A. Obama is president.
B. The economy is weak.
C. Therefore: A must be the cause of B. We need a Republican president.
But the underlying logic to that leads to some uncomfortable conclusions for Republicans. Here is a graph of job growth/loss per month during (and a bit before) Obama's presidency:
He became president, and quickly passed the stimulus, in February 2009. Soon afterward, we went from economic free-fall back toward recovery. One thing to notice from the graph is, whenever you hear claims about the total number of job losses under Obama, clearly that's the result of those first few months. And despite what many try to imply, the recession has been over for a while. I'm not trying to claim "everything is fine!" though. Job growth has been slow, especially when you consider how long it will take to bring back all those jobs that were lost, and it's important to have real debates as a country around identifying the causes and cures of this slow recovery.
However, if your economic argument right now is pretty much the one I summarized earlier, then what does that underlying logic really say about the job chart above? Do you prefer slow recoveries or severe recessions? And if you think economic growth during a presidency is a good way to know which policies are best, then there's a lot more history to look at besides just Obama and Bush...
Before Bush & Obama, here's the break-down of how the economy has done on average under Democratic and Republican presidents. In all categories, we've done significantly better with a Democratic president.
In fact, for as long as we started officially tracking recessions, we've entered a recession in the first term of every Republican president we've elected (link). If Romney is elected, let's hope he's the very first one to break that trend.
And if you also consider congress, we've had the strongest economic growth when Democrats have both the presidency and congress (link). You also see some of that from the job chart during Obama's presidency. The employment situation improved rapidly in the early months of Obama's presidency when there was a Democratic super-majority. You don't really see great results coming from the loss of that super-majority in 2010 or the Republicans gaining the House in 2011.
That being said, this doesn't necessarily prove anything; there are a lot of factors involved in how the economy does other than the president. I don't think Obama is to blame in any significant way for the slow recovery. And I don't think the Great Recession is Bush's fault just because it happened while he was president. But if you do hold the simple view of "economy bad, Obama president, therefore Obama bad", then the last thing that analysis of the economy should do is lead you to vote for a Republican president.