Stable Seasonal Pattern SSP Model Thomas B Fomby Department of Economics Southern Methodist University August 2010 I Introduction Sometimes forecasters have to work with very small data sets For example suppose we owned a Wholesale Toy Distribution Company the XYZ Toy Company and that after 6 months of developing the full scale of our company we have the following 12 months of sales figures in dollars to work with Jan Feb Mar Apr May Jun Jul Aug 964 977 2 699 324 884 494 1 035 007 1 930 143 1 124 814 1 098 136 1 812 798 Sep Oct Nov Dec 1 095 294 1 163 039 1 920 424 1 000 743 Suppose we want to forecast our sales for the 12 months of next year How do we go about doing this One approach is to use the Stable Seasonal Pattern SSP Model Approach Here is the logic of this approach The total sales for the year January December were 16 729 193 The proportion of the total sales for the year as distributed out to each month is as follows Jan Feb Mar Apr May Jun Jul Aug 0 057682 0 161354 0 052871 0 061868 0 115376 0 067237 0 065642 0 108361 Sep Oct Nov Dec 0 065472 0 069522 0 114795 0 059820 Notice that these proportions add to one A plot of these proportions is displayed below 1 Graph of Monthly Proportions Used in SSP Method As you can see the proportions are not evenly distributed over the 12 months of the year i e the proportions are not 0 08333 per month Therefore one might say that the above proportions and thus the sales figures display a certain seasonal pattern In the present case the months of February May August and November appear to be stronger months while the remaining months seems to reflect about the same amount of lesser business per month So if we are going to forecast ahead the sales of our company we should probably take this seasonal pattern into account One of the major assumptions of the SSP model is that the seasonal pattern represented in the graph above will be maintained stable in the future and therefore we could apply it to our forecasting problem for next year and into the future as well All that remains for us to do is to come up with our best guess as to what next year s total sales are going to be and then we could use the above proportions to distribute the total among the various months of next year Given no more information than we currently have we are going to have to rely on someone s domain specific knowledge about our industry Suppose we contact the Executive Director of Research of the Wholesale Toy Sales Association and ask her What is the typical growth rate in the sales of first year Toy Wholesalers If she comes back with the answer that of the 10 Toy Wholesalers that are comparable to XYZ Inc the average first years sales growth is 5 then we have a way of projecting next year s total sales for our company We simply multiply our last year s total sales 16 729 193 by 1 05 to get the figure of 17 565 653 Once we have the next year s projection of the total year s sales we can use the above proportions to distribute the forecasted sales for next year month by month Doing this we get the monthly forecasts for next year of 2 Jan Feb Mar Apr May Jun Jul Aug 1 013 226 2 834 290 928 719 1 086 757 2 026 650 1 181 055 1 153 043 1 903 438 Sep Oct Nov Dec 1 150 059 1 221 191 2 016 445 1 050 780 Of course when we add up the above monthly projections we get our predicted next year s total of 17 565 653 because the proportions that we used in making the monthly projections add to one II Notation and Implementation of the SSP Model Now let us generalize the above simple example to the case where we have n years of monthly data where Consider the following tabular form of the data Months 1 Year 1 2 3 n 2 3 10 11 12 Here represents the sales of a company in the j th month of the i th year Further let us represent the totals of each of the n years as follows That is the i th year s total is represented by 1 In the spirit of the SSP approach the proportion of the total year s sales from month j in the i th year is given by 2 Of course we can use these proportions to judge roughly speaking if there is seasonality in our data or not If for each month j in all years i then there isn t a 3 significant amount of seasonality in the data We will talk about some graphical methods for detecting seasonality namely Buys Ballot plots and a non parametric statistical test Friedman s test for seasonality later On the other hand if there is a significant difference in the proportions across months within each year and if this pattern is fixed and persists from one year to the next stable seasonal pattern then we can use this seasonal pattern to our advantage when forecasting future values of y If in fact the monthly seasonal pattern of our data is stable over time we can get better estimates of the stable monthly proportions say by estimating them with the average of the monthly proportions the averages taken for given month j over the n years as in for 3 Notice that across the 12 months these averages add to 1 i e 4 Now to begin let s assume we have only two years of monthly data and What we need to do is predict next year s total by say and then distribute this predicted total to next year s months by using the estimated stable monthly proportions calculated over the two years of data One way to predict next year s total is to use what is called last year s change rule That is last year s change in the total from year 1 to year 2 when added to year two s total will give us a prediction of next s year total via the formula 5 Then the j th month s predicted value for the following year can be computed as Obviously since 6 we can see that 7 and the monthly predictions add up to the predicted total and the predictions are congruent internally consistent For example consider the following 1991 1993 monthly data taken from the Plano Sales Tax Revenue data base Let s call this the in sample data set 4 In Sample Observations January 1991 December 1992 Suppose we wish to predict the monthly tax revenues for Plano for the next year 1994 The actual sales that were realized in 1994 are plotted below How well will the SSP model …