Question

Consider two least-squares regressions

y=X 1β~1 + e~y=X 1β^1 +X2β² + e^​

(a) Show that β^1 =(X1 M² X 1 ) −1 X 1′​ M ²​ y where M 2 =I n−X² (X² X² ) −1 X²

(b) Under what conditions does β~1 =β^​1
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(c) Let R12​ and R2 be the R-squared from the regression (1) and (2) respectively.
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Answer 1

The answer explains the concepts related to two least-squares regressions and the conditions under which they are equal.

Step-by-step explanation:

In the given question, we are asked to consider two least-squares regressions and perform certain operations.

(a) To show that β^1 = (X1 M² X1 ) −1 X1′ M ² y, we need to substitute the values of β^1 and M². We can show this by substituting the values and simplifying the equation.

(b) β~1 = β^1 under certain conditions, such as when X1 is orthogonal to the error term e.

(c) R12 and R2 are the R-squared values from two regression models. R2 will always be greater than or equal to R12. If R2 is equal to R12, it means that the additional variables in the second regression model have no effect on the prediction of y.