Forecast Friday Topic: Forecasting with Seasonally-Adjusted Data

(Twenty-first in a series)

Last week, we introduced you to fictitious businesswoman Billie Burton, who puts together handmade gift baskets and care packages for customers. Billie is interested in forecasting gift basket orders so that she can get a better idea of how much time she’s going to need to set aside to assemble the packages; how much supplies to have on hand; how much revenue – and cost – she can expect; and whether she will need assistance. Gift-giving is seasonal, and Billie’s business is no exception. The Christmas season is Billie’s busiest season, and a few other months are much busier than others, so she must adjust for these seasonal factors before doing any forecasts.

Why is it important to adjust data for seasonal factors? Imagine trying to do regression analysis on monthly retail data that hasn’t been adjusted. If sales during the holiday season are much greater than at all other times of the year, there will be significant forecast errors in the model because the holiday period’s sales will be outliers. And regression analysis places greater weight on extreme values when trying to determine the least-squares equation.

Billie’s Orders, Revisited

Recall from last week that Billie has five years of monthly gift basket orders, from January 2005 to December 2009. The orders are shown again in the table below:

Month 

TOTAL GIFT BASKET ORDERS 

2005 

2006 

2007 

2008 

2009 

January 

15 

18 

22 

26 

31 

February 

30 

36 

43 

52 

62 

March

25 

18 

22 

43 

32 

April 

15 

30 

36 

27 

52 

May 

13 

16 

19 

23 

28 

June 

14 

17 

20 

24 

29 

July 

12 

14 

17 

20 

24 

August 

22 

26 

31 

37 

44 

September 

20 

24 

29 

35 

42 

October 

14 

17 

20 

24 

29 

November 

35 

42 

50 

60 

72 

December 

40 

48 

58 

70 

84 

 

Billie would like to forecast gift basket orders for the first four months of 2010, particularly February and April, for Valentine’s Day and Easter, two other busier-than-usual periods. Billie must first adjust her data.

Seasonal Adjustment

When we decomposed the time series, we computed the seasonal adjustment factors for each month. They were as follows:

Month 

Factor 

January 

0.78 

February 

1.53 

March 

0.89 

April 

1.13 

May 

0.65 

June 

0.67 

July 

0.55 

August 

1.00 

September 

0.91 

October 

0.62 

November 

1.53 

December 

1.75 

 

Knowing these monthly seasonal factors, Billie adjusts her monthly orders by dividing each month’s orders by its respective seasonal factor (e.g., each January’s orders is divided by 0.78; each February’s orders by 1.53, and so on). Billie’s seasonally-adjusted data looks like this:

Month 

Orders 

Adjustment Factor 

Seasonally Adjusted Orders 

Time Period 

Jan-05 

15 

0.78  

19.28  

1 

Feb-05 

30 

1.53  

19.61  

2 

Mar-05 

25 

0.89  

28.15  

3 

Apr-05 

15 

1.13  

13.30  

4 

May-05 

13 

0.65  

19.93  

5 

Jun-05 

14 

0.67  

21.00  

6 

Jul-05 

12 

0.55  

21.81  

7 

Aug-05 

22 

1.00  

22.10  

8 

Sep-05 

20 

0.91  

21.93  

9 

Oct-05 

14 

0.62  

22.40  

10 

Nov-05 

35 

1.53  

22.89  

11 

Dec-05 

40 

1.75  

22.92  

12 

Jan-06 

18 

0.78  

23.13  

13 

Feb-06 

36

1.53  

23.53  

14 

Mar-06 

18 

0.89  

20.27  

15 

Apr-06 

30 

1.13  

26.61  

16 

May-06 

16 

0.65  

24.53  

17 

Jun-06 

17 

0.67  

25.49  

18 

Jul-06

14 

0.55  

25.44  

19 

Aug-06 

26 

1.00  

26.12  

20 

Sep-06 

24 

0.91  

26.32  

21 

Oct-06 

17 

0.62  

27.20  

22 

Nov-06 

42 

1.53  

27.47  

23 

Dec-06 

48 

1.75  

27.50  

24 

Jan-07 

22 

0.78  

28.27  

25 

Feb-07 

43 

1.53  

28.11  

26 

Mar-07 

22 

0.89  

24.77  

27 

Apr-07 

36 

1.13  

31.93  

28 

May-07 

19 

0.65  

29.13  

29 

Jun-07 

20 

0.67  

29.99  

30 

Jul-07 

17 

0.55  

30.90  

31 

Aug-07 

31 

1.00  

31.14  

32 

Sep-07 

29 

0.91  

31.80

33 

Oct-07 

20 

0.62  

32.01  

34 

Nov-07 

50 

1.53  

32.70  

35 

Dec-07 

58 

1.75  

33.23  

36 

Jan-08 

26 

0.78  

33.42  

37 

Feb-08 

52 

1.53  

33.99  

38 

Mar-08 

43 

0.89  

48.41  

39 

Apr-08 

27 

1.13  

23.94  

40 

May-08 

23 

0.65  

35.26  

41 

Jun-08 

24 

0.67  

35.99  

42 

Jul-08 

20 

0.55  

36.35  

43 

Aug-08 

37 

1.00  

37.17  

44 

Sep-08 

35 

0.91  

38.38  

45 

Oct-08 

24 

0.62  

38.41  

46 

Nov-08 

60 

1.53  

39.24  

47 

Dec-08 

70 

1.75  

40.11  

48 

Jan-09 

31 

0.78  

39.84  

49 

Feb-09 

62 

1.53  

40.53  

50 

Mar-09 

32 

0.89  

36.03  

51 

Apr-09 

52 

1.13  

46.12  

52 

May-09 

28 

0.65

42.93  

53 

Jun-09 

29 

0.67  

43.49  

54 

Jul-09 

24 

0.55  

43.62  

55 

Aug-09 

44 

1.00  

44.20  

56 

Sep-09 

42 

0.91  

46.06  

57 

Oct-09 

29 

0.62  

46.41  

58 

Nov-09 

72 

1.53  

47.09  

59 

Dec-09 

84 

1.75  

48.13  

60 

 

Notice the seasonally adjusted gift basket orders in the fourth column. It is the quotient of the second and third columns. Notice that in the months where the seasonal adjustment factor is greater than 1, the seasonally adjusted orders will be lower than actual orders; in months where the factor is less than 1, the seasonally adjusted orders will be greater than actual. This is intended to normalize the data set. (Note: August has a seasonal factor of 1.00, suggesting it is an average month. However, that is due to rounding. Notice that August 2008’s actual orders are 37 baskets, but its adjusted orders are 37.17. That’s due to rounding). Also, the final column is the sequential time period number for each month, from 1 to 60.

Perform Regression Analysis

Now Billie runs regression analysis. She is going to do a simple regression, using the time period, t, in the last column as her independent variable and the seasonally adjusted orders as her dependent variable. Recall that last week, we ran a simple regression on the actual sales to isolate the trend component, and we identified an upward trend; however, because of the strong seasonal factors in the actual orders, the regression model didn’t fit the data well. By factoring out these seasonal variations, we should expect a model that better fits the data.

Running her regression of the seasonally adjusted orders, Billie gets the following output:

Ŷ = 0.47t +17.12

And as we expected, this model fits the data better, with an R2 of 0.872. Basically, in a baseline month, each passing month increases basket orders by about half an order.

Forecasting Orders

Now Billie needs to forecast orders for January through April 2010. January 2010 is period 61, so she plugs that into her regression equation:

Ŷ = 0.47(61) + 17.12

=45.81

Billie plugs in the data for the rest of the months and gets the following:

Month

Period

Ŷ

Jan-10

61

45.81

Feb-10

62

46.28

Mar-10

63

46.76

Apr-10

64

47.23

Remember, however, that this is seasonally-adjusted data. To get the forecasts for actual orders for each month, Billie now needs to convert them back. Since she divided each month’s orders by its seasonal adjustment factor, she must now multiply each of these months’ forecasts by those same factors. So Billie goes ahead and does that:

Month

Period

Ŷ

Seasonal Factor

Forecast Orders

Jan-10

61

45.81

0.78

35.65

Feb-10

62

46.28

1.53

70.81

Mar-10

63

46.76

0.89

41.53

Apr-10

64

47.23

1.13

53.25

 

So, Billie forecasts 36 gift basket orders in January; 71 in February, 42 in March, and 53 in April.

Next Forecast Friday Topic: Qualitative Variables in Regression Modeling

You’ve just learned how to adjust for seasonality when forecasting. One thing you’ve noticed through all of these forecasts we have built is that all variables have been quantitative. Yet sometimes, we need to account for qualitative, or categorical factors in our explanation of events. The next two Forecast Friday posts will discuss a simple approach for introducing qualitative information into modeling: “dummy” variables. Dummy variables can be helpful in determining differences in predictive estimates by region, gender, race, political affiliation, etc. As you will also find, dummy variables can even be used for a faster, more simplified approach to gauging seasonality. You’ll find our discussion on dummy variables highly useful.

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