Joint tests for dynamic and spatial effects in short panels with fixed effects and heteroskedasticity

Abstract

Simple and reliable tests are proposed for testing the existence of dynamic and/or spatial effects in fixed-effects panel data models with small T and possibly heteroskedastic errors. The tests are constructed based on the adjusted quasi scores (AQS), which correct the conditional quasi scores given the initial differences to account for the effect of initial values. To improve the finite sample performance, standardized AQS tests are also derived, which are shown to have much improved finite sample properties. All the proposed tests are robust against nonnormality, but some are not robust against cross-sectional heteroskedasticity (CH). A different type of adjustments is made on the AQS functions, leading to a set of tests that are fully robust against unknown CH. Monte Carlo results show excellent finite sample performance of the standardized versions of the AQS tests.

Appendices

Appendix A: Some useful lemmas

The development and the proofs of theoretical results reported in this paper depend critically on the following lemmas.

Lemma A.1

(Kelejian and Prucha 1999; Lee 2002). Let $\{A_n\}$ and $\{B_n\}$ be two sequences of $n\times n$ matrices that are uniformly bounded in both row and column sums. Let $C_n$ be a sequence of conformable matrices whose elements are uniformly $O(\iota _{n}^{-1})$. Then,

• (i) the sequence $\{A_n B_n\}$ is uniformly bounded in both row and column sums,

• (ii) the elements of $A_n$ are uniformly bounded and tr$(A_n) = O(n)$, and

• (iii) the elements of $A_n C_n$ and $C_n A_n$ are uniformly $O(\iota _{n}^{-1})$.

Lemma A.2

(Lee 2004, p.1918). For $W_r$ and $B_r$, $r=1,3$, defined in Model (1.1), if $\Vert W_r\Vert$ and $\Vert B_{r0}^{-1}\Vert$ are uniformly bounded, where $\Vert \cdot \Vert$ is a matrix norm, then $\Vert B_{r}^{-1}\Vert$ is uniformly bounded in a neighborhood of $\lambda _{r0}$.

Lemma A.3

(Lee 2004, p.1918). Let $X_n$ be an $n\times p$ matrix. If the elements $X_n$ are uniformly bounded and $\lim _{n\rightarrow \infty }\frac{1}{n}X^{\prime }_n X_n$ exists and is nonsingular, then $P_n = X_n(X^{\prime }_n X_n)^{-1}X^{\prime }_n$ and $M_n = I_n - P_n$ are uniformly bounded in both row and column sums.

Lemma A.4

(Li and Yang 2020b) Let $\{A_n\}$ be a sequence of $n \times n$ matrices that are uniformly bounded in either row or column sums. Suppose that the elements $a_{n,ij}$ of $A_n$ are $O(\iota ^{-1})$ uniformly in all i and j. Let $v_n$ be a random n-vector of inid elements satisfying Assumption B, and $b_n$ a constant n-vector of elements of uniform order $O(\iota ^{-{1}/{2}})$. Then,

$$\begin{aligned} \begin{array}{ll} (i) \; \mathrm{E}(v^{\prime }_n A_n v_n) = O(\frac{n}{\iota _n}), &{} (ii) \mathrm{Var}(v^{\prime }_n A_n v_n) = O(\frac{n}{\iota _n}), \\ (iii) \mathrm{Var}(v^{\prime }_n A_n v_n + b_n^{\prime } v_n) = O(\frac{n}{\iota _n}), &{} (iv) v^{\prime }_n A_n v_n = O_p(\frac{n}{\iota _n}), \\ (v) \; v^{\prime }_n A_n v_n -\mathrm{E}(v^{\prime }_n A_n v_n) = O_p((\frac{n}{\iota _n})^{\frac{1}{2}}), &{} (vi) v^{\prime }_n A_n b_n = O_p((\frac{n}{\iota _n})^{\frac{1}{2}}), \end{array} \end{aligned}$$

and (vii), the results (iii) and (vi) remain valid if $b_n$ is a random n-vector independent of $v_{n}$ such that $\{\mathrm{E}(b_{ni}^{2})\}$ are of uniform order $O(\iota _{n}^{-1})$.

Lemma A.5

(Li and Yang 2020b): Let $\{\Phi _{n}\}$ be a sequence of $n\times n$ matrices with row and column sums being uniformly bounded, and elements of uniform order $O(\iota _{n}^{-1})$. Let $v_{n}$ be a random n-vector satisfying Assumption B. Let $b_{n}=\{b_{ni}\}$ be an $n\times 1$ random vector, independent of $v_{n}$, such that $(i)\ \{\mathrm{E}(b_{ni}^2)\}$ are of uniform order $O(\iota _{n}^{-1})$, (ii) sup$_{i}E|b_{ni}|^{2+\epsilon _{0}} < \infty$, (iii) $\frac{\iota _n}{n}\sum _{i=1}^{n}[\phi _{n,ii}(b_{ni}-\mathrm{E}b_{ni})]=o_p(1)$ where $\{\phi _{n,ii}\}$ are the diagonal elements of $\Phi _{n}$, and (iv) $\frac{\iota _n}{n}\sum _{i=1}^{n}[b_{ni}^{2}-\mathrm{E}(b_{ni}^{2})]=o_p(1)$. Let ${{\mathcal {H}}}_n$= $\mathrm{diag}(h_{n1},\ldots , h_{nn})$. Define the bilinear-quadratic form:

$$\begin{aligned} Q_{n} = b_{n}'v_{n} + v_{n}^{\prime }\Phi _{n}v_{n} - \sigma _{v}^{2}tr(\Phi _{n} {{\mathcal {H}}}_n), \end{aligned}$$

and let $\sigma _{Q_{n}}^{2}$ be the variance of $Q_n$. If $\lim _{n\rightarrow \infty }\iota _{n}^{1+2/\epsilon _{0}}/n=0$ and $\{\frac{\iota _n}{n}\sigma _{Q_{n}}^{2}\}$ are bounded away from zero, then $Q_{n}/\sigma _{Q_{n}}\overset{d}{\longrightarrow }N(0,1)$.

The following lemma extends the formulations in Sect. 2.2 to allow for unknown CH. Its proof follows the results of Theorems 3.2 and 3.3 of Li and Yang (2020b). Recall: $A^{u}$, $A^{l}$ and $A^{d}$ denote the upper-triangular, lower-triangular, and diagonal matrix of a square matrix A; $\Pi _{t}$, $\Phi _{ts}$ and $\Psi _{ts}$ the submatrices of $\Pi$, $\Phi$, and $\Psi$ partitioned according to $t, s = 2, \ldots , T$; $\Psi _{t+}=\sum _{s=2}^{T}\Psi _{ts}$, $\Theta =\Psi _{2+}(B_{30}B_{10})^{-1}$, $\Delta y_{1}^{\circ }=B_{30}B_{10}\Delta y_{1}$, and $\Delta y_{1t}^{*}=\Psi _{t+}\Delta y_{1}$.

Lemma A.6

Suppose Assumptions A, B$^{*}$, C-E hold for Model (2.1). Consider the linear, quadratic, and bilinear forms, $Q(\psi _{0})=\{(\Pi \Delta ^{\prime }v)^{\prime }, \Delta v'\Phi \Delta v, (\Delta v'\Psi \Delta {{\mathbf {y}}}_1)^{\prime }\}^{\prime }$, associated with the model. Assume the elements of $\Pi$ ($N\times 1$) are uniformly bounded, and the matrices $\Phi$ and $\Psi$ ($N\times N$) are uniformly bounded in both row and column sums. Define

$$\begin{aligned}\begin{array}{lll} g_{1i} &{}=&{} \sum _{t=2}^{T} \Pi '_{it} \Delta v_{it}, \\ g_{2i} &{}=&{} \sum _{t=2}^{T}(\Delta v_{it}\Delta \xi _{it} + \Delta v_{it}\Delta v_{it}^{*}-\sigma _{v0}^{2}d_{it}),\\ g_{3i} &{}=&{} \Delta v_{2i}\Delta \zeta _{i} + \Theta _{ii}(\Delta v_{2i}\Delta y_{1i}^{\circ }+\sigma _{v0}^{2}h_{ni}) + \sum _{t=3}^{T}\Delta v_{it}\Delta y_{1i t}^{*}, \end{array} \end{aligned}$$

where $\xi _{t} = \sum _{s=2}^{T}(\Phi _{st}^{u \prime } + \Phi _{ts}^{l}) \Delta v_{s}$, $\Delta v_{t}^{*} = \sum _{s=2}^{T}\Phi _{ts}^{d}\Delta v_{s}$, and $\{d_{it}\}$ are the diagonal elements of $\Phi (C\otimes {{\mathcal {H}}}_n)$, $\{\Delta \zeta _i\} = \Delta \zeta = (\Theta ^{u}+\Theta ^{l})\Delta y_{1}^{\circ }$, and $\mathrm{diag}\{\Theta _{ii}\}=\Theta ^{d}$. Then, we have,

$(i)\;\;\; Q(\psi _{0})-\mathrm{E}[Q(\psi _{0})] = \sum _{i=1}^{n}{} \mathbf{g}_{i}$, where $\mathbf{g}_{i} = (g_{1i}, g_{2i}, g_{3i})'$,

$(ii)\;\; \frac{1}{N}[Q(\psi _{0})-\mathrm{E}(Q(\psi _{0}))] \overset{D}{\longrightarrow } N(0, \lim _{n\rightarrow \infty }\frac{1}{N}\Gamma )$, where $\Gamma =\mathrm{Var}(Q(\psi _{0}))$.

$(iii)\ \mathrm{Var}[Q(\psi _{0})] = \sum _{i=1}^{n}\mathrm{E}(\mathbf{g}_{i}{} \mathbf{g}'_{i})$, and $(iv)\; \frac{1}{N} \sum _{i=1}^{n}[\mathbf{g}_{i}{} \mathbf{g}'_{i}-\mathrm{E}(\mathbf{g}_{i}\mathbf{g}'_{i})] = o_p(1)$.

Appendix B: Some technical details

We sketch the proofs of the theorems, corollaries, and lemmas. Details are given in Supplementary Appendix available at http://www.mysmu.edu/faculty/zlyang/, including the detailed expressions for the derivative matrices of the three AQS functions, which is referred to loosely as the Hessian matrix in this paper.

Proof Theorem 2.1

The proof of Theorem 2.1 follows closely the proofs of Theorems 3.2 and 3.3 of Yang (2018a), and is typically simpler as under $H_{0}^\mathtt{M}$ the model becomes simpler. The Hessian matrix $\frac{\partial }{\partial \psi ^{\prime }}S^{*}(\psi )$ used to estimate $\Sigma ^{*}(\psi _{0})$ can be easily derived based on the expression of $S^{*}(\psi )$ given in (2.2). It can found in Yang (2018a, Proof of Theorem 3.2), and also in the Supplementary Appendix to this paper containing additional ‘asymmetric components’ that did not appear in Yang (2018a).

Proof of Corollary 2.1

The quantities needed for evaluating the AQS function defined in (2.4) become: $\Pi _{1}=\frac{1}{\sigma _{v0}^2}{{\mathbf {C}}}^{-1}\Delta X$, $\Pi _{2}=\frac{1}{\sigma _{v0}^2}{{\mathbf {C}}}^{-1}{{\mathbb {B}}}_{-1}\Delta X\beta$, $\Pi _{3}=\frac{1}{\sigma _{v0}^2}{{\mathbf {C}}}^{-1}{{\mathbf {W}}}_{1}\Delta X\beta$, $\Pi _{4}=\frac{1}{\sigma _{v0}^2}{{\mathbf {C}}}^{-1}{{\mathbf {W}}}_{2}{{\mathbb {B}}}_{-1}\Delta X\beta$, $\Phi _{1}= \frac{1}{2\sigma _{v0}^4}{{\mathbf {C}}}^{-1}$, $\Phi _{2}$=$\frac{1}{\sigma _{v0}^2}{{\mathbf {C}}}^{-1}{{\mathbb {B}}}_{-1}$, $\Phi _{3}=\frac{1}{\sigma _{v0}^2}{{\mathbf {C}}}^{-1}{{\mathbf {W}}}_{1}$, $\Phi _{4}=\frac{1}{\sigma _{v0}^2}{{\mathbf {C}}}^{-1}{{\mathbf {W}}}_{2}{{\mathbb {B}}}_{-1}$, $\Phi _{5}=\frac{1}{2\sigma _{v0}^2}[C^{-1}\otimes (W'_{3}+W_{3})]$, $\Psi _{1}=\frac{1}{\sigma _{v0}^2}{{\mathbf {C}}}^{-1}{{\mathbb {R}}}_{-1}$, $\Psi _{2}=0$, $\Psi _{3}=\frac{1}{\sigma _{v0}^2}{{\mathbf {C}}}^{-1}{{\mathbf {W}}}_{2}{{\mathbb {R}}}_{-1}$, ${{\mathbb {R}}}_{-1}=\mathrm{blkdiag}(I_n, 0,\ldots , 0)$, ${{\mathbb {B}}}_{-1} = I_{T-1}^{*} \otimes I_n$, and $I_{T-1}^{*}$ is a $(T-1)\times (T-1)$ matrix with elements 1 on the positions immediately below the diagonal elements, and zero elsewhere. Further, ${{\mathcal {B}}}_0 = 0_n$, and hence ${{\mathbf {D}}}_0=-C\otimes I_n$ and ${{\mathbf {D}}}_{-10}=-C_{-1}\otimes I_n$, where

$$\begin{aligned} C_{-1} = \left( \begin{array}{ccccccc} -1&{}\quad 0&{}\quad 0&{}\quad \cdots &{}\quad 0&{}\quad 0&{}\quad 0\\ 2&{}\quad -1&{}\quad 0&{}\quad \cdots &{}\quad 0&{}\quad 0&{}\quad 0\\ -1&{}\quad 2&{}\quad -1&{}\quad \cdots &{}\quad 0&{}\quad 0&{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots \\ 0&{}\quad 0&{}\quad 0&{}\quad \cdots &{}\quad 0&{}\quad 2&{}\quad -1\\ \end{array} \right) _{(T-1)\times (T-1)}. \end{aligned}$$

These show that with $\psi _0=(\beta '_{0},\sigma _{v0},0,0,0,0)'$, all the $\Phi$ and $\Psi$ matrices are either of the form $A\otimes I_n$ or $A\otimes W$ for some $(T-1)\times (T-1)$ matrix A and a spatial weight matrix W satisfying Assumption D. Thus, $\mathrm{E}[S^{*}(\psi _0)]=0$ even when the errors are heteroskedastic. Hence by Lemma A.5, we have $\frac{1}{\sqrt{N}}S^*(\psi _{0}) \overset{D}{\longrightarrow } N[0, \lim _{n\rightarrow \infty }\frac{1}{N}\Gamma ^{*}(\psi _{0})]$.

By the mean value theorem (MVT), one easily shows that $\frac{1}{\sqrt{N}}[S_{\delta }^*({\tilde{\psi }})-S_{\delta }^*(\psi _{0})]=o_p(1)$, where ${\tilde{\psi }}=({\tilde{\beta }}', {\tilde{\sigma }}_{v0}^{2},0,0,0,0)'$ and we note that the OLS estimators ${\tilde{\beta }}$ and ${\tilde{\sigma }}_{v0}^{2}$ are robust against unknown heteroskedasticity $\{h_{ni}\}$. Now, since by (2.12) $S^*(\psi _{0})=\sum _{i=1}^{n} \mathbf{g}_{i}$, where $\{\mathbf{g}_{i},{{\mathcal {F}}}_{n,i}\}$ form a vector MD sequence, we have $\frac{1}{N}\sum _{i=1}^{n}[\mathbf{g}_{i}{} \mathbf{g}'_{i}-\mathrm{E}(\mathbf{g}_{i}{} \mathbf{g}'_{i})]=o_p(1)$ by Lemma A.6. By MVT and the consistency of ${\tilde{\beta }}$ and ${\tilde{\sigma }}_{v0}^{2}$, one shows that $\frac{1}{N}\sum _{i=1}^{n}(\tilde{\mathbf{g}}_{i}\tilde{\mathbf{g}}'_{i}-\mathbf{g}_{i}{} \mathbf{g}'_{i})=o_p(1)$ under heteroskedasticity. Finally, it is easy to show that $\mathrm{plim}_{n\rightarrow \infty }(\widetilde{\Lambda }-{\Lambda })=0$, using the simplified expression of $H^{*}(\psi )$ and MVT. $\square$

Proof of Lemma 2.1

Consider the AQS vector $S^{*}(\beta ,\sigma _{v}^{2},\rho )$ for the DPD model, and the concentrated AQS function which defines ${\tilde{\rho }}$ under $H_{0}^\mathtt{DPD}$:

$$\begin{aligned} \textstyle S_\mathtt{DPD}^{*c}(\rho )=\frac{1}{{\tilde{\sigma }}_v^{2}(\rho )}\Delta {\tilde{v}}'(\rho ){{\mathbf {C}}}^{-1}\Delta Y_{-1} + n\big (\frac{1}{1-\rho }-\frac{1-\rho ^T}{T(1-\rho )^2}\big ), \end{aligned}$$

where $\Delta {\tilde{v}}(\rho )=\Delta Y-\rho \Delta Y_{-1}-\Delta X{\tilde{\beta }}(\rho )$, ${\tilde{\beta }}(\rho ) = (\Delta {X}^{\prime }{{\mathbf {C}}}^{-1}\Delta {X})^{-1}\Delta {X}^{\prime }{{\mathbf {C}}}^{-1}(\Delta Y-\rho \Delta Y_{-1})$ and ${\tilde{\sigma }}_v^2(\rho ) = \textstyle \frac{1}{N}\Delta {\tilde{v}}^{\prime }(\rho ){{\mathbf {C}}}^{-1}\Delta {\tilde{v}}(\rho )$.

Define ${\bar{S}}^{*}(\beta ,\sigma _{v}^{2},\rho )=\mathrm{E}[S^{*}(\beta ,\sigma _{v}^{2},\rho )]$. Given $\rho$, ${\bar{S}}^{*}(\beta ,\sigma _{v}^{2},\rho ) = 0$ is partially solved at ${\bar{\beta }}(\rho )=(\Delta {X}^{\prime }{{\mathbf {C}}}^{-1}\Delta {X})^{-1}\Delta {X}^{\prime }{{\mathbf {C}}}^{-1}(\mathrm{E}\Delta Y-\rho \mathrm{E}\Delta Y_{-1})$ and ${\bar{\sigma }}_{v}^{2}(\rho )=\frac{1}{N}\mathrm{E}[\Delta {\bar{v}}(\rho )'{{\mathbf {C}}}^{-1}\Delta {\bar{v}}(\rho )]$, where $\Delta {\bar{v}}(\rho )=\Delta Y-\rho \Delta Y_{-1}-\Delta X{{\bar{\beta }}}(\rho )$. Substituting ${\bar{\beta }}(\rho )$ and ${\bar{\sigma }}_{v}^{2}(\rho )$ back into ${\bar{S}}^{*}(\beta ,\sigma _{v}^{2},\rho )$ gives the population counterpart of $S_\mathtt{DPD}^{*c}(\rho )$ as

$$\begin{aligned} \textstyle {\bar{S}}_\mathtt{DPD}^{*c}(\rho )=\frac{1}{{{\bar{\sigma }}}_v^{2}(\rho )}\mathrm{E}[\Delta {\bar{v}}'(\rho ){{\mathbf {C}}}^{-1}\Delta Y_{-1}] + n\big (\frac{1}{1-\rho }-\frac{1-\rho ^T}{T(1-\rho )^2}\big ). \end{aligned}$$

By Theorem 5.9 of van der Vaart (1998), ${\tilde{\rho }}$ will be consistent if $(i) \inf _{\rho : |\rho -\rho _0|\ge \epsilon }|{\bar{S}}_\mathtt{DPD}^{*c}(\rho )| > 0$ for every $\epsilon > 0$, and $(ii) \sup _{\rho \in \Upsilon }\frac{1}{\sqrt{N}}|S_\mathtt{DPD}^{*c}(\rho )-{\bar{S}}_\mathtt{DPD}^{*c}(\rho )| \overset{p}{\longrightarrow } 0$, which are straightforward. The asymptotic normality can be proved using Lemma A.5. $\square$

Proof

First, with $\psi _{0}=(\beta _{0},\sigma _{v0}^{2},\rho _{0},0'_{3})'$ it is easy to show that $\mathrm{E}[S^{*}(\psi _0)]=0$ under the general heteroskedasticity $\{h_{ni}\}$. By Lemma A.5, one shows that $\frac{1}{\sqrt{N}}S^{*}(\psi _{0}) \overset{D}{\longrightarrow } N(0, \Gamma ^{*}(\psi _0))$. By Lemma A.6, one shows that $\frac{1}{N}\sum _{i=1}^{n}[\mathbf{g}_{n,i}{} \mathbf{g}'_{n,i}-\mathrm{E}(\mathbf{g}_{i}{} \mathbf{g}'_{i})] \overset{p}{\longrightarrow } 0$. By the mean value theorem, and $\sqrt{N}$ consistency and robustness of ${\tilde{\beta }}$, ${\tilde{\sigma }}_{v}^{2}$ and ${\tilde{\rho }}$ against unknown heteroskedasticity $\{h_{ni}\}$ as shown in Lemma 2.1, we have $\frac{1}{\sqrt{N}}[S_{\lambda }^{*}({\tilde{\psi }})-S_{\lambda }^{*}(\psi _{0})] \overset{p}{\longrightarrow } 0$ where ${\tilde{\psi }}=({\tilde{\beta }}', {\tilde{\sigma }}_{v}^{2},{\tilde{\rho }},0_3)'$, and $\frac{1}{N}\sum _{i=1}^{n}(\tilde{\mathbf{g}}_{n,i}\tilde{\mathbf{g}}'_{n,i}-\mathbf{g}_{i}{} \mathbf{g}'_{i}) \overset{p}{\longrightarrow } 0$. Finally, using the simplified expression of $H^{*}(\psi )$ and MVT, we show $\mathrm{plim}_{n\rightarrow \infty }(\widetilde{\Lambda }-{\Lambda })=0$.

Proof of Theorem 2.2

The proof is similar to that of Theorem 2.1. The partial derivatives of $S^{\diamond }(\delta )$ required to estimate the components of $\Sigma _{\varphi \pi }^{\diamond }(\delta _{0})$ and $\Sigma _{\varphi \varphi }^{\diamond }(\delta _{0})$ can be easily obtained from the expression $S^{\diamond }(\delta )$ given in (2.19). The full expression of $\frac{\partial }{\partial \delta ^{\prime }}S^{\diamond }(\delta )$ is given in the Supplementary Appendix to this paper. $\square$

Proof of Theorem 3.1

The proof of Theorem 3.1 follows closely the proofs of Theorems (3.2) and (3.3) of Li and Yang (2020b). The Hessian matrix $\frac{\partial }{\partial \psi ^{\prime }}S_\mathtt{H}^{*}(\psi )$ used to estimate $\Sigma _\mathtt{H}^{*}(\psi _{0})$ is given in Li and Yang (2020b, Proof of Theorem 3.2), and can also be found in the Supplementary Appendix to this paper, where the ‘asymmetric components’ that did not appear in Li and Yang (2020b) are also given. $\square$