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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 10

Question

Math 6010 • Linear Models Spring 2009 • Question 10

Consider the simple linear regression model \(Y_i=\beta _0+\beta _1x_i+\epsilon _i\) (\(i=1,2,\ldots ,n\)). Assume independent \(N(0,\sigma ^2)\) errors.

(a) Find the least squares estimators \(\hat \beta _0\) of \(\hat \beta _1\) of \(\beta _0\) and \(\beta _1\).

(b) Calculate the covariance matrix of \((\hat \beta _0,\hat \beta _1)\).

Raw LaTeX

Consider the simple linear regression model
\(Y_i=\beta_0+\beta_1x_i+\epsilon_i\) (\(i=1,2,\ldots,n\)). Assume independent
\(\N(0,\sigma^2)\) errors.

\textbf{(a)} Find the least squares estimators \(\hat\beta_0\) of \(\hat\beta_1\)
of \(\beta_0\) and \(\beta_1\).

\textbf{(b)} Calculate the covariance matrix of \((\hat\beta_0,\hat\beta_1)\).

Topic Tags

Covariance and Independence Estimator Distributions Generalized Least Squares Linear Models