计量经济学 答案

ˆ ,β ˆ = ∑u ˆ −β ˆ X ˆ i2 = ∑ Yi − β Minimize RSS β 1 2 i 1 2
i =1 i=1
(
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N
N
(
)
2
ˆ } {β j
(4 marks) • State the OLS normal equations.
(4 marks)
The first OLS normal equation can be written in any one of the following forms:
The exam consists of FIVE (5) questions. Students are required to answer ALL FIVE (5) questions. Answer all questions in the exam booklets provided. Be sure your name and student number are printed clearly on the front of all exam booklets used. Do not write answers to questions on the front page of the first exam booklet. Please label clearly each of your answers in the exam booklets with the appropriate number and letter. Please write legibly. The marks for each question are indicated in parentheses immediately above each question. Total marks for the exam equal 100.
(
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(1)
∂RSS = ˆ ∂β
2
ˆi ∑ 2u
i =1 N
N
∂u ˆi ˆ ∂β 2
=
ˆ i (− X i ) = ∑ 2u
i =1
N
ˆi −2 ∑ X i u
i =1
N
= −2 = −2
(4 marks)
ˆ −β ˆ X ) ∑ X i (Yi − β 1 2 i
i =1 N
(
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ˆ −β ˆ X ˆ i = Yi − β u 1 2 i
⇒
ˆi ∂u = −1 ˆ ∂β
1
and
ˆi ∂u = −X i . ˆቤተ መጻሕፍቲ ባይዱ∂β
2
N N ∂u ˆ N ∂RSS N ˆ −β ˆ X ˆ i i = ∑ 2u ˆ i (− 1) = −2 ∑ u ˆ i = −2 ∑ Yi − β = ∑ 2u 1 2 i ˆ ˆ ∂β1 i =1 i =1 i =1 ∂β1 i =1
ˆ ∑ X −β ˆ ∑ X2 = 0 ∑ i X i Yi − β 1 i i 2 i i ˆ ˆ − β1 ∑ i X i − β 2 ∑ i X i2 = − ∑ i X i Yi ˆ ∑ X +β ˆ ∑ X2 = ∑ X Y β 1 i i 2 i i i i i
(N2)
ECON 351 -- Fall Term 2003: Answers to Mid-Term Exam
Yi = β 1 + β 2 X i + u i
(1)
where Yi and Xi are observable variables, β 1 and β 2 are unknown (constant) regression coefficients, and ui is an unobservable random error term. The Ordinary Least Squares (OLS) sample regression equation corresponding to regression equation (1) is
MARKING: GOOD LUCK!
All questions pertain to the simple (two-variable) linear regression model for which the population regression equation can be written in conventional notation as:
ˆ −β ˆ X ˆ i = Yi − β since u 1 2 i
(2)
ˆ X −β ˆ X 2 ). ∑ (X i Yi − β 1 i 2 i
i =1
Step 2: Obtain the first-order conditions (FOCs) for a minimum of the RSS function by setting the partial derivatives (1) and (2) equal to zero and then dividing each equation by −2 and re-arranging:
ˆ and β ˆ that The OLS coefficient estimators are those formulas or expressions for β 1 2 minimize the sum of squared residuals RSS for any given sample of size N. The OLS estimation criterion is therefore:
ECON 351 -- Fall Term 2003: Answers to Mid-Term Exam
Page 1 of 16 pages
ECONOMICS 351
ANSWERS to Mid-Term Exam -- Fall 2003
Page 2 of 16 pages
(15 marks) 1. State the Ordinary Least Squares (OLS) estimation criterion. State the OLS normal equations. Derive the OLS normal equations from the OLS estimation criterion. ANSWER: (3 marks) • State the Ordinary Least Squares (OLS) estimation criterion. (3 marks)
(1) (2) (3)
V1 ~ χ2[m1] V2 ~ χ2[m2] V1 and V2 are independent.
Then the random variable F =
V1 m1 ~ F[m1, m2] V2 m 2
where F[m1, m2] denotes the F-distribution with m1 numerator degrees of freedom and m2 denominator degrees of freedom.
Page 1 of 16 pages
Fall Term 2003 DATE: TIME: INSTRUCTIONS:
MID-TERM EXAM: ANSWERS Tuesday October 28, 2003. 80 minutes; 1:00 p.m. - 2:20 p.m.
M.G. Abbott
ECONOMICS 351
ANSWERS to Mid-Term Exam -- Fall 2003 QUEEN'S UNIVERSITY AT KINGSTON Department of Economics ECONOMICS 351* - Fall Term 2003 Introductory Econometrics
ˆ −β ˆ ∑ X =0 ∑ i Yi − Nβ 1 2 i i ˆ −β ˆ ∑ X = −∑ Y − Nβ 1 2 i i i i ˆ +β ˆ ∑ X =∑ Y Nβ
1 2 i i i i
(N1)
The second OLS normal equation can be written in any one of the following forms:
Page 2 of 16 pages
ECONOMICS 351
Question 1 (continued)
ANSWERS to Mid-Term Exam -- Fall 2003
Page 3 of 16 pages
(8 marks) • Show how the OLS normal equations are derived from the OLS estimation criterion. (4 marks) ˆ ,β ˆ function with respect to β ˆ and β ˆ , using Step 1: Partially differentiate the RSS β 1 2 1 2
∂RSS =0 ˆ ∂β
1
ˆ −β ˆ X =0 ˆ i = 0 ⇒ − 2 ∑ i Yi − β ⇒ − 2 ∑i u 1 2 i ⇒
i i 1 2 i
( ˆ ∑ (Y − β
) ˆ X )= 0 −β
ˆ −β ˆ ∑ X =0 ⇒ ∑ i Yi − Nβ 1 2 i i ˆ +β ˆ ∑ X ⇒ ∑ i Yi = Nβ 1 2 i i
(
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(
(
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ˆ ∑ X +β ˆ ∑ X2 ⇒ ∑ i X i Yi = β 1 i i 2 i i
(N2)