Robust estimation of dynamic fixed-effects panel data models

Abstract

This paper extends an existing outlier-robust estimator of linear dynamic panel data models with fixed effects, which is based on the median ratio of two consecutive pairs of first-order differenced data. To improve its precision and robustness properties, a general procedure based on higher-order pairwise differences and their ratios is designed. The asymptotic distribution of this class of estimators is derived. Further, the breakdown point properties are obtained under contamination by independent additive outliers and by the patches of additive outliers, and are used to select the pairwise differences that do not compromise the robustness properties of the procedure. The proposed estimator is additionally compared with existing methods by means of Monte Carlo simulations.

Appendix

Let us first derive formula (7) for $r_{{\varvec{j}}}$ given its definition in (3). The covariance between $\varDelta ^sy_{it}$ and $\varDelta ^py_{it-q}$ can be decomposed as

$$\begin{aligned} \mathrm{{cov}}(\varDelta ^sy_{it},\varDelta ^py_{it-q})&= \mathrm{{cov}}(y_{it},y_{it-q}) -\mathrm{{cov}}(y_{it},y_{it-q-p})\\ \nonumber&-\,\mathrm{{cov}}(y_{it-s},y_{it-q}) +\mathrm{{cov}}(y_{it-s},y_{it-q-p}). \end{aligned}$$

(18)

Next, a general expression for $\mathrm{{cov}}(y_{it},y_{it-q})$ has to be derived, which is time independent due to the stationarity of $y_{it}$. Given Assumptions A.1 and A.2, let $\sigma _{\eta }^2$ and $\sigma _{y}^2$ denote the variance of $\eta _i$ and $y_{it}$, respectively, and let $\sigma _{y\eta }=\mathrm{{cov}}(y_{it},\eta _i)$ (this covariance is again time independent because of Assumption A.2). Thus each of the terms in (18) can be generically expressed as [using recursive substitution for $y_{it}$ from the model Eq. (1)]

$$\begin{aligned} \mathrm{{cov}}(y_{it},y_{it-q})&= \mathrm{{cov}} \left\{ \left[ \left( \sum _{k=0}^{q-1}\alpha ^k\right) \eta _i+\alpha ^qy_{it-q}+\sum _{k=0}^{q-1}\alpha ^k\varepsilon _{it-k} \right] ,y_{it-q}\right\} \nonumber \\&= \left( \sum _{k=0}^{q-1}\alpha ^k\right) \sigma _{y\eta }+\alpha ^q\sigma _{y}^2 =\frac{\sigma _{y\eta }^2}{1-\alpha } +\left( \sigma _{y}^2-\frac{\sigma _{y\eta }}{1-\alpha }\right) \alpha ^q. \end{aligned}$$

(19)

It follows that $\mathrm{{cov}}(\varDelta ^sy_{it},\varDelta ^py_{it-q})=c\left( \alpha ^q-\alpha ^{q+p}-\alpha ^{|s-q|}+\alpha ^{|s-p-q|}\right)$, where $c:=\sigma _{y}^2-\sigma _{y\eta }/(1-\alpha )$. The result in (7) now follows directly by the application of the same steps to $\mathrm{{var}}(\varDelta ^py_{it-q})$:

$$\begin{aligned} \mathrm{{var}}(\varDelta ^py_{it-q})&= \mathrm{{cov}}(y_{it-q},y_{it-q})-\mathrm{{cov}}(y_{it-q},y_{it-q-p})-\mathrm{{cov}}(y_{it-q-p},y_{it-q})\nonumber \\&+\,\mathrm{{cov}}(y_{it-q-p},y_{it-q-p}) = c(1 - \alpha ^p - \alpha ^p + 1) = 2c(1-\alpha ^p). \nonumber \\ \end{aligned}$$

(20)

1.1 Asymptotic distribution

Lemma 1

Let the moment conditions in (9) be considered for a finite set $\mathcal{J }$ of indices and Assumptions A.1–A.4 hold. Then for a fixed $T$ and $n\rightarrow \infty$

$$\begin{aligned} \sqrt{n}\varvec{g}_n(\alpha )\rightarrow \mathrm{{N}}(0,\varvec{V}), \end{aligned}$$

(21)

where the length $l({\varvec{j}})={\varvec{A}}x\{s,p+q\}$ and the square matrix $\varvec{V}$ has a typical element with indices ${\varvec{j}},{\varvec{j}}^{\prime }\in \mathcal{J }$

$$\begin{aligned} v_{{\varvec{j}}{\varvec{j}}^{\prime }}&= \frac{\pi ^2\sqrt{ (1 \!-\! \alpha ^s \!-\! \frac{1}{4}[\alpha ^q \!-\! \alpha ^{p+q} \!-\! \alpha ^{|s-q|} + \alpha ^{|s-p-q|}]^2 ) (1 - \alpha ^{s^{\prime }} -\frac{1}{4} [\alpha ^{q^{\prime }} - \alpha ^{p^{\prime }+q^{\prime }} - \alpha ^{|s^{\prime }-q^{\prime }|} + \alpha ^{|s^{\prime }-p^{\prime }-q^{\prime }|}]^2 ) }}{\sqrt{[T-l(\varvec{j})][T-l(\varvec{j}^{\prime })]}}\nonumber \\&\quad \times \, \mathrm{{E}}\left[ \left( \sum _{t=l({\varvec{j}})+1}^T\mathrm{{sgn}}(\varDelta ^sy_{it}-r_{{\varvec{j}}}\varDelta ^py_{it-q})\mathrm{{sgn}}(\varDelta ^py_{it-q})\right) \right. \nonumber \\&\quad \times \left. \left( \sum _{t=l({\varvec{j}}^{\prime })+1}^T\mathrm{{sgn}}(\varDelta ^{s^{\prime }}y_{it}-r_{{\varvec{j}}^{\prime }}\varDelta ^{p^{\prime }}y_{it-q^{\prime }})\mathrm{{sgn}}(\varDelta ^{p^{\prime }}y_{it-q^{\prime }}) \right) \right] . \nonumber \\ \end{aligned}$$

(22)

Proof

To prove the lemma, note that $\varvec{g}_{n{\varvec{j}}}(\alpha )$ can be rewritten in the following way for any index ${\varvec{j}}$:

$$\begin{aligned} \sqrt{n}g_{n{\varvec{j}}}(\alpha )&= \sqrt{n}\left[ 2(1-\alpha ^p)r_{{\varvec{j}}}-\alpha ^q+\alpha ^{q+p}+\alpha ^{|s-q|}-\alpha ^{|s-p-q|}\right] \nonumber \\&= \sqrt{n}2(1-\alpha ^p) \left( \hat{r}_{n{\varvec{j}}}-\frac{\alpha ^q-\alpha ^{q+p}-\alpha ^{|s-q|}+\alpha ^{|s-p-q|}}{2(1-\alpha ^p)}\right) \nonumber \\&= \sqrt{n}2(1-\alpha ^p) \left( \hat{r}_{n{\varvec{j}}}-r_{{\varvec{j}}}\right) \nonumber \\&= \frac{2(1-\alpha ^p) }{\sqrt{T}} \sqrt{nT}(\hat{r}_{n{\varvec{j}}}-r_{{\varvec{j}}}). \end{aligned}$$

(23)

Since the proof of Dhaene and Zhu (2009, Lemma 1) is valid not only for the first differences, but for any $s$th difference, we can use the results derived in the proof of Lemma 1 in Dhaene and Zhu (2009) to state that $( \sqrt{nT} [\hat{r}_{n{\varvec{j}}}-r_{{\varvec{j}}}])_{{\varvec{j}} \in \mathcal{J }}$ has the same asymptotic distribution as

$$\begin{aligned} \left( \frac{\sqrt{T}}{\sqrt{T-l({\varvec{j}})}}\frac{a_{\varvec{j}}^{-1}(r_{\varvec{j}})}{\sqrt{n[T-l({\varvec{j}})]}} \sum _{i=1}^n \sum _{t=l({\varvec{j}})+1}^T \mathrm{{sgn}}(\varDelta ^sy_{it}-r_{{\varvec{j}}}\varDelta ^py_{it-q})\mathrm{{sgn}}(\varDelta ^py_{it-q}) \right) _{{\varvec{j}} \in \mathcal{J }}, \end{aligned}$$

where the constant $a_{\varvec{j}}(r_{\varvec{j}}) = 2f_{i{\varvec{j}}}(0) \mathrm{{E}}|\varDelta ^p y_{it-q}| =2\pi ^{-1}\sqrt{\mathrm{{var}}(\varDelta ^py_{it-q}) / \mathrm{{var}}(\varDelta ^sy_{it}-r_{{\varvec{j}}}\varDelta ^py_{it-q})}$ and $f_{i{\varvec{j}}}$ denotes the density function of $\varDelta ^sy_{it}-r_{{\varvec{j}}}\varDelta ^py_{it-q}$.

To derive the expression for $a_{\varvec{j}}(r_{\varvec{j}})$, note that the variables $\varDelta ^sy_{it}-r_{{\varvec{j}}}\varDelta ^py_{it-q}$ and $\varDelta ^py_{it-q}$ are uncorrelated because of definition (2), and by Assumption A.3, they are also independent and normally distributed around zero. From Eq. (20), it follows that

$$\begin{aligned} \left( \begin{array}{cc} \varDelta ^sy_{it}-r_{{\varvec{j}}}\varDelta ^py_{it-q}\\ \varDelta ^py_{it-q} \end{array}\right) \sim \mathrm{{N}} \left[ \mathbf{0}, 2c \left( \begin{array}{ll} 1-\alpha ^s-r_{{\varvec{j}}}^2(1-\alpha ^p) &{}\quad 0\\ 0 &{}\quad 1-\alpha ^p \end{array}\right) \right] \end{aligned}$$

(24)

because Eq. (3) implies $\mathrm{{cov}}(\varDelta ^sy_{it},\varDelta ^py_{it-q}) = r_{\varvec{j}}\mathrm{{var}}(\varDelta ^py_{it-q})$ and thus

$$\begin{aligned} \mathrm{{var}}(\varDelta ^sy_{it}\!-\!r_{{\varvec{j}}}\varDelta ^py_{it-q})&= \mathrm{{var}}(\varDelta ^sy_{it})\!+\!r_{{\varvec{j}}}^2\mathrm{{var}}(\varDelta ^py_{it-q}) -2r_{\varvec{j}}\mathrm{{cov}}(\varDelta ^sy_{it},\varDelta ^py_{it-q}) \\&= \mathrm{{var}}(\varDelta ^sy_{it})-r_{{\varvec{j}}}^2\mathrm{{var}}(\varDelta ^py_{it-s}). \end{aligned}$$

Hence, substituting from (7) for $r_{\varvec{j}}$ implies $a_{\varvec{j}}(r_{\varvec{j}}) = 2\pi ^{-1} \sqrt{ (1-\alpha ^p)^2 / (1 - \alpha ^s - [\alpha ^q - \alpha ^{p+q} - \alpha ^{|s-q|} + \alpha ^{|s-p-q|}]^2/4 )}$.

The claim of the lemma now follows from the application of the central limit theorem to (24) with respect to $n\rightarrow \infty$ because the nominator of $a_{\varvec{j}}(r_{\varvec{j}})$ cancels with the coefficient $2(1-\alpha ^p)$ in (23). $\square$

Proof

(Proof of Theorem 1) The estimator $\hat{\alpha }_n$ is defined by the sample analogs of Eq. (9), which are deterministic functions of $\hat{r}_{n{\varvec{j}}}$. Thus the stochastic behavior of the moment equations is fully determined by the asymptotic properties of $\hat{r}_{n{\varvec{j}}}$, which are given in Lemma 1. To derive the asymptotic distribution of $\hat{\alpha }_n$, we will thus use general consistency and asymptotic normality theorems given, for e.g., in Arellano (2003, Appendix A.4 and A.5).

Because the moment conditions $\varvec{g}(\alpha )=0$ are continuous functions of the parameter, $\hat{r}_{n{\varvec{j}}}$ converges to $r_{\varvec{j}}$ (uniformly as it is independent of $\alpha$ and the set $\mathcal{J }$ of indices ${\varvec{j}}$ is finite), and Assumption A.4 holds, the consistency of $\hat{\alpha }_n$ follows from the standard consistency theorem (e.g., Arellano 2003, Appendix A.4) if the true parameter $\alpha$ is uniquely identified. This however directly follows from $(1,1,1)^{\prime }\in \mathcal{J }$ because this first equation corresponding to $g_{111}(\alpha ) = (1-\alpha )(2r_{111}+1-\alpha )=0$ has only one solution $\alpha =1+2r_{111}$ on $(-1,1)$.

Further, the true parameter $\alpha \in (-1,1)$, moment conditions $\varvec{g}_n(\alpha )$ are differentiable with respect to $\alpha$, and their derivatives $\varvec{d}_n(\alpha ) = \partial \varvec{g}_n(\alpha )/\partial \alpha$ converge to $\varvec{d}(\alpha )=\partial \varvec{g}(\alpha )/\partial \alpha$ uniformly in $\alpha$ because their only stochastic element $\hat{r}_{n{\varvec{j}}}$ is independent of $\alpha$. Moreover, Lemma 1 guarantees the asymptotic normality of the moment conditions $\varvec{g}_n(\alpha )$. We can thus follow the asymptotic normality theorem (e.g., Arellano 2003, Appendix A.5, p. 187), which is based on the Taylor expansion of the first-order conditions of (10):

$$\begin{aligned} \frac{\partial \varvec{g}_n^{^{\prime }}(\hat{\alpha }_n)}{\partial \alpha } {{\varvec{A}}}_n \varvec{g}_n(\hat{\alpha }_n) = \varvec{d}_n^{^{\prime }}(\hat{\alpha }_n){\varvec{A}}_n \varvec{g}_n(\hat{\alpha }_n) = 0. \end{aligned}$$

Consequently, it holds for some $\xi _n \in (\hat{\alpha }_n,\alpha )$ that

$$\begin{aligned} \varvec{d}_n^{^{\prime }}(\hat{\alpha }_n){\varvec{A}}_n [ \varvec{g}_n(\alpha ) + \varvec{d}_n(\xi _n) (\hat{\alpha }_n - \alpha ) ] = 0. \end{aligned}$$

Given the uniform convergence of $\varvec{d}_n(\alpha ) \rightarrow \varvec{d}(\alpha )$, $\varvec{d}_n(\xi _n)\rightarrow \varvec{d}(\alpha )$, and ${\varvec{A}}_n\rightarrow {\varvec{A}}$ as $n\rightarrow \infty$, it follows from Assumption A.4 that

$$\begin{aligned} \sqrt{n} (\hat{\alpha }_n - \alpha ) = [ \varvec{d}^{\prime }(\alpha ){\varvec{A}} \varvec{d}(\alpha ) (1+o_p(1))]^{-1} (1+o_p(1)) \varvec{d}^{\prime }(\alpha ) {\varvec{A}} \cdot \sqrt{n} \varvec{g}_n(\alpha ). \end{aligned}$$

Finally, Lemma 1 implies that, in distribution,

$$\begin{aligned} \sqrt{n}(\hat{\alpha }_n-\alpha ) \rightarrow \mathrm{{N}}(0, (\varvec{d}^{\prime }{\varvec{A}}\varvec{d})^{-1} \varvec{d}^{\prime }{\varvec{A}} \varvec{V} {\varvec{A}} \varvec{d} (\varvec{d}^{\prime }{\varvec{A}}\varvec{d})^{-1} ) \end{aligned}$$

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as $n\rightarrow \infty$, where $\varvec{V}$ is derived in Lemma 1 and the abbreviated notation $\varvec{d}=\varvec{d}(\alpha )$ is used ($\alpha$ represents the true parameter value). $\square$

1.2 Robustness properties

Proof

(Theorems 2 and 3) The breakdown properties stated in Theorems 2 and 3 are a direct consequence of Theorems 5 and 8 stated in Dhaene and Zhu (2009).

Since $r_{\varvec{j}}$ is considered only for ${\varvec{j}}=(s,s,p)^{\prime }$, where both $s$ and $p$ are odd, $r_{\varvec{j}} = -(1-\alpha ^s)/2$. This mapping of $\alpha$ to $r_{\varvec{j}}=-(1-\alpha ^s)/2$ has the same important properties for $s=1$ and any odd $s>1$: it maps interval $(-1,0)$ to $(-1,-1/2)$ and interval $(0,1)$ to $(-1/2,0)$, it is continuous, and it is strictly increasing on $(-1,1)$. The proofs of Dhaene and Zhu (2009, Theorems 5 and 8) thus apply not only to the case of $s=p=1$, but any odd $s$ and $p$ with one main adjustment: the variables $\varDelta ^sy_{it}-r_{{\varvec{j}}}\varDelta ^py_{it-s}$ and $\varDelta ^py_{it-s}$ have to be standardized Dhaene and Zhu (2009, Eq. (17)) and their variances generally depend on the values of $s$ and $p$.

By Eq. (2), the variables $\varDelta ^sy_{it}-r_{{\varvec{j}}}\varDelta ^py_{it-s}$ and $\varDelta ^py_{it-s}$ are uncorrelated, and by Assumption A.3, they are independent and normally distributed around zero. From Eq. (20), it follows that

$$\begin{aligned} \left( \begin{array}{cc} \varDelta ^sy_{it}-r_{{\varvec{j}}}\varDelta ^py_{it-s}\\ \varDelta ^py_{it-s} \end{array}\right) \sim \mathrm{{N}} \left[ 0, 2c \left( \begin{array}{ll} 1-\alpha ^s-r_{{\varvec{j}}}^2(1-\alpha ^p) &{}\quad 0\\ 0 &{}\quad 1-\alpha ^p \end{array}\right) \right] \end{aligned}$$

(26)

because Eq. (3) implies $\mathrm{{cov}}(\varDelta ^sy_{it},\varDelta ^py_{it-s}) = r_{\varvec{j}}\mathrm{{var}}(\varDelta ^py_{it-s})$ and thus

$$\begin{aligned} \mathrm{{var}}(\varDelta ^sy_{it}\!-\!r_{{\varvec{j}}}\varDelta ^py_{it-s})&= \mathrm{{var}}(\varDelta ^sy_{it})\!+\!r_{{\varvec{j}}}^2\mathrm{{var}}(\varDelta ^py_{it-s}) -2r_{\varvec{j}}\mathrm{{cov}}(\varDelta ^sy_{it},\varDelta ^py_{it-s})\\ \nonumber&=\mathrm{{var}}(\varDelta ^sy_{it})-r_{{\varvec{j}}}^2\mathrm{{var}}(\varDelta ^py_{it-s}). \end{aligned}$$

Consequently, the standardization of $b^{\prime }$ in Dhaene and Zhu (2009, Eq. (17)) by the ratio of the two variances in (26) corresponds to the factor

$$\begin{aligned} \frac{1-\alpha ^s}{1-\alpha ^p} - r_{\varvec{j}}^2 = \frac{1-\alpha ^s}{1-\alpha ^p} - \frac{1}{4} (1-\alpha ^s)^2 \end{aligned}$$

(27)

instead of to factor $1-r_{111}^2$ in the original paper. This concludes the proof of Theorem 2 once we substitute $-(1-\alpha ^s)/2$ for $r_{\varvec{j}}$ in Dhaene and Zhu (2009, Theorems 5 and 8). Regarding the proof of Theorem 3, the above standardization applies as well, but the exact dependence of the contamination probabilities on the orders $s$ and $p$ follows after a long technical derivation, which extends that of Dhaene and Zhu (2009, Theorem 8) and is done by Aquaro (2013). $\square$

Proof

(Theorem 4) The result follows directly from the definition of the breakdown point. The GMM estimator $\hat{\alpha }_n$ minimizes $\sum _{\varvec{j}}\in \mathcal{J } A_{\varvec{j}}{\varvec{j}} g_{n{\varvec{j}}}^2(\alpha )$. As this objective function depends on the data only by means of the estimates $\hat{r}_{n{\varvec{j}}}$, the estimator $\hat{\alpha }_n$ breaks down, that is, becomes fully independent of the data-generating process, if and only if all estimates $\hat{r}_{n{\varvec{j}}}$ break down. In other words, as long as the limit of $\hat{r}_{n{\varvec{j}}}$ corresponds to some $\alpha ^{\prime }<c$ (if $\alpha <c$) or to some $\alpha ^{\prime }>c$ (if $\alpha >c$) for some triplet ${\varvec{j}}^{\prime }$, the GMM estimator—averaging across different ratios $\hat{r}_{n{\varvec{j}}}$—will not break down to the fixed $c$. $\square$