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# Effect of rescaling on the intercept term

EMETRC-XYGWEE

Consider the Simple Linear Regression Model:

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

Let $\hat{\beta}_0^x$ and $\hat{\beta}_1^x$ be OLS estimates for this model. Next suppose the explanatory variable is rescaled as $w_i = c x_i,$ implying the rescaled model:

$$y_i = \beta_0^w + \beta_1^w w_i + \varepsilon_i$$

Indicate the OLS estimates for this rescaled model as $\hat{\beta}_0^w$ and $\hat{\beta}_1^w$.

What is an expression for the estimate of the intercept term for the rescaled model?

A

$\hat{\beta}_0^w = (1/c)\hat{\beta}_1^x$

B

$\hat{\beta}_0^w = \hat{\beta}_0^x$

C

$\hat{\beta}_0^w = c\hat{\beta}_0^x$

D

$\hat{\beta}_0^w = (1/c)\hat{\beta}_0^x$

E

$\hat{\beta}_0^w = c\hat{\beta}_1^x$