Intro. Math. Statistics & Econometrics
1
Nee Yildiz
s
Problem Set 8 - Solution
1. log (wage) = 0 + 1 married + 2 educ + z + u and E (u|married, educ, z ) = 0.
(a) The above equation means that wage = eu e0 +1 married+2 educ+z . Since u is
independent of a
Intro. Econometric Theory (Fall 06/07)
1-1
Nee Yildiz
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Midterm Exam
Note questions 1 through 4 are taken from Introduction to Statistics and Econometrics,
Amemiya, T., Harvard University Press (1994).
1. Suppose X and Y are independent Bernoulli random v
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
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Problem Set 2
Due September 22
1. Suppose X and Y are independent random variables with the same distribution
function, F ().
(i) Find the distribution functions of the random variables: Z1 = maxcfw_
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
s
Problem Set 8
Due December 5
1. log wage = 0 + 1 married + 2 educ + z + u, E (u|married, educ, z ) = 0 where
z contains factors other than marital status and education that can aect wage.
When 1 is s
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
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Problem Set 1
Due September 14
1. (Amemiya, Ch. 2, Ex. 10) A die is rolled successively until the ace turns up. How
many rolls are necessary before the probability is at least 0.5 that the ace will t
Intro. Econometric Theory (Fall 08/09)
1-1
Nee Yildiz
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Problem Set 1 - Solution
Due September 14
1. With n rolls, the probability of getting the ace at least once is 1
number we want is the smallest n satisfying
1
5
6
n
1
1
2
2
5
6
5n
.
6
The
n
.
Thus t
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
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Midterm Solution
Note question 4 is taken from Introduction to Statistics and Econometrics , Amemiya,
T., Harvard University Press (1994), and question 3 is taken from Casella, G. and R. L.
Berger (2
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
s
Problem Set 9
Due December 12
1. Consider the system
yi1 = xi1 1 + ui1
yi2 = xi2 2 + ui2 ,
with the instrument matrix
Zi =
ziT 0
1
0 ziT
2
,
which is a 2 (L1 + L2 ) dimensional matrix, so that E [zi1
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
s
Problem Set 7
Due November 14
1. (Amemiya, Ch. 9, Ex. 2) Suppose X has the following probability distribution:
P (X = 1) = , P (X = 2) = 2, and P (X = 3) = 1 3, where [0, 1/3]. We
are to test H0 : =
Intro. Math. Statistics & Econometrics
1-1
Problem Set 6
Due October 27
1. Solve Exercise 3 of Chapter 8 of Amemiya (1994).
2. Solve Exercise 6 of Chapter 8 of Amemiya (1994).
3. Solve Exercise 12 of Chapter 8 of Amemiya (1994).
4. Solve Exercise 14 of Ch
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
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Problem Set 5 - Solution
1. (Ch 7, Ex: 16) The likelihood function is given by L = n=1 1 exp (|Xi |).
i
2
(a) log L = n log 2 n=1 |Xi |. Thus, maximizer of the likelihood function
i
is the same as th
Intro. Math. Statistics & Econometrics
1-1
Problem Set 5
Due October 14
1. Solve Exercise 16 of Chapter 7 of Amemiya (1994).
2. Solve Exercise 17 of Chapter 7 of Amemiya (1994).
3. Solve Exercise 32 of Chapter 7 of Amemiya (1994).
4. Solve Exercise 36 of
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
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Problem Set 4 - Solution
1. (Ch 5, Ex: 1) The probability that an ace will turn up is 1/6. So X is Binomial(5, 1/6).
Then EX = 5/6 and V ar(X ) = 25/36. Moreover, P (X 4) = P (X = 4)+ P (X =
5!
5) =
Intro. Math. Statistics & Econometrics
1-1
Problem Set 4
Due October 6
1. Solve Exercise 1 of Chapter 5 of Amemiya (1994).
2. Solve Exercise 2 of Chapter 5 of Amemiya (1994).
3. Solve Exercise 5 of Chapter 5 of Amemiya (1994).
4. Solve Exercise 6 of Chapt
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
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Problem Set 3 - Solution
1. (Ch3, Ex: 8) P (U + V < y ) =
y
0
y v
0
ev eu dudv = 1 ey (1 + y ) = F (y )
f (y ) = yey
u
u+v
To get the joint density of X and Y , let (u, v ) =
1 (x, y ) =
D1 (x, y ) =
Intro. Math. Statistics & Econometrics
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Problem Set 3
Due September 29
1. Solve Exercise 8 of Chapter 3 of Amemiya (1994).
2. Solve Exercise 4 of Chapter 4 of Amemiya (1994).
3. Solve Exercise 11 of Chapter 4 of Amemiya (1994).
4. Solve Exercise 19 of
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
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Problem Set 2 - Solution
1. For each P (maxcfw_X, Y a) = P (X a, Y a). By independence of X and Y
this is equal to F 2 (a). Similarly, P (mincfw_X, Y a) = 1 P (mincfw_X, Y > a) =
1 P (X > a)P (Y >
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
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Final Exam Solution
1. (a) The log-likelihood function is given by
Xi
.
i
n log ()
The FOC is given by
n
+
i
Xi
2
Xi
.
n
= 0 =
i
The second derivative of the objective function evaluated at yields n
Intro. Math. Statistics & Econometrics
1-1
Nee Yildiz
s
Final Exam Solution
1. (a) The log-likelihood function is given by
Xi
.
i
n log ()
The FOC is given by
n
+
i
Xi
2
Xi
.
n
= 0 =
i
The second derivative of the objective function evaluated at yields n