QUESTION 1 [25] Consider the model y₁ = B₁ + B₁₁₁ + B2 x 12 + ··· + Bp-1 x ₁₁p-1 +&,,i = 1,2,...,n, where the &, are independent, normally distributed random variables with mean 0 and variance σ². The model can be written in matrix form as Y = XB+ε. ) ) 1.1 Write down the elements of the matrix X and show that the least squares estimator of B is B (XX)XTY. 1.2 Consider BAY any linear estimator of B. (3. (i) Show that for ẞ to be an unbiased estimator of ẞ the condition AX = I must hold. (2 (ii) Prove that cov (ẞ') = σ² AA". (2) (iii) Prove the identity AAT = [(XX)'X] [(XX)¯'X']+[A-(XX)'X] x [A-(XX)"'X']". (5 ) Hint: Set (XX)"'X = B and use (b)(i). (iv) By making use of results (i), (ii) and (iii) prove that B = (XX)"'XY is the linear unbiased estimator of ẞ that has minimum variance. 1.3 Let = Y-XB. Show that SSE = " & => =YY-BXTY. (2) (4) 1.4 Suppose the hypotheses Ho: B₁ = B₁₂ to be tested. == B-10 versus H₁: Not all B's zero are (i) Explain why the error sum of squares under the assumption that His true is given by SSEH =(y-5)². i-l (2) (ii) By combining the results in part(c) and d(i), explain how an F-statistic for testing these hypotheses can be constructed. Summarize your results in the form of an Analysis of Variance (ANOVA) table. (5)
QUESTION 1 [25] Consider the model y₁ = B₁ + B₁₁₁ + B2 x 12 + ··· + Bp-1 x ₁₁p-1 +&,,i = 1,2,...,n, where the &, are independent, normally distributed random variables with mean 0 and variance σ². The model can be written in matrix form as Y = XB+ε. ) ) 1.1 Write down the elements of the matrix X and show that the least squares estimator of B is B (XX)XTY. 1.2 Consider BAY any linear estimator of B. (3. (i) Show that for ẞ to be an unbiased estimator of ẞ the condition AX = I must hold. (2 (ii) Prove that cov (ẞ') = σ² AA". (2) (iii) Prove the identity AAT = [(XX)'X] [(XX)¯'X']+[A-(XX)'X] x [A-(XX)"'X']". (5 ) Hint: Set (XX)"'X = B and use (b)(i). (iv) By making use of results (i), (ii) and (iii) prove that B = (XX)"'XY is the linear unbiased estimator of ẞ that has minimum variance. 1.3 Let = Y-XB. Show that SSE = " & => =YY-BXTY. (2) (4) 1.4 Suppose the hypotheses Ho: B₁ = B₁₂ to be tested. == B-10 versus H₁: Not all B's zero are (i) Explain why the error sum of squares under the assumption that His true is given by SSEH =(y-5)². i-l (2) (ii) By combining the results in part(c) and d(i), explain how an F-statistic for testing these hypotheses can be constructed. Summarize your results in the form of an Analysis of Variance (ANOVA) table. (5)
Chapter1: Financial Statements And Business Decisions
Section: Chapter Questions
Problem 1Q
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![QUESTION 1 [25]
Consider the model y₁ = B₁ + B₁₁₁ + B2 x 12 + ··· + Bp-1 x ₁₁p-1 +&,,i = 1,2,...,n, where the
&, are independent, normally distributed random variables with mean 0 and variance σ².
The model can be written in matrix form as Y = XB+ε.
)
)
1.1 Write down the elements of the matrix X and show that the least squares estimator of
B is B (XX)XTY.
1.2 Consider BAY any linear estimator of B.
(3.
(i) Show that for ẞ to be an unbiased estimator of ẞ the condition AX = I must hold. (2
(ii) Prove that cov (ẞ') = σ² AA".
(2)
(iii) Prove the identity
AAT = [(XX)'X] [(XX)¯'X']+[A-(XX)'X] x [A-(XX)"'X']".
(5
)
Hint: Set (XX)"'X = B and use (b)(i).
(iv) By making use of results (i), (ii) and (iii) prove that B = (XX)"'XY is the linear
unbiased estimator of ẞ that has minimum variance.
1.3 Let = Y-XB. Show that SSE = " & => =YY-BXTY.
(2)
(4)
1.4 Suppose the hypotheses Ho: B₁ = B₁₂
to be tested.
==
B-10 versus H₁: Not all B's zero are
(i) Explain why the error sum of squares under the assumption that His true is given by
SSEH
=(y-5)².
i-l
(2)
(ii) By combining the results in part(c) and d(i), explain how an F-statistic for testing these
hypotheses can be constructed. Summarize your results in the form of an Analysis of
Variance (ANOVA) table.
(5)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe465c21d-7ae0-4b51-9640-0eb1732fa2d3%2F3038e22a-de2b-4619-8279-b2476e91e30a%2Fb3nvdpf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:QUESTION 1 [25]
Consider the model y₁ = B₁ + B₁₁₁ + B2 x 12 + ··· + Bp-1 x ₁₁p-1 +&,,i = 1,2,...,n, where the
&, are independent, normally distributed random variables with mean 0 and variance σ².
The model can be written in matrix form as Y = XB+ε.
)
)
1.1 Write down the elements of the matrix X and show that the least squares estimator of
B is B (XX)XTY.
1.2 Consider BAY any linear estimator of B.
(3.
(i) Show that for ẞ to be an unbiased estimator of ẞ the condition AX = I must hold. (2
(ii) Prove that cov (ẞ') = σ² AA".
(2)
(iii) Prove the identity
AAT = [(XX)'X] [(XX)¯'X']+[A-(XX)'X] x [A-(XX)"'X']".
(5
)
Hint: Set (XX)"'X = B and use (b)(i).
(iv) By making use of results (i), (ii) and (iii) prove that B = (XX)"'XY is the linear
unbiased estimator of ẞ that has minimum variance.
1.3 Let = Y-XB. Show that SSE = " & => =YY-BXTY.
(2)
(4)
1.4 Suppose the hypotheses Ho: B₁ = B₁₂
to be tested.
==
B-10 versus H₁: Not all B's zero are
(i) Explain why the error sum of squares under the assumption that His true is given by
SSEH
=(y-5)².
i-l
(2)
(ii) By combining the results in part(c) and d(i), explain how an F-statistic for testing these
hypotheses can be constructed. Summarize your results in the form of an Analysis of
Variance (ANOVA) table.
(5)
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