# Collinearity in multivariate regression with huge amounts of data

Take the following example:

I wish to predict physical performance as a function of height and weight. I already know weight negatively affects performance. Height also negatively affects performance, but I don't know if this is only because of the weight, or if there is a positive correlation between height and performance when disregarding the correlated weight. So, I measure the heights and weights of thousands of individuals, and measure their performance. I then model performance with multivariate regression: $p = (a * \text{height}) + (b * \text{weight}$). The estimated coefficients $\hat{a}$ and $\hat{b}$ turn out to be 0.2 and -1 respectively when using "lm" in R. The individual significances of these estimated coefficients are very high according to the t-tests (as is shown in a table when using the R command "summary" of a linear model made with the command "lm"), and their std. errors are very low. I therefore conclude that height is positively correlated with performance when disregarding the negative effects of weight. When is this conclusion a mistake?

To rephrase, if I have plenty of data and the std. error is very low for the estimated coefficients of both correlated variables (say $x_1$ and $x_2$) in the regression summary produced by the R command "summary", is there any reason (in general) why I should distrust the estimated coefficients for $x_1$ and $x_2$? I.e. can the collinearity mean that these estimated coefficients do not necessarily reflect the real coefficients even though the standard errors shown by the regression summary are very low?

• Why not check for collinearity and see? – Peter Flom Jan 6 '15 at 22:28
• I already know these variables are highly correlated, but that is not the question here. The question concerns how correlated variables affect inference when there is a large amount of data – Datoraki Jan 6 '15 at 22:36
• Correlation is not the same as colinear. Look at the condition indexes to see if there is a severe effect. – Peter Flom Jan 6 '15 at 22:38
• @PeterFlom (+1) for the useful CI remark.. – Analyst Jan 7 '15 at 8:51
• @PeterFlom Thanks for pointing that out. I don't really think it answers the question though. Perhaps I was a bit unclear in my formulation – Datoraki Jan 12 '15 at 21:33

To add to the other excellent answers; "It depends" If multicollinearity is the only problem with the model, its effect would asymptotically vanish, when $n$ gets large enough. But, in practice, it might well be other problems with the model, which might interfere with multicolinearity!
Very specifically, if (some of) the regressor variables (the $x$'s) are measured with error, the problems caused by that will be amplified by multicollinearity, and I'm afraid that might happen in such a way that larger sample sizes will not help to much. I will try to add some simulations to this answer to illustrate/investigate that aspect.