Discussion
In a multiple linear regression model \(y = X\beta + \varepsilon\), the covariance between y and \(\hat{y}\), the predicted value of y using least square estimator of \(\beta\) is :
Answer: A
The covariance is Σ X(X'X)−1X'.
The fitted values are \(\hat{y} = Hy\) with \(H = X(X'X)^{-1}X'\). Writing \(\Sigma = Cov(y) = Cov(\varepsilon)\),
\[ Cov(y, \hat{y}) = Cov(y, Hy) = \Sigma H' = \Sigma H \]
since H is symmetric. So the answer is \(\Sigma X(X'X)^{-1}X'\); with \(\Sigma = \sigma^{2}I\) this is \(\sigma^{2}H\).
\(\Sigma(I - H)\) is the covariance between y and the residuals, and \(\Sigma\) alone is the covariance of y with itself.
No comment is present. Be the first to comment.
Loading…
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development