Test for difference in coefficients: same sample, same outcome, but different explanatory variable I do Ordinary Least Squares regressions and want to test if the difference between two estimated coefficients is statistically significant. I use the same sample, the same outcome, and only the explanatory variable of interest differs between the two models, i.e.: 
$y= b_0+b_1x_1+b_2x_3$
and
$y= a_0+a_1x_2+a_2x_3$
and I want to see if $b_1$ is statistically different from $a_1$, since $x_1$ and $x_2$ are measuring something slightly different and I want to see, if this leads to different effect sizes. 
Which statistical test can I use for that? Can I just look at the confidence intervals and see if they overlap or not? Or is there some statistical "trap" if I do this?
 A: As user2974951 mentioned in the comments, you cannot directly compare the two coefficients $b_1$ and $a_1$ (the comparison would be meaningless). You could either put them in the same model, although if $x_1$ and $x_2$ are redundant (correlated) you might end up having multicollinearity, and the standard error of the coefficients might become very large (so much that you cannot rely on them to test each coefficient against zero). The best thing to do depends on what you want to achieve with this analysis: if you just need to pick one model you could compare the $R^2$ (variance explained) of the two models, and choose the best one. If you need to pick the model with best predictive ability (i.e. the one that is expected to perform better in predicting new data) it might be worth to test both models, as well as a model with both predictors $x_1$ and $x_2$, with a cross-validation procedure such as leave-one-out cross-validation. (Multicollinearity per se is not a problem for extrapolating the model to new data, and by picking the model that perform better in the cross-validation you also control for overfitting.)
