Computes the coefficients \((a, b, c)\) of the quadratic inequality \(a\beta^2 + b\beta + c \leq 0\)
used to construct confidence intervals for \(\beta\) in general instrumental variable designs.
This function supports asymmetric weighting matrices (e.g., UJIVE with continuous covariates)
by using the fully generalized _iloop variance estimators.
GetCIcoef_iloop(
df,
P,
G,
X,
Y,
MX,
MY,
Z,
W,
q = qnorm(0.975)^2,
noisy = FALSE
)Data frame. Contains the observable variables \(X, Y\) and their projections.
Matrix of dimension n x n. The full projection matrix \(P\).
Matrix of dimension n x n. The UJIVE weighting matrix \(G\).
Column name (unquoted). The endogenous regressor.
Column name (unquoted). The outcome variable.
Column name (unquoted). Leverage-adjusted regressor (\(M X\)).
Column name (unquoted). Leverage-adjusted outcome (\(M Y\)).
Matrix of instruments.
Matrix of covariates.
Numeric scalar. Critical value for the test inversion (typically \(1.96^2\)).
Defaults to qnorm(.975)^2 (approx. 3.84) for a 95 percent confidence interval.
Logical. If TRUE, prints progress dots during calculation.
Defaults to FALSE.
Numeric vector of length 3 containing c(a, b, c).
This is the most general version of the confidence interval coefficient calculator. It does not assume symmetry of the weighting matrix \(G\). Consequently, it computes all five distinct variance components (\(A_1\) through \(A_5\)) for each term in the polynomial expansion of \(\hat{V}(\beta)\).
The returned coefficients define the curvature and position of the confidence set parabola:
The function explicitly constructs the diagonal adjustments for the UJIVE signal calculation using the matrices \(Z\) and \(W\).
Yap, L. (2025). "Inference with Many Weak Instruments and Heterogeneity". Working Paper.