Q:

Inferencing from Pearson correlation and multiple linear regression

My hypothesis is: Company size (CSIZE) is positively correlated with Return on Assets (ROA).
I create a Pearson correlation matrix in Stata using pwcorr csize roa and it returns a correlation of 0.2 with a 5% significance level.
Now I also want create a multiple regression to test Company Age (CAGE) and Company Size (CSIZE) on Return on Assets (ROA).
I create a linear regression model in Stata using regress roa cage csize
I returns low R^2=0,007 and low F-statistic 1,08. Neither independent variable had significant T-statistic either.
My question is how to interpret such a result: The low R^2 means my model does not explain much of the variation. Plus it is not significant. But is that only for the combined effect of both independent variables (CAGE+CSIZE) on the dependent variable (ROA)? The individual t-test was also insigificant, so what about the result from my Pearson correlation?
I am having a hard time seeing how linear regression and Pearson correlation relates to each other, and what implications the results of both have on my hypothesis.

A:

First, I think regress is the multiple linear regression in stata, not correlate. Second, "theoretically," the relationship (or model) between your response and covariates cannot weaken (lower $R^2$) when you add variables (for more reading on this, I recommend google searching "degrees of freedom"). Third, if in the first case you mentioned, if correlation is .2, then $$R^2 = (.2)^2 = .04 < .07$$ meaning the response was explained somewhat better when you added the aditional covariate. 
Nevertheless, if the above doesn't apply (because you made up the numbers for the sake of example), the behavior you observed can occur if your two covariates (age and size) are highly correlated, resulting in some numerical instability. Certainly company size and age may be highly correlated, particularly in more traditional sectors of the economy. For more on dealing with this issue, Google "colinearity" and perhaps even "pre-conditioning."

