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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:

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

A

$SE(\hat{\beta}_1^z) = SE(\hat{\beta}_1^x) + c \overline{x}$

B

$SE(\hat{\beta}_1^z) = SE(\hat{\beta}_1^x) - c \overline{x}$

C

$SE(\hat{\beta}_1^z) = c SE(\hat{\beta}_1^x) + c \overline{x}$

D

$SE(\hat{\beta}_1^z) = c SE(\hat{\beta}_1^x)$

E

$SE(\hat{\beta}_1^z) = (1/c)SE(\hat{\beta}_1^x)$

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