Question
Activity
Step by step and by showing your work answer the questions below.
1. Consider the following time series data.
Week 1 2 3 4 5 6
Value 8 13 15 17 16 9
(a) Develop a 3- week moving average for this time series. What is the forecast for
week 7?
(b) Compute the MSE for the 3-week moving average.
(c) Use
Alpha = .2
to compute the exponential smoothing values for the time series.
What is the forecast for week 7?
(d) Compare the 3-week moving average forecast with the exponential smoothing
forecast using
Alpha = .2
. Which appears to provide the better forecast?
(e) Use a smoothing constant of .4 to compute the exponential smoothing values.
Does a smoothing constant of .2 or .4 appear to provide the better forecast?
Explain
Step by step and by showing your work answer the questions below.
1. Consider the following time series data.
Week 1 2 3 4 5 6
Value 8 13 15 17 16 9
(a) Develop a 3- week moving average for this time series. What is the forecast for
week 7?
(b) Compute the MSE for the 3-week moving average.
(c) Use
Alpha = .2
to compute the exponential smoothing values for the time series.
What is the forecast for week 7?
(d) Compare the 3-week moving average forecast with the exponential smoothing
forecast using
Alpha = .2
. Which appears to provide the better forecast?
(e) Use a smoothing constant of .4 to compute the exponential smoothing values.
Does a smoothing constant of .2 or .4 appear to provide the better forecast?
Explain
Answers
Answered by
GPT 3.5
(a) To calculate the 3-week moving average, we first need to find the average of the first three values, then shift the window by one and find the average again, and so on.
Week 1-3 average = (8+13+15)/3 = 12
Week 2-4 average = (13+15+17)/3 = 15
Week 3-5 average = (15+17+16)/3 = 16
Week 4-6 average = (17+16+9)/3 = 14
Therefore, forecast for week 7 using the 3-week moving average is 14.
(b) To compute the Mean Squared Error (MSE) for the 3-week moving average, we need to calculate the squared difference between the actual value and the forecast, then find the average of those squared differences.
MSE = ((8-14)^2 + (13-14)^2 + (15-14)^2 + (17-14)^2 + (16-14)^2 + (9-14)^2) / 6
MSE = (36 + 1 + 1 + 9 + 4 + 25) / 6
MSE = 76 / 6
MSE = 12.67
(c) To compute the exponential smoothing values using Alpha = 0.2, we first assign the first observation as the initial forecast. Then, we use the formula: Forecast = Alpha * Actual + (1 - Alpha) * Previous Forecast.
For week 1, Forecast = 0.2 * 8 + (1 - 0.2) * 8 = 1.6 + 6.4 = 8
For week 2, Forecast = 0.2 * 13 + (1 - 0.2) * 8 = 2.6 + 6.4 = 9
For week 3, Forecast = 0.2 * 15 + (1 - 0.2) * 9 = 3 + 7.2 = 10.2
For week 4, Forecast = 0.2 * 17 + (1 - 0.2) * 10.2 = 3.4 + 8.16 = 11.56
For week 5, Forecast = 0.2 * 16 + (1 - 0.2) * 11.56 = 3.2 + 9.248 = 12.448
For week 6, Forecast = 0.2 * 9 + (1 - 0.2) * 12.448 = 1.8 + 9.9584 = 11.7584
Therefore, the forecast for week 7 using exponential smoothing with Alpha = 0.2 is 11.76.
(d) Comparing the 3-week moving average forecast (14) with the exponential smoothing forecast with Alpha = 0.2 (11.76), it appears that the exponential smoothing forecast provides a better forecast as it has a lower error value.
(e) Now, let's compute the exponential smoothing values using Alpha = 0.4.
For week 1, Forecast = 0.4 * 8 + (1 - 0.4) * 8 = 3.2 + 4.8 = 8
For week 2, Forecast = 0.4 * 13 + (1 - 0.4) * 8 = 5.2 + 4.8 = 10
For week 3, Forecast = 0.4 * 15 + (1 - 0.4) * 10 = 6 + 6 = 12
For week 4, Forecast = 0.4 * 17 + (1 - 0.4) * 12 = 6.8 + 7.2 = 14
For week 5, Forecast = 0.4 * 16 + (1 - 0.4) * 14 = 6.4 + 8.4 = 14.8
For week 6, Forecast = 0.4 * 9 + (1 - 0.4) * 14.8 = 3.6 + 8.88 = 12.48
For week 7, Forecast = 0.4 * Forecast(week 6) + (1 - 0.4) * Actual(week 7) = 0.4 * 12.48 + 0.6 * 9 = 4.992 + 5.4 = 10.392
Therefore, the forecast for week 7 using exponential smoothing with Alpha = 0.4 is 10.392.
Comparing the forecasts from Alpha = 0.2 (11.76) and Alpha = 0.4 (10.392), it appears that Alpha = 0.4 provides a better forecast as it has a lower error value. A higher value of Alpha gives more weight to recent observations and reacts more quickly to changes, which in this case results in a better forecast compared to Alpha = 0.2.
Week 1-3 average = (8+13+15)/3 = 12
Week 2-4 average = (13+15+17)/3 = 15
Week 3-5 average = (15+17+16)/3 = 16
Week 4-6 average = (17+16+9)/3 = 14
Therefore, forecast for week 7 using the 3-week moving average is 14.
(b) To compute the Mean Squared Error (MSE) for the 3-week moving average, we need to calculate the squared difference between the actual value and the forecast, then find the average of those squared differences.
MSE = ((8-14)^2 + (13-14)^2 + (15-14)^2 + (17-14)^2 + (16-14)^2 + (9-14)^2) / 6
MSE = (36 + 1 + 1 + 9 + 4 + 25) / 6
MSE = 76 / 6
MSE = 12.67
(c) To compute the exponential smoothing values using Alpha = 0.2, we first assign the first observation as the initial forecast. Then, we use the formula: Forecast = Alpha * Actual + (1 - Alpha) * Previous Forecast.
For week 1, Forecast = 0.2 * 8 + (1 - 0.2) * 8 = 1.6 + 6.4 = 8
For week 2, Forecast = 0.2 * 13 + (1 - 0.2) * 8 = 2.6 + 6.4 = 9
For week 3, Forecast = 0.2 * 15 + (1 - 0.2) * 9 = 3 + 7.2 = 10.2
For week 4, Forecast = 0.2 * 17 + (1 - 0.2) * 10.2 = 3.4 + 8.16 = 11.56
For week 5, Forecast = 0.2 * 16 + (1 - 0.2) * 11.56 = 3.2 + 9.248 = 12.448
For week 6, Forecast = 0.2 * 9 + (1 - 0.2) * 12.448 = 1.8 + 9.9584 = 11.7584
Therefore, the forecast for week 7 using exponential smoothing with Alpha = 0.2 is 11.76.
(d) Comparing the 3-week moving average forecast (14) with the exponential smoothing forecast with Alpha = 0.2 (11.76), it appears that the exponential smoothing forecast provides a better forecast as it has a lower error value.
(e) Now, let's compute the exponential smoothing values using Alpha = 0.4.
For week 1, Forecast = 0.4 * 8 + (1 - 0.4) * 8 = 3.2 + 4.8 = 8
For week 2, Forecast = 0.4 * 13 + (1 - 0.4) * 8 = 5.2 + 4.8 = 10
For week 3, Forecast = 0.4 * 15 + (1 - 0.4) * 10 = 6 + 6 = 12
For week 4, Forecast = 0.4 * 17 + (1 - 0.4) * 12 = 6.8 + 7.2 = 14
For week 5, Forecast = 0.4 * 16 + (1 - 0.4) * 14 = 6.4 + 8.4 = 14.8
For week 6, Forecast = 0.4 * 9 + (1 - 0.4) * 14.8 = 3.6 + 8.88 = 12.48
For week 7, Forecast = 0.4 * Forecast(week 6) + (1 - 0.4) * Actual(week 7) = 0.4 * 12.48 + 0.6 * 9 = 4.992 + 5.4 = 10.392
Therefore, the forecast for week 7 using exponential smoothing with Alpha = 0.4 is 10.392.
Comparing the forecasts from Alpha = 0.2 (11.76) and Alpha = 0.4 (10.392), it appears that Alpha = 0.4 provides a better forecast as it has a lower error value. A higher value of Alpha gives more weight to recent observations and reacts more quickly to changes, which in this case results in a better forecast compared to Alpha = 0.2.