744 Microeconometrics: Methods and Developments
14.4 Statistical inference
There have been considerable advances in statistical inference methods, even for
the standard LS, IV and ML estimators (MLEs).
The most notable change is the use of robust statistical inference, notably robust
standard error computation, that relies on distributional assumptions that are as
weak as possible. Various model specification tests have been developed, such as
the Hausman test and White’s information matrix test. The bootstrap provides an
alternative way to perform statistical inference that can be simpler than conven-
tional asymptotic methods. Furthermore, the bootstrap can produce more accurate
asymptotic approximations for test statistics that lead to tests with actual size close
to nominal size in small samples.
14.4.1 Robust inference for Wald tests
Analysis begins with the cross-section case of independent observations, before
moving to clustered observations, which includes short panels.
Consider an m-estimator̂θthat maximizes with respect toθthe objective func-
tionQN(θ)=N−^1
∑
iq(yi,xi,θ). For ML estimationq(·)is the log-density, and for
least squares estimationq(·)is minus the squared error (or a rescaling of this). The
m-estimator solves the first-order conditions:
N−^1∑
ih(wi,θ)=^0 , (14.19)whereh(wi,θ)=∂q(yi,xi,θ)/∂θ. Under suitable assumptions, notably that E[h(wi,
θ)]= 0 in the population, it can be shown that̂θis
√
N−consistent, with limit
distribution:
√
N(̂θ−θ 0 )
d
→N[ 0 ,A− 01 B 0 A 0 ′−^1 ], (14.20)
whereA 0 =plimN−^1
∑
i∂h(wi,θ)/∂θ
′
∣
∣∣
θ 0andB 0 =plimN−^1
∑
ih(wi,θ 0 )h(wi,θ 0 )′,andθ 0 is the value ofθin the DGP.
Inference is based on̂θbeing asymptotically normally distributed with meanθ 0
and estimated asymptotic variance matrix of sandwich form:
̂V[̂θ]=N−^1 ̂A−^1 ̂B̂A′−^1 , (14.21)wherêAand̂Bare consistent estimates ofA 0 andB 0. The Wald test statistic for
H 0 :θj=ris thenW=(̂θj−r)/sj, wheresjis thejth diagonal entry of̂V[̂θ], and
W
a
∼N[0, 1]underH 0. The more general hypothesisH 0 :c(θ)= 0 is tested using
W=c(̂θ)′(̂R̂V[̂θ]̂R′)−^1 c(̂θ), wherêR=∂c(θ)/∂θ′
∣
∣∣
̂θandW
∼aχ^2 (rank[R])underH
0.
There are several possible ways to form̂Aand̂B, depending in part on the
strength of the distributional assumptions made. Robust variance estimates are
those that rely on minimal distributional assumptions, providedN→∞.