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Please describe the trade-off of including a new independent variable into the regression model, especially for...

Please describe the trade-off of including a new independent variable into the regression model, especially for the unbiasedness and variance of the OLS estimator.

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Answer #1

Tradeoff of including a new independent variable is that it may lead to overfitting of the model.
(1) With overfitting the variances would be larger, thereby standard errors would be large
(2) The OLS estimators would be inefficient due to larger variaces.
(3) Confidence Interval and  hypothesis testing results may not be realiable (due to larger variances) as calculated t-value and f-value would be lower and we will not reject the null hypothesis more often than not.

(4) The OLS estimators for the model (with additional independent variable) would be unbiased as long as the new independent variable is not correlated with error term. That is,

Other problems with overfitting:
(5) there will be loss of degrees of freedom
(6) It may create multicollinearity problem

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