## ⓘ First-difference estimator

The first-difference estimator is an approach used to address the problem of omitted variables in econometrics and statistics with panel data. The estimator is obtained by running a pooled OLS estimation for a regression of Δ y i t {\displaystyle \Delta y_{it}} on Δ x i t {\displaystyle \Delta x_{it}}.

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

y i t = x i t β + c i + u i t, t = 1. T, {\displaystyle y_{it}=x_{it}\beta +c_{i}+u_{it},t=1.T,} y i t − 1 = x i t − 1 β + c i + u i t − 1, t = 2. T. {\displaystyle y_{it-1}=x_{it-1}\beta +c_{i}+u_{it-1},t=2.T.}Differencing both equations, gives:

Δ y i t = y i t − y i t − 1 = Δ x i t β + Δ u i t, t = 2. T, {\displaystyle \Delta y_{it}=y_{it}-y_{it-1}=\Delta x_{it}\beta +\Delta u_{it},t=2.T,}which removes the unobserved c i {\displaystyle c_{i}}.

The FD estimator β ^ F D {\displaystyle {\hat {\beta }}_{FD}} is then simply obtained by regressing changes on changes using OLS:

β ^ F D = Δ X ′ Δ X − 1 Δ X ′ Δ y {\displaystyle {\hat {\beta }}_{FD}=\Delta X\Delta X^{-1}\Delta X\Delta y}Note that the rank condition must be met for Δ X ′ Δ X {\displaystyle \Delta X\Delta X} to be invertible (r a n k ^{-1}{\hat {u}}{\hat {u}}.}

## 1. Properties

Under the assumption of E =\beta } and p l i m β ^ = β {\displaystyle plim{\hat {\beta }}=\beta }. Note that this assumption is less restrictive than the assumption of strict exogeneity required for unbiasedness using the fixed effects FE estimator. If the disturbance term u i t {\displaystyle u_{it}} follows a random walk, the usual OLS standard errors are asymptotically valid.

## 2. Relation to fixed effects estimator

For T = 2 {\displaystyle T=2}, the FD and fixed effects estimators are numerically equivalent.

Under the assumption of homoscedasticity and no serial correlation in u i t {\displaystyle u_{it}}, the FE estimator is more efficient than the FD estimator. If u i t {\displaystyle u_{it}} follows a random walk, however, the FD estimator is more efficient as Δ u i t {\displaystyle \Delta u_{it}} are serially uncorrelated.

Latent variable model |

Doubly stochastic model |

Dynamic unobserved effects model |

Factor regression model |

Nuisance variable |

Partial least squares regression |

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