part © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in Business Analytics: Data Analysis and Chapter Decision Making 12 Time Series Analysis and Forecasting 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 © 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: Judgmental methods Extrapolation (or time series) methods Econometric (or causal) methods The first method is basically nonquantitative; the last two are quantitative © 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 © 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) © 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 © 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 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 © 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 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 © 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 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 © 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 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 © 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 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 © 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 of 2)  The graph below shows the forecast series superimposed on the original series © 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, Tt, and a corresponding smoothing constant β  The interpretation of L is exactly as before t  The interpretation of T is that it represents an estimate of the change in t the series from one period to the next  The equations for Holt’s model are shown below: © 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 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 © 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 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 © 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  In a multiplicative seasonal model, a base forecast is multiplied by an appropriate seasonal index  These indexes, one for each season, typically average to © 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: © 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 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 of 1997 through quarter 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) © 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 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 © 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 © 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 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 © 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 of 2)  The deseasonalized data, with forecasts superimposed, are shown below, on the left  The results of reseasonalizing are shown below, on the right © 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 as the reference quarter) and estimate the additive equation:  Then the coefficients of the dummy variables, b1, b2, and b3, 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 © 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 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 © 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 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 © 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 ... 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... residuals visually— although this is not always reliable © 2015 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole... 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