1. Times Series Forecasting and Index Numbers Chapter 16 Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

2. Time Series Forecasting 16.1Time Series Components and Models 16.2Time Series Regression 16.3Multiplicative Decomposition 16.4Simple Exponential Smoothing 16.5Holt-Winter’s Models 16.6The Box Jenkins Methodology (Optional Advanced Section) 16.7Forecast Error Comparisons 16.8Index Numbers 16-2

3. 16.1 Time Series Components and Models TrendLong-run growth or decline CycleLong-run up and down fluctuation around the trend level SeasonalRegular periodic up and down movements that repeat within the calendar year IrregularErratic very short-run movements that follow no regular pattern LO16-1: Identify the components of a times series. 16-3

4. Time Series Exhibiting Trend, Seasonal, and Cyclical Components LO16-1 Figure 16.1 16-4

5. Seasonality Some products have demand that varies a great deal by period ◦ Coats, bathing suits, bicycles This periodic variation is called seasonality ◦ Constant seasonality: the magnitude of the swing does not depend on the level of the time series ◦ Increasing seasonality: the magnitude of the swing increases as the level of the time series increases Seasonality alters the linear relationship between time and demand LO16-1 16-5

6. 16.2 Time Series Regression Within regression, seasonality can be modeled using dummy variables Consider the model: y t =  0 +  1 t +  Q2 Q 2 +  Q3 Q 3 +  Q4 Q 4 +  t ◦ For Quarter 1, Q 2 = 0, Q 3 = 0 and Q 4 = 0 ◦ For Quarter 2, Q 2 = 1, Q 3 = 0 and Q 4 = 0 ◦ For Quarter 3, Q 2 = 0, Q 3 = 1 and Q 4 = 0 ◦ For Quarter 4, Q 2 = 0, Q 3 = 0 and Q 4 = 1 The  coefficient will then give us the seasonal impact of that quarter relative to Quarter 1 ◦ Negative means lower sales, positive lower sales LO16-2: Use time series regression to forecast time series having linear, quadratic, and certain types of seasonal patterns. 16-6

7. Transformations Sometimes, transforming the sales data makes it easier to forecast ◦ Square root ◦ Quartic roots ◦ Natural logarithms While these transformations can make the forecasting easier, they make it harder to understand the resulting model LO16-3: Use data transformations to forecast time series having increasing seasonal variation. 16-7

8. 16.3 Multiplicative Decomposition We can use the multiplicative decomposition method to decompose a time series into its components: Trend Seasonal Cyclical Irregular LO 4: Use multiplicative decomposition and moving averages to forecast time series having increasing seasonal variation. 16-8

9. 16.4 Simple Exponential Smoothing Earlier, we saw that when there is no trend, the least squares point estimate b 0 of β 0 is just the average y value ◦ y t = β 0 +  t That gave us a horizontal line that crosses the y axis at its average value Since we estimate b 0 using regression, each period is weighted the same If β 0 is slowly changing over time, we want to weight more recent periods heavier Exponential smoothing does just this LO 16-5: Use simple exponential smoothing to forecast a time series. 16-9

10. 16.5 Holt–Winters’ Models Simple exponential smoothing cannot handle trend or seasonality Holt–Winters’ double exponential smoothing can handle trended data of the form y t = β 0 + β 1 t +  t ◦ Assumes β 0 and β 1 changing slowly over time ◦ We first find initial estimates of β 0 and β 1 ◦ Then use updating equations to track changes over time  Requires smoothing constants called alpha and gamma LO16-6: Use double exponential smoothing to forecast a time series. 16-10

11. Multiplicative Winters’ Method Double exponential smoothing cannot handle seasonality Multiplicative Winters’ method can handle trended data of the form y t = (β 0 + β 1 t) · SN t +  t ◦ Assumes β 0, β 1, and SN t changing slowly over time ◦ We first find initial estimates of β 0 and β 1 and seasonal factors ◦ Then use updating equations to track over time  Requires smoothing constants alpha, gamma and delta LO16-7: Use multiplicative Winters’ method to forecast a time series. 16-11

12. 16.6 The Box–Jenkins Methodology (Optional Advanced Section) Uses a quite different approach Begins by determining if the time series is stationary ◦ The statistical properties of the time series are constant through time Plots can help If non-stationary, will transform series LO16-8: Use the Box–Jenkins methodology to forecast a time series. 16-12

13. 16.7 Forecast Error Comparison Forecast errors: e t = y t - y ̂ t Error comparison criteria ◦ Mean absolute deviation (MAD) ◦ Mean squared deviation (MSD) LO16-9: Compare time series models by using forecast errors. 16-13

14. 16.8 Index Numbers Index numbers allow us to compare changes in time series over time We begin by selecting a base period Every period is converted to an index by dividing its value by the base period and then multiplying the results by 100 ◦ Simple (quantity) index LO16-10: Use index numbers to compare economic data over time. 16-14