How Much Does Yesterday Explain Today?

A small bakery records how many loaves it sells on each of $n = 8$ consecutive days:

Day $t$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
Loaves $y_t$	$34$	$36$	$39$	$42$	$43$	$45$	$42$	$39$

To judge how strongly one day's sales carry over into the next, the owner computes the lag-1 sample autocorrelation. Use the standard estimator, in which a single overall mean $\bar{y}$ (taken over all $8$ observations) is subtracted everywhere:

$r_1 = \frac{\displaystyle\sum_{t=2}^{n} (y_t - \bar{y})(y_{t-1} - \bar{y})}{\displaystyle\sum_{t=1}^{n} (y_t - \bar{y})^2}$

What is the value of $r_1$ ? Give it as a decimal (not a percentage), rounded to 3 decimal places.

How Much Does Yesterday Explain Today?

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