How much influence does the president really have on US economy? We hear a lot about how the economy changed under a certain president, but how much does the incumbent actually affect the economy? Are the fluctuations we see in economic growth due to natural market fluctuations or can they be attributed to policy changes?
Here is one paper that may be of interest:


  Analyses of the effects of election outcomes on the economy have been
  hampered by the problem that economic outcomes also influence
  elections. We sidestep these problems by analyzing movements in
  economic indicators caused by clearly exogenous changes in
  expectations about the likely winner during Election Day. Analyzing
  high frequency financial fluctuations following the release of flawed
  exit poll data on Election Day 2004, and then during the vote count,
  we find that markets anticipated higher equity prices, interest rates
  and oil prices and a stronger dollar under a Bush presidency than
  under Kerry. A similar Republican-Democrat differential was also
  observed for the 2000 Bush-Gore contest. Prediction market based
  analyses of all Presidential elections since 1880 also reveal a
  similar pattern of partisan impacts, suggesting that electing a
  Republican President raises equity valuations by 2-3 percent, and that
  since Reagan, Republican Presidents have tended to raise bond yields.


Partisan Impacts on the Economy: Evidence from Prediction Markets and Close Elections Snowberg, Wolfers, and Zitzewitz (2006) 

This might give some indication of an upper bound on these effects since only some of the increase in stock market wealth is from an increase in aggregate wealth. The rest comes from a change in transfers and so not really an effect on the economy. 

Here's a similar paper from Germany:


  Rational partisan theory suggests that firms perform better under
  right- than left-leaning governments. In the pre-election time,
  investors should anticipate these effects of government partisanship.
  This is the first study to investigate such anticipated partisan
  effects in Germany. Applying conditional volatility models we analyze
  the impact of expected government partisanship on stock market
  performance in the 2002 German federal election. Our results show that
  small-firm stock returns were positively (negatively) linked to the
  probability of a right- (left-) leaning coalition winning the
  election. Moreover, we find that volatility increased as the electoral
  prospects of right-leaning parties improved, while greater electoral
  uncertainty had a volatility-reducing effect.


Partisan politics and stock market performance: The effect of expected government partisanship on stock returns in the 2002 German federal election   (Füss, Bechtel (2002))

It also depends on the baseline way of measuring impact. A really terrible president might be able to wreck serious damage on the economy while the other branches and electorate would take time to undo that damage and remove him. Should presidential performance be measured against the very worst that an incompetent or malicious person could deliver? Or are they best compared with other likely serious contenders? These papers above are much more measures of the latter than the former.