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Chapter 13. Time Series Forecasting. Time Series Forecasting. 13.1 Time Series Components and Models 13.2 Time Series Regression: Basic Models 13.3 Time Series Regression: More Advanced Models ( Optional ) 13.4 Multiplicative Decomposition 13.5 Exponential Smoothing

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Chapter 13

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Chapter 13

Chapter 13

Time Series Forecasting

Time series forecasting

Time Series Forecasting

13.1Time Series Components and Models

13.2Time Series Regression: Basic Models

13.3Time Series Regression: More Advanced Models (Optional)

13.4Multiplicative Decomposition

13.5Exponential Smoothing

13.6Forecast Error Comparisons

13.7Index Numbers

Time series components and models

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

No trend

No Trend

  • When there is no trend, the least squares point estimate b0 of b0 is just the average y value

    • yt = b0 + et

  • That is, we have a horizontal line that crosses the y axis at its average value



  • 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



  • 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

Modeling seasonality

Modeling Seasonality

  • Within regression, seasonality can be modeled using dummy variables

  • Consider the model:yt = b0 + b1t + bQ2Q2 + bQ3Q3 + bQ4Q4 + et

    • For Quarter 1, Q2 = 0, Q3 = 0, and Q4 = 0

    • For Quarter 2, Q2 = 1, Q3 = 0, and Q4 = 0

    • For Quarter 3, Q2 = 0, Q3 = 1, and Q4 = 0

    • For Quarter 4, Q2 = 0, Q3 = 0, and Q4 = 1

  • The b coefficient will then give us the seasonal impact of that quarter relative to Quarter 1

    • Negative means lower sales

    • Positive means higher sales

Time series regression more advanced models

Time Series Regression: MoreAdvanced 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



  • 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

    • et = φet-1 + at

      • φ is the correlation coefficient that measures the relationship between the error terms

      • at is an error term, often called a random shock

Autocorrelation continued

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 modelet = φet-1 + at

Autoregressive model

Autoregressive Model

  • The error term et can be related to more than just the previous error term et-1

    • This is often the case with seasonal data

  • The autoregressive error term model of order q:et = φet-1 + φet-2 + … + φet-q + atrelates the error term to any number of past error terms

  • The Box-Jenkins methodology can be used to systematically a model that relates et to an appropriate number of past error terms

Multiplicative decomposition

Multiplicative Decomposition

  • We can use the multiplicative decomposition method to decompose a time series into its components:

    • Trend

    • Seasonal

    • Cyclical

    • Irregular

Steps to multiplicative decomposition 1

Steps to Multiplicative Decomposition#1

  • Compute a moving average

    • This eliminates the seasonality

    • Averaging period matches the seasonal period

  • 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

Steps to multiplicative decomposition 2

Steps to Multiplicative Decomposition#2

  • 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

  • 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

Steps to multiplicative decomposition 3

Steps to Multiplicative Decomposition#3

  • 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

  • A forecast is prepared using the deseasonalized observations

    • This is usually simple regression

  • 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

Steps to multiplicative decomposition 4

Steps to Multiplicative Decomposition#4

  • 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

  • We use a three-period moving average to average out the irregular component

  • 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

Exponential smoothing

Exponential Smoothing

  • Earlier, we saw that when there is no trend, the least squares point estimate b0 of b0 is just the average y value

    • yt = b0 + et

  • That gave us a horizontal line that crosses the y axis at its average value

  • Since we estimate b0 using regression, each period is weighted the same

  • If b0 is slowly changing over time, we want to weight more recent periods heavier

  • Exponential smoothing does just this

Exponential smoothing continued

Exponential Smoothing Continued

  • Exponential smoothing takes on the form:ST = ayT + (1 – a)ST-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

Holt winters double exponential smoothing

Holt–Winters’ Double Exponential Smoothing

  • Simple exponential smoothing cannot handle trend or seasonality

  • Holt–Winters’ double exponential smoothing can handle trended data of the formyt = b0 + b1t + et

    • Assumes b0 and b1 changing slowly over time

    • We first find initial estimates of b0 and b1

    • 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

Multiplicative winters method

Multiplicative Winters’ Method

  • Double exponential smoothing cannot handle seasonality

  • Multiplicative Winters’ method can handle trended data of the formyt = (b0 + b1t) · SNt + et

    • Assumes b0,b1, and SNt changing slowly over time

    • We first find initial estimates of b0 and b1 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

Forecast error comparison

Forecast Error Comparison

Forecast Errors

Error Comparison Criteria

Mean Absolute Deviation (MAD)

Mean Squared Deviation (MSD)

Index numbers

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

Aggregate price index

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

Weighted aggregate price index

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

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