Inference in Two-Step Panel Data Models with Time-Invariant Regressors: Bootstrap Versus Analytic Estimators

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.

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

$$\displaystyle{ \hat{\beta }_{FE}^{w} = \hat{\beta }_{ FE} +{\biggl (\sum _{i}\mathbf{X}_{i}^{\prime}\mathbf{Q}_{T}\mathbf{X}{_{i}\biggr )}}^{-1}\sum _{ i}\mathbf{X}_{i}^{\prime}\mathbf{Q}_{T}\boldsymbol{\xi }_{i}^{w}, }$$

(5.33)

where $\boldsymbol{\xi }_{i}^{w}$ is a (T × 1) vector. Then

$$\displaystyle{ E(\hat{\beta }_{FE}^{w}\vert \mathbf{X}_{ i}) = \hat{\beta }_{FE} +{\biggl (\sum _{i}\mathbf{X}_{i}^{\prime}\mathbf{Q}_{T}\mathbf{X}{_{i}\biggr )}}^{-1}\sum _{ i}\mathbf{X}_{i}^{\prime}\mathbf{Q}_{T}E(\boldsymbol{\xi }_{i}^{w}\vert \mathbf{X}_{ i}) = \hat{\beta }_{FE}, }$$

(5.34)

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

$$\displaystyle{ u_{i}^{w} =\bar{ y}_{ i}^{w} -\bar{\mathbf{x}}_{ i}\hat{\beta }_{FE}^{w} -\mathbf{z}_{ i}\hat{\gamma }_{FE}. }$$

(5.35)

Now average (5.16) over t to obtain

$$\displaystyle{ \bar{y}_{i}^{w} =\bar{ \mathbf{x}}_{ i}\hat{\beta }_{FE} + \mathbf{z}_{i}\hat{\gamma }_{FE} +\bar{\xi }_{ i}^{w} }$$

(5.36)

and substitute (5.36) into (5.35) to yield

$$\displaystyle{ u_{i}^{w} =\bar{ \mathbf{x}}_{ i}(\hat{\beta }_{FE} -\hat{\beta }_{FE}^{w}) +\bar{\xi }_{ i}^{w}. }$$

(5.37)

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:

$$\displaystyle\begin{array}{rcl} E(\hat{\gamma }_{FE}^{w}\vert \mathbf{z}_{ i},\bar{\mathbf{x}}_{i})& =& \hat{\gamma }_{FE} +{\biggl (\sum _{i}\mathbf{z}_{i}^{^{\prime}}\mathbf{z}_{ i}^{{}\biggr )}}^{-1}{\biggl (\sum _{ i}\mathbf{z}_{i}^{^{\prime}}E(u_{ i}^{w}\vert \mathbf{z}_{ i},\bar{\mathbf{x}}_{i})\biggr )} \\ & =& \hat{\gamma }_{FE} +{\biggl (\sum _{i}\mathbf{z}_{i}^{^{\prime}}\mathbf{z}_{ i}^{{}\biggr )}}^{-1}{\biggl (\sum _{ i}\mathbf{z}_{i}^{^{\prime}}E\{[\bar{\mathbf{x}}_{ i}(\hat{\beta }_{FE} -\hat{\beta }_{FE}^{w}) +\bar{\xi }_{ i}^{w}]\vert \mathbf{z}_{ i},\bar{\mathbf{x}}_{i}\}\biggr )} \\ & =& \hat{\gamma }_{FE}, {}\end{array}$$

(5.38)

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,

$$\displaystyle\begin{array}{rcl} E(\mathbf{Q}_{T}\boldsymbol{\hat{\xi }}_{i}\vert v_{i},\mathbf{X}_{i})& =& E(\mathbf{M}_{i}\mathbf{Q}_{T}\boldsymbol{\xi }_{i}\vert v_{i},\mathbf{X}_{i}) \\ & =& E(\mathbf{Q}_{T}\boldsymbol{\xi }_{i}\vert v_{i},\mathbf{X}_{i})-\mathbf{Q}_{T}\mathbf{X}_{i}{\biggl (\sum _{i}\mathbf{X}_{i}^{\prime}\mathbf{Q}_{T}\mathbf{X}{_{i}\!\biggr )}}^{-1}\mathbf{X}_{ i}^{\prime}\mathbf{Q}_{T}E(\mathbf{Q}_{T}\boldsymbol{\xi }_{i}\vert v_{i},\mathbf{X}_{i}) \\ & =& 7E\{\mathbf{Q}_{T}[(\mathbf{j}_{T} \otimes c_{i}) + \mathbf{e}_{i}]\vert v_{i},\mathbf{X}_{i}\} \\ & & -\,\mathbf{Q}_{T}\mathbf{X}_{i}{\biggl (\sum _{i}\mathbf{X}_{i}^{\prime}\mathbf{Q}_{T}\mathbf{X}{_{i}\!\biggr )}}^{-1}\mathbf{X}_{ i}^{\prime}\mathbf{Q}_{T}E\{\mathbf{Q}_{T}[(\mathbf{j}_{T} \otimes c_{i})+\mathbf{e}_{i}] \\ & & \times \,\vert v_{i},\mathbf{X}_{i}\},\quad {}\end{array}$$

(5.39)

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

$$\displaystyle\begin{array}{rcl} E(\hat{\beta }_{FE}^{p}\vert v_{ i},\mathbf{X}_{i})& =& \hat{\beta }_{FE} +{\biggl (\sum _{i}v_{i}\mathbf{X}_{i}^{\prime}\mathbf{Q}_{T}\mathbf{X}{_{i}\biggr )}}^{-1}\sum _{ i}v_{i}\mathbf{X}_{i}^{\prime}\mathbf{Q}_{T}E(\mathbf{Q}_{T}\boldsymbol{\hat{\xi }}_{i}\vert v_{i},\mathbf{X}_{i}) \\ & =& \hat{\beta }_{FE}, {}\end{array}$$

(5.40)

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

$$\displaystyle{ \hat{\gamma }_{FE}^{p} = \hat{\gamma }_{ FE} +{\biggl (\sum _{i}v_{i}\mathbf{z}_{i}^{^{\prime}}\mathbf{z}_{ i}^{{}\biggr )}}^{-1}{\biggl (\sum _{ i}v_{i}\mathbf{z}_{i}^{^{\prime}}\hat{u}_{ i}\biggr )}. }$$

(5.41)

We can relate $\hat{u}_{i}$ to u _i as follows:

$$\displaystyle{ \hat{u}_{i} = u_{i} -\mathbf{z}_{i}{\biggl (\sum _{i}\mathbf{z}_{i}^{\prime}\mathbf{z}{_{i}\biggr )}}^{-1}\sum _{ i}\mathbf{z}_{i}^{\prime}u_{i}. }$$

(5.42)

Then substitute (5.42) into (5.41) to obtain

$$\displaystyle{ \hat{\gamma }_{FE}^{p} = \hat{\gamma }_{ FE} +{\biggl (\sum _{i}v_{i}\mathbf{z}_{i}^{^{\prime}}\mathbf{z}_{ i}^{{}\biggr )}}^{-1}\sum _{ i}v_{i}\mathbf{z}_{i}^{^{\prime}}u_{ i} -{\biggl (\sum _{i}\mathbf{z}_{i}^{\prime}\mathbf{z}{_{i}\biggr )}}^{-1}\sum _{ i}\mathbf{z}_{i}^{\prime}u_{i} }$$

(5.43)

Conditioning on $(\mathbf{z}_{i},\bar{\mathbf{x}}_{i},v_{i})$, we use (5.8) and take the expectation of both sides to obtain

$$\displaystyle{ E(u_{i}\vert \mathbf{z}_{i},\bar{\mathbf{x}}_{i},v_{i}) = E(\bar{\xi }_{i}\vert \mathbf{z}_{i},v_{i}) +\bar{ \mathbf{x}}_{i}E[(\hat{\beta }_{FE}-\beta )\vert \mathbf{z}_{i},\bar{\mathbf{x}}_{i},v_{i}]. }$$

(5.44)

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.