Homework 6 Solution

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This homework is to practice more on multiple linear regression. Attach the complete R codes for Problem 2 at the end of the homework. Total: 90 points.

  1. (25 points) Understanding the general linear regression model. For each of the following models, indicate whether we can use the techniques of multiple linear regression model to estimate the coe cients i’s or not. Explain. (Here, we assume that all Xi’s are non-random, and “i’s in each model are i.i.d. with E(“i) = 0 and Var(“i) = 2.)

    1. Yi = 0 + 1Xi1 + 2 log Xi2 + 3Xi21 + “i.

    1. Yi = log( 1Xi1) + 2Xi2 + “i, where 1 > 0; Xi1 > 0; 8i:

    1. Yi = log( 1 + Xi1) + 2Xi2 + “i, where 1 > 0; Xi1 > 0; 8i:

    2. Yi = exp( 0 + 1Xi1) + “i.

    3. Yi = exp( 0 + 1Xi1 + 2Xi1Xi2 + “i).

  1. (45 points) Data analysis: general testing framework. Consider the multiple linear regression

iid 2

model: Yi = 0 + 1Xi1 + 2Xi2 + 3Xi3 + 4Xi4 + i; i = 1; : : : ; n; “i N(0; ):

Describe how you would test (at 0.05 signi cance level):

  1. H0 : 1 = 2 = 0: vs. Ha : either 1 or 2 not equal to 0:

  2. H0 : 1 = 1; 2 = 2: vs. Ha : not both equalities in H0 holds.

  1. H0 : 2 = 3: vs. Ha : 2 6= 3.

Perform the test for the dataset ‘HW6Q2.txt”.

  1. (20 points) Rigorous deviation. Consider the multiple linear regression model in the matrix form Y = X + ” with E(“) = 0 and Var(“) = 2In. Let H = X(XT X) 1XT be the hat matrix. Show that

    1. HT =H,H2=H.

    1. The diagonal elements of H are all between 0 and 1.

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