1. Time Series Forecasting
Chapter 16
Time Series Forecasting

2. Time Series Forecasting
16.1 Time Series Components and Models
16.2 Time Series Regression: Basic Models
16.3 Time Series Regression: More Advanced Models (Optional)
16.4 Multiplicative Decomposition

3. Time Series Forecasting
16.5 Simple Exponential Smoothing
16.6 Holt-Winter’s Models
16.7 Forecast Error Comparisons
16.8 Index Numbers

4. 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

5. Trend

6. Cycle

7. Seasonal Variation

8. 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

9. Example 16.1: The Cod Catch Case
Bay City Seafood Co. operates fishing trawlers and a fish processing plant
Wishes to forecast its minimum and maximum possible revenues
Wishes to make point estimate and prediction interval forecast of monthly cod catch

10. Example 16.1: Cod Catch (In Tons)
Month
Year 1
Year 2
January
362
276
February
381
334
March
317
394
April
297
May
309
384
June
402
314
July
375
344
August
349
337
September
386
345
October
328
November
389
December
343
365

11. Example 16.1: Plot of Cod Catch versus Time

12. Example 16.1: The Cod Catch Case #4
Company believes this data pattern will continue
It seems reasonable to use the “no trend” regression model yt = β0 + εt
Point estimate of b0 is y̅
ŷ = εt

13. Example 16.1: The Cod Catch Case #5
A prediction interval for the “no trend” model is
t/2 is based on n-1 degrees of freedom
A 95 percent prediction interval is:

14. 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

15. Example 16.2: Calculator Sales Case
For two years, Smith’s Department Sore has carried a Bismark X-12 calculator
Sales are increasing
Smith’s wishes to forecast demand
Both point estimate and prediction interval

16. Example 16.2: Calculator Sales Data
Month
Year 1
Year 2
January
197
296
February
211
276
March
203
305
April
247
308
May
239
356
June
269
393
July
363
August
262
386
September
258
443
October
256
November
261
358
December
288
384

17. Example 16.2: Plot of Calculator Sales versus Time

18. Example 16.2: The Calculator Sales Case #4
Smith’s believes the trend will continue
Reasonable to use the “linear trend” regression model yt = β0 + β1t + εt y25 = (25) = y26 = (26) = 408.0
An software package can show the prediction intervals are [328.6,471.2] and [336.0,479.9] respectively

19. 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

20. Types of Seasonality
Constant seasonal variation is where the magnitude of the swing does not depend on the level of the time series
Increasing seasonal variation is where the magnitude of the swing increases as the level of the time series increases

21. 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 lower sales

22. Example 16.3: The Bike Sales Case
A bicycle shop in Switzerland has sold the TRK-50 mountain bike for four years
Wish to describe sales using a regression model

23. Example 16.3: Quarterly Sales Data
Year
Quarter t
Sales 1 10 3 9 13 2 31 34 43 11 48 4 16 12 19 5 15 6 33 14 37 7 45 51 8 17 21

24. Example 16.3: MINITAB Plot of Bike Sales

25. Example 16.3: MINITAB Output of Dummy Variable Regression

26. Example 16.3: The Bike Sales Case #5
ŷ17 = b0 + b1(17) + bQ2(0) + bQ3(0) + bQ4(0) ŷ17 = (17) =
ŷ18 = b0 + b1(18) + bQ2(1) + bQ3(0) + bQ4(0) ŷ18 = (18) + 21 =
ŷ19 = b0 + b1(19) + bQ2(0) + bQ3(1) + bQ4(0) ŷ19 = (19) =
ŷ20 = b0 + b1(20) + bQ2(0) + bQ3(0) + bQ4(1) ŷ20 = (20) =

27. Time Series Regression: More Advanced 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

28. Example 16.4: Traveler’s Rest Case #1
Seasonal Regression

29. Example 16.4: MegaStat Regression Output

30. Example 16.4: MegaStat Output of Predicted Values

31. 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

32. First-Order Autocorrelation
Error term in time period t is related to the one in t-1
t = φt-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

33. Autocorrelation Continued
We can test for first-order correlation using Durbin-Watson
Covered in Chapter 15
One approach to dealing with first-order correlation is predict future values of the error term using the model et = φet-1 + at

34. 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
Autoregressive error term model of order q: t = φt-1 + φt-2 + … + φt-q + at relates the error term to any number of past error terms
The Box-Jenkins methodology can be used to systematically build a model that relates t to an appropriate number of past error terms

35. Multiplicative Decomposition
We can use the multiplicative decomposition method to decompose a time series into its components:
Trend
Seasonal
Cyclical
Irregular

36. Steps to Multiplicative Decomposition #1
Compute a moving average
This eliminates the seasonality
Averaging period matches seasonal period

37. Steps to Multiplicative Decomposition #2
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 3.5 is 3
This matches up with period 3
Step 2 not needed if Step 1 uses odd number of periods

38. Steps to Multiplicative Decomposition #3
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

39. Steps to Multiplicative Decomposition #4
The original demand for each period is divided by the appropriate seasonal factor for that period
This gives us the deseasonalized observation
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

40. Steps to Multiplicative Decomposition #5
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

41. Example 16.5: The Tasty Cola Case
Discount Soda Shop owns ten drive-in soft drink scores
Have been selling Tasty Cola for three years
They wish to forecast monthly demand
Data has seasonal variation

42. Example 16.5: Monthly Sales of Tasty Cola (In Hundreds of Cases)
Year
Month t
Sales 1
189 2 7 19
831
229 8 20
960 3
249 9 21
1,152 4
289 10 22
759 5
260 11 23
607 6
431 12 24
371
660 25
298
777 26
378
915 27
373
613 28
443
485 29
374
277 30 13
244 31
1,004 14
296 32
1,153 15
319 33
1,388 16
370 34
904 17
313 35
715 18
556 36
441

43. Example 16.5: Monthly Sales of Tasty Cola Graphically

44. Example 16.5: The Tasty Cola Case #4
Seasonal variation seems to be increasing
It is reasonable to use the multiplicative model yt = TRt x SNt x CLt x IRt
TRt x SNt x CLt x IRt represent the trend, seasonal, cyclical, and irregular components in time period t

45. Example 16.5: Estimation of the Seasonal Factors

46. Example 16.5: Plot of Tasty Cola Sales and Deseasonalized Data

47. Example 16.5: Forecasts of Future Values of Tasty Cola Sales

48. Example 16.5: A Plot of the Observed and Forecast Tasty Cola Sales

49. Simple Exponential Smoothing
Earlier, we saw that when there is no trend, the least squares point estimate b0 of β0 is just the average y value
yt = β0 + t
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 β0 is slowly changing over time, we want to weight more recent periods heavier
Exponential smoothing does just this

50. Exponential Smoothing Continued
Exponential smoothing takes on the form: ST = yT + (1 – )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

51. Example 16.6: The Cod Catch Case
Example 16.1 suggested that the no trend model may approximate the cod catch series
It is also possible that β0 is slowly changing over time
Will estimate β0 using first six observations
Denoted as S0
After of first six periods
S0 =

52. Example 16.6: The Cod Catch Case #2
Assume that at the end of T-1, we have ST-1 to estimate β0
Assume that in T, we obtain a new observation yT, we can update ST-1 to ST
We compute the updated estimate using the smoothing equation ST = yT + (1-)ST-1
 is between zero and one

53. Example 16.6: One-Period-Ahead Forecast Using Exponential Smoothing

54. Example 16.7: The Cod Catch Case
Example 16.6 showed that =0.02 was a good value
We can use =0.02 to forecast future monthly cod catches
Example 16.6 estimated S24=356.13
The point forecast made in month 24 of any future monthly cod catch is

55. Example 16.7: The Cod Catch Case #2
Assume the cod catch in January of year 3 is y25=384
S25 = y25 + (1-)S24 S25 = 0.02(384) (356.13) =
is the point forecast made in month 25 for any future monthly cod catch

56. Example 16.7: MINITAB Output of Using Exponential Smoothing to Forecast Cod

57. Holt–Winters’ Models
Simple exponential smoothing cannot handle trend or seasonality
Holt–Winters’ double exponential smoothing can handle trended data of the form yt = β0 + β1t + t
Assumes β0 and β1 changing slowly over time
We first find initial estimates of β0 and β1
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

58. Multiplicative Winters’ Method
Double exponential smoothing cannot handle seasonality
Multiplicative Winters’ method can handle trended data of the form yt = (β0 + β1t) · SNt + t
Assumes β0, β1, and SNt changing slowly over time
We first find initial estimates of β0 and β1 and seasonal factors
Then use updating equations to track over time
Requires smoothing constants alpha, gamma and delta
Updating equations in Appendix K of the CD-ROM

59. MINITAB Output of Double Exponential Smoothing for Calculator Sales Data

60. MINITAB Output of Double Exponential Smoothing for Calculator Sales Data

61. MINITAB Output of Graphical Forecasts When =0.496 and ƴ=0.142

62. MINITAB Output Using Winters’ Method to Forecast Tasty Cola Sales

63. Forecast Error Comparison
Forecast errors: et = yt - ŷt
Error comparison criteria
Mean absolute deviation (MAD)
Mean squared deviation (MSD)

64. Example: The Tasty Cola Case

65. 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 then multiplying the results by 100
Simple (quantity) index

66. Example: Installment Credit
Installment Credit Outstanding (in billions of dollars):

67. 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

68. Example: Market Basket of Groceries

69. 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

70. Example: Laspeyres Index
(base quantity weights)
1992 (base) and 1997 Prices for a Market Basket of Grocery Items

71. Example: Paasche Index
(current quantity weights)
1992 (base) and 1997 Prices for a Market Basket of Grocery Items