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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

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

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

• 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

• 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

• 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

• 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

• 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

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

• Trend

• Seasonal

• Cyclical

• Irregular

• 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

• 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

• 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

• 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

• 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

• 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 Errors

Error Comparison Criteria

Mean Squared Deviation (MSD)

• 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

• 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

• 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