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

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

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

Time Series Exhibiting Trend, Seasonal, and Cyclical Components LO16-1 Figure

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 LO

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

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

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

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

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

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

16.6 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

16.7 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

16.8 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