Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Time Series Forecasting Chapter 16

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

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 LO 1: Identify the components of a times series.

Time Series Regression: More Advanced Models 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 LO 2: Use time series regression to forecast time series having linear, quadratic, and certain types of seasonal patterns.

Time Series Regression: More Advanced Models (Optional) One of the assumptions of regression is that the error terms are independent With time series data, that assumption is often violated Positive or negative autocorrelation is common One type of autocorrelation is first-order autocorrelation LO 3: Use data transformations to forecast time series having increasing seasonal variation (optional).

16-6 First-Order Autocorrelation Error term in time period t is related to the one in t-1  t = φ  t-1 + a t φ is the correlation coefficient that measures the relationship between the error terms a t is an error term, often called a random shock LO3

16-7 Autocorrelation Continued We can test for first-order correlation using Durbin- Watson Covered in Chapter 15 One approach to dealing with first-order correlation is predict future values of the error term using the model  t = φ  t-1 + a t LO3

16-8 Autoregressive Model The error term et can be related to more than just the previous error term e t-1 This is often the case with seasonal data Autoregressive error term model of order q:  t = φ  t-1 + φ  t-2 + … + φ  t-q + a t relates the error term to any number of past error terms The Box-Jenkins methodology can be used to systematically build a model that relates  t to an appropriate number of past error terms LO3

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.

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 5: Use simple exponential smoothing to forecast a time series that exhibits a slowly changing level.

16-11 Exponential Smoothing Continued Exponential smoothing takes on the form: S T =  y T + (1 –  )S T-1 Alpha is a smoothing constant between zero and one Alpha is typically between 0.02 and 0.30 Smaller values of alpha represent slower change We want to test the data and find an alpha value that minimizes the sum of squared forecast errors LO5

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 Updating equations in Appendix K of the CD-ROM LO 6: Use double exponential smoothing to forecast a time series.

16-13 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 Updating equations in Appendix K of the CD-ROM LO 7: Use multiplicative Winters’ method to forecast a time series.

Forecast Error Comparison Forecast errors: e t = y t - y ̂ t Error comparison criteria Mean absolute deviation (MAD) Mean squared deviation (MSD) LO 8: Compare time series models by using forecast errors.

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 LO 9: Use index numbers to compare economic data over time.