Compute L3O Variance for Score Statistic (Grouped Design)

Computes the Leave-Three-Out (L3O) variance estimator ($\hat{V}_{LM}$) for the score statistic in a grouped instrument design with covariates. This function simultaneously calculates all five variance components by exploiting the block-diagonal symmetry of the projection matrices inherent to discrete instruments.

L3Ovar_gloop_cov(df, group, groupW, X, e, MX, Me)

Arguments

df: Data frame. Contains the variables used in estimation.
group: Column name (unquoted). The instrument grouping variable.
groupW: Column name (unquoted). The covariate stratification 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 Limited Information Maximum Likelihood (LIML) or UJIVE estimator. It computes: $$\hat{V} = A_1 + 2A_2 + A_3 - A_4 - A_5$$

Because the design implies symmetric weighting matrices ($G_{ij} = G_{ji}$), the function simplifies the asymmetric components.

Components:

A1, A2, A3: Variance and covariance of the score statistic. Calculated via 5 sub-components each.
A4, A5: Bias correction terms for "own-observation" variance contributions. Calculated via 4 sub-components each.