Compute L3O Variance for Score Statistic (Grouped Design, No Covariates)

Computes the Leave-Three-Out (L3O) variance estimator ($\hat{V}_{LM}$) for the score statistic in a grouped instrument design without covariates. This function leverages the block-diagonal structure of the projection matrix to compute all variance components locally within each group.

L3Ovar_gloop_nocov(df, group, X, e, MX, Me)

Arguments

df: Data frame. Contains the variables used in estimation.
group: Column name (unquoted). The instrument grouping variable.
X: Column name (unquoted). The endogenous regressor.
e: Column name (unquoted). Residuals under the null hypothesis ($Y - X\beta_0$).
MX: Column name (unquoted). Leverage-adjusted regressor ($M_i X_i$).
Me: Column name (unquoted). Leverage-adjusted residual ($M_i e_i$).

Value

Scalar. The estimated variance $\hat{V}_{LM}$.

Details

This function implements the variance estimator $\hat{V}_{LM}$ for the specific case where $G = P$ (symmetric weights) and the design matrix is block-diagonal (no covariates).

Instead of operating on $N \times N$ matrices, the function iterates through groups. Within each group subset, the projection matrix is dense but small. The function computes: $$\hat{V} = \sum_g (A_{1g} + 2A_{2g} + A_{3g} - A_{4g} - A_{5g})$$

Components:

A1, A2, A3: Signal, covariance, and error variance terms calculated using the symmetric weighting matrix $G=P$.
A4, A5: Bias correction terms utilizing squared weights $G_{ij}^2$.