For a yearly time series, \(y_t\) denotes the value for the year \(t\). Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | 3-year moving average placed against year \(t\) | (I) | \( \displaystyle \frac{y_{t-2} + 2y_{t-1} + 2y_t + 2y_{t+1} + y_{t+2}}{8} \) |
| (B) | 5-year moving average placed against year \(t\) | (II) | \( \displaystyle \frac{y_{t-1} + y_t + y_{t+1}}{3} \) |
| (C) | 4-year moving average placed between years \(t\) and \(t+1\) | (III) | \( \displaystyle \frac{y_{t-1} + y_t + y_{t+1} + y_{t+2}}{4} \) |
| (D) | 4-year centred moving average placed against year \(t\) | (IV) | \( \displaystyle \frac{y_{t-2} + y_{t-1} + y_t + y_{t+1} + y_{t+2}}{5} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(IV), (C)-(III), (D)-(I)
(A) 3-year moving average placed against year \(t\)
This matches (II). Year \(t\) is the middle of the three years \(t - 1\), \(t\), \(t + 1\).
(B) 5-year moving average placed against year \(t\)
This matches (IV). Year \(t\) is the middle of the five years \(t - 2\) to \(t + 2\).
(C) 4-year moving average placed between years \(t\) and \(t+1\)
This matches (III). A 4-year average is placed between its 2nd and 3rd years, so its four years are \(t - 1\), \(t\), \(t + 1\), \(t + 2\).
(D) 4-year centred moving average placed against year \(t\)
This matches (I). It is the mean of the two 4-year averages on either side of year \(t\):
\( \displaystyle \frac{1}{2}\left(\frac{y_{t-2} + y_{t-1} + y_t + y_{t+1}}{4} + \frac{y_{t-1} + y_t + y_{t+1} + y_{t+2}}{4}\right) \)
\( \displaystyle = \frac{y_{t-2} + 2y_{t-1} + 2y_t + 2y_{t+1} + y_{t+2}}{8} \)