© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 3: Forecasting trends: Exponential Smoothing 3.1 Methods or Models? 3.2 Extrapolation Methods 3.3 Simple Exponential Smoothing 3.4 Linear Exponential Smoothing 3.5 Exponential Smoothing with a Damped Trend 3.6 Other Approaches To Trend Forecasting 3.7 Prediction Intervals 3.8 The Use of Transformations 3.9 Model Selection 3.10 Principles for Extrapolative Models 2

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.1: Methods or Models? A forecast function is a mathematical expression for deriving the forecasts over the forecast horizon. A forecasting method is a (numerical) procedure for generating a forecast. That is, it involves the direct use of a forecast function. When such methods are not based upon an underlying statistical model, they are termed heuristic. A statistical (forecasting) model is a statistical description of the data generating process from which a forecasting method may be derived. Forecasts are made by using a forecast function that is derived from the model. A statistical model is a necessary foundation for the construction of prediction intervals. 3

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.1: Methods or Models? Forecasting method: Statistical (forecasting) model: o Plus assumptions about the distribution of the random error term. o The estimated model provides the forecast function, along with the framework to make statements about model uncertainty. 4

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.2 Extrapolation Methods 5 Figure 3.1: Past, present & future

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.2 Extrapolation Methods 6 Figure 3.2: Record the past, forecast the future

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.2 Extrapolation Methods 7

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.2: Extrapolation Methods 8 Locally constant forecasts A series is locally constant if the mean level changes gradually over time but there is no reason to expect a systematic increase or decrease. Series may change level / behavior over time. o Global averages apply weight (1/n) to each observation We may apply greater weight to the recent past to capture such changes. Trade off: o catching new trends [quick adjustment] against o spurious movements in response to random movements [slow adjustment]

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use : Use of moving averages A moving average of order K calculates the average for the last K periods: At each point in time, we drop the oldest observation and add a new one. Large values of K produce smoother plots but are slower to adapt to changes. The forecast for period (t+1) is 9

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Table 3.2: Calculation of moving average 10 Data shown is from file WFJ_sales_MA.xlsx. © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.3: WFJ Sales 11

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.4: WFJ Sales, first 26 weeks 12

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.5: MA for WFJ Sales 13

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.3: Simple Exponential Smoothing 14 Updating the usual average A verbal description of this expression is: © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.3: Simple Exponential Smoothing Simple (or Single) Exponential Smoothing follows the same general idea but makes a smooth transition from one period to the next. We select a smoothing constant , such that 0<  <1 making for a partial adjustment. We denote the mean level at time t by L t. 15

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.3: Simple Exponential Smoothing By repeated substitution, we obtain the extended form showing the declining weights that give rise to the name exponential smoothing: 16

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.3: Simple Exponential Smoothing This expression simplifies to the Error Correction form: o where  is known as the smoothing constant o e t is the 1-step ahead forecast error = Y t – F t A smaller value of  gives a smoother trend line: o  = 0.1 gives slow adjustment o  = 0.9 gives rapid adjustment 17

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.6 Effect of different weights 18 © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Table 3.3: SES calculations using Y 1 as starting value and α= © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Discussion Question If you had to make a subjective choice for the value of the smoothing constant, what value would you choose for (a) a product with long-term steady sales and (b) a stock price index? A.0.8 for (a) and 0.3 for (b) B.0.8 for (a) and 0.3 for (b) C.0.5 for (a) and 0.5 for (b) D.0.2 for (a) and 1.0 for (b) E.1.0 for (a) and 0.2 for (b) 20

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.7: SES for WFJ Sales 21 © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.3: Simple Exponential Smoothing Developed by R.G. Brown in 1940s & 50s. For each series, only need to record last forecast, latest observation and smoothing constant. Useful for short-term forecasting [especially for a large number of series]. Local level model, so h-step ahead point forecast is same as 1-step ahead: Uncertainty increases as the forecasting horizon increases. 22

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use : The Use of Hold-Out Samples For model development purposes, a time series is typically divided into two parts: o The first part, the estimation or fitting sample, is used to estimate the parameters of the forecast function. o The second part, the hold-out or test sample is used to check the performance of the forecasting method. Measures of performance based upon the estimation sample are known as in-sample measures. Measures of performance based upon the holdout sample are known as out-of-sample measures. 23

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.3: Simple Exponential Smoothing Estimation Minimize RMSE, MAE or MAPE for the estimation sample; results tend to be similar. o Text uses RMSE unless stated otherwise. (R)MSE corresponds to the use of Least Squares. For SES, we need to estimate both the starting value and the smoothing parameter. o The SES procedure in Excel uses the first observation and requires the user to specify smoothing parameter. o The macro EMS [Exponential Smoothing Macro] provided on the book’s website allows use of the first K observations, where K may be selected by the user. 24

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.3: Simple Exponential Smoothing 25 MethodMPEMAERMSEMAPE MA(3) MA(8) SES(0.2) SES(0.5) SES(opt) Table 3.5: Summary error measures for WFJ Sales (in-sample = periods 1-26; out-of-sample = periods 27-62) © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.3: Simple Exponential Smoothing 26 Q: Does the method of estimation make much difference to the parameter estimates?

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Table 3.6: Effects of fitting using different methods 27 Q: Does the method of estimation make much difference to the parameter estimates? Data: WFJ_sales.xlsx. © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.4: Linear Exponential Smoothing When a time series has a long-term trend (e.g. increases in GDP or sales) the forecasting method must accommodate such features. There are two main approaches: o Convert the series to rates of change (growth rates, either absolute or percentage) then predict the rate of change, OR o Develop forecasting methods that account for trends 28

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.8A: Linear trend fitted to Quarterly Sales 29 Q: A high R 2 but would you rely on this model to provide good forecasts? Data: Netflix_1.xlsx. © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.8B: Quadratic trend fitted to Quarterly Sales 30 Q: A high R 2 but would you rely on this model to provide good forecasts? Data: Netflix_1.xlsx. © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.4: Linear Exponential Smoothing 31 Holt’s Linear Method Forecast function includes local trend: One-step-ahead: o L t is local estimate of level at time t o T t is local estimate of trend at time t SES corresponds to T t = 0

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Initial one-step-ahead forecast made at time t-1 is: Then we observe the new value, The observed forecast error is: This latest observed error (or equivalently the new observation and the previous forecast) provide the new information that is used to update the forecasts : Updating Relationships

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. New forecasts Update local level, allowing for a one-period change due to the trend: Update local trend as a combination of the previous trend and the change in level: New forecast, one-step-ahead is: New forecast, h-steps-ahead: : Updating Relationships

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Error correction form Initial one-step-ahead forecast made at time t-1 is: Forecast error: Update local level, in terms or one-step-ahead error: Update local trend after substituting for level: New forecasts, as before: : Updating Relationships

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.9A: Netflix with SES 35 Q: What you do observe?

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.9A: Netflix with SES 36 Q: What you do observe? Data: Netflix_1.xlsx. © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.9A: Netflix with SES (continued) 37 Q: Does this look better? Why?

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.5: Exponential Smoothing with a Damped Trend 38 Use ‘Damped trend’ in the macro ESM Use the updating equations: New forecasts: The effect of using damped trend is that the long-term forecast levels out and approaches the steady value:

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Table 3.11: Forecasts for Netflix using the damped trend method ( ϕ = 0.95) 39 PeriodNetflix SalesForecast RMSE = MAE = MAPE = α = β = φ = 0.95 Data: Netflix_1.xlsx. © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 40 Figure 3.10: Possible Trend Patterns

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.6: Other Approaches to Trend Forecasting (*) Brown’s method of Double Exponential Smoothing: Use LES with SES with constant drift: Use LES with Linear moving averages: combine a moving average for the level with another for the trend. Again, exponential smoothing is likely to provide better forecasts. 41

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use : Tracking Signals(*) Use time dependent smoothing parameters. For SES, use the parameter: E t and M t are smoothed values of the error and the absolute error: 42

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.7: Prediction Intervals Use same structure as before: 43 Q: How do we justify this?

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.7: Prediction Interval for Netflix Example 3.8 Prediction Interval for Netflix From Table 3.11, the point forecast for the next period (2003.4) was with RMSE = The resulting (approximate) 95 percent prediction interval is: 44

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.8: The Use of Transformations Compute growth rates (Figure 3.11) Transform to logarithms (Figure 3.12) Use a power transform [C transform; see Figure 3.12] o Growth rates and differences are ways to remove trends o Logarithmic and power transforms are ways to stabilize the variance 45

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use : Use of Growth Rates Define the growth rate as: After forecasting the growth rate as g t+1, then convert back to the original series: 46

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Figure 3.11: Netflix growth rates 47 © Cengage Learning 2013.

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.8: The Use of Transformations 48

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use : The Box-Cox Transformations (*) 49 Use ‘C Transform’ in the macro ESM The Box-Cox transform has the general form: Use C in range [0, 1]; The macro ESM selects C to minimize RMSE Log transform corresponds to C approaching zero Reverse transform: Error measures are computed after transforming back to the original units

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.9: Model Selection Does the series display a trend? Do we expect the trend to continue into the future? If so, then o Use a method that includes a trend. Do the observations tend to be more (or less) variable over time? If so, then o Transform the data to obtain roughly constant variability. 50

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3.10: Principles for Extrapolative Methods 1.Plot the Series. 2.Clean the Data. 3.Use Transformations as Required by Expectations About the Process. 4.Select Simple Methods, Unless Convincing Empirical Evidence calls for Greater Complexity. 5.Evaluate Alternative Methods, Preferably Using Out-of- Sample Data. 6.Update the Estimates Frequently. 51

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Take-Aways Always start by plotting (at least a subset of) the data series to identify past and likely future behavior as well as unusual observations. The adaptive features of local methods are typically preferable for short-term forecasting. Keep it simple, unless the data indicate the need for a more complex approach. Always evaluate forecasting performance, ideally using a hold-out sample. 52

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Mini-Case 3.1: The Growth of Netflix 53 The Netflix annual report for 2009 indicates that, as of the end of 2009, the company had over 12 million subscribers and annual total rental revenues in excess of $1.6 billion. As a market analyst, you would like to know how the company is faring in the marketplace and whether it is likely to continue to grow at the same rate. Table 3.20 [Netflix_2.xlsx] provides the following quarterly data over the period from quarter 1 of 2000 to quarter 4 of 2009: o S = quarterly revenue (or sales, $ million) o G = sales growth rate over previous quarter (percent) o N = number of subscribers at end of quarter (000s)

© 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Mini-Case 3.1: The Growth of Netflix To assess the company’s prospects, carry out the following analyses and comment on the results: o Develop forecasting methods for each of the series S, G, and N, using data through quarter 4 of Then use your results to generate forecasts for 2009 and o What other factors would you wish to take into account before making forecasts one quarter ahead? Eight quarters ahead? 54