The vector of regressors is x_t = (y_t−1 y_t−2…y_t−k)′. Let γ_j = γ_−j = E[y_t y_t−j] and Γ_j = (γ_j γ_j−1…γ_{j−[ell ]+1})′. The matrix of cross-covariances of x_t and z_t is

where I_m denotes m × m identity matrix and O_m1×m2 – zero m₁ × m₂ matrix. The covariance matrix of z_tε_t is

The efficient generalized method of moments estimator based on the instrumental vector z_t effectively uses the instrument optimal in the class of linear transformations of z_t. This optimal instrument is Q_xzV_zε⁻¹z_t (see, e.g., West, 2001). We will show that in the matrix Q_xzV_zε⁻¹ the right k × ([ell ] − k) submatrix is zero, so that the optimal combination of elements of z_t involves only the first k entries.

Recall that the Yule–Walker equations for an AR(k) model contain the following recursion:

This implies that

where

and 0_m denotes zero m × 1 vector. Thus

It follows that Q_xzV_zε⁻¹ equals σ⁻²(I_k O_{k×([ell ]−k)})P^p−1, a matrix whose right k × ([ell ] − k) submatrix is indeed zero.