A question on OLS biasedness

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Just a question like this:

A sample correlation coefficient of 0.95 between two independent variables both inculded in the model, if in this case, can the OLS estimators be biased?

The answer is not unless there is a perfect linear relationship among two or more variables.

My question is: If so, why should we deal with collinearity problems in empirical studies?

Thanks!

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关键词：biasedness question Biased Quest edn OLS question biasedness

beta_hat=((X`*X)^-1)*X`*Y
E[beta_hat]=E[((X`*X)^-1)*X`*Y]=((X`*X)^-1)*X`*E[Y]=]=((X`*X)^-1)*(X`*X)*beta=beta
So, the estimator is unbiased. But there might be some precision problems to calculate the inverse of X`*X. If two independent variables are perfect linear related, the matrix might be sigular.
In regression problem, if there are strong lienarity among two independent variables, we try to get the parsimony in the model. If there is no strong evidence that we should keep two variables, we'd better retain one in the model.

2# zj_ocean

Thanks for your answer! You mean that collinearity can lead to precision problem instead of biasedness. So when dealing with collinearity problem in our empirical studies, what we concern is whether it could improve the precision of estimators?

No exactly. In fact, if two variables are perfectly correlated, the inversion does not exist, the system is not identifiable. Intuitively, all the information has been collected from one of the variables, it's no use to include both of them. In empirical study, first if the two varibles are perfectly linear correlated, we can't calculate the coefficients. If the  sample correlation coefficient of 0.95, then since the calculation is often done by computers, the exists the precision problem as mentioned by zj_ocean.

I agree with cbqywl. If there is great collinearity among two variables, why you must keep all of them in the model?