In reference to the above data, which of the following statements are correct? (where $x = t – 2013$)

(A) If the equation of the straight line trend is $y = 12 + 2.29 x$, then the trend value for the year 2017 is 21.16.

(B) If the equation of the straight line trend is $y = 12 + 2.29 x$, then the trend value for the year 2013 is 11.

(C) If the equation of the straight line trend is $y = 12 + 2.29 x$, then the trend value for the year 2015 is 17.

(D) If $(t_1, y_1), (t_2, y_2), (t_3, y_3).......... (t_n, y_n)$ denote the time series and $y_t$ are the trend values of the variable y, then $\sum(y-y_t)= 0$.

Choose the correct answer from the options given below:

(A) and (D) only

The correct answer is Option (2) → (A) and (D) only

$x=t-2013$

Data: $(-3,6),\ (-2,8),\ (-1,9),\ (0,11),\ (1,13),\ (2,17),\ (3,20)$

Trend: $y_{t}=12+2.29x$

$x=4\ (\text{year }2017)\Rightarrow y_{2017}=12+2.29\times 4=12+9.16=21.16\quad\text{(A true)}$

$x=0\ (\text{year }2013)\Rightarrow y_{2013}=12+2.29\times 0=12\neq 11\quad\text{(B false)}$

$x=2\ (\text{year }2015)\Rightarrow y_{2015}=12+2.29\times 2=12+4.58=16.58\neq 17\quad\text{(C false)}$

Trend values and residuals:

$y_{t}:\ 5.13,\ 7.42,\ 9.71,\ 12,\ 14.29,\ 16.58,\ 18.87$

Residuals $y-y_{t}:\ 0.87,\ 0.58,\ -0.71,\ -1,\ -1.29,\ 0.42,\ 1.13$

$\sum (y-y_{t})=0.87+0.58-0.71-1-1.29+0.42+1.13=0\quad\text{(D true)}$