Abstract
The primary advantage of panel data is the ability they afford to control for unobserved heterogeneity. The fixed-effects (FE) estimator is by far the most popular technique for exploiting this advantage, but it eliminates any time-invariant regressors in the model along with the unobserved effects. Their partial effects can be easily recovered in a second-step regression of residuals constructed from the FE estimator and the group means of the time-invariant variables. In this paper, we reconsider such a two-step estimation procedure, derive its correct asymptotic covariance matrix, and compare conventional inference based on the asymptotic formula to bootstrap alternatives. Bootstrapping has a natural appeal, because of the complications associated with estimating the asymptotic covariance matrix and the inherent finite-sample bias of the resulting standard errors. We adapt the pairs and wild bootstrap to our two-step panel-data setup and show that both procedures are unbiased. Using Monte Carlo methods, we compare the error in rejection probability (ERP) of t-tests, measured as the difference between their actual and nominal size, and the power of such tests for the asymptotic and bootstrap estimators. Bootstrap estimators show small ERPs, even with a small number of cross-sectional observations, N, and clearly dominate the asymptotic method based on ERP and power. Bootstrap ERPs drop to nearly zero with N = 250, while those of the asymptotic methods are still substantial. This dominance diminishes but continues until N = 1, 000 and the number of time-series observations equals 20, so that inference with smaller samples is problematic using the asymptotic formula.
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Notes
- 1.
Although the model setup assumes a balanced panel, this is not necessary. The asymptotic covariance matrix and bootstrap procedures can readily accommodate settings in which the number of time-series observations varies with the cross-section unit.
- 2.
Atkinson and Cornwell (2013) extend the analysis here to allow some of the elements of z i to be correlated with the unobserved effect.
- 3.
As discussed in Atkinson and Cornwell (2013), allowing some of the elements of z i to be correlated with the unobserved effect leads to the two-step “simple, consistent” instrumental variables estimator of Hausman and Taylor (1981). From this perspective, you can view our two-step estimator as an instrumental variables estimator using \([\mathbf{Q}_{T}\mathbf{X}_{i}, (j_{T} \otimes \mathbf{z}_{i})]\) as instruments.
- 4.
Also, see Kapetanios (2008), who shows that if the data do not exhibit cross-sectional dependence but exhibit temporal dependence, then cross-sectional resampling is superior to block bootstrap resampling. Further, he shows that cross-sectional resampling provides asymptotic refinements. Monte Carlo results using these assumptions indicate the superiority of the cross-sectional method.
- 5.
Further, this transformation is needed to obtain a heteroskedastic-consistent covariance matrix as explained above.
- 6.
As indicated above, Davidson and Flachaire (2008) find that many other factors in addition to bias, especially heteroskedasticity, can increase the ERP of bootstrap and asymptotic estimators.
- 7.
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Appendix
Appendix
5.1.1 Unbiasedness of the Wild First-Step Estimator, \(\hat{\beta }_{FE}^{w}\)
Lemma 1:
Since ε i is drawn independently and E(ε i = 0), \(E(\boldsymbol{\xi }_{i}^{w}\vert \mathbf{X}_{i}) = 0.\)
Proof of Lemma 1: From (5.17), \(\boldsymbol{\xi }_{i}^{w} =\boldsymbol{\hat{\xi }} _{i}\epsilon _{i}\vartheta\). Thus, \(E(\boldsymbol{\xi }_{i}^{w}\vert \mathbf{X}_{i}) = E(\boldsymbol{\hat{\xi }}_{i}\epsilon _{i}\vartheta \vert \mathbf{X}_{i})\) = \(E(\boldsymbol{\hat{\xi }}_{i}\vert \mathbf{X}_{i})\vartheta E(\epsilon _{i}\vert \mathbf{X}_{i}) = E(\boldsymbol{\hat{\xi }}_{i}\vert \mathbf{X}_{i})\vartheta E(\epsilon _{i}) = 0,\) since ε i is independent of \(\boldsymbol{\hat{\xi }}_{i}\) and X i and in addition E(ε i ) = 0 by definition in (5.15).
Theorem 1:
Given the FE conditional-mean assumption in (5.3) and Lemma 1, the wild bootstrap first-step estimator \(\hat{\beta }_{FE}^{w}\) is unbiased for \(\hat{\beta }_{FE}\) .
Proof of Theorem 1: Writing the vector form of (5.16) as \(\mathbf{y}_{i}^{w} = \mathbf{X}_{i}\hat{\beta }_{FE} + \mathbf{z}_{i}\hat{\gamma }_{FE} +\boldsymbol{\xi }_{ i}^{w}\) and substituting into (5.18), the first-step wild estimator can be written as
where \(\boldsymbol{\xi }_{i}^{w}\) is a (T × 1) vector. Then
using Lemma 1. Further, \(E[E(\hat{\beta }_{FE}^{w}\vert \mathbf{X}_{i})] = E(\hat{\beta }_{FE}^{w}) = \hat{\beta }_{FE}\).
5.1.2 Unbiasedness of the Wild Second-Step Estimator, \(\hat{\gamma }_{FE}^{w}\)
To show that the second-step wild estimator is unbiased, we substitute (5.19) into (5.20) to obtain
Now average (5.16) over t to obtain
and substitute (5.36) into (5.35) to yield
Lemma 2:
Since ε i is drawn independently and E(ε i = 0), \(E(\bar{\xi }_{it}^{w}\vert \mathbf{z}_{i},\bar{\mathbf{x}}_{i}) = 0.\)
Proof of Lemma 2: Use the definition of \(\xi _{it}^{w}\) in (5.17) and condition on \(\mathbf{z}_{i},\bar{\mathbf{x}}_{i}\). Then use the independence of ε i from \(\mathbf{z}_{i},\bar{\mathbf{x}}_{i}\).
Theorem 2:
Given Theorem 1 and Lemma 2, the wild second-step estimator, \(\hat{\gamma }_{FE}^{w}\) , is unbiased for \(\hat{\gamma }_{FE}\) .
Proof of Theorem 2:
after substituting from (5.37) for u i w and then applying Theorem 1 and Lemma 2. Finally, \(E[E(\hat{\gamma }_{FE}^{w}\vert \mathbf{z}_{i},\bar{\mathbf{x}}_{i})] = E(\hat{\gamma }_{FE}^{w}) = \hat{\gamma }_{FE}\).
5.1.3 Unbiasedness of the Pairs First-Step Estimator, \(\hat{\beta }_{FE}^{p}\)
To show the unbiasedness of the pairs first-step estimator, we need (5.3).
Lemma 3:
Given (5.3), \(E(\mathbf{Q}_{T}\boldsymbol{\hat{\xi }}_{i}\vert v_{i},\mathbf{X}_{i}) = 0\) .
Proof of Lemma 3: First,
using \(\mathbf{M}_{i} = \mathbf{I}_{T} -\mathbf{Q}_{T}\mathbf{X}_{i}{\biggl (\sum _{i}\mathbf{X}_{i}^{\prime}\mathbf{Q}_{T}\mathbf{X}{_{i}\biggr )}}^{-1}\mathbf{X}_{i}^{\prime}\mathbf{Q}_{T}\) and \(\boldsymbol{\xi }_{i} = (\mathbf{j}_{T} \otimes c_{i}) + \mathbf{e}_{i}\). Then, using (5.3) and the fact that Q T eliminates c i completes the proof.
Theorem 3:
Given Lemma 3, the bootstrap pairs first-step estimator, \(\hat{\beta }_{FE}^{p},\) is unbiased for \(\hat{\beta }_{FE}.\)
Proof of Theorem 3: Substitute \(\mathbf{Q}_{T}\mathbf{y}_{i}\) in (5.28) and take expectations. Then
using Lemma 3.
5.1.4 Unbiasedness of the Pairs Second-Step Estimator, \(\hat{\gamma }_{FE}^{p}\)
Theorem 4:
Given (5.3), (5.4), and Theorem 3, the pairs second-step estimator, \(\hat{\gamma }_{FE}^{p}\) , is unbiased for \(\hat{\gamma }_{FE}\) .
Proof of Theorem 4: Substituting (5.29) into (5.30) we obtain
We can relate \(\hat{u}_{i}\) to u i as follows:
Then substitute (5.42) into (5.41) to obtain
Conditioning on \((\mathbf{z}_{i},\bar{\mathbf{x}}_{i},v_{i})\), we use (5.8) and take the expectation of both sides to obtain
The first term on the right-hand-side of (5.44) is zero due to (5.3) and (5.4), while \(\hat{\beta }_{FE}\) is unbiased for β from Theorem 3.
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Atkinson, S.E., Cornwell, C. (2014). Inference in Two-Step Panel Data Models with Time-Invariant Regressors: Bootstrap Versus Analytic Estimators. In: Sickles, R., Horrace, W. (eds) Festschrift in Honor of Peter Schmidt. Springer, New York, NY. https://doi.org/10.1007/978-1-4899-8008-3_5
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