Chapter 13

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# Chapter 13 - PowerPoint PPT Presentation

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

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

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
• 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
• 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
• 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
• 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: 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
• 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
• 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
• 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
• We can use the multiplicative decomposition method to decompose a time series into its components:
• Trend
• Seasonal
• Cyclical
• Irregular
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
• 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
• 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
• 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
• 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 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
• 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
• 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 Errors

Error Comparison Criteria

Mean Absolute Deviation (MAD)

Mean Squared Deviation (MSD)

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
• 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
• 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