(Twentyfirst 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. Giftgiving 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 leastsquares 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 busierthanusual 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 seasonallyadjusted data looks like this:
Month 
Orders 
Adjustment Factor 
Seasonally Adjusted Orders 
Time Period 
Jan05 
15 
0.78 
19.28 
1 
Feb05 
30 
1.53 
19.61 
2 
Mar05 
25 
0.89 
28.15 
3 
Apr05 
15 
1.13 
13.30 
4 
May05 
13 
0.65 
19.93 
5 
Jun05 
14 
0.67 
21.00 
6 
Jul05 
12 
0.55 
21.81 
7 
Aug05 
22 
1.00 
22.10 
8 
Sep05 
20 
0.91 
21.93 
9 
Oct05 
14 
0.62 
22.40 
10 
Nov05 
35 
1.53 
22.89 
11 
Dec05 
40 
1.75 
22.92 
12 
Jan06 
18 
0.78 
23.13 
13 
Feb06 
36 
1.53 
23.53 
14 
Mar06 
18 
0.89 
20.27 
15 
Apr06 
30 
1.13 
26.61 
16 
May06 
16 
0.65 
24.53 
17 
Jun06 
17 
0.67 
25.49 
18 
Jul06 
14 
0.55 
25.44 
19 
Aug06 
26 
1.00 
26.12 
20 
Sep06 
24 
0.91 
26.32 
21 
Oct06 
17 
0.62 
27.20 
22 
Nov06 
42 
1.53 
27.47 
23 
Dec06 
48 
1.75 
27.50 
24 
Jan07 
22 
0.78 
28.27 
25 
Feb07 
43 
1.53 
28.11 
26 
Mar07 
22 
0.89 
24.77 
27 
Apr07 
36 
1.13 
31.93 
28 
May07 
19 
0.65 
29.13 
29 
Jun07 
20 
0.67 
29.99 
30 
Jul07 
17 
0.55 
30.90 
31 
Aug07 
31 
1.00 
31.14 
32 
Sep07 
29 
0.91 
31.80 
33 
Oct07 
20 
0.62 
32.01 
34 
Nov07 
50 
1.53 
32.70 
35 
Dec07 
58 
1.75 
33.23 
36 
Jan08 
26 
0.78 
33.42 
37 
Feb08 
52 
1.53 
33.99 
38 
Mar08 
43 
0.89 
48.41 
39 
Apr08 
27 
1.13 
23.94 
40 
May08 
23 
0.65 
35.26 
41 
Jun08 
24 
0.67 
35.99 
42 
Jul08 
20 
0.55 
36.35 
43 
Aug08 
37 
1.00 
37.17 
44 
Sep08 
35 
0.91 
38.38 
45 
Oct08 
24 
0.62 
38.41 
46 
Nov08 
60 
1.53 
39.24 
47 
Dec08 
70 
1.75 
40.11 
48 
Jan09 
31 
0.78 
39.84 
49 
Feb09 
62 
1.53 
40.53 
50 
Mar09 
32 
0.89 
36.03 
51 
Apr09 
52 
1.13 
46.12 
52 
May09 
28 
0.65 
42.93 
53 
Jun09 
29 
0.67 
43.49 
54 
Jul09 
24 
0.55 
43.62 
55 
Aug09 
44 
1.00 
44.20 
56 
Sep09 
42 
0.91 
46.06 
57 
Oct09 
29 
0.62 
46.41 
58 
Nov09 
72 
1.53 
47.09 
59 
Dec09 
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 R^{2} 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 
Jan10 
61 
45.81 
Feb10 
62 
46.28 
Mar10 
63 
46.76 
Apr10 
64 
47.23 
Remember, however, that this is seasonallyadjusted 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 
Jan10 
61 
45.81 
0.78 
35.65 
Feb10 
62 
46.28 
1.53 
70.81 
Mar10 
63 
46.76 
0.89 
41.53 
Apr10 
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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Tags: Analysights, components of a time series, Forecast Friday, Forecasting, forecasts, regression analysis, seasonal adjustment, seasonal component, simple regression, time series, time series analysis
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