Difficult# Rescaling the Slope Term and the Effect on the Standard Error

EMETRC-7XTEX4

Consider the Simple Linear Regression Model with homoskedastic errors:

$$y_i = \beta_0^x + \beta_1^x x_i + u_i$$

Let $\hat{\beta}_0^x$ and $\hat{\beta}_1^x$ be OLS estimates for this model, and let $SE(\hat{\beta}_1^x)$ be the standard error of the slope coefficient estimator.

Suppose the explanatory variable is rescaled as $z_i = c x_i,$ implying the rescaled model:

$$y_i = \beta_0^z + \beta_1^z z_i + \varepsilon_i$$

...where the OLS estimates are indicated as $\hat{\beta}_0^z$ and $\hat{\beta}_1^z$, with $SE(\hat{\beta}_1^z)$ being the standard error of $\hat{\beta}_1^z$.

What is an expression for $SE(\hat{\beta}_1^z)$ in terms of:

$$SE(\hat{\beta}_1^x)?$$