Which of the following statements are correct?
A. If \( x_1,x_2,x_3,\ldots,x_n \) is the given annual time series, then 3-yearly moving averages are \( \displaystyle\frac{x_1+x_2+x_3}{3},\displaystyle\frac{x_2+x_3+x_4}{3},\ \ldots\ldots \)
B. In the method of the least squares, if \( (t_1,y_1),(t_2,y_2),\ldots(t_n,y_n) \) denote the time series and \( y_t \) are trend values of \( y \), then \( \sum(y-y_t)=0 \)
C. In the method of the least squares, the sum of squares of the deviations of the values of \( y \) from their corresponding trend values is least.
D. The rise in prices before Diwali is an example of a cyclical trend.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A, B, and C only
A 3-yearly moving average is the mean of three consecutive values. The least squares trend line is the line for which
\( \displaystyle \sum(y-y_t)^2 \text{ is least} \)
and for this line
\( \displaystyle \sum(y-y_t)=0 \)
So the moving averages are means of three consecutive values, the least squares trend gives \( \sum(y-y_t)=0 \) with \( \sum(y-y_t)^2 \) least, and a rise that comes at the same time every year is a seasonal variation.
Checking the statements
(A) If \( x_1,x_2,x_3,\ldots,x_n \) is the given annual time series, then 3-yearly moving averages are \( \displaystyle\frac{x_1+x_2+x_3}{3},\displaystyle\frac{x_2+x_3+x_4}{3},\ \ldots\ldots \) — Correct. Each average drops the first value and adds the next one.
(B) In the method of the least squares, if \( (t_1,y_1),(t_2,y_2),\ldots(t_n,y_n) \) denote the time series and \( y_t \) are trend values of \( y \), then \( \sum(y-y_t)=0 \) — Correct. The deviations from the least squares trend always add up to zero.
(C) In the method of the least squares, the sum of squares of the deviations of the values of \( y \) from their corresponding trend values is least. — Correct. This is the condition that gives the method its name.
(D) The rise in prices before Diwali is an example of a cyclical trend. — Incorrect. A rise before a festival every year is a seasonal variation, not a cyclical one.