In statistics and econometrics, the first-difference (FD) estimator is an estimator used to address the problem of omitted variables with panel data. It is consistent under the assumptions of the fixed effects model. In certain situations it can be more efficient than the standard fixed effects (or "within") estimator.
The estimator requires data on a dependent variable,
yit
xit
i=1,...,N
t=1,...,T
\Deltayit
\Deltaxit
The FD estimator avoids bias due to some unobserved, time-invariant variable
ci
yit=xit\beta+ci+uit,t=1,...T,
yit-1=xit-1\beta+ci+uit-1,t=2,...T.
\Deltayit=yit-yit-1=\Deltaxit\beta+\Deltauit,t=2,...T,
ci
The FD estimator
\hat{\beta}FD
\hat{\beta}FD=(\DeltaX'\DeltaX)-1\DeltaX'\Deltay=\beta+(\DeltaX'\DeltaX)-1\DeltaX'\Deltau
Where
X,y,
u
\DeltaX'\DeltaX
rank[\DeltaX'\DeltaX]=k
k
Let
\DeltaXi=[\DeltaXi2,\DeltaXi3,...,\DeltaXiT]
\Deltaui
E[uit|xi1,xi2,..,xiT]=0
E[\DeltaXi'\Delta
-1 | |
X | |
i] |
E[\DeltaXi'\Deltaui\Deltaui'\DeltaXi]E[\DeltaXi'\Delta
-1 | |
X | |
i] |
Var(\Deltau|
2 | |
X)=\sigma | |
\Deltau |
\widehat{Avar}(\hat{\beta}FD
2 | |
)=\hat{\sigma} | |
\Deltau |
(\DeltaX'\DeltaX)-1,
2 | |
\hat{\sigma} | |
u |
2 | |
\hat{\sigma} | |
\Deltau |
=[n(T-1)-K]-1
T | |
\sum | |
t=2 |
\widehat{\Deltauit
and
\widehat{\Deltauit
To be unbiased, the fixed effects estimator (FE) requires strict exogeneity,
E[uit|xi1,xi2,..,xiT]=0
E[(uit-uit-1)(xit-xit-1)]=0
For
T=2
Under the assumption of homoscedasticity and no serial correlation in
uit
uit
\Deltauit