Computes the roots of the quadratic equation \(a\beta^2 + b\beta + c = 0\). These roots serve as the boundaries for the confidence sets constructed by inverting the UJIVE/LIML score test.
GetCIvals(CIcoef)Numeric vector of length 3. The coefficients \((a, b, c)\) of the quadratic
inequality, typically obtained from GetCIcoef.
Numeric vector of length 2. Contains the two roots.
This function applies the standard quadratic formula: $$\beta_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Note: This function does not check the sign of the discriminant. It is intended
to be called by a wrapper function (like GetCItypebd) that first verifies
the existence of real roots (i.e., \(b^2 - 4ac \ge 0\)).
Depending on the sign of \(a\) (convexity), these values represent either the endpoints of a bounded confidence interval or the inner boundaries of a disjoint ("donut") confidence set.
Yap, L. (2025). "Inference with Many Weak Instruments and Heterogeneity". Working Paper.