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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

Chapter 13

Time Series Forecasting

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

13.6 Forecast Error Comparisons

13.7 Index 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
trend
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
seasonality
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
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
autocorrelation
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
    • 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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