First-difference estimator explained

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

, and independent variables,

xit

, for a set of individual units

i=1,...,N

and time periods

t=1,...,T

. The estimator is obtained by running a pooled ordinary least squares (OLS) estimation for a regression of

\Deltayit

on

\Deltaxit

.

Derivation

The FD estimator avoids bias due to some unobserved, time-invariant variable

ci

, using the repeated observations over time:

yit=xit\beta+ci+uit,t=1,...T,

yit-1=xit-1\beta+ci+uit-1,t=2,...T.

Differencing the equations, gives:

\Deltayit=yit-yit-1=\Deltaxit\beta+\Deltauit,t=2,...T,

which removes the unobserved

ci

.

The FD estimator

\hat{\beta}FD

is then obtained by using the differenced terms for x and u in OLS:

\hat{\beta}FD=(\DeltaX'\DeltaX)-1\DeltaX'\Deltay=\beta+(\DeltaX'\DeltaX)-1\DeltaX'\Deltau

Where

X,y,

and

u

, are notation for matrices of relevant variables. Note that the rank condition must be met for

\DeltaX'\DeltaX

to be invertible (

rank[\DeltaX'\DeltaX]=k

) where

k

is the number of regressors.

Let

\DeltaXi=[\DeltaXi2,\DeltaXi3,...,\DeltaXiT]

and define

\Deltaui

analogously. If

E[uit|xi1,xi2,..,xiT]=0

, by the Central limit theorem, Law of large numbers, and Slutsky's theorem, the estimator is distributed normally with asymptotic variance of

E[\DeltaXi'\Delta

-1
X
i]

E[\DeltaXi'\Deltaui\Deltaui'\DeltaXi]E[\DeltaXi'\Delta

-1
X
i]

.Under the assumption of homoskedasticity and no serial correlation, mathematically that,

Var(\Deltau|

2
X)=\sigma
\Deltau
, the asymptotic variance can be estimated with

\widehat{Avar}(\hat{\beta}FD

2
)=\hat{\sigma}
\Deltau

(\DeltaX'\DeltaX)-1,

where
2
\hat{\sigma}
u
is given by
2
\hat{\sigma}
\Deltau

=[n(T-1)-K]-1

T
\sum
t=2

\widehat{\Deltauit

}^2

and

\widehat{\Deltauit

}=\Delta y_-\hat_\Delta x_.

Properties

To be unbiased, the fixed effects estimator (FE) requires strict exogeneity,

E[uit|xi1,xi2,..,xiT]=0

. The first difference estimator is also unbiased under this assumption. Under the weaker assumption that

E[(uit-uit-1)(xit-xit-1)]=0

, the FD estimator is consistent. Note that this assumption is less restrictive than the assumption of strict exogeneity which is required for consistency using the FE estimator when T is fixed. If T goes to infinity, then both FE and FD are consistent with the weaker assumption of contemporaneous exogeneity.

Relation to fixed effects estimator

For

T=2

, the FD and fixed effects estimators are numerically equivalent.

Under the assumption of homoscedasticity and no serial correlation in

uit

, the FE estimator is more efficient than the FD estimator. This is because the FD estimator induces no serial correlation when differencing the errors. If

uit

follows a random walk, however, the FD estimator is more efficient as

\Deltauit

are serially uncorrelated.

See also

References