Palgrave Handbook of Econometrics: Applied Econometrics

736 Microeconometrics: Methods and Developments

With no model the only constraint is that probabilities sum to one. This leads
to maximum EL estimateŝπi = 1 /N, so the estimated density function̂f(y|x)
has mass 1/Nat each of the realized valuesyi,i=1,...,N, and the resulting
distribution function estimate is just the usual empirical distribution function.
With a model introduced, attention focuses on the estimates for parameters of
that model. In the simplest case of estimation of a common population meanμ,
the maximum EL estimate can be shown to be the sample mean. A more general
example is to specify a model that imposesrmoment conditions:

E[h(wi,θ)]= 0 , (14.6)

the same condition as in (14.1) for MM or GMM estimation. The EL approach
maximizes the empirical likelihood functionN−^1
∑
∑ ilnπisubject to the constraint
iπi=1, since probabilities sum to one, and the additional sample constraint
based on the population moment condition (14.6) that:

∑N

i= 1

πih(wi,θ)= 0. (14.7)

Thus maximize with respect toπ=[π 1 ...πN]′,η,λandθthe Lagrangian:

LEL(π,η,λ,θ)=

1

N

∑N

i= 1

lnπi−η

⎛

⎝

∑N

i= 1

πi− 1

⎞

⎠−λ′

∑N

i= 1

πih(wi,θ), (14.8)

where the Lagrangian multipliers are a scalarηand anr×1 column vectorλ.
This maximization is not straightforward. First concentrate out theNparameters
π 1 ,...,πN. DifferentiatingL(π,η,λ,θ)with respect toπiyields 1/(Nπi)−η−λ′hi=

0. Then findη=1 by multiplying byπiand summing overiand using

∑
iπihi=^0.
It follows that the Lagrangian multipliersλsolveπi(θ,λ)= 1 /[N( 1 +λ′h(wi,θ))].
The problem is now reduced to a maximization problem with respect to(r+q)vari-
ablesλandθ, the Lagrangian multipliers associated with thermoment conditions
(14.7) and theqparametersθ. Solution at this stage requires numerical methods,
even for just-identified models withr=q. After some algebra, the log-likelihood
function evaluated atθis:

LEL(θ)=−N−^1

∑N

i= 1

ln[N( 1 +λ(θ)′h(wi,θ))]. (14.9)

The maximum empirical likelihood (MEL) estimator̂θMELmaximizes this function
with respect toθ.
Qin and Lawless (1994) show that the MEL estimator has the same limit distri-
bution as the optimal GMM estimator. In finite samples, however,̂θMELdiffers
from̂θGMM. Furthermore, inference can be based on sample estimatesĜ =
∑
îπi∂hi/∂θ

′∣∣∣

̂θand

̂S=∑îπihi(̂θ)hi(̂θ)′that weight by the estimated probabili-

tieŝπirather than the proportions 1/N. Newey and Smith (2004) show that MEL