Problem 4. (Scaling) Consider the linear regression model in which Y = XB +€ under our standard assumptions with p – 1 predictors. Suppose that we scale the explanatory variables s that rij = kjwij, where i = 1,., n, j = 1,. p. By expressing X in terms of W, prove that Ý remain unchanged under this change of scale. How do the new coefficients BW with W; as predictors, compare t coefficients BX, with X as predictors?
Problem 4. (Scaling) Consider the linear regression model in which Y = XB +€ under our standard assumptions with p – 1 predictors. Suppose that we scale the explanatory variables s that rij = kjwij, where i = 1,., n, j = 1,. p. By expressing X in terms of W, prove that Ý remain unchanged under this change of scale. How do the new coefficients BW with W; as predictors, compare t coefficients BX, with X as predictors?
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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![**Problem 4. (Scaling)**
Consider the linear regression model in which
\[
Y = X\boldsymbol{\beta} + \boldsymbol{\varepsilon}
\]
under our standard assumptions with \(p - 1\) predictors. Suppose that we scale the explanatory variables so that \(x_{ij} = k_jw_{ij}\), where \(i = 1, ..., n\), \(j = 1, ..., p\). By expressing \(\mathbf{X}\) in terms of \(\mathbf{W}\), prove that \(\mathbf{Y}\) remains unchanged under this change of scale. How do the new coefficients \(\boldsymbol{\beta}^W\) with \(W_j\) as predictors compare to coefficients \(\boldsymbol{\beta}^X\), with \(\mathbf{X}\) as predictors?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d64a72c-ef4f-4748-9866-4f4da42ec416%2F9351ba0b-77dd-4e2a-bc2b-2c93345500c5%2Fk47k6s_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4. (Scaling)**
Consider the linear regression model in which
\[
Y = X\boldsymbol{\beta} + \boldsymbol{\varepsilon}
\]
under our standard assumptions with \(p - 1\) predictors. Suppose that we scale the explanatory variables so that \(x_{ij} = k_jw_{ij}\), where \(i = 1, ..., n\), \(j = 1, ..., p\). By expressing \(\mathbf{X}\) in terms of \(\mathbf{W}\), prove that \(\mathbf{Y}\) remains unchanged under this change of scale. How do the new coefficients \(\boldsymbol{\beta}^W\) with \(W_j\) as predictors compare to coefficients \(\boldsymbol{\beta}^X\), with \(\mathbf{X}\) as predictors?
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