Homework SourseThe following table reports the percentages of stocks in a t
The following table reports the percentages of stocks in a typical portfolio in nine quarters from 2005 to 2007.
Quarter Stock %
1st-2005 29.8
2nd-2005 31.0
3rd-2005 29.9
4th-2005 30.1
1st-2006 32.2
2nd-2006 31.5
3rd-2006 32.0
4th-2006 31.9
1st-2007 30.0
A. use exponential smoothing to forecast this time series. Consider smoothing constants of a=.2, .3, and .4. What value of the smoothing constant provides the best forcast?
Solution
We see that the mean squared error is least for alpha=0.2. Hence alpha=0.2 gives the best forecast.
| Simple Exponential Smoothing | ||||||
| Alpha | ||||||
| .20 | ||||||
| t | Data | Smoothed | Forecast | % error | ||
| 30.75 | * | |||||
| 1 | 29.8 | 30.56 | 30.8 | -3.4 | ||
| 2 | 31.0 | 30.65 | 30.6 | 1.3 | ||
| 3 | 29.9 | 30.50 | 30.6 | -2.3 | ||
| 4 | 30.1 | 30.42 | 30.5 | -1.3 | ||
| 5 | 32.2 | 30.77 | 30.4 | 5.6 | ||
| 6 | 31.5 | 30.92 | 30.8 | 2.2 | ||
| 7 | 32.0 | 31.14 | 30.9 | 3.4 | ||
| 8 | 31.9 | 31.29 | 31.1 | 2.5 | ||
| 9 | 30.0 | 31.03 | 31.3 | -4.3 | ||
| 31.0 | ||||||
| 1.01 | Mean Squared Error | |||||
| 2.9% | Mean Absolute Percent Error | |||||
| 55.6% | Percent Positive Errors | |||||
| * initial value - mean of first six data values | ||||||
| Simple Exponential Smoothing | ||||||
| Alpha | ||||||
| .30 | ||||||
| t | Data | Smoothed | Forecast | % error | ||
| 30.75 | * | |||||
| 1 | 29.8 | 30.47 | 30.8 | -3.4 | ||
| 2 | 31.0 | 30.63 | 30.5 | 1.6 | ||
| 3 | 29.9 | 30.41 | 30.6 | -2.3 | ||
| 4 | 30.1 | 30.32 | 30.4 | -1.0 | ||
| 5 | 32.2 | 30.88 | 30.3 | 5.9 | ||
| 6 | 31.5 | 31.07 | 30.9 | 1.9 | ||
| 7 | 32.0 | 31.35 | 31.1 | 2.8 | ||
| 8 | 31.9 | 31.51 | 31.3 | 1.9 | ||
| 9 | 30.0 | 31.06 | 31.5 | -5.0 | ||
| 31.1 | ||||||
| 1.02 | Mean Squared Error | |||||
| 2.9% | Mean Absolute Percent Error | |||||
| 55.6% | Percent Positive Errors | |||||
| * initial value - mean of first six data values | ||||||
| Simple Exponential Smoothing | ||||||
| Alpha | ||||||
| .40 | ||||||
| t | Data | Smoothed | Forecast | % error | ||
| 30.75 | * | |||||
| 1 | 29.8 | 30.37 | 30.8 | -3.4 | ||
| 2 | 31.0 | 30.62 | 30.4 | 1.9 | ||
| 3 | 29.9 | 30.33 | 30.6 | -2.3 | ||
| 4 | 30.1 | 30.24 | 30.3 | -0.7 | ||
| 5 | 32.2 | 31.02 | 30.2 | 6.2 | ||
| 6 | 31.5 | 31.21 | 31.0 | 1.6 | ||
| 7 | 32.0 | 31.53 | 31.2 | 2.5 | ||
| 8 | 31.9 | 31.68 | 31.5 | 1.3 | ||
| 9 | 30.0 | 31.01 | 31.7 | -5.7 | ||
| 31.0 | ||||||
| 1.09 | Mean Squared Error | |||||
| 2.8% | Mean Absolute Percent Error | |||||
| 55.6% | Percent Positive Errors | |||||
| * initial value - mean of first six data values |
