Color correction is often posed as a linear regression problem either from camera RGB to XYZ - where the aim is use a camera for color measurement - or to a display color space such as sRGB for image reproduction. While linear regression is simple and also ensures exposure independence, the mapping found through regressing RGB to XYZ is not optimal in terms of perceived color. In this paper, we begin by observing that the best linear transform for mapping RGB to XYZ to minimize a color difference metric, such as CIE LAB, is not separable. In particular we show that the best fitted Y channel should be different depending on whether L*, a* or b* error is minimized. Consequently, we develop an extended linear regression framework for CIE LAB where we solve for Y - we map RGB onto Y - three times - once for L*, once for a* and once for b*. As in conventional regression we solve for X and Z only once. Experiments demonstrate that compared to our new extended linear regression method the mean, 95% quantile and CIE LAB error afforded by simple linear least-squares is respectively 30%, 50% and 70% larger. Extended linear regression delivers leading color correction performance with fewer parameters than competing methods.