# Studentization ad pivotal quantity for the slope in simple linear regression

Take the usual linear model: $$Y=\alpha+\beta X + e$$

In good hypothesis (IID sample, normal error assumption $e \sim N(0,\sigma_e^2)$ and homoschedasticity) the distribution of the sample slope $\beta_n=\frac{S_{xy}}{S_x^2}$ in a simple linear regression is known to be: $$\hat\beta_n \sim N\left(\beta,\frac{1-\rho^2}{n-3}\frac{\sigma_Y^2}{\sigma_X^2} \right)$$

Knowing $\sigma_Y$ and $\sigma_X$ it's straightforward to setup a CI on $\beta,$ due to normality of $\hat\beta_n$. 

If $(1-\rho^2)\sigma_Y^2$ is not known, due to it's identification with $\sigma_e^2$

(in fact $(1-\rho^2)\sigma_Y^2 = V[Y|X]=E[e^2|X]=\sigma_e^2)$,we can studentize (providing non trivial consideration) by knowing that the residual sum of squares is $rss \sim\sigma_e^2 \chi^2_{n-2}$ distributed and by it's consistency for $\sigma_e^2$.

From this we can find the pivotal quantity $t_{n-2}=\frac{Z}{\sqrt{\frac{K}{n-2}}}$ which clearly does not depends on $\sigma_e$ and allows us to set a t-student based CI on $\beta$.

My question is:

If instead $\sigma_e^2$ is known and $\sigma_X^2$ is not, with a similar procedure I can't get rid of the parameter due to the fact that it appears in the denominator of the variance of $\hat\beta_n$. Which pivotal quantity should I use in order to put a proper CI in this case?

• regression conditions on the X's – Glen_b Jul 11 '18 at 5:52
• Only one regressor and normally jointed with Y – omega Jul 11 '18 at 8:25