ABSTRACT

This paper extends Staiger and Stock's (1997) weak instrument asymptotic approximations to the case of many weak instruments by modeling the number of instruments as increasing slowly with the number of observations. It is shown that the resulting “many weak instrument” approximations can be calculated sequentially by letting first the sample size, and then the number of instruments, tend to infinity. The resulting distributions are given for k-class estimators and test statistics.

NTRODUCTION

Most of the literature on the distribution of statistics in instrumental variables (IV) regression assumes, either implicitly or explicitly, that the number of instruments (K2) is small relative to the number of observations (T); see Rothenberg's (1984) survey of Edgeworth approximations to the distributions of IV statistics. In some applications, however, the number of instruments can be large; for example, Angrist and Krueger (1991) had 178 instruments in one of their specifications. Sargan (1975), Kunitomo (1980), and Morimune (1983) provided early asymptotic treatments of many instruments. More recently, Bekker (1994) obtained first-order distributions of various IV estimators under the assumptions that K2 → ∞, T → ∞, and K2/T → c, 0 ≤ c < 1, when the so-called concentration parameter (μ2) is proportional to the sample size and the errors are Gaussian. Chao and Swanson (2002) have explored the consistency of IV estimators with weak instruments when the number of instruments is large, in the sense that K2 is also modeled as increasing to infinity, but more slowly than T.

This paper continues this line of research on the asymptotic distribution of IV estimators when there are many instruments.