Refer to the following table which contains the sales (in $,000) for a department store for the first ten months of the year.
Month Sales
January 440
February 480
March 590
April 400
May 500
June 550
July 470
August 500
September 600
October 520
Using a three period moving average [i.e. MA(3)] as a forecasting method, what is the MSE for this forecasting model?
Using a simple exponential smoothing (with a smoothing constant of 0.2) as a forecasting method, what is the MAPE for this forecast model ________ % ?
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To calculate the Mean Squared Error (MSE) for the three-period moving average (MA(3)) method, follow these steps:
1. Calculate the forecasted sales for each month by taking the average of the sales for that month and the two previous months. Start from March since you need three periods of data to calculate the moving average.
March: (440 + 480 + 590) / 3 = 503.33
April: (480 + 590 + 400) / 3 = 490
May: (590 + 400 + 500) / 3 = 496.67
June: (400 + 500 + 550) / 3 = 483.33
July: (500 + 550 + 470) / 3 = 506.67
August: (550 + 470 + 500) / 3 = 506.67
September: (470 + 500 + 600) / 3 = 523.33
October: (500 + 600 + 520) / 3 = 540
2. Calculate the squared difference between the actual sales and the forecasted sales for each month.
March: (590 - 503.33)² = 7517.78
April: (400 - 490)² = 8100
May: (500 - 496.67)² = 10.89
June: (550 - 483.33)² = 4422.22
July: (470 - 506.67)² = 1338.89
August: (500 - 506.67)² = 4.44
September: (600 - 523.33)² = 5850.22
October: (520 - 540)² = 384.44
3. Sum up all the squared differences:
7517.78 + 8100 + 10.89 + 4422.22 + 1338.89 + 4.44 + 5850.22 + 384.44 = 27648.88
4. Calculate the MSE by dividing the sum of squared differences by the number of months. Since there are 8 months in the calculation, divide by 8.
MSE = 27648.88 / 8 = 3456.11
Therefore, the Mean Squared Error (MSE) for the MA(3) method is 3456.11.
To calculate the Mean Absolute Percentage Error (MAPE) for the simple exponential smoothing method with a smoothing constant of 0.2, follow these steps:
1. Calculate the forecasted sales for each month using simple exponential smoothing. Start with the first month (January) with no previous forecast available. Assume the forecast equals the actual sales for the first month.
January forecast = January sales = 440
For the remaining months, use the formula:
Forecasted sales = Smoothing constant * Actual sales + (1 - Smoothing constant) * Previous forecast
February forecast = 0.2 * 480 + (1 - 0.2) * 440 = 464
March forecast = 0.2 * 590 + (1 - 0.2) * 464 = 501.2
April forecast = 0.2 * 400 + (1 - 0.2) * 501.2 = 431.36
May forecast = 0.2 * 500 + (1 - 0.2) * 431.36 = 457.68
June forecast = 0.2 * 550 + (1 - 0.2) * 457.68 = 481.94
July forecast = 0.2 * 470 + (1 - 0.2) * 481.94 = 479.56
August forecast = 0.2 * 500 + (1 - 0.2) * 479.56 = 486.72
September forecast = 0.2 * 600 + (1 - 0.2) * 486.72 = 511.38
October forecast = 0.2 * 520 + (1 - 0.2) * 511.38 = 516.1
2. Calculate the absolute percentage error for each month by taking the absolute difference between the actual sales and the forecasted sales, divided by the actual sales, multiplied by 100.
January MAPE = |(440 - 440) / 440| * 100 = 0%
February MAPE = |(480 - 464) / 480| * 100 ≈ 3.33%
March MAPE = |(590 - 501.2) / 590| * 100 ≈ 15.08%
April MAPE = |(400 - 431.36) / 400| * 100 ≈ 7.84%
May MAPE = |(500 - 457.68) / 500| * 100 ≈ 8.47%
June MAPE = |(550 - 481.94) / 550| * 100 ≈ 12.43%
July MAPE = |(470 - 479.56) / 470| * 100 ≈ 2.04%
August MAPE = |(500 - 486.72) / 500| * 100 ≈ 2.65%
September MAPE = |(600 - 511.38) / 600| * 100 ≈ 14.77%
October MAPE = |(520 - 516.1) / 520| * 100 ≈ 0.75%
3. Calculate the average of the absolute percentage errors.
MAPE = (0% + 3.33% + 15.08% + 7.84% + 8.47% + 12.43% + 2.04% + 2.65% + 14.77% + 0.75%) / 10
≈ 6.46%
Therefore, the Mean Absolute Percentage Error (MAPE) for the simple exponential smoothing method with a smoothing constant of 0.2 is approximately 6.46%.