first-difference estimator

The FD estimator avoids bias due to some unobserved, time-invariant variable $c\_\{i\}$, using the repeated observations over time:

:$y\_\{it\}=x\_\{it\}\backslash beta\; +\; c\_\{i\}+\; u\_\{it\},\; t=1,...T\; ,$

:$y\_\{it-1\}=x\_\{it-1\}\backslash beta\; +\; c\_\{i\}+u\_\{it-1\},\; t=2,...T\; .$

Differencing the equations, gives:

:$\backslash Delta\; y\_\{it\}=y\_\{it\}-y\_\{it-1\}=\backslash Delta\; x\_\{it\}\backslash beta\; +\; \backslash Delta\; u\_\{it\},\; t=2,...T\; ,$

which removes the unobserved $c\_\{i\}$ and eliminates the first time period.Wooldridge 2013, p. 461.Wooldridge 2001, p. 279.

The FD estimator $\backslash hat\{\backslash beta\}\_\{FD\}$ is then obtained by using the differenced terms for $x$ and $u$ in OLS:

:$\backslash hat\{\backslash beta\}\_\{FD\}\; =\; (\backslash Delta\; X\text{'}\backslash Delta\; X)^\{-1\}\backslash Delta\; X\text{'}\; \backslash Delta\; y=\backslash beta\; +\; (\backslash Delta\; X\text{'}\backslash Delta\; X)^\{-1\}\backslash Delta\; X\text{'}\; \backslash Delta\; u$

where $X,y,$ and $u$, are notation for matrices of relevant variables. Note that the rank condition must be met for $\backslash Delta\; X\text{'}\backslash Delta\; X$ to be invertible ($\backslash text\{rank\}[\backslash Delta\; X\text{'}\backslash Delta\; X]=k$), where $k$ is the number of regressors.

Let

:$\backslash Delta\; X\_i\; =[\backslash Delta\; X\_\{i2\},\; \backslash Delta\; X\_\{i3\},\; ...,\; \backslash Delta\; X\_\{iT\}]$,

and, analogously,

:$\backslash Delta\; u\_i\; =[\backslash Delta\; u\_\{i2\},\; \backslash Delta\; u\_\{i3\},\; ...,\; \backslash Delta\; u\_\{iT\}]$.

If the error term is strictly exogenous, i.e. $E[u\_\{it\}|x\_\{i1\},\; x\_\{i2\},\; ..,\; x\_\{iT\}]=0$, by the central limit theorem, the law of large numbers, and the Slutsky's theorem, the estimator is distributed normally with asymptotic variance of

:$\backslash widehat\backslash text\{Avar\}(\backslash hat\{\backslash beta\}\_\{FD\})=E[\backslash Delta\; X\_i\text{'}\; \backslash Delta\; X\_i]^\{-1\}E[\backslash Delta\; X\_i\text{'}\; \backslash Delta\; u\_i\; \backslash Delta\; u\_i\text{'}\; \backslash Delta\; X\_i]E[\backslash Delta\; X\_i\text{'}\backslash Delta\; X\_i]^\{-1\}$.

Under the assumption of homoskedasticity and no serial correlation, $\backslash text\{Var\}(\backslash Delta\; u\; |\; X)=\backslash sigma^2\_\{\backslash Delta\; u\}$, the asymptotic variance can be estimated as

:$\backslash widehat\{\backslash text\{Avar\}\}(\backslash hat\{\backslash beta\}\_\{FD\})=\backslash hat\{\backslash sigma\}^\{2\}\_\{\backslash Delta\; u\}(\backslash Delta\; X\text{'}\backslash Delta\; X)^\{-1\}\; ,$

where $\backslash hat\{\backslash sigma\}^\{2\}\_\{u\}$, a consistent estimator of $\backslash sigma^\{2\}\_\{u\}$, is given by

:$\backslash hat\{\backslash sigma\}^\{2\}\_\{\backslash Delta\; u\}\; =\; [n(T-1)-K]^\{-1\}\backslash sum\_\{i=1\}^n\backslash sum\_\{t=2\}^T\; \backslash widehat\{\backslash Delta\; u\_\{it\}\}^2$

and

:$\backslash widehat\{\backslash Delta\; u\_\{it\}\}=\backslash Delta\; y\_\{it\}-\backslash hat\{\backslash beta\}\_\{FD\}\backslash Delta\; x\_\{it\}$.Wooldridge 2001, p. 281.

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

:$E[u\_\{it\}|x\_\{i1\},\; x\_\{i2\},\; ..,\; x\_\{iT\}]=0$.

The first difference estimator (FD) is also unbiased under this assumption.

If strict exogeneity is violated, but the weaker assumption

:$E[(u\_\{it\}-u\_\{it-1\})(x\_\{it\}-x\_\{it-1\})]=0$

holds, then 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 $T\; \backslash rightarrow\; \backslash infty$, then both FE and FD are consistent under the weaker assumption of contemporaneous exogeneity.

The Hausman test can be used to test the assumptions underlying the consistency of the FE and FD estimators.Wooldridge 2001, p. 285.

For $T=2$, the FD and fixed effects estimators are numerically equivalent.Wooldridge 2001, p. 284.

Under the assumption of homoscedasticity and no serial correlation in $u\_\{it\}$, 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 $u\_\{it\}$ follows a random walk, however, the FD estimator is more efficient as $\backslash Delta\; u\_\{it\}$ are serially uncorrelated.Wooldridge 2001, p. 284.

• Factor analysis
• Panel analysis
• Fixed effect model

{{reflist| 2}}

• {{cite book |first=Jeffrey M. |last=Wooldridge |title=Econometric Analysis of Cross Section and Panel Data |url=https://archive.org/details/econometricanaly0000wool |year=2001 |publisher=MIT Press |isbn=978-0-262-23219-7 |pages=[https://archive.org/details/econometricanaly0000wool/page/279 279]–291|access-date=30 August 2024}}

• {{cite book |last1=Wooldridge |first1=Jeffrey M. |title=Introductory Econometrics: A Modern Approach |date=2013 |publisher=South-Western Cengage Learning |isbn=978-1-111-53104-1 |edition=5th |url=https://cbpbu.ac.in/userfiles/file/2020/STUDY_MAT/ECO/2.pdf |access-date=30 August 2024}}

Category:Estimator

Category:Latent variable models