1) Please clearly mention questions (a) or (b) in your solution not to be confused. 2) Please add explanation/sentences to understand equations. 3) Please use the following equation typing function of...

1a`
The projection matrix P and the annihilator M are defined as
P = X(X’X)-1X’
P2 = (X(X’X)-1X)2
= (    X(X’X)-1X)
=    P
Thus P is symmetric
1b
Yes, R2 can be negative in classical linear regression.
1c)
A is a symmetric matrix.
Prove: ββAβ = 2 Aβ
Proof:
A is symmetric, let A = ( ) and let β = ()
then,
ββAβ = (β12a11+2a12β1β2 +β22a22)
    
=[ ]
    = [ ]
    = 2A
1d
In the fixed regressor, the independent variables are deterministic and fixed. We already know the properties of the OLS estimators.
While, in the random regressors, the independent variables are stochastic or random, they can take any values under given conditions. The properties of OLS estimators are not known beforehand.
2a
Assumptions are
· N ( sample size ) > K ( predictor variables )
· Homoscedasticity
· Error term is normally distributed with 0 mean and variance 2 In
· Error term has to satisfy K normal equations.
PROOF.
Since s2 = e’ e / ( n – K ), we can suffice the proof by showing that E ( e’ e | X ) = ( n – K ) 2
We have e’ e = є’ M є , where M is the annihilator. The proof will consist of...