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Consider the following multiple linear regression model y = XB+u where y is nx1, X is nxk, and u is nx1 such that ulx~N(0,0²In). Write y = ŷ + û, where ŷ = XÂ is the least squares predicted value. a. Show that (B- B) = Au and û = Mu, what is your A and M? b. Show that y = the mean of the predicted values y C. Show that X'û = 0, y'û = 0 d. Derive R² for the model where the first column of X has a constant.

Consider the following multiple linear regression model y = XB+u where y is nx1, X is nxk, and u is nx1 such that ulx~N(0,0²In). Write y = ŷ + û, where ŷ = XÂ is the least squares predicted value. a. Show that (B- B) = Au and û = Mu, what is your A and M? b. Show that y = the mean of the predicted values y C. Show that X'û = 0, y'û = 0 d. Derive R² for the model where the first column of X has a constant.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Solve part c and d. Thank you.

### Multiple Linear Regression Model Consider the following multiple linear regression model: \[ y = X \beta + u \] where \( y \) is \( n \times 1 \), \( X \) is \( n \times k \), and \( u \) is \( n \times 1 \) such that \( u \sim \mathcal{N}(0, \sigma^2 I_n) \). Write \( y = \hat{y} + \hat{u} \), where \( \hat{y} = X \hat{\beta} \) is the least squares predicted value. #### Tasks: a. Show that \( (\hat{\beta} - \beta) = Au \) and \( \hat{u} = Mu \). What are your \( A \) and \( M \)? b. Show that \( \bar{y} = \text{the mean of the predicted values } \hat{y} \). c. Show that \( X' \hat{u} = 0, \hat{y}' \hat{u} = 0 \). d. Derive \( R^2 \) for the model where the first column of \( X \) has a constant. #### Detailed Explanations and Graphs Currently, the problem does not include any graphs or diagrams. The steps provided involve algebraic manipulations and proofs which are generally solved through sequential steps. Here's an initial walk-through of the tasks: ##### Part (a) To show \( (\hat{\beta} - \beta) = Au \), and \( \hat{u} = Mu \): - Solve for \( \hat{\beta} \) which is obtained via the Ordinary Least Squares (OLS) estimator. - Demonstrate the residual properties and derive matrices \( A \) and \( M \). ##### Part (b) To show \( \bar{y} \): - Use properties of the OLS estimators and mean properties. ##### Part (c) To demonstrate orthogonality: - Use the normal equations derived from the OLS estimator properties. ##### Part (d) To derive \( R^2 \): - Use the definition of \( R^2 \) in the context of the regression model. This problem enables an understanding of the multiple linear regression model and associated properties, specifically focusing on derivations involving estimators and their residuals.
Transcribed Image Text:### Multiple Linear Regression Model Consider the following multiple linear regression model: \[ y = X \beta + u \] where \( y \) is \( n \times 1 \), \( X \) is \( n \times k \), and \( u \) is \( n \times 1 \) such that \( u \sim \mathcal{N}(0, \sigma^2 I_n) \). Write \( y = \hat{y} + \hat{u} \), where \( \hat{y} = X \hat{\beta} \) is the least squares predicted value. #### Tasks: a. Show that \( (\hat{\beta} - \beta) = Au \) and \( \hat{u} = Mu \). What are your \( A \) and \( M \)? b. Show that \( \bar{y} = \text{the mean of the predicted values } \hat{y} \). c. Show that \( X' \hat{u} = 0, \hat{y}' \hat{u} = 0 \). d. Derive \( R^2 \) for the model where the first column of \( X \) has a constant. #### Detailed Explanations and Graphs Currently, the problem does not include any graphs or diagrams. The steps provided involve algebraic manipulations and proofs which are generally solved through sequential steps. Here's an initial walk-through of the tasks: ##### Part (a) To show \( (\hat{\beta} - \beta) = Au \), and \( \hat{u} = Mu \): - Solve for \( \hat{\beta} \) which is obtained via the Ordinary Least Squares (OLS) estimator. - Demonstrate the residual properties and derive matrices \( A \) and \( M \). ##### Part (b) To show \( \bar{y} \): - Use properties of the OLS estimators and mean properties. ##### Part (c) To demonstrate orthogonality: - Use the normal equations derived from the OLS estimator properties. ##### Part (d) To derive \( R^2 \): - Use the definition of \( R^2 \) in the context of the regression model. This problem enables an understanding of the multiple linear regression model and associated properties, specifically focusing on derivations involving estimators and their residuals.
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