1. Chapter © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. BUSINESS ANALYTICS: DATA ANALYSIS AND DECISION MAKING Time Series Analysis and Forecasting 12

2. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Introduction  Forecasting is a very difficult task, both in the short run and in the long run.  Analysts search for patterns or relationships in historical data and then make forecasts.  There are two problems with this approach: It is not always easy to undercover historical patterns or relationships. It is often difficult to separate the noise, or random behavior, from the underlying patterns. Some forecasts may attribute importance to patterns that are in fact random variations and are unlikely to repeat themselves. There are no guarantees that past patterns will continue in the future.

3. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Forecasting Methods: An Overview  There are many forecasting methods available, and there is little agreement as to the best forecasting method.  The methods can be divided into three groups: 1. Judgmental methods 2. Extrapolation (or time series) methods 3. Econometric (or causal) methods  The first method is basically nonquantitative; the last two are quantitative.

4. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Extrapolation Models  Extrapolation models are quantitative models that use past data of a time series variable to forecast future values of the variable.  Many extrapolation models are available:  Trend-based regression  Autoregression  Moving averages  Exponential smoothing  All of these methods look for patterns in the historical series and then extrapolate these patterns into the future.  Complex models are not always better than simpler models.  Simpler models track only the most basic underlying patterns and can be more flexible and accurate in forecasting the future.

5. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Econometric Models  Econometric models, also called causal or regression-based models, use regression to forecast a time series variable by using other explanatory time series variables.  Prediction from regression equation:  Causal regression models present mathematical challenges, including:  Determining the appropriate “lags” for the regression equation  Deciding whether to include lags of the dependent variable as explanatory variables  Autocorrelation (correlation of a variable with itself) and cross- correlation (correlation of a variable with a lagged version of another variable)

6. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Combining Forecasts  This method combines two or more forecasts to obtain the final forecast.  The reasoning is simple: The forecast errors from different forecasting methods might cancel one another.  Forecasts that are combined can be of the same general type, or of different types.  The number of forecasts to combine and the weights to use in combining them have been the subject of several research studies.

7. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Components of Time Series Data (slide 1 of 4)  If observations increase or decrease regularly through time, the time series has a trend.  Linear trend—occurs if the observations increase by the same amount from period to period.  Exponential trend—occurs when observations increase at a tremendous rate.  S-shape trend—occurs when it takes a while for observations to start increasing, but then a rapid increase occurs, before finally tapering off to a fairly constant level.

8. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Components of Time Series Data (slide 2 of 4)  If a time series has a seasonal component, it exhibits seasonality—that is, the same seasonal pattern tends to repeat itself every year.

9. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Components of Time Series Data (slide 3 of 4)  A time series has a cyclic component when business cycles affect the variables in similar ways.  The cyclic component is more difficult to predict than the seasonal component, because seasonal variation is much more regular.  The length of the business cycle varies, sometimes substantially.  The length of a seasonal cycle is generally one year, while the length of a business cycle is generally longer than one year and its actual length is difficult to predict.

10. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Components of Time Series Data (slide 4 of 4)  Random variation (or noise) is the unpredictable component that gives most time series graphs their irregular, zigzag appearance.  A time series can be determined only to a certain extent by its trend, seasonal, and cyclic components; other factors determine the rest.  These other factors combine to create a certain amount of unpredictability in almost all time series.

11. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Measures of Accuracy (slide 1 of 2)  The forecast error is the difference between the actual value and the forecast. It is denoted by E with appropriate subscripts.  Forecasting software packages typically report several summary measures of the forecast errors:  MAE (Mean Absolute Error):  RMSE (Root Mean Square Error):  MAPE (Mean Absolute Percentage Error):  One other measure of forecast errors is the average of the errors.

12. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Measures of Accuracy (slide 2 of 2)  Some forecasting software packages choose the best model from a given class by minimizing MAE, RMSE, or MAPE.  However, small values of these measures guarantee only that the model tracks the historical observations well.  There is still no guarantee that the model will forecast future values accurately.  Unlike residuals from the regression equation, forecast errors are not guaranteed to always average to zero.  If the average of the forecast errors is negative, this implies a bias, or that the forecasts tend to be too high.  If the average is positive, the forecasts tend to be too low.

13. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Randomness (slide 1 of 2)  All forecasting models have the general form shown in the equation below:  The fitted value is the part calculated from past data and any other available information.  The residual is the forecast error.  The fitted value should include all components of the original series that can possibly be forecast, and the leftover residuals should be unpredictable noise.  The simplest way to determine whether a time series of residuals is random noise is to examine time series graphs of residuals visually—although this is not always reliable.

14. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Testing for Randomness (slide 2 of 2)  Some common nonrandom patterns are shown below.

15. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Runs Test  The runs test is a quantitative method of testing for randomness. It is a formal test of the null hypothesis of randomness.  First, choose a base value, which could be the average value of the series, the median value, or even some other value.  Then a run is defined as a consecutive series of observations that remain on one side of this base level. If there are too many or too few runs in the series, the null hypothesis of randomness can be rejected.

16. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.1: Stereo Sales.xlsx (slide 1 of 2)  Objective: To use StatTools’s Runs Test procedure to check whether the residuals from this simple forecasting model represent random noise.  Solution: Data file contains monthly sales for a chain of stereo retailers from the beginning of 2009 to the end of 2012, during which there was no upward or downward trend in sales and no clear seasonality.  A simple forecast model of sales is to use the average of the series, 182.67, as a forecast of sales for each month.  The residuals for this forecasting model are found by subtracting the average from each observation.  Use the runs test to see whether there are too many or too few runs around the base of 0.  Select Runs Test for Randomness from the StatTools Time Series and Forecasting dropdown, choose Residual as the variable to analyze, and choose Mean of Series as the cutoff value.

17. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.1: Stereo Sales.xlsx (slide 2 of 2)  The resulting output is shown below:

18. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Autocorrelation  Another way to check for randomness of a time series of residuals is to examine the autocorrelations of the residuals.  An autocorrelation is a type of correlation used to measure whether values of a time series are related to their own past values. In positive autocorrelation, large observations tend to follow large observations, and small observations tend to follow small observations.  The autocorrelation of lag k is essentially the correlation between the original series and the lag k version of the series. Lags are previous observations, removed by a certain number of periods from the present time. To lag a time series in a spreadsheet by one month, “push down” the series by one row, as shown below.

19. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.1 (Continued): Stereo Sales.xlsx (slide 1 of 2)  Objective: To examine the autocorrelations of the residuals from the forecasting model for evidence of nonrandomness.  Solution: Use StatTools’s Autocorrelation procedure, found on the StatTools Time Series and Forecasting dropdown list.  Specify the times series variable (Residual), the number of lags you want, and whether you want a chart of the autocorrelations, called a correlogram. It is common practice to ask for no more lags than 25% of the number of observations.  Any autocorrelation that is larger than two standard errors in magnitude is worth your attention.  One measure of the lag 1 autocorrelation is provided by the Durbin-Watson (DW) statistic. A DW value of 2 indicates no lag 1 autocorrelation. A DW value less than 2 indicates positive autocorrelation. A DW value greater than 2 indicates negative autocorrelation.

20. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.1 (Continued): Stereo Sales.xlsx (slide 2 of 2)  The autocorrelations and correlogram of the residuals are shown below.

21. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Regression-Based Trend Models  Many time series follow a long-term trend except for random variation.  This trend can be upward or downward.  A straightforward way to model this trend is to estimate a regression equation for Y t, using time t as the single explanatory variable.  The two most frequently used trend models are the linear trend and the exponential trend.

22. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Linear Trend  A linear trend means that the time series variable changes by a constant amount each time period.  The equation for the linear trend model is:  The interpretation of b is that it represents the expected change in the series from one period to the next. If b is positive, the trend is upward. If b is negative, the trend is downward.  The intercept term a is less important: It literally represents the expected value of the series at time t = 0.  A graph of the time series indicates whether a linear trend is likely to provide a good fit.

23. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.2: US Population.xlsx (slide 1 of 2)  Objective: To fit a linear trend line to monthly population and examine its residuals for randomness.  Solution: Data file contains monthly population data for the United States from January 1952 to December 2011. During this period, the population has increased steadily from about 156 million to about 313 million.  To estimate the trend with regression, use a numeric time variable representing consecutive months 1 through 720.  Then run a simple regression of Population versus Time.

24. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.2: US Population.xlsx (slide 2 of 2)  Use Excel’s ® Trendline tool to superimpose a trend line on the time series graph.  Then plot the residuals.

25. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Exponential Trend  An exponential trend is appropriate when the time series changes by a constant percentage (as opposed to a constant dollar amount) each period.  The appropriate regression equation contains a multiplicative error term u t :  This equation is not useful for estimation; for that, a linear equation is required.  You can achieve linearity by taking natural logarithms of both sides of the equation, as shown below, where a = ln(c) and e t = ln(u t ). The coefficient b (expressed as a percentage) is approximately the percentage change per period. For example, if b = 0.05, then the series is increasing by approximately 5% per period.  If a time series exhibits an exponential trend, then a plot of its logarithm should be approximately linear.

26. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.3: PC Device Sales.xlsx (slide 1 of 2)  Objective: To estimate the company’s exponential growth and to see whether it has been maintained during the entire period from 1999 until the end of 2013.  Solution: Data file contains quarterly sales data for a large PC device manufacturer from the first quarter of 1999 through the fourth quarter of 2013.  First, estimate and interpret an exponential trend for the years 1999 through 2008.  Use Excel’s Trendline tool, with the Exponential option, to superimpose an exponential trend line and the corresponding equation on the time series graph through 2008.

27. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.3: PC Device Sales.xlsx (slide 2 of 2)  To use this equation for forecasting the future, substitute later values of Time into the regression equation, so that each future forecast is about 6.54% larger than the previous forecast.  Check whether the exponential growth continued beyond 2008 by creating the Forecast column shown below (by substituting into the regression equation for the entire period through Q4-13).  Then use StatTools to create a time series graph of the two series Sales and Forecast, also shown below.

28. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Random Walk Model  The random walk model is an example of using random series as building blocks for other time series models.  In this model, the series itself is not random.  However, its differences—that is, changes from one period to the next— are random.  This type of behavior is typical of stock price data and other similar data.  The equation for the random walk model is shown below, where m (mean difference) is a constant, and e t is a random series (noise) with mean 0 and a standard deviation that remains constant through time.  A series that behaves according to this random walk model has random differences, and the series tends to trend upward (if m > 0), or downward (if m < 0) by an amount m each period.  If you are standing in period t and want to forecast Y t+1, then a reasonable forecast is given by the equation below:

29. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.4: Stock Prices.xlsx (slide 1 of 2)  Objective: To check whether the company’s monthly closing prices follow a random walk model with an upward trend and to see how future prices can be forecast.  Solution: The monthly closing prices of the manufacturing company’s stock from January 2006 through April 2012 are shown to the right.  To check for the adequacy of a random walk model, a series of differences is required.  Calculate this series with an Excel formula or generate it automatically by selecting Difference from the StatTools Data Utilities dropdown menu.

30. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.4: Stock Prices.xlsx (slide 2 of 2)  Next, plot the differences, as shown below.  To forecast future closing prices, multiply the mean difference by the number of periods ahead, and add this to the final closing price.

31. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Moving Averages Forecasts  One of the simplest and the most frequently used extrapolation models is the moving averages model.  A moving average is the average of the observations in the past few periods, where the number of terms in the average is the span.  If the span is large, extreme values have relatively little effect on the forecasts, and the resulting series of forecasts will be much smoother than the original series. For this reason, this method is called a smoothing method.  If the span is small, extreme observations have a larger effect on the forecasts, and the forecast series will be much less smooth.  Using a span requires some judgment: If you believe the ups and downs in the series are random noise, use a relatively large span. If you believe each up and down is predictable, use a smaller span.

32. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.5: House Sales.xlsx (slide 1 of 3)  Objective: To see whether a moving averages model with an appropriate span fits the housing sales data and to see how StatTools implements this method.  Solution: Data file contains monthly data on the number of new one-family houses sold in the U.S. from January 1991 through December 2011.  Select Forecast from the StatTools Time Series and Forecasting dropdown list.  Then select the time period on the Time Scale tab, and Moving Average on the Forecast Settings tab.

33. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.5: House Sales.xlsx (slide 2 of 3)  The output consists of several parts, with the summary measures MAE, RMSE, and MAPE of the forecast errors included.

34. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.5: House Sales.xlsx (slide 3 of 3)  The graphs below show the behavior of the forecasts. The first is with span 3; the second is with span 12.

35. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Exponential Smoothing Forecasts  Exponential smoothing bases its forecasts on a weighted average of past observations, with more weight on the more recent observations.  There are many variations of exponential smoothing, including:  Simple exponential smoothing—appropriate for a series with no pronounced trend or seasonality  Holt’s method—appropriate for a series with trend but no seasonality  Winters’ method—appropriate for a series with seasonality (and possibly trend)

36. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Simple Exponential Smoothing  Every exponential model has at least one smoothing constant, which is always a number between 0 and 1.  Simple exponential smoothing has a single smoothing constant denoted by α.  The level of the series at time t (L t ) is an estimate of where the series would be at time t if there were no random noise.  The simple exponential method is defined by the following two equations:  The second equation says that the k-period-ahead forecast, F t+k, made of Y t+k in period t is essentially the most recently estimated level, L t.

37. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.5 (Continued): House Sales.xlsx (slide 1 of 2)  Objective: To see how well a simple exponential smoothing model, with an appropriate smoothing constant, fits the housing sales data, and to see how StatTools implements this method.  Solution: Select Forecast from the StatTools Time Series and Forecasting dropdown list.  Then select the simple exponential smoothing option in the Forecast Settings tab, and choose a smoothing constant.  The results are shown to the right.

38. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.5 (Continued): House Sales.xlsx (slide 2 of 2)  The graph below shows the forecast series superimposed on the original series.

39. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Holt’s Model for Trend  When there is a trend in the series, Holt’s method deals with it explicitly by including a trend term, T t, and a corresponding smoothing constant β.  The interpretation of L t is exactly as before.  The interpretation of T t is that it represents an estimate of the change in the series from one period to the next.  The equations for Holt’s model are shown below:

40. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.5 (Continued): House Sales.xlsx (slide 1 of 2)  Objective: To see whether Holt’s method, with appropriate smoothing constants, captures the trends in the housing sales data better than simple exponential smoothing (or moving averages).  Solution: Implement Holt’s method in StatTools almost exactly as for simple exponential smoothing.  The only difference is that you now choose two smoothing constants.  The output is very similar to the simple exponential smoothing output, except that there is now a trend column.

41. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.5 (Continued): House Sales.xlsx (slide 2 of 2)  Now perform a second run of Holt’s method, using the Optimize Parameters option.  The forecasts with nonoptimal smoothing constants are shown below, on the left. The forecasts with optimal smoothing constants are shown below, on the right.

42. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Seasonal Models  Seasonality is the consistent month-to-month (or quarter-to-quarter) differences that occur each year.  The easiest way to check for seasonality is graphically: Look for a regular pattern of ups and/or downs in particular months or quarters.  There are three basic methods for dealing with seasonality:  Winters’ exponential smoothing model  Deseasonalizing the data (then use any forecasting method to model the deseasonalized data and finally “reseasonalize” these forecasts)  Multiple regression with dummy variables for the seasons  Seasonal models are classified as additive or multiplicative.  In an additive seasonal model, an appropriate seasonal index is added to a base forecast. The indexes, one for each season, typically average to 0.  In a multiplicative seasonal model, a base forecast is multiplied by an appropriate seasonal index. These indexes, one for each season, typically average to 1.

43. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Winters’ Exponential Smoothing Model  Winters’ exponential smoothing model is very similar to Holt’s model, but it also has seasonal indexes and a corresponding smoothing constant γ.  This new smoothing constant controls how quickly the method reacts to observed changes in the seasonality pattern. If the constant is small, the method reacts slowly. If it is large, the method reacts more quickly.  The equations for this method are shown below:

44. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.6: Soft Drink Sales.xlsx (slide 1 of 2)  Objective: To see how well Winters’ method, with appropriate smoothing constants, can forecast the company’s seasonal soft drink sales.  Solution: Data file contains quarterly sales for a large soft drink company from quarter 1 of 1997 through quarter 4 of 2012.  There has been an upward trend in sales during this period, and there is also a fairly regular seasonal pattern: sales in the warmer quarters are consistently higher than in the colder quarters.  Proceed in StatTools exactly as with the other exponential smoothing methods, but hold out some of the data for validation.  Fill in the Forecast Settings tab, selecting Winters’ method, basing the model on the data through Q4-2010, holding out eight quarters of data (Q1-2011 through Q4-2012), and forecasting four quarters into the future (all of 2013).

45. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.6: Soft Drink Sales.xlsx (slide 2 of 2)  Parts of the output are shown below, on the left.  The plot of the forecasts superimposed on the original series is shown below, on the right.

46. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Deseasonalizing: The Ratio-to-Moving-Averages Method  Most methods for deseasonalizing time series data are variations of the ratio-to-moving-averages method.  To deseasonalize an observation (assuming a multiplicative model of seasonality), divide it by the appropriate seasonal index.  To find the seasonal index for a particular month, divide the month’s observation by the average of the 12 observations surrounding it. There is a minor problem with this approach: Any one month is not in the middle of any 12-month sequence. Compromise by averaging the two possible averages. (For June, this would be the January-to-December and December-to-November averages.) This is called a centered average.  The usual way to combine all of the indexes for a specific month (if the series covers several years) is to average them.

47. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.6 (Continued): Soft Drink Sales.xlsx (slide 1 of 2)  Objective: To use the ratio-to-moving-averages method to deseasonalize the soft drink data and then forecast the deseasonalized data.  Solution: In StatTools, proceed as with the other exponential smoothing methods, but check the Deseasonalize option in the Time Scale tab of the Forecast dialog box.  Selected outputs are shown below.

48. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.6 (Continued): Soft Drink Sales.xlsx (slide 2 of 2)  The deseasonalized data, with forecasts superimposed, are shown below, on the left.  The results of reseasonalizing are shown below, on the right.

49. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Estimating Seasonality with Regression  A regression approach to forecasting seasonal data uses dummy variables for the seasons.  Depending on how the regression equation is written, you can create either an additive or a multiplicative seasonal model. For example, for quarterly data, create three dummy variables for the first three quarters (using quarter 4 as the reference quarter) and estimate the additive equation: Then the coefficients of the dummy variables, b 1, b 2, and b 3, indicate how much each quarter differs from the reference quarter, and the coefficient b represents the trend.  It is also possible to estimate a multiplicative model using dummy variables for seasonality (and possibly time for trend). An advantage of this approach is that it provides a model with multiplicative seasonal factors and is fairly easy to interpret.

50. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.6 (Continued): Soft Drink Sales.xlsx (slide 1 of 2)  Objective: To use a multiplicative regression equation, with dummy variables for seasons and a time variable for trend, to forecast soft drink sales.  Solution: The data setup is shown below, with dummy variables for three of the four quarters and a Log(Sales) variable.  Then use multiple regression, with Log(Sales) as the dependent variable, and Time, Q1, Q2, and Q3 as the explanatory variables.

51. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 12.6 (Continued): Soft Drink Sales.xlsx (slide 2 of 2)  The regression output is shown on the top right.  A plot of observations versus forecasts for this model is shown on the bottom right.