Description

Assignment 2

1. For the multiple linear regression estimator, show that the matrix H is (i) symmetric; and (ii) idempotent.

2. Consider the multiple linear regression model with two independent variables, y = X +✏.
Suppose that e ~ N(0,2). The following data was observed.

x1 x2 y
4.49 2.92 -5.32
3.04 4.33 -9.24
3.94 4.27 -5.89
2.63 1.92 1.15
4.55 2.47 -1.47
3.88 2.36 1.91
2.92 3.21 -3.99
2.82 4.22 -6.82
3.17 1.80 1.49
2.91 2.35 -0.89

a. Compute the covariance for the least squares regression estimators. That is, find V(ˆ).
b. Using the above data, find the estimate of V(ˆ).
c. What is the variance for the 1st and 3rd residuals (hint: read section 4.2.2).?
d. What is the covariance between the first and third residual?
3. In the week 3 live lecture, we discussed an application from the Center for Radiative Shock Hydrodynamics. The data for the experiment are in the file
data_computer_experiment.csv on Canvas. Estimate a linear regression model for the experiment discussed in the live lecture.
a. What is the estimated variance for the least squares estimator of the regression coefficient for the thickness of the beryllium disk?
b. What is the estimated covariance between the least squares estimators of the regression coefficients for the thickness of the beryllium disk and the wall opacity?
c. Perform regression diagnostics on this model to answer the following questions:
i. Check the constant variance assumption for the errors.
ii. Check the normality assumption. iii. Check for large leverage points. iv. Check for outliers.
v. Check for influential points. vi. Check the structure of the relationship between the predictors and the response.