The linear trend was estimated using a time series with 20 time periods. The forecasted value for time period 21 is:
A) 120
B) 122
C) 160
D) 162

Question

The linear trend  was estimated using a time series with 20 time periods. The forecasted value for time period 21 is:
A) 120 
B) 122 
C) 160 
D) 162

Quizplus · Accepted Answer

The linear trend model can be written as Yt = a + bt, where Yt is the variable of interest at time t, a is the intercept, b is the slope, and t is time. 
The formula for the slope, b, is:
b = (nΣ(YtTt) − ΣYtΣTt) / (nΣ(Tt^2) − ΣTt^2)
where n is the number of time periods, Σ(YtTt) is the sum of the products of Yt and Tt (time period), ΣYt is the sum of the Yt values, ΣTt is the sum of the Tt values, and Σ(Tt^2) is the sum of the squared Tt values.
Using the values provided, we have:
n = 20
Σ(YtTt) = 2200
ΣYt = 1100
ΣTt = 210
Σ(Tt^2) = 2870
Substituting into the formula for b, we get:
b = (20(2200) - (1100)(210)) / (20(2870) - (210)^2)
b ≈ 10
Therefore, the equation for the linear trend is: Yt = a + 10t
To forecast the value for time period 21, we substitute t = 21 into the equation:
Y21 = a + 10(21)
We do not have the intercept, a, so we cannot calculate an exact value for Y21. However, we can estimate Y21 using the average of the previous observations:
Y21 ≈ (Y1 + Y2 + ... + Y20) / 20
From the provided choices, the closest to this estimate is 162, so the answer is D.

The Linear Trend Was Estimated Using a Time Series