We have a following inequality:
$$ \sum(x_i - \bar{x})^2 (\beta - \hat{\beta})^2 + 2\sum(x_i-\bar{x})(z_i-\bar{z})(\beta-\hat{\beta})(\gamma-\hat{\gamma})+\sum(z_i-\bar{z})^2(\gamma-\hat{\gamma})^2 \le 2{\hat{\sigma}}^2F $$
Only $\beta$ and $\gamma$ are unknown terms, whereas all other values are known ($x_i$, $\bar{x}$, $\hat{\beta}$, $z_i$, $\bar{z}$, $\hat{\gamma}$, ${\hat{\sigma}}^2$, $F$).
This is supposed to denote an elliptical confidence region around some parameters.
I was trying to play around with the formula and I first moved the expression $2{\hat{\sigma}}^2F$ to the left and then changing the inequality sign to the equality sign so that I have:
$$ \sum(x_i - \bar{x})^2 (\beta - \hat{\beta})^2 + 2\sum(x_i-\bar{x})(z_i-\bar{z})(\beta-\hat{\beta})(\gamma-\hat{\gamma})+\sum(z_i-\bar{z})^2(\gamma-\hat{\gamma}) -2{\hat{\sigma}}^2F = 0 $$
Additionaly from the previous literature I know that:
$$ \gamma = -A\sqrt{1-\frac{\beta^2}{A^2}}$$
where the value of $A$ is also known. Thus the previous equation can be rewritten as:
$$ \sum(x_i - \bar{x})^2 (\beta - \hat{\beta})^2 + 2\sum(x_i-\bar{x})(z_i-\bar{z})(\beta-\hat{\beta})(-A\sqrt{1-\frac{\beta^2}{A^2}}-\hat{\gamma})+\sum(z_i-\bar{z})^2(-A\sqrt{1-\frac{\beta^2}{A^2}}-\hat{\gamma}) -2{\hat{\sigma}}^2F = 0 $$
Trying to solve this with some help from online sources has led me to the fact that this will be a quartic equation with four solutions, $\beta_1$, $\beta_2$, $\beta_3$ and $\beta_4$. I will therefore also have $\gamma_1$, $\gamma_2$, $\gamma_3$ and $\gamma_4$.
I am wondering what will these numbers tell me. Remember: I took the inequality defining the which points fall into an elliptical region on the graph and then changed it to the equality and rearranged the equation so that the right-hand term is zero.
Four values per $\beta$ and $\gamma$ would seem to form four points in the coordinate system. Would I be correct to assume that these four points would be the four points where the minor and the major axis touch the ellipsis? Or would these points represent something else?
