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Statistical Modelling Exam 03/09/2024 Exercise 1 Consider the following models, for i = 1,...,n 1. Y₁ = B1+ẞ2x1,2 + ẞ3 log10 £1,3 + ẞ4x²+; and ε; ~ N(0,0²) independent. 2. Y₁ = 3. log(Y)= = ẞ1 + B21,2 B3xi,1 Bi1B3 log(1,3) I1,2 + and E~N(0,0²) independent. +; and E~N(0,0²) independent. 4. Y=1,2 exp{i} and ~ N(0, 1) independent. Answer the following questions: a) For each model, indicate whether it is a linear regression model. If it is not, explain why and whether it can be expressed in the form Y₁ = B₁₁,1 + B₂xi,2 ++ ẞpi,p + ɛ; by a suitable transformation and write explicitly such transformation. b) Consider model 4 appropriately transformed, denoting with Y*, 2, (3, 3) and the transformed quantities. Express it in the matrix form Y* = x**+*, explicitly stating Y(and its distribution), X*, *, and *. c) Write the expression of the maximum likelihood estimator 3* and its exact distribution. d) Let e = y* - X** be the vector of the residuals. State which of the following identities are satisfied and motivate the answer: 12 72 Σε =0 i=1 Σe; log(1,2) = 0 i=1 72 Σεπ1,2 = 0 i=1 Π Σe; log(2) = 0 i=1

Statistical Modelling Exam 03/09/2024 Exercise 1 Consider the following models, for i = 1,...,n 1. Y₁ = B1+ẞ2x1,2 + ẞ3 log10 £1,3 + ẞ4x²+; and ε; ~ N(0,0²) independent. 2. Y₁ = 3. log(Y)= = ẞ1 + B21,2 B3xi,1 Bi1B3 log(1,3) I1,2 + and E~N(0,0²) independent. +; and E~N(0,0²) independent. 4. Y=1,2 exp{i} and ~ N(0, 1) independent. Answer the following questions: a) For each model, indicate whether it is a linear regression model. If it is not, explain why and whether it can be expressed in the form Y₁ = B₁₁,1 + B₂xi,2 ++ ẞpi,p + ɛ; by a suitable transformation and write explicitly such transformation. b) Consider model 4 appropriately transformed, denoting with Y*, 2, (3, 3) and the transformed quantities. Express it in the matrix form Y* = x**+*, explicitly stating Y(and its distribution), X*, *, and *. c) Write the expression of the maximum likelihood estimator 3* and its exact distribution. d) Let e = y* - X** be the vector of the residuals. State which of the following identities are satisfied and motivate the answer: 12 72 Σε =0 i=1 Σe; log(1,2) = 0 i=1 72 Σεπ1,2 = 0 i=1 Π Σe; log(2) = 0 i=1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 9T
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For context, the image provided below is a quesion from a Sepetember, 2024 past paper in statistical modeling 

Statistical Modelling Exam 03/09/2024 Exercise 1 Consider the following models, for i = 1,...,n 1. Y₁ = B1+ẞ2x1,2 + ẞ3 log10 £1,3 + ẞ4x²+; and ε; ~ N(0,0²) independent. 2. Y₁ = 3. log(Y)= = ẞ1 + B21,2 B3xi,1 Bi1B3 log(1,3) I1,2 + and E~N(0,0²) independent. +; and E~N(0,0²) independent. 4. Y=1,2 exp{i} and ~ N(0, 1) independent. Answer the following questions: a) For each model, indicate whether it is a linear regression model. If it is not, explain why and whether it can be expressed in the form Y₁ = B₁₁,1 + B₂xi,2 ++ ẞpi,p + ɛ; by a suitable transformation and write explicitly such transformation. b) Consider model 4 appropriately transformed, denoting with Y*, 2, (3, 3) and the transformed quantities. Express it in the matrix form Y* = x**+*, explicitly stating Y(and its distribution), X*, *, and *. c) Write the expression of the maximum likelihood estimator 3* and its exact distribution. d) Let e = y* - X** be the vector of the residuals. State which of the following identities are satisfied and motivate the answer: 12 72 Σε =0 i=1 Σe; log(1,2) = 0 i=1 72 Σεπ1,2 = 0 i=1 Π Σe; log(2) = 0 i=1
Transcribed Image Text:Statistical Modelling Exam 03/09/2024 Exercise 1 Consider the following models, for i = 1,...,n 1. Y₁ = B1+ẞ2x1,2 + ẞ3 log10 £1,3 + ẞ4x²+; and ε; ~ N(0,0²) independent. 2. Y₁ = 3. log(Y)= = ẞ1 + B21,2 B3xi,1 Bi1B3 log(1,3) I1,2 + and E~N(0,0²) independent. +; and E~N(0,0²) independent. 4. Y=1,2 exp{i} and ~ N(0, 1) independent. Answer the following questions: a) For each model, indicate whether it is a linear regression model. If it is not, explain why and whether it can be expressed in the form Y₁ = B₁₁,1 + B₂xi,2 ++ ẞpi,p + ɛ; by a suitable transformation and write explicitly such transformation. b) Consider model 4 appropriately transformed, denoting with Y*, 2, (3, 3) and the transformed quantities. Express it in the matrix form Y* = x**+*, explicitly stating Y(and its distribution), X*, *, and *. c) Write the expression of the maximum likelihood estimator 3* and its exact distribution. d) Let e = y* - X** be the vector of the residuals. State which of the following identities are satisfied and motivate the answer: 12 72 Σε =0 i=1 Σe; log(1,2) = 0 i=1 72 Σεπ1,2 = 0 i=1 Π Σe; log(2) = 0 i=1
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