Q.Fit a linear equation from the following data for variable $(y)$ of a time series $n = 4$, $\Sigma y = 270$, $\Sigma ty = 734$.

Even $n$: code $t = -3,-1,1,3$ ($\sum t = 0$, $\sum t^2 = 20$); $a = 270/4 = 67.5$, $b = 734/20 = 36.7$; $\hat y = 67.5 + 36.7t$.

The straight-line trend is $\hat{y} = a + bt$. For $n = 4$ (an even number of years) we take the time origin at the middle and use a half-year unit, so the coded values are

$t = -3,\ -1,\ 1,\ 3 \implies \sum t = 0,\quad \sum t^2 = 9+1+1+9 = 20.$

Because $\sum t = 0$, the normal equations simplify to:

$a = \frac{\sum y}{n} = \frac{270}{4} = 67.5$

$b = \frac{\sum t y}{\sum t^2} = \frac{734}{20} = 36.7$

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