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1   Let y, X, β and u be a n × 1 vector, a n × k matrix, a k × 1 vector and a n × 1 vector, respectively, where X is nonstochastic, u is an error term, and β is a parameter.

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Homework (Due: July 28, 2016 AM10:20)

1   Let y, X, β and u be a n × 1 vector, a n × k matrix, a k × 1 vector and a n × 1 vector, respectively, where X is nonstochastic, u is an error term, and β is a parameter.

Consider the regression model: y = + u, where E(u) = 0 and V(u) = σ

2

I

n

are assumed.

(1) Let β b be the OLS estimator. Show that β b is a consistent estimator of β.

(2) As n goes to infinity, derive the distribution of 1

n X

0

u. What is the assumption(s) for the derivatiion?

(3) As n goes to infinity, derive the distribution of

n( β b β).

2   Suppose that X

1

, X

2

, · · · , X

n

are mutually independent. The density function of X

i

is given by f (x

i

; θ), where θ is the parameter to be estimated. For simplicity of discussion, suppose that θ is a scalar (i.e., 1 × 1).

(4) Construct the likelihood function of θ, denoted by L(θ).

(5) Let ˆ θ be an unbiased estimator of θ. Show the following inequality:

V(ˆ θ) 1 I(θ) ,

where I(θ) = E

(

2

log L(θ)

∂θ

2

)

.

(6) Let ˜ θ be the maximum likelihood estimator of θ. Derive the distribution of

n(˜ θ θ) as n goes

to infinity. What is the assumption(s) for the derivatiion?

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