The results are all developed for the case of a balanced panel. This is done primarily to streamline the notation and calculations; extension of the results to the unbalanced panel is left to future research.
Our first theoretical result is that the estimator £t given in (2.10) is uniformly (in t) consistent for a linear combination of the true factors Fy, at the rate 5^y.

All proofs are given in Appendix A.

Several remarks are in order. First, the consistency of the estimated factors is obtained by averaging over a very large number of cross sectional observations, relative to the number of time series observations. Technically this is reflected in the condition that p > 2 (so T=o(N )), and in the exponent b being an increasing function of p. In contrast to conventional factor model estimation approaches which require a large number of time series observations and a small number of variables, a larger N relative to T improves the asymptotic performance in the sense that consistency is achieved at a faster rate.
Second, for the interpretation of the matrices H and R it is useful to consider separately the three cases of ky < ry, ky=ry, and ky > г у. When ky < ry, the rows of R span the space of the first ky eigenvectors of the positive definite ryXry matrix A = E p yDE p’y. When ky = ry, R is a full rank square matrix with R’R^I^ so asymptotically Pt equals F^ up to the full rank transformation matrix H. When ky > ry, the row rank of R and thus H is only ry, so £t contains ky-ry redundant estimates of the factors that are just linear combinations of the ry elements of Ft.