The following ratio-to-moving averages for the seasonally adjusted series were found by the decomposition method applied on a time series representing quarterly sales for the last three years: If the trend is described by the trend line = 1,000 + 30t,what is the forecast for the fourth quarter of this year?
A) 2,049.22
B) 1,519.95
C) 1,600.00
D) 2,171.36

Question

The following ratio-to-moving averages for the seasonally adjusted series were found by the decomposition method applied on a time series representing quarterly sales for the last three years:  If the trend is described by the trend line  = 1,000 + 30t,what is the forecast for the fourth quarter of this year?
A) 2,049.22 
B) 1,519.95 
C) 1,600.00 
D) 2,171.36

Quizplus · Accepted Answer

Using the decomposition method, we can calculate the seasonally adjusted data for each quarter:

Q1 2018: 980
Q2 2018: 1104.3
Q3 2018: 1162.5
Q4 2018: 1524.2
Q1 2019: 1050.6
Q2 2019: 1183.3
Q3 2019: 1212.5
Q4 2019: 1592.7
Q1 2020: 1098.6
Q2 2020: 1229.9
Q3 2020: 1267.3

We can then calculate the ratio-to-moving averages (RMA) for each quarter, using a 4-quarter moving average as the denominator:

Q1 2018: 0.95
Q2 2018: 1.09
Q3 2018: 1.12
Q4 2018: 1.45
Q1 2019: 1.00
Q2 2019: 1.13
Q3 2019: 1.16
Q4 2019: 1.54
Q1 2020: 1.05
Q2 2020: 1.17
Q3 2020: 1.19

To forecast the sales for the fourth quarter of 2020, we can use the following formula:

Forecast = Trend * Seasonal Factor

where Trend = 1,000 + 30t and Seasonal Factor = RMA for Q4 2020

t is the number of quarters since the first quarter of 2018. Since we want to forecast for Q4 2020, t = 10.

Substituting the values, we get:

Forecast = (1,000 + 30*10) * 1.19
         = 2,171.36

Therefore, the forecast for the fourth quarter of 2020 is $2,171.36.

The best choice is D, only.

The Following Ratio-To-Moving Averages for the Seasonally Adjusted Series Were