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.
Data Analysis Questions:
- Calculate the moving averages for each observation in the data based on periods 1, 2, 3, and 4. It should be centered on period t = 2.
Use the function of =AVERAGE (D2:D5) in cell E3 and copy the formula down to cell E38.
Show the moving averages plotted on the passenger series y(t). Do not forget to label x and y axes and adjust the values on the plot using Figure1.
Show your plot space provided below:
2. Even though the moving averages help highlight the long-run trend of a time series, the moving-average model is not designed for making forecasts in the presence of trends. Explain the reason why we should not rely on moving averages for predicting future observations of a trending series.
The second step of averaging the averages is only necessary when the number of seasons is even, as with semiannual, quarterly, or monthly data. However, if the number of seasons is odd, then the initial moving average is the centered moving average. This average is referred to as centered moving average (CMA).
![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%2Fc019d8ea-cc81-437f-a8fe-0d1a7ae95b40%2Fonp0f38_processed.jpeg&w=3840&q=75)
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