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1. (a) Suppose we have the simple linear model Yi = Bo + B1x; + ei, i = 1, ...,n, E[e;] = 0, var(e;) = o², cov(e;, e;) = 0, i + j. Write down the model using suitable matrix notation, and provide a condition for both Bo and B, to be linearly estimable. (b) Show that the least squares estimate of 31 under the model when B, = 0 is known sgven by β - Ση/ ΣL ή (c) Find the bias of Bị when in fact B, # 0. (d) Derive expressions for the variance of Bi and the variance of the least squares estimate B1 obtained when 3o is not known. Comment on which estimator has the smallest variance, whether or not this contradicts the Gauss-Markov Theorem.

1. (a) Suppose we have the simple linear model Yi = Bo + B1x; + ei, i = 1, ...,n, E[e;] = 0, var(e;) = o², cov(e;, e;) = 0, i + j. Write down the model using suitable matrix notation, and provide a condition for both Bo and B, to be linearly estimable. (b) Show that the least squares estimate of 31 under the model when B, = 0 is known sgven by β - Ση/ ΣL ή (c) Find the bias of Bị when in fact B, # 0. (d) Derive expressions for the variance of Bi and the variance of the least squares estimate B1 obtained when 3o is not known. Comment on which estimator has the smallest variance, whether or not this contradicts the Gauss-Markov Theorem.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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1. (a) Suppose we have the simple linear model Yi = Bo + Bix; + e;, i = 1,..., n, E[e:] = 0, var(e;) = o², cov(e;, e;) = 0, i j. Write down the model using suitable matrix notation, and provide a condition for both Bo and B, to be linearly estimable. (b) Show that the least squares estimate of 3, under the model when 3, = 0 is known is given by β - Σau/ ΣL (c) Find the bias of 3i when in fact Bo # 0. (d) Derive expressions for the variance of i and the variance of the least squares estimate B1 obtained when Bo is not known. Comment on which estimator has the smallest variance, whether or not this contradicts the Gauss-Markov Theorem.
Transcribed Image Text:1. (a) Suppose we have the simple linear model Yi = Bo + Bix; + e;, i = 1,..., n, E[e:] = 0, var(e;) = o², cov(e;, e;) = 0, i j. Write down the model using suitable matrix notation, and provide a condition for both Bo and B, to be linearly estimable. (b) Show that the least squares estimate of 3, under the model when 3, = 0 is known is given by β - Σau/ ΣL (c) Find the bias of 3i when in fact Bo # 0. (d) Derive expressions for the variance of i and the variance of the least squares estimate B1 obtained when Bo is not known. Comment on which estimator has the smallest variance, whether or not this contradicts the Gauss-Markov Theorem.
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