Easy# 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?