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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.
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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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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,
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 ]
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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