Last updated: 2018-08-23

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Suppose we have model \[ \mathbf{y} = \mathbf{x}\beta + \epsilon \quad \epsilon \sim N(0,\sigma^2) \]

From simple linear regression, the estimations are \[ \hat{\beta} = \frac{\mathbf{x}^{T}\mathbf{y}}{\mathbf{x}^{T}\mathbf{x}} \quad \hat{\sigma}^2 = \frac{(\mathbf{y} - \mathbf{x}\hat{\beta})^{T}(\mathbf{y} - \mathbf{x}\hat{\beta})}{n-1} \quad s^{2} = \frac{\hat{\sigma}^2}{\mathbf{x}^{T}\mathbf{x}} \] \[ z = \frac{\mathbf{x}^{T}\mathbf{y}}{\hat{\sigma}\sqrt{\mathbf{x}^{T}\mathbf{x}}} \]

If we scale the \(\mathbf{y}\) and \(\mathbf{x}\) to different variance separately \[ \mathbf{y}_s = \alpha_{y}\mathbf{y} \quad \mathbf{x}_s = \alpha_{x}\mathbf{x} \]

Then \[ \hat{\beta}_{s} = \frac{\alpha_{y}\mathbf{x}^{T}\mathbf{y}}{\alpha_{x}\mathbf{x}^{T}\mathbf{x}} \]

\[ \hat{\sigma}_{s}^{2} = \frac{(\mathbf{y}_s - \mathbf{x}_s\hat{\beta}_s)^{T}(\mathbf{y}_s - \mathbf{x}_s\hat{\beta}_s)}{n-1} = \alpha_{y}^{2} \hat{\sigma}^2 \] The scaled z score is \[ z_s = \frac{\mathbf{x}_{s}^{T}\mathbf{y}_{s}}{\hat{\sigma}_{s}\sqrt{\mathbf{x}_{s}^{T}\mathbf{x}_{s}}} = \frac{\mathbf{x}^{T}\mathbf{y}}{\hat{\sigma}\sqrt{\mathbf{x}^{T}\mathbf{x}}} = z \]

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