When the middle time period is taken as the origin in fitting a straight-line trend \(y = a + bX\) by the method of least squares, which of the following is always true?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\sum X = 0\)
The time periods are equally spaced, so with the middle time period as the origin the values of X lie symmetrically about 0. For five years, for example, X = −2, −1, 0, 1, 2, and the negative and positive values cancel:
\(\sum X = -2 - 1 + 0 + 1 + 2 = 0\)
This is why the normal equations reduce to
\( \displaystyle a = \frac{\sum y}{n}, \quad b = \frac{\sum Xy}{\sum X^2} \)
Why the other options are incorrect
Option (1) \(\sum y = 0\) — Incorrect. \(\sum y\) is the total of the actual values; it does not depend on where the origin of time is placed.
Option (2) \(\sum X^2 = 0\) — Incorrect. Every \(X^2\) is 0 or positive and not all X are 0, so \(\sum X^2 \gt 0\).
Option (3) \(\sum Xy = 0\) — Incorrect. This sum depends on the data; it is zero only when the trend line is horizontal (b = 0).