Example 13.1 Forecasting Monthly Stereo Sales Testing for Randomness

| 13.1a | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Objective To use StatPro’s runs test procedure to check whether the residuals from this simple forecasting model represent random noise.

| 13.1a | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b STEREO.XLS n Monthly sales for a chain of stereo retailers are listed in this file. They cover the period from the beginning of 1998 to then end of 2001, during which there was no upward or downward trend in sales and no clear seasonal peaks or valleys. n This behavior is apparent in the time series plot of sales shown on the next slide. n Therefore, a simple forecast model of sales is to use the average of the series, , as a forecast of sales for each month. n Do the resulting residuals represent random noise?

| 13.1a | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Time Series Plot of Stereo Sales

| 13.1a | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Runs Tests n A runs test is one of the formal methods of testing for randomness. n The runs test is a formal test of the null hypothesis of randomness. n If there are too many or too few runs in this series, then we conclude that the series is not random. n We first choose a base value, then we define a run as a consecutive series of observations that remain on one side of this base level.

| 13.1a | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Solution n To obtain the residuals for this forecasting model, we subtract a constant, , from each observation. n Therefore, the plot of the residuals, shown on the next slide, has exactly the same shapes as the plot of sales. n The only difference is that it is shifted down by and has mean 0. n We now use the runs test to check whether there are too many or too few runs around the base value of 0 in this residual plot.

| 13.1a | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Plot of Residuals

| 13.1a | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Solution -- continued n To do so, we use the StaPro/Statistical Inference/Runs Test for Randomness menu item, choose Residual as the variable to analyze, and choose “mean of the series” as the cutoff value. n This produces the output in columns F through K of runs test results shown here.

| 13.1a | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Solution -- continued n StatPro inserts columns F and G to count the runs. n You can check the formulas in these columns, but basically, a 1 in column G indicates the beginning of a new run. n Column J and K then report the results of the runs test. n The important points are as follows.

| 13.1a | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Solution -- continued n The number of observed runs is 20, in cell K8. n The number of runs expected under an assumption of randomness is , in cell K10. Therefore, the series of residuals has too few runs. Positive values tend to follow positive values, and negative values tend to follow negative values. n The z-value in cell K12, -1.42, indicates how many standard errors the observed number of runs is below the expected number of runs. The corresponding p- value indicates how extreme this z-value is.

| 13.1a | 13.2 | 13.3 | 13.4 | 13.5 | 13.6 | 13.6a | 13.6b | 13.7 | 13.7a | 13.7b13.1a a13.6b a13.7b Solution -- continued n It should be interpreted just like other p-values for hypothesis tests. n If it is small, say, less than 0.05, then we can reject the null hypothesis of randomness and conclude that the series of residuals is not random noise. n However, the p-value for this example is only Therefore, there is not convincing evidence of non- randomness in the residuals, and we can conclude that the residuals represent noise.