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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

- The average from Step 1 needs to be matched up with a specific period

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

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