A straight-line trend \(y_t = a + bX\) is fitted to a time series \(y\) by the method of least squares, taking the middle year as the origin. Which of the following statements are correct?
(A) \(\sum (y - y_t) = 0\)
(B) \( \displaystyle a = \frac{\sum y}{n}\), where \(n\) is the number of years.
(C) \( \displaystyle b = \frac{\sum Xy}{n}\)
(D) Forecasts made from this line are based only on the trend values of the series.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (D) only
By the principle of least squares, the line makes \(\sum (y - y_t)^2\) least, and for this line the deviations themselves add up to zero. With the middle year as origin, \(\sum X = 0\), and the normal equations reduce to
\(\sum y = na\)
\(\sum Xy = b\sum X^2\)
so that
\( \displaystyle a = \frac{\sum y}{n}, \quad b = \frac{\sum Xy}{\sum X^2} \)
The line describes only the long-term trend, so a forecast made from it ignores seasonal, cyclical and irregular changes.
So \(\sum (y - y_t) = 0\), \(a = \sum y / n\), \(b = \sum Xy / \sum X^2\), and forecasts from the line use only the trend.
Checking the statements
(A) \(\sum (y - y_t) = 0\) — Correct. The deviations from the least-squares line add up to zero.
(B) \( \displaystyle a = \frac{\sum y}{n}\), where \(n\) is the number of years. — Correct. It comes from \(\sum y = na\) when \(\sum X = 0\).
(C) \( \displaystyle b = \frac{\sum Xy}{n}\) — Incorrect. The divisor is \(\sum X^2\), not \(n\).
(D) Forecasts made from this line are based only on the trend values of the series. — Correct. The line contains only the trend component.