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2020/05/14

Econometrics I’s Homework

Deadline: May 20, 2020, PM23:59:59

• The answer should be written in English or Japanese.

• Your name and student ID number should be included in your answer sheet.

• Send your answer to the email address: tanizaki@econ.osaka-u.ac.jp.

• The subject should be Econome 1 or

計量

1. Otherwise, your mail may go to the trash box.

１

Consider the following regression model:

y = Xβ + u

where y, X , β and u denote T × 1, T ×k, k ×1 and T ×1 matrices. k and T are the numberof explanatory variables and the sample size. u

1

, u

2

, · · ·, u

T

are mutually independently and normally distributed with mean zero and variance σ

2

, i.e., u ∼ N (0, σ

2

I

T

). β is a vector of unknown parameters to be estimated.

Let ˆ β be the ordinary least squares estimator of β.

(1) Show that s

2

= 1

T − k (y − X β) ˆ

0

(y − X β ˆ ) is an unbiased estimator of σ

2

. (2) Show that (T − k)s

2

σ

2

is distributed as a χ

2

random variavble with T − k degrees of freedom.

(3) Show that ˆ β is a best linear unbiased estimator.

参照

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