Consider the following time series data.
|
b)
Value | Qtr1 | Qtr2 | Qtr3 |
4 | 1 | 0 | 0 |
2 | 0 | 1 | 0 |
3 | 0 | 0 | 1 |
5 | 0 | 0 | 0 |
6 | 1 | 0 | 0 |
3 | 0 | 1 | 0 |
5 | 0 | 0 | 1 |
7 | 0 | 0 | 0 |
7 | 1 | 0 | 0 |
6 | 0 | 1 | 0 |
6 | 0 | 0 | 1 |
8 | 0 | 0 | 0 |
SUMMARY OUTPUT | ||||||
Regression Statistics | ||||||
Multiple R | 0.6310547 | |||||
R Square | 0.3982301 | |||||
Adjusted R Square | 0.1725664 | |||||
Standard Error | 1.6832508 | |||||
Observations | 12 | |||||
ANOVA | ||||||
df | SS | MS | F | Significance F | ||
Regression | 3 | 15 | 5 | 1.764706 | 0.231425 | |
Residual | 8 | 22.66667 | 2.833333 | |||
Total | 11 | 37.66667 | ||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
Intercept | 6.6666667 | 0.971825 | 6.859943 | 0.00013 | 4.425633 | 8.9077 |
Qtr1 | -1 | 1.374369 | -0.72761 | 0.4876 | -4.1693 | 2.1693 |
Qtr2 | -3 | 1.374369 | -2.18282 | 0.060595 | -6.1693 | 0.1693 |
Qtr3 | -2 | 1.374369 | -1.45521 | 0.183698 | -5.1693 | 1.1693 |
Estimated regression equation:
ŷ = 6.667 + (-1)Qtr1 + (-3)Qtr2 + (-2)Qtr3
c)
Quarter 1 forecast: x1 = 1, x2 = 0, x3 = 0
ŷ = 6.667 + (-1)*1 + (-3)*0 + (-2)*0 = 5.667
Quarter 2 forecast: x1 = 0, x2 = 1, x3 = 0
ŷ = 6.667 + (-1)*0 + (-3)*1 + (-2)*0 = 3.667
Quarter 3 forecast: x1 = 0, x2 = 0, x3 = 1
ŷ = 6.667 + (-1)*0 + (-3)*0 + (-2)*1 = 4.667
Quarter 4 forecast: x1 = 0, x2 = 0, x3 = 0
ŷ = 6.667 + (-1)*0 + (-3)*0 + (-2)*0 = 6.667
--------------------
d)
value | t | Qtr1 | Qtr2 | Qtr3 |
4 | 1 | 1 | 0 | 0 |
2 | 2 | 0 | 1 | 0 |
3 | 3 | 0 | 0 | 1 |
5 | 4 | 0 | 0 | 0 |
6 | 5 | 1 | 0 | 0 |
3 | 6 | 0 | 1 | 0 |
5 | 7 | 0 | 0 | 1 |
7 | 8 | 0 | 0 | 0 |
7 | 9 | 1 | 0 | 0 |
6 | 10 | 0 | 1 | 0 |
6 | 11 | 0 | 0 | 1 |
8 | 12 | 0 | 0 | 0 |
SUMMARY OUTPUT | ||||||
Regression Statistics | ||||||
Multiple R | 0.9793216 | |||||
R Square | 0.9590708 | |||||
Adjusted R Square | 0.9356827 | |||||
Standard Error | 0.4692953 | |||||
Observations | 12 | |||||
ANOVA | ||||||
df | SS | MS | F | Significance F | ||
Regression | 4 | 36.125 | 9.03125 | 41.00676 | 6.04E-05 | |
Residual | 7 | 1.541667 | 0.220238 | |||
Total | 11 | 37.66667 | ||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
Intercept | 3.4166667 | 0.428406 | 7.9753 | 9.3E-05 | 2.403647 | 4.429686 |
t | 0.40625 | 0.04148 | 9.79382 | 2.45E-05 | 0.308165 | 0.504335 |
Qtr1 | 0.21875 | 0.402878 | 0.542968 | 0.604002 | -0.73391 | 1.171406 |
Qtr2 | -2.1875 | 0.392056 | -5.57956 | 0.000834 | -3.11456 | -1.26044 |
Qtr3 | -1.59375 | 0.385417 | -4.13514 | 0.004376 | -2.50512 | -0.68238 |
Estimated regression equation:
ŷ = 3.417 + (0.406)t + (0.219)Qtr1 + (-2.188)Qtr2 + (-1.594)Qtr3
e)
Quarter 1 forecast: x1 = 1, x2 = 0, x3 = 0, t = 13
ŷ = 3.417 + (0.406)*13 + (0.219)*1 + (-2.188)*0 + (-1.594)*0 = 8.917
Quarter 2 forecast: x1 = 0, x2 = 1, x3 = 0, t = 14
ŷ = 3.417 + (0.406)*14 + (0.219)*0 + (-2.188)*1 + (-1.594)*0 = 6.917
Quarter 3 forecast: x1 = 0, x2 = 0, x3 = 1, t = 15
ŷ = 3.417 + (0.406)*15 + (0.219)*0 + (-2.188)*0 + (-1.594)*1 = 7.917
Quarter 4 forecast: x1 = 0, x2 = 0, x3 = 0, t = 16
ŷ = 3.417 + (0.406)*16 + (0.219)*0 + (-2.188)*0 + (-1.594)*0 = 9.917
f)
MSE fr part b) = 2.833
MSE for part d) = 0.220
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