A.     What is the least squares estimator of á?

B.     What is the least squares estimator of â₁, the coefficient of X₁?

C.    What is the least squares estimator of â₂, the coefficient of X₂?

Show the formulas and the computations. You can check your work with Excel or other statistical software.

Can you please explain the following:
1) If two explanatory variables are highly correlated, does adding the second explanatory
variable raise or lower the estimated standard deviation of the residuals of the regression?
2) If two explanatory variables are uncorrelated and each is correlated with the dependent
variable, does adding the second explanatory variable raise or lower the estimated standard deviation of the residuals?

[NEAS: The issues are important but subtle. Fox has good explanations.

(1) Adding explanatory variables raises the R² but not necessarily the adjusted (corrected) R².

See Section 5.2.3 on pages 92-94.

(2) Adding a correlated explanatory variable may raise the standard error of the estimator (not the regression).

See Section 6.2.2 on pages 106-110 and Section 6.3 on pages 110-112.

Illustration: Suppose we regress the personal auto loss cost trend on inflation. If we regress on a single inflation index, such as the CPI or the medical CPI, and we have extensive data with little random fluctuation, the standard error of the regression is low and the standard error of the estimator is low. If we regress on both inflation indices (which are highly correlated), the standard error of the regression will probably decline a bit but the standard error of the estimators will increase.]