Compute Quadratic Coefficients for Confidence Sets (Grouped Data)

Estimates the coefficients $a$, $b$, and $c$ for the quadratic inequality $a\beta^2 + b\beta + c \le 0$, which defines the $1-\alpha$ confidence set for the structural parameter $\beta$. This function is highly optimized for large-scale datasets, relying on block-diagonal geometries and scalar algebra.

GetCIcoef(df, groupW, group, X, Y, MX, MY, q = qnorm(0.975)^2, noisy = FALSE)

Arguments

df: Data frame. Contains the observable variables and their projections.
groupW: Column name (unquoted). The covariate stratification variable.
group: Column name (unquoted). The instrument grouping variable.
X: Column name (unquoted). The endogenous regressor.
Y: Column name (unquoted). The outcome variable.
MX: Column name (unquoted). Leverage-adjusted regressor ($M X$).
MY: Column name (unquoted). Leverage-adjusted outcome ($M Y$).
q: Numeric scalar. Critical value for the test statistic inversion (e.g., $\chi^2_{1, 1-\alpha}$). Defaults to qnorm(.975)^2 (approx. 3.84) for a 95 percent confidence interval.
noisy: Logical. If TRUE, prints progress dots during calculation. Defaults to FALSE.

Value

Numeric vector of length 3: c(a, b, c).

Details

The confidence set is constructed by inverting a test statistic based on the quadratic form $Q(\beta) = (\mathbf{Y} - \beta \mathbf{X})' G (\mathbf{Y} - \beta \mathbf{X})$. The coefficients are derived from the variance estimator $\hat{V}(\beta)$ of this quadratic form, decomposed into interactions between the outcome and the regressor.

Algorithmic Implementation: To achieve high performance, the function processes the data using a two-level nested loop over covariate strata (groupW) and instrument groups (group). The heavy $N \times N$ geometry matrices (such as the projection matrices $P$, $M$, and the leverage-adjusted weight matrix $G$) are pre-computed exactly once per stratum.

Furthermore, the target inner products ($P_{XY}$ and $P_{XX}$) are aggregated inline during the stratum loop to avoid redundant passes over the data. Within each instrument group, unique Leave-Three-Out (L3O) variance interactions ($A_1$ and $A_4$ terms) are evaluated using exact scalar algebra helpers. This approach minimizes computational complexity to $O(N)$ at the group level and completely eliminates redundant matrix allocations.

The returned coefficients correspond to: $$a = P_{XX}^2 - q \cdot C_2$$ $$b = -2 P_{XY} P_{XX} - q \cdot C_1$$ $$c = P_{XY}^2 - q \cdot C_0$$

Where $P_{XY}$ and $P_{XX}$ are the UJIVE estimators for the cross-products, and $C_0, C_1, C_2$ are the variance components compiled via the L3O adjustment framework.

References

Yap, L. (2025). "Inference with Many Weak Instruments and Heterogeneity". Working Paper.