Piecewise linear model: Let x be a known constant, and

Chapter 8, Problem 21SE

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Piecewise linear model: Let \(\tilde{x}\) be a known constant, and suppose that a dependent variable  is related to an independent variable \(x_1\) as follows:

          \(y= \begin{cases}\beta_{0}+\beta_{1} x_{1}+\varepsilon & \text { if } x_{1} \leq \tilde{x} \\ \beta_{0}^{*}+\beta_{1}^{*} x_{1}+\varepsilon & \text { if } x_{1}>\tilde{x}\end{cases}\)

In other words,  and \(x_1\) are linearly related, but different lines are appropriate depending on whether \(x_{1} \leq \tilde{x}\) or \(x_{1}>\tilde{x}\). Define a new independent variable \(x_2\) by

           \(x_{2}= \begin{cases}0 & \text { if } x_{1} \leq \tilde{x} \\ 1 & \text { if } x_{1}>\tilde{x}\end{cases}\)

Also define \(\beta_{2}=\beta_{0}^{*}-\beta_{0}\) and \(\beta_{3}=\beta_{1}^{*}-\beta_{1}\). Find a multiple regression model involving \(y, x_{1}, x_{2}, \beta_{0}, \beta_{1}, \beta_{2}, \text { and } \beta_{3}\) that expresses the relationship described here.

Equation Transcription:

Text Transcription:

tilde{x}

x_1

y=beta_0+beta_1x_1+varepsilon   if x_1{</=} tilde{x}

y=beta_0^*+beta_1^*x_1+varepsilon   if x_1>tilde{x}

x_1

x_1{</=}tilde{x}

x_1>tilde{x}

x_2

x_2=0   if x_1{</=}tilde{x}

x_2=1   if x_1>tilde{x}

beta_2=beta_0^*-beta_0

beta_3=beta_1^*-beta_1

y,x_1,x_2,beta_0,beta_1,beta_2 and beta_3