Year Quarter t y(t) 2015 1 1 110569 2 2 113433 3 3 118183 4 4 114932 2016 1 5 112337 2 6 117224 3 7 118863 4 8 116554 2017 1 9 116287 2 10 124077 3 11 126540 4 12 123559 2018 1 13 122607.4 2 14 129549.7 3 15 128105.9 4 16 126818.9 2019 1 17 123020.3 2 18 130158.9 3 19 133584.3 4 20 132147.9 2020 1 21 126932.4 2 22 123051.92] 3 23 134041.56 4 24 135031.20 2021 1 25 130020.840 2 26 128010.480 3 27 138000.12 4 28 139989.76 2022 1 29 140979.40! 2 30 135969.04 3 31 130958.68! 4 32 142948.32! 2023 1 33 144321.12! 2 34 139782.059 3 35 135372.74 4 36 147611.86 2024 123 37 149334.83 38 145124.004 39 141022.359 Passengers (thousands) 150000 145000 140000 135000 130000 125000 120000 115000 110000 Time Series plot of quarterly light rail usage (first quarter 2015 through third quarter 2024) wwww 0 8 12 16 20 Time (Quarters) 24 28 32 36 40 40
Figure 1 below displays the quarterly number of U.S. passengers (in thousands) using light rail as a mode of transportation. The series begins in the first quarter of 2015 and ends with the third quarter of 2024.
We can see a regularity to the series: the first quarter’s ridership tends to be lowest; then there is a progressive rise in ridership going into the second and third quarters, followed by a decline in the fourth quarter. Superimposed on the series are the moving-average
Notice that the seasonal pattern in the time series is not present in the moving averages. The moving averages are a smoothed-out version of the original time series, reflecting only the general trend in the series, which is upward.
Question:
1. When you divide the original series by the seasonal component, you are then left with the trend component with no seasonality. This adjusted series does not show the regular ups and downs of the original unadjusted data. At this stage, you can come up with a trend line equation.
a) Regress adjusted seasonality (deseasonality component) to time component. Show the trend line equation using regression for trend-and-season predictive model here and substitute the intercept and slope using the coefficient from the simple linear regression output:
= [a + b (t)] × SR
Show the regression output here:
![Year
Quarter
t
y(t)
2015
1
1
110569
2
2
113433
3
3
118183
4
4
114932
2016
1
5
112337
2
6
117224
3
7
118863
4
8
116554
2017
1
9
116287
2
10
124077
3
11
126540
4
12
123559
2018
1
13
122607.4
2
14
129549.7
3
15
128105.9
4
16 126818.9
2019
1
17
123020.3
2
18
130158.9
3
19 133584.3
4
20
132147.9
2020
1
21 126932.4
2
22 123051.92]
3
23 134041.56
4
24 135031.20
2021
1
25 130020.840
2
26 128010.480
3
27 138000.12
4
28 139989.76
2022
1
29 140979.40!
2
30 135969.04
3
31 130958.68!
4
32 142948.32!
2023
1
33 144321.12!
2
34 139782.059
3
35 135372.74
4
36 147611.86
2024
123
37 149334.83
38 145124.004
39 141022.359](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3c20e04-668d-4e7f-ab67-18ddfb8e3211%2Fd3477653-8c48-449e-8f1e-e5a2c86e1b94%2Ffngh2lk_processed.jpeg&w=3840&q=75)
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