Problem Set 1

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Kyoo il Kim Department of Economics University of Minnesota

Introduction to Econometrics, Econ4261 Spring 2013

P ROBLEM S ET 1 (D UE F EBRUARY 7, T HURSDAY IN CLASS )

Answer the Followings 1. Calculate: a) E[X], E[Y ] b) V ar[X], V ar[Y ] c) Cov[X, Y ] d) ρ(X, Y ) from the distribution below Y =1 X = 5000 X = 10, 000 X = 15, 000 0 1/8 1/3 Y =0 1/4 1/8 1/6 (1)

2. Suppose E[X] = 1 and E[Y ] = 2 and suppose X and Y are independent. Evaluate: a) E[2X + 1] b) E[X + Y ] c) E[X − 2Y ] d) E[XY + 1] 3. Suppose V ar[X] = 2, V ar[Y ] = 1, Cov[X, Y ] = 0. Evaluate: a) V ar[X + 2Y ] b) V ar[X − Y ] c) Cov[2X − Y, X − 1] ￿n

i=1 [Xi

4. Suppose c)

￿n

i=1 Xi

= 2 and

+ 2]

￿n

i=1 Yi

= 3. Evaluate: a)

￿n

i=1 [Xi

+ Yi ]

b)

￿n

i=1 [Xi

− 2Yi ]
￿

5. Let X = 1 with n = 100. What is

6. Show that V ar[X] = E[X 2 ] when E[X] = 0.

￿n

i=1 Xi ? What is

￿n

i=1 [Xi −X]? and What is

￿n

i=1

￿

X +1 ?

7. Show that Cov[X, Y ] = E[XY ] when E[X] = 0 or E[Y ] = 0. 8. Suppose V ar[X] = 1, V ar[Y ] = 2, V ar[Z] = 2, Cov[X, Y ] = 0, Cov[X, Z] = −1, and Cov[Y, Z] = 2. Obtain V ar[X − Y + 3Z] 1

9. Show that (a) (b)
￿n ￿n
i=1 (Xi i=1 (Xi

− X)2 =

− X)(Yi − Y ) =

￿n

2 i=1 Xi

10. Let b be an estimator of β. The Mean Squared Error is defined as M SE(b) = E[(b − β)2 ]. Show that M SE(b) = V ar(b) + bias2 (b) = E[(b − E[b])2 ] + (E[b] − β)2

￿n

−X

i=1 Xi Yi

￿n

i=1 Xi

=

−X

￿n

￿n

2 i=1 Xi

i=1 Yi

=

￿n

− nX

2

i=1 Xi Yi

− nXY

11. Do questions 2.6, 2.10, 2.15, 2.24 in Stock and Watson 12. Do questions 3.3, 3.6, 3.10, 3.13, 3.19 in Stock and Watson

2

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