Author Question: Unbiasedness and small variance are desirable properties of estimators. However, you can imagine ... (Read 112 times)

skymedlock · **on:** Jun 29, 2018

Unbiasedness and small variance are desirable properties of estimators.

However, you can imagine situations where a trade-off exists between the two: one estimator may be have a small bias but a much smaller variance than another, unbiased estimator. The concept of mean square error estimator combines the two concepts. Let be an estimator of . Then the mean square error (MSE) is defined as follows: MSE( ) = E( )2. Prove that MSE( ) = bias2 + var( ). (Hint: subtract and add in E( ) in E( )2.)
What will be an ideal response?

Question 2

In a multiple regression framework, the slope coefficient on the regressor X2i

A) takes into account the scale of the error term.
B) is measured in the units of Yi divided by units of X2i.
C) is usually positive.
D) is larger than the coefficient on X1i.

jonathanballen97 · **Reply #1 on:** Jun 29, 2018

Answer to Question 1

Answer:
MSE ( ) = E( - E( ) + E( ) - )2 = E( - E( )) + (E( ) - )2
= E( - E( ))2 + (E( ) - )2 + 2( - E( ))(E( ) - )
Next, moving through the expectation operator results in
E - E( )2 + EE( ) - )2 + 2E( ) - E( ))( E( ) - ).
The first term is the variance, and the second term is the squared bias, since
EE( ) - )2 = E( ) - )2. This proves MSE ( ) = bias2 + var( ) if the last term equals zero. But
E( - E( ))(E( ) - ) = EE( ) - - (E( ))2 + E( )
= E( ) E( ) - E( ) - (E( ))2 + E( ) = 0.

Answer to Question 2

Answer: B