The Fed Model (not endorsed by the Federal Reserve) hypothesizes that the market is in equilibrium when the earnings yield on the S&P 500 matches the yield on the 10 year Treasury note. Any dissonance in the relationship would show that equity valuations are out of whack.
The Fed model was developed by economist Ed Yardeni. In it, he compared two key metrics: the yield of the 10-year U.S. Treasury note and the forward operating earnings per share of the S&P 500. Note that this is not the traditional P/E ratio, which commonly uses trailing net income per share. Operating earnings are before taxes and interest charges or income and are also known as EBIT (earnings before interest and taxes). That said, some people do substitute in PE on the S&P 500 side of the comparison.
According to this model, the S&P 500 is "fairly valued" if the inverse of the earnings per share (the earnings yield) is equivalent to the rate on the 10-year note.
For example, if the 10 year Treasury note yields 4%, the S&P 500 average P/E ratio should be 25 (because the inverse of 25 is 0.04 or 4% earnings yield) to be in equilibrium. If the S&P earnings yield is higher, then the market is considered undervalued and stock prices should be bid up until the earnings yield drops to 4%.
The Fed Model has proven valuable in some cases (1990s) and completely bunk in others (2000-2002).