Ways to Address Collinearity in Regression Models
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learnitdaily.com
Leaving all the variables in a regression leads to large standard errors. These large standard errors accurately reflect the uncertainty of having highly related explanatory variables. That is, the data literally can't show which variable is causing the outcome.
Removing covariates leads to smaller standard errors, giving the illusion of certainty.
Obviously, if you are just trying to get p-values less than 0.05, this isn't going to help. Or if you are presenting the data to someone who doesn't understand standard errors, than having standard errors that accurately reflect inferential uncertainty isn't important. And, if the end goal is prediction, this whole point is irrelevant.
But "leave everything in there" is frequently written off when it's the most appropriate approach for representing relationships in the data.
I don't see how "do nothing" is a good option. If the co-linearity between two variables is high enough, then the linear regression equation will not have a solution at all. But this is an edge case. The more common problem with co-linearity is that it leads to unstable regression parameters. For example, all variables in a dataset could have very high, positive correlation, but one of the regression parameters could come out as a negative number due to co-linearity (which would counter-intuitive). If you are basing decisions on the regression parameters then it could lead to some very silly decisions. Moreover, if you resample the data and re-fit the regression model, you could get very different regression parameters (i.e. the parameters will be unstable). Stability of the parameters over time (or over different training sets) is one of the most important properties of a good model.
To see why "do nothing" is not a logical option, you can imagine that your data points are plotted within a 3-dimensional cube (1 response variable, and 2 predictor variables). If the data is co-linear, then your data points will lie along a straight line that cuts across the diagonal of the cube. If you "do nothing," then it is directly equivalent to trying to fit a PLANE through a LINE in 3 dimensions. The problem is that points along a LINE do not uniquely identify a plane. The slope along one axis of the plane will be stable, but the slope along the other axis will be fitted arbitrarily by the regression model and will be unstable. The proper thing to do here is to model the line in 2 dimensions. In order to do this, you can either eliminate one of the two predictor variables from the model, or you can project the line in 3-dimensions onto a 2-dimensional subspace (which is what PCA does). Of course, a line formed by data points in 2 dimensions (x and y) is very nicely modeled by ordinary linear regression and it will give very a stable slope (Beta) parameter.
It is explained in the rest of the comment. I would suggest you read that. For a more in-depth explanation, this also explained in most econometrics textbooks.
Stability of the parameters over time (or over different training sets) is one of the most important properties of a good model.
In some cases it is, in many cases unbiasedness or consistency are more important.
When stability is a priority, collinearity clearly needs to be addressed.
If unbiasedness and/or consistency is a priority, doing nothing is the best option (since removing variables leads to omitted variable bias, PCA does not yield the parameter of interest, and regularization techniques such as ridge regression are both biased and inconsistent).
False. PLS (Partial Least Squares) is an ad-hoc algorithm that was invented by people working in chemometrics. PLS was not based on any rigorous statistical theory, it is merely a heuristic. It might appear to work reasonably well on some data sets, but asserting that "PLS is the correct choice" is cargo cult science.
the same general argument you make can be said of PCA ... but it works. We know it works becuase it can be used to predict the real world.