Correlation doesn't prove causation. Saying, "Obama was president and the economy recovered; therefore he caused the economy to recover," as the sum total of one's argument is only slightly more compelling than saying "Obama was president and the sun rose every day; therefore he caused the sun to rise."
It's actually very hard to determine the effect of any policy on the economy for a simple reason: we don't know what would have happened if that action hadn't been taken (the counter factual). Because of this, economists and social scientists develop complex models to try to resolve these issues. To have a reasonable chance at discerning truth from all the noise, good data scientists look for a natural experiment or statistical instrument. They find something exogenous that triggers the policy to a random part of a larger group group. For example, we could assess the impact of immigration by looking at a program that randomly assigned various numbers of migrants to different cities.
In the absence of a natural experiment, many statisticians simply try to control for all possible variables in the aim of seeing the relationship of one variable and one other one. The problem with this approach for a large complex system like the US economy is that it basically assumes that the writer of the model has most of the relevant data to explain any changes not caused by what he or she is trying to measure. This is an almost impossible assumption for trying to empirically understand something as complex as the entire US economy. It is far beyond the capabilities of modern data analysis to understand the sum total and interacting effects of all the executive actions of a president, on something as complex as the entire US economy.
Since we can't see the counter factual world of a Nobma presidency, Obama's influence wasn't randomly hoisted on separate parts of the country, and we don't have all the relevant data to understand the world economy, we don't have a strong basis for saying one way or another what the sum total impact of all the President's policies has been.