Question

In: Statistics and Probability

Consider the following time series: Quarter Year 1 Year 2 Year 3 1 69 66 60...

Consider the following time series:

Quarter Year 1 Year 2 Year 3
1 69 66 60
2 45 37 47
3 55 57 50
4 83 86 77
(a) Choose a time series plot.
(i)
(ii)
(iii)
(iv)
- Select your answer -Graph (i)Graph (ii)Graph (iii)Graph (iv)Item 1
What type of pattern exists in the data? Is there an indication of a seasonal pattern?
- Select your answer -Positive trend pattern, no seasonalityHorizontal pattern, no seasonalityNegative trend pattern, no seasonalityPositive trend pattern, with seasonalityHorizontal pattern, with seasonalityItem 2
(b) Use a multiple linear regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data: Qtr1 = 1 if quarter 1, 0 otherwise; Qtr2 = 1 if quarter 2, 0 otherwise; Qtr3 = 1 if quarter 3, 0 otherwise. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
ŷ =   +   Qtr1 +   Qtr2 +   Qtr3
(c) Compute the quarterly forecasts for next year.
Year Quarter Ft
4 1
4 2
4 3
4 4

Solutions

Expert Solution

a)

with seasonality Horizontal pattern

b)

year Qurter Y Q1 Q2 Q3
1 1 69 1 0 0
1 2 45 0 1 0
1 3 55 0 0 1
1 4 83 0 0 0
2 1 66 1 0 0
2 2 37 0 1 0
2 3 57 0 0 1
2 4 86 0 0 0
3 1 60 1 0 0
3 2 47 0 1 0
3 3 50 0 0 1
3 4 77 0 0 0

Excel > Data > Data Analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.968245837
R Square 0.9375
Adjusted R Square 0.9140625
Standard Error 4.55521679
Observations 12
ANOVA
df SS MS F Significance F
Regression 3 2490 830 40 3.65996E-05
Residual 8 166 20.75
Total 11 2656
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 82 2.62995564 31.17923313 1.21753E-09 75.93531142 88.06468858 75.93531142 88.06468858
Q1 -17 3.719318934 -4.570729292 0.001824043 -25.57676484 -8.423235158 -25.57676484 -8.423235158
Q2 -39 3.719318934 -10.48579073 5.95026E-06 -47.57676484 -30.42323516 -47.57676484 -30.42323516
Q3 -28 3.719318934 -7.528260011 6.74509E-05 -36.57676484 -19.42323516 -36.57676484 -19.42323516

Y = 82-17*Q1-39*Q2-28*Q3

c)

Year Quarter Ft = 82-17*Q1-39*Q2-28*Q3 Q1 Q2 Q3
4 1 65 1 0 0
4 2 43 0 1 0
4 3 54 0 0 1
4 4 82 0 0 0

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