The tasks below require you to use maximum likelihood to estimate regression coefficients. the...

Question

The tasks below require you to use maximum likelihood to estimate regression coefficients. the Standard error of the regression and the standard error of the estimates. Specifically, assume that the errd term in the following regression model Is drstrkuted normally with mean zero and constant variance:
= r• - a Px( t -11(0.42) You need to maximize the likelihood function:
E(a.P.a2) h {:11-0.1 — a - 1S—Tto exP I 2a2
and use the first order conditions to estimate aft and ? .
Then use the second order conditions to demonstrate that the likelihood function has been maximized land is not at a saddle point or a minimum point). Finally, estimate the standard error of your estimates.
1. (20 points) Take the natural logarithm of the likelihood function to obtain the log-likelihood function. Then derive the first-order conditions for a maximum of the log kkelitiood function with respect to it.0 and o2 .
2. (20 points' Define: r is Ex, and y E yi . Stony that the first order conditions Imply that: a - - g A E(X1 2)(Yt - E(xl - Aa
a RDA - a — Oxi)2
3. (30 points) Show that the second order conditions for a maximum are satisfied.
• Set up the Hessian matrix of second partials at ea} and . • Show that the own-partials are negative at 6.0 and o?. • Show that the determinant of the Restian is negative at a.p and ? .

005_kffoazo-nr55cw1g.pdf

Robert · Accepted Answer

1) 
   
)3........(0
22
ln
)2....(..........0
1ln
)1...(..........02
2
1ln
2
2ln
2
ln
1
4^
2
^^
2^2
1
^^
2^
1
^^
2^
1
2
2
2







































N
i
ii
N
i
iii
N
i
ii
N
i
ii
xy
NL
xxy
L
xy
L
xyN
L













 
Equations (1), (2) and (3) are the first order conditions for the maximum of log likelihood function 
with respect to 
^^
,   and  
2^
 .  
2)  
Solving equation (1) we get, 
xy
xNNyN
xNy
N
i
i
N
i
i
^^
^^
1
^^
1
0
0





 

 
Solving equation (2) we get, 




















N
i
i
N
i
ii
N
i
i
N
i
ii
N
i
i
N
i
ii
N
i
i
N
i
i
N
i
ii
xx
yyxx
xNx
yxNxy
xxNxyxy
xxxy
1
2
1
2
1
2
1
^
1
2
^^
1
1
2
^
1
^
1
)(
))((
0)(
0



 
Solving equation (3) we get, 







N
i
ii
N
i
ii
xy
N
N
xy
1
2
^^2^
2^4^
1
2
^^
)(
1
22
)(



 
3) 
Hessian matrix for second order conditions is as follows: 
2
2 ln

 L
                    

 Lln2
                         
2
2 ln

 L
 

 Lln2
                    
2
2 ln

 L
                         
2
2 ln

 L
 
 

2
2 ln L
                   
 

2
2 ln L
                       
22
2
)(
ln

 L

On solving for the log likelihood function we get following matrix:

The tasks below require you to use maximum likelihood to estimate regression coefficients. the Standard error of the regression and the standard error of the estimates. Specifically, assume that the...

Answer To: The tasks below require you to use maximum likelihood to estimate regression coefficients. the...

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