Consider a simple linear regression model which satisfies Gauss Markov assumptions Y = f0 + fi1 X + u

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Ql. [26 points]

Consider a simple linear regression (SLR) model which satisfies Gauss Markov assumptions:

Y = f0 + fi1 X + u, where the error term u has an unconditional variance of a.2.

Let ì be a linear estimator of ß1, i.e. can be expressed as  wY1, where n is the sample size and the w coefficients are functions of sample data of the independent variable X. Let n 3 in this question for simplicity.

(a) (6 points) Derive an expression for the conditional expectation of . What condition(s) must w1, w2, and w3 satisfy for  to be an unbiased estimator? (sec note below)

(b) (6 points) Derive an expression for the conditional variance of . (see note below)

(c) (2 points) Prove the identity 2X — X — Xk = 3 (X — 2), where I can be 1, 2, or 3, and k will take on the other two values. For example, if j = I, then j and k take on values of 2 and 3, hence 2X1 — X2 — X3 = 3 (X1 —2). Or if j = 2, then] and k take on values of 1 and 3, hence 2X2 — X1 — X3 = 3(X2—X).

(d) (12 points) Make use of your answer in parts (a), (b), and (c), find the formula of w1, w2, and w3 that makes  the estimator with the minimum variance among all linear unbiased estimators. Try your best to simplify the expressions for w1, w2, and w3.

Hint for part (d): This is essentially an optimization problem under constraint in which you want to minimize the conditional variance under the constraint that the linear estimator is unbiased. Use of basic calculus is needed (e.g. =k X’). Although you will encounter some long expressions in the process of solving this problem, you will discover that many terms cancel out and the solution has a rather simple structure. Be patient.

Note: “Conditional” here means conditional on sample data of the independent variable X.

Please identify which Gauss Markov assumptions are needed where appropriate in your answer.

Attachments:

Related Questions

. The fundamental operations of create, read, update, and delete (CRUD) in either Python or Java

CS 340 Milestone One Guidelines and Rubric  Overview: For this assignment, you will implement the fundamental operations of create, read, update,

. Develop a program to emulate a purchase transaction at a retail store. This  program will have two classes, a LineItem class and a Transaction class

Retail Transaction Programming Project  Project Requirements:  Develop a program to emulate a purchase transaction at a retail store. This

. The following program contains five errors. Identify the errors and fix them

7COM1028   Secure Systems Programming   Referral Coursework: Secure

. Accepts the following from a user: Item Name Item Quantity Item Price Allows the user to create a file to store the sales receipt contents

Create a GUI program that:Accepts the following from a user:Item NameItem QuantityItem PriceAllows the user to create a file to store the sales receip

. The final project will encompass developing a web service using a software stack and implementing an industry-standard interface. Regardless of whether you choose to pursue application development goals as a pure developer or as a software engineer

CS 340 Final Project Guidelines and Rubric  Overview The final project will encompass developing a web service using a software stack and impleme