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What is the weak-instruments (WI) problem and what causes it? Universal agreement does not exist on these questions. We define weak instruments by two features: (i) two-stage least squares (2SLS) analysis is badly biased toward the ordinary least-squares (OLS) estimate, and alternative "unbiased" estimators such as limited-information maximum likelihood (LIML) may not solve the problem; and (ii) the standard (first-order) asymptotic distribution does not give an accurate framework for inference.1 Thus, a researcher may estimate "bad results" and not be aware of the outcome. The cause of WI is often stated to be a low R^sup 2^ or F statistic of the reduced-form equation, in the most commonly occurring situation of one right-hand-side endogenous variable. We find the situation is more complex with an additional factor, the correlation between the stochastic disturbances of the structural equation and the reduced form, that needs to be taken into account. We discuss in this paper a specification test (Hahn and Hausman, 2002a) for WI, a caution against using "no moments" estimators such as LIML in the WI situation, and suggestions for different estimators, an approach to inference of Frank Kleibergen (2002) for WI. We end with a caution of how "small biases" can become "large biases" in the WI situation.
We begin with the limited-information structural model under the assumptions of Hausman (1983):
(1) Y^sub 1^ = Y^sub 2^[beta] + Z^sub 1^[gamma] + u
Y^sub 2^ = Z^sub 1^[pi]^sub 1^ + Z^sub 2^[pi]^sub 2^ + V
where we assume that Y^sub 1^ and Y^sub 2^ are each single jointly endogenous variables. Without loss of generality, we "partial out" the Z^sub 1^ variables by multiplying through each equation by the complementary projection Q^sub Z^sub 1^^ = I Z'^sub 1^(Z'^sub 1^Z^sub 1^)^sup -1^Z'^sub 1^ = I - P^sub Z^sub 1^^. We write the resulting equations as
(2) y^sub 1^ = y^sub 2^[beta] + [epsilon]^sub 1^
y^sub 2^ = Z[pi]^sub 2^ + [nu]^sub 2^
where dim([pi]^sub 2^) = K and the sample size is n. We also assume homoscedasticity:
We initially assume the presence of valid instruments, E[z' [epsilon]/n] = 0 and [pi]^sub 2^ [not =] 0. Without loss of generality we use the normalization (rescaling of units) [sigma]^sub [epsilon][epsilon]^ = [sigma]^sub [nu][nu]^ = 1 so that Var(y^sub 2^) = 1/(1...