Construct UJIVE Weighting and Projection Matrices

Generates the weighting matrix $G$ and projection matrix $P$ required for the Unbiased Jackknife Instrumental Variables Estimator (UJIVE). It constructs these matrices based on group/cluster indicators for instruments ($Z$) and stratification covariates ($W$).

GetGP(group, groupW, n)

Arguments

group: Integer vector. Primary grouping variable used to define the instrument structure (e.g., judge or examiner IDs).
groupW: Integer vector. Grouping variable for stratification or covariates (e.g., time periods or court locations).
n: Integer. The total sample size (number of observations).

Value

A list containing four matrices:

G: The $N \times N$ UJIVE weighting matrix.
P: The $N \times N$ full projection matrix on $[Z, W]$.
Z: The $N \times K$ matrix of instrument indicators.
W: The $N \times L$ matrix of covariate indicators.

Details

The function constructs the design matrices for instruments ($Z$) and covariates ($W$) based on the provided grouping vectors. It creates dummy variable matrices where $Z_{ij} = 1$ if observation $i$ belongs to group $j$.

It calculates the standard projection matrices: $$P_Z = Z(Z'Z)^{-1}Z'$$ $$P_W = W(W'W)^{-1}W'$$ $$P = P_{[Z,W]} \quad (\text{Projection onto both } Z \text{ and } W)$$

The UJIVE weighting matrix $G$ is then computed as the difference between the "leave-one-out" adjusted projection of the full set and the covariates: $$G = D_P^{-1}(P - \text{diag}(P)) - D_W^{-1}(P_W - \text{diag}(P_W))$$ where $D_P$ and $D_W$ are diagonal matrices containing the annihilator diagonals ($1 - P_{ii}$) for the respective projections.

Note: This function assumes that the resulting design matrices are full rank.

References

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