Regression on a constant
So basically the question is: 
If I know the average ($\hat{\mu}$) of the daily temperatures ($y_i$) of last year, does that tell me anything about how many people were born ($x_i$) each day?

Unsurprisingly the answer is no.

The most you can get is the average of the $x_i$ series if you have the parameters of an unbiased regression between $x_i$ and $y_i$.