Calculates the UJIVE "signal" or cross-product term \(X' G e\) for a design where instruments are nested within discrete covariate strata (e.g., Judges within Years). This function iterates through covariate groups to compute the quadratic form locally, handling the centering of instruments within each block.
GetLM(df, X, e, groupW, group, noisy = FALSE)Data frame. Contains the observable variables and grouping indicators.
Column name (unquoted). The first variable (e.g., endogenous regressor).
Column name (unquoted). The second variable (e.g., outcome or residual).
Column name (unquoted). The covariate stratification variable (defines blocks).
Column name (unquoted). The instrument grouping variable (defines treatments).
Logical. If TRUE, prints progress of the stratum iteration.
Defaults to FALSE.
Numeric scalar. The sum of stratum-specific quadratic forms.
This function implements the estimator for the signal component \(S\) in a stratified design: $$S = \sum_{s} e_s' [ U(P_{Q,s}) - U(P_{W,s}) ] X_s$$
Within each stratum \(s\):
\(P_{Q,s}\) is the projection onto the instrument groups.
\(P_{W,s}\) is the projection onto the stratum intercept (local mean).
\(U(P)\) denotes the projection matrix with its diagonal elements removed, rescaled by the inverse of the annihilator diagonal \((1 - P_{ii})^{-1}\).
This corresponds to the numerator terms (\(P_{XY}, P_{XX}\)) for test inversion in designs with discrete controls.
Algorithmic Implementation: To ensure high performance and low memory overhead on large datasets, this function computes the projection transformation \(G_s X_s\) using strictly \(O(N)\) vector operations. It computes group means via optimized aggregation, derives the diagonal leverage adjustments inline, and directly applies the leave-one-out transformation without ever constructing the dense \(N \times N\) projection matrices.
Yap, L. (2025). "Inference with Many Weak Instruments and Heterogeneity". Working Paper.