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Regression - Exam 2005 - Statistics and Economics, Exams of Economic statistics

Professor Peter Clarkson, University of Kent, Statistics and Economics, Regression, Exam 2005, least squares estimates, FACULTY OF SCIENCE, TECHNOLOGY AND MEDICAL STUDIES, linear regression model, p-parameter linear regression model, forward selection, backward elimination.

Typology: Exams

2010/2011

Uploaded on 10/04/2011

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MA632/05
UNIVERSITY OF KENT AT CANTERBURY
FACULTY OF SCIENCE, TECHNOLOGY AND MEDICAL STUDIES
LEVEL I EXAMINATION
REGRESSION
Thursday, 26 May 2005: 2.00 4.00
This paper contains FIVE questions. Candidates must not
attempt more than THREE questions. All questions will be
marked out of 40.
Copies of the New Cambridge Statistical Tables are provided.
Approved calculators may be used.
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Download Regression - Exam 2005 - Statistics and Economics and more Exams Economic statistics in PDF only on Docsity!

UNIVERSITY OF KENT AT CANTERBURY

FACULTY OF SCIENCE, TECHNOLOGY AND MEDICAL STUDIES

LEVEL I EXAMINATION

REGRESSION

Thursday, 26 May 2005: 2.00 – 4.

This paper contains FIVE questions. Candidates must not attempt more than THREE questions. All questions will be marked out of 40.

Copies of the New Cambridge Statistical Tables are provided. Approved calculators may be used.

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  1. (a) Thirteen specimens of 90/10 Cu-Ni alloys, each with specific iron content (X), were tested in a corrosion-wheel setup. The wheel was rotated in salt water at 30ft/sec for 60 days. The corrosion (Y ) was measured as weight loss in milligrams/square decimetre/day, giving the following data:

X Y 0.01 127. 0.48 124. 0.71 110. 0.95 103. 1.19 101. 0.01 130. 0.48 122. 1.44 92. 0.71 113. 1.96 83. 0.01 128. 1.44 91. 1.96 86.

Assume these data follow a simple linear regression model:

Yi = α + βXi + ei

where the random error terms, ei, are independent N (0, σ^2 ) random variables. In answering the following questions you may quote the relevant formulae without derivation.

(i) Calculate the least squares estimates of the slope and intercept of the regression line. [ 8 marks ] (ii) Calculate an estimate of σ^2 , and find 95% confidence intervals for the slope and intercept of the regression line. [ 12 marks ] (b) Suppose we have the following samples (Xi 1 , Xi 2 , Yi), i = 1, 2 , 3 , 4, from (X 1 , X 2 , Y ),

X 1 X 2 Y 1 1 3. 1 -1 -0. -1 0 -0. -1 0 -0.

We assume the linear model

Yi = α 0 + α 1 Xi 1 + α 2 Xi 2 + ei,

  1. Consider the matrix representation of a p-parameter linear regression model (p > 0)

Y = Xθ + e,

where Y is the response vector for the n observations, X is the n × p matrix of explanatory variables, θ is the parameter vector and e is a vector of independent and identically distributed error terms, each with the N (0, σ^2 ) distribution. X is full rank. The least squares estimator of θ is ˆθ = (XT^ X)−^1 XT^ Y. (a) (i) Find E(θˆ) and var(ˆθ). [ 10 marks ] (ii) Let P = X(XTX)−^1 XT. Express the residual sum squares (RSS) in terms of Y and P, and find E(RSS). [ 10 marks ] (b) Suppose that the parameters are partitioned into two subsets of size p 1 and p 2 (p 1 > 0 , p 2 > 0), such that Xθ = X 1 θ 1 + X 2 θ 2 where X 1 is an n × p 1 matrix, X 2 is an n × p 2 matrix, and θ 1 and θ 2 are vectors of dimension p 1 and p 2 respectively. (i) If θ 2 is now omitted from the model, show that θˆ 1 , the least squares estimator of θ 1 , is still unbiased if XT 1 X 2 = 0. [ 10 marks ] (ii) If θ 2 is now omitted from the model, what is the residual sum squares? Represent the difference between this residual sum squares and the residual sum squares for the full model in terms of Y, P and P 1 , where P 1 = X 1 (XT 1 X 1 )−^1 XT 1. [ 10 marks ]

  1. (a) Draw flow diagrams for the forward selection and backward elimination methods of selecting a subset of explanatory variables in multiple regression. [ 16 marks ] (b) The response variable y is thought to be linearly related to a subset of the four explana- tory variables x 1 , x 2 , x 3 , x 4. The results of fitting all possible regressions to a set of 40 observations are summarized below.

Explanatory Residual Explanatory Residual variables Sum of Squares variables Sum of Squares

None 160 · 8 x 2 , x 3 58 · 9 x 1 160 · 7 x 2 , x 4 77 · 5 x 2 83 · 3 x 3 , x 4 58 · 5 x 3 64 · 2 x 1 , x 2 , x 3 58 · 2 x 4 160 · 4 x 1 , x 2 , x 4 77 · 5 x 1 , x 2 83 · 3 x 1 , x 3 , x 4 55 · 5 x 1 , x 3 62 · 7 x 2 , x 3 , x 4 51 · 9 x 1 , x 4 160 · 2 x 1 , x 2 , x 3 , x 4 50 · 2

Use the methods of forward selection and backward elimination, with the critical value F = 3 at each step, to select subsets of the explanatory variables. [ 24 marks ]

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