Is there a correlation between a President's approval rating on Election day and their popular vote margin during re-election?

Question

I was looking 538's historical approval ratings and I noticed that on the last day of a two-term president's last term, the approval rating was about twice the popular vote margin nationwide. This result was consistent, going all the way back to Reagan's 1984 reelection. But, this is at the end of a term after the popular vote result is fully released and finalized.

When a President is running for re-election, is there a correlation between their approval rating on Election Day and the share of the popular vote they receive? If so, is it possible to use the net approval on November 3rd, 2020 to estimate the approximate popular vote margin for Trump vs Biden?

Answer 1

There is a correlation, but there are not enough data points to predict the result with much certainty.

There are only five Presidents where the question of approval rating have been close enough to their re-election to be meaningful for this purpose, the first of whom being Reagan.

+-----------+--------------+---------------------+
| President | Net Approval | Popular vote margin |
+-----------+--------------+---------------------+
| Reagan    |          +26 |               +18.2 |
| Bush Sr   |          -21 |                -5.6 |
| Clinton   |          +18 |                +8.5 |
| Bush Jr   |           +4 |                +1.4 |
| Obama     |           +6 |                +3.9 |
+-----------+--------------+---------------------+

If you had to draw a trend line through this, it would have a slope of around 0.5% popular vote per 1% approval. However, this produces some large errors for some cases.

Source for approval rating data

(Usual caveat: popular vote means precisely nothing in US Presidential elections.)