Palgrave Handbook of Econometrics: Applied Econometrics

Thorsten Beck 1187

the critical values of this F-test for IV estimation, however, are larger than for OLS
estimation; for the case of a single endogenous variable, Staiger and Stock (1997)
show, using Monte Carlo simulations, that for most specifications and indepen-
dent of the degrees of freedom, a critical value of 10 is sufficient to reject the null
hypothesis, and Stock and Yogo (2005) derive critical values for this F-test for the
case of several endogenous variables, with the critical values increasing with the
number of instruments.^10 Second, one can use a partial R^2 of the first-stage regres-
sion (25.9) that takes into account the intercorrelation among the instruments
(Shea, 1997). Specifically, Godfrey (1999) shows that this statistic for endogenous
regressoriis:

σˆiOLS

σˆiIV

⎡

⎣

(

1 −R^2 IV

)

(

1 −R^2 OLS

)

⎤

⎦,

whereσˆiis the estimated asymptotic variance of the coefficienti. This measure
thus tests for the relevance of the individual instruments, unlike the F-test, which
tests for the overall relevance.
Weak instruments can bias the IV results towards OLS and turn them inconsis-
tent. Further, weak instruments can result in an over-rejection of the overidentifi-
cation test discussed above. If instruments are both invalid and irrelevant, the bias
thus increases in a multiplicative way.^11
Most of the cross-country finance and growth papers utilizing IVs find that the
IV estimator ofβ 1 is higher than the OLS estimator.^12 Manipulating regressions
(25.8), (25.9) and (25.10), one can show that this implies:

δˆ 2 +ˆρ

σ(υ)̂

σ(ε)̂ <

βˆ 1

(

1 −βˆ 1 δˆ 2

)σ(̂u)

σ(ε)̂, (25.11)

whereρis the correlation betweenεandυ, and the other parameters are taken from
regressions (25.8), (25.9) and (25.10). There are several possible explanations for
this finding and thus for inequality (25.11) to hold (Kraay and Kaufman, 2002).
First, there could be negative reverse causation (δ 2 <0), which would bias the
OLS estimator of theβ 1 coefficient downwards. Given empirical studies show-
ing the positive relationship between economic and financial development, this
explanation seems rather unlikely (Harrison, Sussman and Zeira, 1999). A sec-
ond explanation that makes inequality (25.11) hold is that omitted variables are
correlated with growth and finance with opposite signs (ρ<0), an explanation
for which, again, little evidence exists. A third – and most commonly adopted –
explanation relies on attenuation bias, where measurement error in financial devel-
opment (σ(̂u))biases the OLS estimate downwards and makes inequality (25.11)
hold. Critically, however, if the IVs are positively correlated with omitted vari-
ables and the exclusion condition is thus violated, the IV estimator ofβ 1 is biased
upwards. This is of concern, as a few IVs, such as historical country traits, have been