# Regressions. Why a and b explains more than a+b?

So I have sample of 1987 observations. I'm checking how accounting measures can explain stock returns. If I do a regression of stock returns on CFO (cash flow) and Accruals, I get $R^2= 0.075$. But if I regress the stock returns on Earnings (CFO+Accruals) I only get $R^2=0.042$. I want to know how's that possible.

• Can you explain more clearly what you mean? How is regression of CFO and Accruals different from a regression of CFO+Accruals? – Sycorax Jan 29 '15 at 20:20
• I mean the first regression is Y=c+CFO+Accruals. And the second one is Y=c+EBEI. But in order to get EBEI, one needs to add CFO and accruals. – Donatas Gudauskas Jan 29 '15 at 20:24
• To what phenomenon does "that" refer? – whuber Jan 29 '15 at 20:27

## 1 Answer

The reason is flexibility.

Option 1: When you regress $X_1, X_2$ on $Y$ you are allowing the coefficients to be different. In other words, your regression equation is $Y = \alpha + \beta_1X_1 + \beta_2X_2 + \epsilon$. Notice $\beta_1$ may equal $\beta_2$ if it wants to - if that's what the data suggests.

Option 2: When you regress $X_3 = X_1 + X_2$ on $Y$ you are forcing the coefficients to be the same - even if the data doesn't want them to be the same. Your regression equation becomes $Y = \alpha^\prime + \beta_1^\prime(X_3) + \epsilon = \alpha + \beta_1^\prime X_1 + \beta_1^\prime X_2+ \epsilon$.

A regression equation will always select values for the coefficients that minimizes SSE (and thus maximizes $R^2$) based on the data you give it. In option 2, it'll do it's best but this will never be better (in terms of $R^2$) then option 1. This is because every conceivable possible value for the vector of coefficients in option 2 is a subset of the conceivable possible value for the vector of coefficients in option 1.

• Note for any possibly confused readers - while most usually the conventional description (at least in English) would say that "we regress $Y$ on $X_1$ and $X_2$" when $Y$ is the DV and the other two are IVs, some people do say it the other way around, just as "plot Y vs X" (to denote Y on the vertical axis) would be said by some people as "plot X vs Y". We must sometimes look for context. In this case, the equations are unambiguous so the meaning is clear (which suggests that we should always consider supplying, as TrynnaDoStat does here, some unambiguous information such as an equation). – Glen_b Jan 29 '15 at 21:28