McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Time Series Forecasting Chapter 13

13-2 Time Series Forecasting 13.1Time Series Components and ModelsTime Series Components and Models 13.2Time Series Regression: Basic ModelsTime Series Regression: Basic Models 13.3Time Series Regression: More Advanced Models (Optional)Time Series Regression: More Advanced Models 13.4Multiplicative DecompositionMultiplicative Decomposition 13.5Exponential SmoothingExponential Smoothing 13.6Forecast Error ComparisonsForecast Error Comparisons 13.7Index NumbersIndex Numbers

13-3 Time Series Components and Models Trend Long-run growth or decline Cycle Long-run up and down fluctuation around the trend level Seasonal Regular periodic up and down movements that repeat within the calendar year Irregular Erratic very short-run movements that follow no regular pattern

13-4 No Trend When there is no trend, the least squares point estimate b 0 of  0 is just the average y value y t =  0 +  t That is, we have a horizontal line that crosses the y axis at its average value

13-5 Trend When sales increase (or decrease) over time, we have a trend Oftentimes, that trend is linear in nature Linear trend is modeled using regression Sales is the dependent variable Time is the independent variable Weeks Months Quarters Years Not only is simple linear regression used, quadratic regression is sometimes used

13-6 Seasonality Some products have demand that varies a great deal by period Coats Bathing suits Bicycles This periodic variation is called seasonality Seasonality alters the linear relationship between time and demand

13-7 Modeling Seasonality 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 means higher sales

13-8 Time Series Regression: More Advanced Models 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

13-9 Autocorrelation 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 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

13-10 Autocorrelation Continued We can test for first-order correlation using Durbin-Watson Covered in Chapters 11 and 12 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

13-11 Autoregressive Model The error term  t can be related to more than just the previous error term  t-1 This is often the case with seasonal data The 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 a model that relates  t to an appropriate number of past error terms

13-12 Multiplicative Decomposition We can use the multiplicative decomposition method to decompose a time series into its components: Trend Seasonal Cyclical Irregular

13-13 Steps to Multiplicative Decomposition #1 1.Compute a moving average This eliminates the seasonality Averaging period matches the seasonal period 2.Compute a two-period centering moving average The average from Step 1 needs to be matched up with a specific period Consider a 4-period moving average The average of 1, 2, 3, and 4 is 2.5 This does not match any period The average of 2.5 and the next term of 3.5 is 3 This matches up with period 3 Step 2 not needed if Step 1 uses odd number of periods

13-14 Steps to Multiplicative Decomposition #2 3.The original demand for each period is divided by the value computed in Step 2 for that same period The first and last few period do not have a value from Step 2 These periods are skipped 4.All of the values from Step 3 for season 1 are averaged together to form seasonal factor for season 1 This is repeated for every season If there are four seasons, there will be four factors

13-15 Steps to Multiplicative Decomposition #3 5.The original demand for each period is divided by the appropriate seasonal factor for that period This gives us the deseasonalized observation for that period 6.A forecast is prepared using the deseasonalized observations This is usually simple regression 7.The deseasonalized forecast for each period from Step 6 is multiplied by the appropriate seasonal factor for that period This returns seasonality to the forecast

13-16 Steps to Multiplicative Decomposition #4 8.We estimate the period-by-period cyclical and irregular component by dividing the deseasonalized observation from Step 5 by the deseasonalized forecast from Step 6 9.We use a three-period moving average to average out the irregular component 10.The value from Step 9 divided by the value from Step 8 gives us the cyclical component Values close to one indicate a small cyclical component We are interested in long-term patterns

13-17 Exponential Smoothing Earlier, we saw that when there is no trend, the least squares point estimate b 0 of  0 is just the average y value y t =  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  0 is slowly changing over time, we want to weight more recent periods heavier Exponential smoothing does just this

13-18 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

13-19 Holt–Winters’ Double Exponential Smoothing Simple exponential smoothing cannot handle trend or seasonality Holt–Winters’ double exponential smoothing can handle trended data of the form y t =  0 +  1 t +  t Assumes  0 and  1 changing slowly over time We first find initial estimates of  0 and  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

13-20 Multiplicative Winters’ Method Double exponential smoothing cannot handle seasonality Multiplicative Winters’ method can handle trended data of the form y t = (  0 +  1 t) · SN t +  t Assumes  0,  1, and SN t changing slowly over time We first find initial estimates of  0 and  1 and seasonal factors Then use updating equations to track over time Requires smoothing constants called alpha, gamma, and delta Updating equations in Appendix K of the CD-ROM

13-21 Forecast Error Comparison Mean Absolute Deviation (MAD) Mean Squared Deviation (MSD) Forecast Errors Error Comparison Criteria

13-22 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 them multiplying times 100 Simple (Quantity) Index

13-23 Aggregate Price Index Often wish to compare a group of items To do this, we compute the total prices of the items over time We then index this total Aggregate Price Index

13-24 Weighted Aggregate Price Index An aggregate price index assumes all items in the basket are purchased with the same frequency A weighted aggregate price index takes into account varying purchasing frequency The Laspeyres index assumes the same mixture of items for all periods as was used in the base period The Paasche index allows the mixture of items in the basket to change over time as purchasing habits change