# Interpretation of simultaneous and independent ordinary least squares regression

I'm using ordinary least squares to regress a noisy overdetermined system.

$$y = \beta_0 x_0 + \beta_1 x_1$$

For comparison, I'm also solving the independent equations

\begin{align} y &= \beta_0 x_0 \\ y &= \beta_1 x_1 \end{align}

I'm surprised to find that sometimes when the independent solutions are all positive, some of the simultaneous solution elements are negative. What does it mean when that occurs? Can I conclude anything about my data set? Can I conclude that my data set violates the no autocorrelation assumption about OLS regression?

Should I use Feasible Generalized Least Squares?

Here is an example of a small data set for which the independent solutions are positive, but the simultaneous solution has negative elements.

#! /usr/bin/env runhaskell

import System.IO
import Data.Functor
import Numeric.LinearAlgebra
import Numeric.LinearAlgebra.Data
import Numeric.LinearAlgebra.HMatrix

main :: IO ()
main = do

putStr "independent  β = "
print $(<\> y) . asColumn <$> toColumns x

putStr "simultaneous β = "