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I am considering two linear models with model matrices related as follows:

\begin{equation*} \text{Model} \ 1 \\ \begin{bmatrix} \mu_1 \\ \mu_2 \\ \vdots \\ \mu_n \end{bmatrix} = \begin{bmatrix} 1 & x_{12} & x_{13} \\ 1 & x_{22} & x_{23} \\ \vdots & \vdots & \vdots \\ 1 & x_{n2} & x_{n3} \end{bmatrix} \begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \end{bmatrix} . \end{equation*}

\begin{equation*} \text{Model} \ 2 \\ \begin{bmatrix} \mu_1 \\ \mu_2 \\ \vdots \\ \mu_n \end{bmatrix} = \begin{bmatrix} 1 & cx_{12} + x_{13} & x_{13} \\ 1 & cx_{22} + x_{23}& x_{23} \\ \vdots & \vdots & \vdots \\ 1 & cx_{n2} + x_{n3} & x_{n3} \end{bmatrix} \begin{bmatrix} \beta^*_0 \\ \beta^*_1 \\ \beta^*_2 \\ \end{bmatrix} . \end{equation*}

$$y\sim N(\mu,I_n \sigma^2)$$

Under what conditions is $\frac{\hat\beta_1}{\hat\sigma_{\beta_1}} = \frac{\hat\beta^*_1}{\hat\sigma_{\beta^*_1}}$, and how can it be proven?

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