3-point moving average series for the gold price data. Plot the smoothed series and discuss the resulting graph.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 6

The data file Housing Starts shows private housing units started per thousand of population in the United States over a period of 24 years. Compute a simple, centered 5-point moving average series for the housing starts data. Draw a time plot of the smoothed series and comment on your results.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 7

The data file Earnings per Share shows earnings per share of a corporation over a period of 28 years. Compute a simple, centered 7-point moving average series for the corporate earnings data. Based on a time plot of the smoothed series, what can be said about its regular components?

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 8

Let
$$
x_t^*=\frac{1}{2 m+1} \sum_{j=-m}^m x_{t+j}
$$
be a simple, centered $(2 m+1)$-point moving average. Show that
$$
x_{t+1}^*=x_t^* \frac{x_{t+m+1}-x_{t-m}}{2 m+1}
$$
How might this result be used in the efficient computation of series of centered moving averages?

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 9

The data file Acme LLC Earnings per Share shows earnings per share of a corporation over a period of 7 years.
a. Draw a time plot of these data. Does your graph suggest the presence of a strong seasonal component in this earnings series?
b. Using the seasonal index method, obtain a seasonally adjusted earnings series. Graph this series and comment on its behavior.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 10

a. Show that the centered s-point moving average series of Section 16.2 can be written as follows:
$$
x_t^*=\frac{x_{t-(s / 2)}+2\left(x_{t-(s / 2)+1}+\cdots+x_{t+(s / 2)-1}\right)+x_{t+(s / 2)}}{2 s}
$$
b. Show that
$$
x_{t+1}^*=x_t^*+\frac{x_{t+(s / 2)+1}+x_{t+(s / 2)}-x_{t-(s / 2)+1}-x_{t-(s / 2)}}{2 s}
$$
Discuss the computational advantages of this formula in the seasonal adjustment of monthly time series.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 11

The data file Inventory Sales shows the inventory-sales ratio for manufacturing and trade in the United States over a period of 12 years. Use the method of simple exponential smoothing to obtain forecasts of the inventory-sales ratio over the next 4 years. Use a smoothing constant of $\alpha=0.6$. Graph the observed time series and the forecasts.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 12

The data file Gold Price shows the year-end price of gold (in dollars) over 14 consecutive years. Use the method of simple exponential smoothing, with a smoothing constant of $\alpha=0.7$, to obtain forecasts of the price of gold in the next 5 years.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 13

The data file Housing Starts shows private housing units started per thousand of population in the United States over a period of 24 years. Using the data, employ the method of simple exponential smoothing with smoothing constant $\alpha=0.5$ to predict housing starts in the next 3 years.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 14

The data file Earnings per Share shows earnings per share of a corporation over a period of 18 years.
a. Using smoothing constants $\alpha=0.8,0.6,0.4$, and 0.2 , find forecasts based on simple exponential smoothing.
b. Which of the forecasts would you choose to use?

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 15

a. If forecasts are based on simple exponential smoothing, with $\hat{x}_t$ denoting the smoothed value of the series at time $t$, show that the error made in forecasting $x_t$, standing at time $(t-1)$, can be written as follows:
$$
e_f=x_t-\hat{x}_{t-1}
$$
b. Hence, show that we can write $\hat{x}_t=x_t-(1-\alpha) e_t$, from which we see that the most recent observation and the most recent forecast error are used to compute the next forecast.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 16

Suppose that in the simple exponential smoothing method, the smoothing constant $\alpha$ is set equal to 1 . What forecasts will result?

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 17

Comment on the following statement: We know that all business and economic time series exhibit variability through time. Yet if simple exponential smoothing is used, the same forecast results for all future values of the time series. Since we know that all future values will not be the same, this is absurd.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 18

The data file Industrial Production Canada shows an index of industrial production for Canada over a period of 15 years. Use the Holt-Winters procedure with smoothing constants $\alpha=0.7$ and $\beta=0.5$ to obtain forecasts over the next 5 years.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 19

The data file Hourly Earnings shows manufacturing hourly earnings in the United States over 24 months. Use the Holt-Winters procedure with smoothing constants $\alpha=0.7$ and $\beta=0.6$ to obtain forecasts for the next 3 months.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 20

The data file Food Prices shows an index of food prices, seasonally adjusted, over a period of 14 months in the United States. Use the Holt-Winters method with smoothing constants $\alpha=0.5$ and $\beta=0.5$ to obtain forecasts for the next 3 months.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 21

The data file Profit Margins shows percentages of profit margins of a corporation over a period of 11 years. Obtain forecasts for the next 2 years, using the Holt-Winters method with smoothing constants $\alpha=0.4$ and $\beta=0.4$.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 23

The data file Quarterly Sales shows quarterly sales of a corporation over a period of 6 years. Use the Holt-Winters seasonal method to obtain forecasts of sales up to eight quarters ahead. Employ smoothing constants $\alpha=0.5, \beta=0.6$, and $\gamma=0.7$. Graph the data and the forecasts.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 24

Using the data in the data file Earnings per Share, estimate a first-order autoregressive model for the earnings per share. Use the fitted model to obtain forecasts for the next 4 days.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 26

Using the data file Housing Starts, estimate autoregressive models of orders 1 through 4 . Use the method of this section to test the hypothesis that the order of the auto regression is $p-1$ against the alternative that the order is $p$, with a significance level of $10 \%$. Select one of these models, and calculate forecasts of housing starts for the next 5 years. Draw a time plot showing the original observations together with the forecasts. Would different forecasts result if a significance level of $5 \%$ was used for the tests of autoregressive order?

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Problem 27

From the data file Earnings per Share on corporate earnings per share, fit autoregressive models of orders 1 through 4 . Use the procedure of this section to test the hypothesis that the order of the auto regression is $p-1$ against the alternative that the true order is $p$, with a $10 \%$ significance level. Choose one of these models, and compute forecasts of earnings per share for the next 5 years. Draw a graph showing the original data along with these forecasts. Would the results differ if a $5 \%$ significance level was used for the tests?

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Problem 28

In Figure 16.10, fitted autoregressive models of orders 1 through 4 are given for annual sales data. We then selected a model by testing the null hypothesis of autoregression of order $p-1$ against the alternative of autoregression of order $p$ at the $5 \%$ significance level. Repeat this procedure, but test at the $10 \%$ significance level.
a. What autoregressive model is now selected?
b. Obtain forecasts of sales for the next 3 years, based on this selected model.

Sneha Ravi

Numerade Educator

Problem 29

For a certain product it was found that annual sales volume could be well described by a third-order autoregressive model. The estimated model obtained was as follows:
$$
x_t=202+1.10 x_{t-1}-0.48 x_{t-2}+0.17 x_{t-3}+\varepsilon_t
$$
For 1993, 1994, and 1995, sales were 867, 923, and 951, respectively. Calculate sales forecasts for the years 1996 through 1998.

Dwijendra Rao

Numerade Educator

Problem 30

For many time series, particularly prices in speculative markets, the random walk model has been found to give a good representation of actual data. This model is written as follows:
$$
x_t=x_{t-1}+\varepsilon_t
$$
Show that, if this model is appropriate, forecasts of $x_{n+h t}$, standing at time $n$, are given by
$$
\hat{x}_{n+h}=x_n \quad(h=1,2,3, \ldots)
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 31

Refer to the data file Hourly Earnings, showing earnings over 24 months. Denote the observations $x_t(t=1,2, \ldots, 24)$. Now, form the series of first differences:
$$
z_t=x_t-x_{t-1}(t=2,3, \ldots, 24)
$$
Fit autoregressive models of orders 1-4 to the series $z_t$. Using the approach of this section for testing the hypothesis that the autoregressive order is $p-1$ against the alternative of order $p$, with a $10 \%$ significance level, select one of these models. Using the selected model, find forecasts for $z_t$, where $t=25,26$, and 27 . Hence, obtain forecasts of earnings for the next 3 months.

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Problem 32

Explain the statement that a time series can be viewed as being made up of a number of components. Provide examples of business and economic time series for which you would expect particular components to be important.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 33

In many business applications, forecasts for future values of time series, such as sales and earnings, are made exclusively on the basis of past information on the time series in question. What features of time-series behavior are exploited in the production of such forecasts?

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 34

A manager in charge of inventory control requires monthly sales records over the past 4 years for each of these products. He decides to use, as forecasts for each of the next 6 months, the average monthly sales over the previous 4 years. Do you think this is a good strategy? Provide reasons.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 35

What is meant by the seasonal adjustment of a time series? Explain why government agencies expend a large amount of effort on the seasonal adjustment of economic time series.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 36

The data file Quarterly Earnings shows quarterly earnings per share of a corporation over 7 years.
a. Draw a time plot of these data. Does this graph suggest the presence of a strong seasonal component?
b. Use the seasonal index method to obtain a seasonally adjusted series.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 37

The data file Product Sales shows 24 annual observations on sales of a product. Use simple exponential smoothing with smoothing constant $\alpha=0.5$ to obtain forecasts of sales for the next 3 years.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 38

Refer to the data file Quarterly Earnings. Use the Holt-Winters seasonal method with smoothing constants $\alpha=0.6, \beta=0.6$, and $\gamma=0.8$ to obtain forecasts of this earnings-per-share series for the next four quarters.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 39

Using the data file Product Sales, estimate autoregressive models of orders 1-4 for product sales. Using the procedure of Section 16.4 for testing the hypothesis that the autoregressive order is $p-1$ against the alternative that the order is $p$, with a significance level of $10 \%$, choose one of these models. Compute forecasts for the next 3 years from the chosen model.

Jai Chadha

Numerade Educator

Problem 40

Using the data in the file Macro2010, develop and autoregressive model for the prime interest rate. First, use the data for the period 1980, first quarter, through 2000, fourth quarter, to forecast for the quarters in years 2001-2003. Then use the data from 1980, first quarter, through 2007, fourth quarter, to forecast the quarters in the years 2008 and 2009. Discuss the differences in the accuracy of the forecasts compared to the actual results and indicate reasons for these differences.

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Problem 41

Using the data in the file Macro2010 develop an autoregressive model for the Personal Consumption Expenditures. First, use the data for the period 1980, first quarter, through 2000, fourth quarter, to forecast for the quarters in years 2001-2003. Then use the data from 1980, first quarter, through 2007, fourth quarter, to forecast the quarters in the years 2008 and 2009. Discuss the differences in the accuracy of the forecasts compared to the actual results and indicate reasons for these differences.

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Problem 42

Using the data in the file Macro2010 develop an autoregressive model for fixed investment. First, use the data for the period 1965, first quarter, through 2000 , fourth quarter, to forecast for the quarters in years 2001-2003. Then use the data from 1965, first quarter, through 2007, fourth quarter, to forecast the quarters in the years 2008 and 2009.

Discuss the differences in the accuracy of the forecasts compared to the actual results and indicate reasons for these differences.

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Problem 43

Using the data in the file Macro2010, develop an autoregressive model for imports. First, use the data for the period 1970, first quarter, through 2000 , fourth quarter, to forecast for the quarters in years 2001-2003. Then use the data from 1970, first quarter, through 2007, fourth quarter, to forecast the quarters in the years 2008 and 2009. Discuss the differences in the accuracy of the forecasts compared to the actual results and indicate reasons for these differences.

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Problem 254

The data file Trading Volume shows the volume of transactions (in hundreds of thousands) in shares of a corporation over a period of 12 weeks. Using these data, estimate a first-order autoregressive model, and use the fitted model to obtain forecasts of volume for the next 3 weeks.

Adriano Chikande

Adriano Chikande

Numerade Educator