Directions: Think of this exam as a game of “Jeopardy!” (the TV show). It gives you the answers to...

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Directions: Think of this exam as a game of “Jeopardy!” (the TV show). It gives you the answers to the maximization problem. You have to find the questions that lead to those answers. The tasks below require you use maximum likelihood to estimate regression coefficients, the standard error of the regression and the standard error of the estimates. Specifically, assume that the error term in the following regression model is distributed normally with mean zero and constant variance: ei = yi -a-ßxi e ~ N 0,s 2  Your assignment is to maximize the likelihood function: £ a,ß,s 2  = ? 1 v 2ps2 exp - (yi -a-ßxi) 2 2s2 ! and use the first-order conditions to estimate ab, bß and sc2 . Then you must use the second-order conditions to demonstrate that the likelihood function has been maximized (and is not at a saddle point or minimum). Finally, you must estimate the standard error of your estimates XXXXXXXXXXpoints) Take the natural logarithm of the likelihood function to obtain the log-likelihood function. Then derive the first-order conditions for a minimum of the log-likelihood function with respect to ab, bß and sc XXXXXXXXXXpoints) Define: ¯x = 1 N ?xi and: ¯y = 1 N ?yi . Show that the first-order conditions imply that: ab = y¯-bßx¯ bß = ?(xi -x¯) (yi -y¯) ?(xi -x¯) 2 sc2 = 1 N ?  yi -ab -bßxi  XXXXXXXXXXpoints) Next, you need to show that the second-order conditions for a maximum are satisfied. • Set up the Hessian matrix of second partials at ab, bß and sc2 . • Show that the own-partials are negative at ab, bß and sc2 . • Show that the determinant of the Hessian is negative at ab, bß and sc XXXXXXXXXXpoints) Define the information matrix as the negative of the inverse of the Hessian matrix at ab, bß and sc2 . Show that the square roots of the information matrix’s diagonal elements imply that: std. error ab = sb s 1 n ?x 2 i ?(xi -x) 2 std. error bß = sb s 1 ?(xi -x) 2 std. error sc2 = sc2 r 2 N XXXXXXXXXXpoints) Use what you have learned about optimization to explain why our estimate of a parameter’s standard error is smaller when the log-likelihood surface comes to a sharp peak along its dimension.

Robert · Accepted Answer

1.  The regression model is given by
                                
iii
yx
abe
=-+
where 
2
(0,)
i
N
es
:
The likelihood function is given by
                         
2
2
2
2
()
1
(,,)exp
2
2
ii
yx
L
ab
abs
s
ps
æö
--
÷
ç
÷
=-
ç
÷
ç
÷
ç
èø
Õ
Then, the log-likelihood function is given by
                
2222
2
1
(,,)log(,,)log(2)log()
222
ii
i
nn
lLyx
absabspsab
s
==----
å
The first order conditions are given by
          
2
2
1
(,,)()
ii
lyx
absab
as
¶
=--
¶
å
2
(,,)0()0
ii
lyx
absab
a
¶
=Þ--=
¶
å
     
                              =>   
ii
ynx
ab
=+
åå
       (1)
2
2
1
(,,)()
iii
lyxx
absab
bs
¶
=--
¶
å
2
(,,)0()0
iii
lyxx
absab
b
¶
=Þ--=
¶
å
                  (2)
  
22
224
1
(,,)()
22
ii
i
n
lyx
absab
sss
¶
=-+--
¶
å
   
22
224
11
(,,)0()0
22
ii
i
lyx
absab
sss
¶
=Þ-+--=
¶
å
2.       
Putting the value of 
a
 from (1) in (2) we get
                      
2
(//)
iiiiii
yxynxnxx
bb
=-+
ååååå
                     => 
22
(())
iiiiii
nyxyxnxx
b
-=-
ååååå
                     => 
22
()
iii
yxnyxxnx
b
-=-
åå
   where 
1
i
xx
N
=
å
, 
1
i
yy
N
=
å
                     => 
2
()()()
iii
yyxxxx
b
--=-
åå
                    => 
2
()()
ˆ
()
ii
i
yyxx
xx
b
--
=
-
å
å
Putting the value of 
ˆ
b
 in (1) we get
                       
ˆ
ˆ
yx
ab
=-
Again,
       
22
224
11
(,,)0()0
22
ii
i
lyx
absab
sss
¶
=Þ-+--=
¶
å
                           
22
1
ˆ
ˆˆ
()
ii
i
yx
N
sab
Þ=--
å
3.
The second-order conditions are
          
2
22
(,,)
N
l
abs
as
¶
=-
¶
2
22
(,,)
i
x
Nx
l
abs
abss
¶
=-=-
¶¶
å
     
2
24
1
(,,)()
ii
lyx
absab
ass
¶
=---
¶¶
å
22
22
1
(,,

Directions: Think of this exam as a game of “Jeopardy!” (the TV show). It gives you the answers to the maximization problem. You have to find the questions that lead to those answers. The tasks below...

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