A J test for dynamic panel model with fixed effects, and nonparametric spatial and time dependence

Abstract

In this paper, we suggest a J test in a dynamic spatial panel framework of a null model against one or more alternatives. The null model we consider has fixed effects, along with nonparametrically specified spatial and time dependence. The alternatives can have either fixed or random effects with the same complications. We implement our procedure to test the specifications of a demand for cigarette model. We find that the most appropriate specification is one that contains the average price of cigarettes in neighboring states, as well as the spatial lag of the dependent variable. Along with formal large sample results, we also give small sample Monte Carlo results. Our large sample results are based on the assumption $N\rightarrow \infty$ and T is fixed. Our Monte Carlo results suggest that our proposed J test has good power and proper size even for small to moderately sized samples.

Appendix

1.1 Proof of Theorem 1 (Table 4)

To simplify notation, we prove Theorem 1 for the case in which $\overbrace{ Q_{0}Y_{J}^{E,A}}$is used, $J=1,\ldots ,G.$ The proof for the case in which $\overbrace{Q_{0}Y_{J}^{E,B}}$ is used is virtually identical.

First note from (10), (14) and (15) that

$$\begin{aligned} \hat{Y}_{1,G}^{A}= & {} [Z_{1},\ldots ,Z_{G}]\,[\hat{\gamma }_{1}^{\prime },\ldots ,\hat{\gamma }_{G}^{\prime }]^{\prime } \nonumber \\= & {} Z_{1,G}\,\hat{\gamma }_{1,G} \end{aligned}$$

(29)

where $Z_{1,G}=[Z_{1},\ldots ,Z_{G}]$ and $\hat{\gamma }_{1,G}=[\hat{\gamma } _{1}^{\prime },\ldots ,\hat{\gamma }_{G}^{\prime }]^{\prime }$ where $\hat{\gamma } _{J}^{\prime }=(\hat{\phi }_{1,J}^{\prime },\hat{\phi }_{2,J}^{\prime },\hat{ \lambda }_{J},\hat{\alpha }_{J}),$ $J=1,\ldots ,G.$ For this case $\hat{\Gamma } _{A}$ in (16) is

$$\begin{aligned} \hat{\Gamma }_{A}= & {} [Z,Z_{1,G}\,\hat{\gamma }_{1,G}] \end{aligned}$$

(30)

and $\Gamma _{A}=[Z,Z_{1,G}$ $\gamma _{1,G}]$. Therefore, recalling that $\tilde{\Phi }_{A}=P_{H}\hat{\Phi }_{A}$ and $\hat{\Phi }_{A}=Q_{0}\hat{\Gamma } _{A}$ it follows from (16) and (18) that

$$\begin{aligned} (NT)^{1/2}[\tilde{\xi }_{i}-\xi _{i}]= & {} \left\{ (NT)^{-1}\hat{\Gamma } _{A}^{\prime }Q_{0}H\,[(NT)^{-1}H^{\prime }H]^{-1}(NT)^{-1}H^{\prime }Q_{0}\hat{\Gamma }_{A}\right\} ^{-1} \nonumber \\&\times (NT)^{-1}\hat{\Gamma }_{A}^{\prime }Q_{0}H\,(NT)(H^{\prime }H)^{-1} \,(NT)^{-1/2}H^{\prime }Q_{0}R\varepsilon \end{aligned}$$

(31)

Consider the term $(NT)^{-1}H^{\prime }Q_{0}\hat{\Gamma }_{A}$ in the inverse on the first line of (31) and note that by Assumption 5, $\hat{\gamma }_{1,G}\overset{P}{\rightarrow }C_{1,G}=(c_{1}^{\prime },\ldots ,c_{G}^{\prime })^{\prime },$ where $c_{J}^{\prime }=$ $(c_{0,J},c_{2,J},L_{J},a_{J}),J=1,\ldots ,G$. Also note that $Q_{0}^{\prime }=Q_{0},Q_{0}^{2}=Q_{0},$ and therefore from (17) $Q_{0}H=H$ and so from Part (b) of Assumption 6, and (29)

$$\begin{aligned} (NT)^{-1}H^{\prime }Q_{0}\hat{\Gamma }_{A}]= & {} p\lim _{N\rightarrow \infty }(NT)^{-1}H^{\prime }[Z,Z_{1,G}\,\hat{\gamma }_{1,G}] \nonumber \\= & {} [p\lim _{N\rightarrow \infty }(NT)^{-1}H^{\prime }Z,\, (p\lim _{N\rightarrow \infty }(NT)^{-1}H^{\prime }Z_{1,G})\, p\lim _{N\rightarrow \infty }\hat{\gamma }_{1,G}] \nonumber \\= & {} [p\lim _{N\rightarrow \infty }(NT)^{-1}H^{\prime }Z,\, (p\lim _{N\rightarrow \infty }(NT)^{-1}H^{\prime }Z_{1,G})\,\gamma _{1,G}] \nonumber \\= & {} p\lim _{N\rightarrow \infty }(NT)^{-1}H^{\prime }\Gamma _{A} \nonumber \\= & {} \Omega _{H\Gamma _{A}} \end{aligned}$$

(32)

Let $\hat{\digamma }$ be the inverse term on the first line of (31) and let

$$\begin{aligned} \tilde{\digamma }=\hat{\digamma }*[(NT)^{-1}\hat{\Gamma } _{A}^{\prime }H]\,[(NT)(H^{\prime }H)^{-1}] \end{aligned}$$

(33)

Then from (32), and parts (a) and (b) of Assumption 6, first note that

$$\begin{aligned} \hat{\digamma }\overset{P}{\rightarrow }\left\{ \Omega _{H\Gamma _{A}}^{\prime }\Omega _{HH}^{-1}\Omega _{H\Gamma _{A}}\right\} ^{-1} \end{aligned}$$

(34)

where $\Omega _{H\Gamma _{A}}^{\prime }\Omega _{HH}^{-1}\Omega _{H\Gamma _{A}}$ is positive definite and, therefore, non-singular since $\Omega _{HH}$ is positive definite and so, therefore, is $\Omega _{HH}^{-1},$ and $\Omega _{H\Gamma _{A}}$ has full column rank. It then follows from (32) to   (34), and parts (a) and (b) of Assumption 6 that

$$\begin{aligned}&\tilde{\digamma }\overset{P}{\rightarrow }\digamma ^{*} \nonumber \\&\digamma ^{*} =\left\{ \Omega _{H\Gamma _{A}}^{\prime }\Omega _{HH}^{-1}\Omega _{H\Gamma _{A}}\right\} ^{-1}\Omega _{H\Gamma _{A}}^{\prime }\Omega _{HH}^{-1} \end{aligned}$$

(35)

Finally, consider the last term on the second line in (31), namely $(NT)^{-1/2}H^{\prime }Q_{0}R\varepsilon .$ Since the row and column sums of both $Q_{0}$ and R are uniformly bounded in absolute value, the row and column sums of $Q_{0}R$ are also uniformly bounded in absolute value. Since by Assumption 4 (a) the elements of H are uniformly bounded in absolute value, it follows that the elements of $H^{\prime }Q_{0}R=H^{\prime }R$ are uniformly bounded in absolute value. Given this, and Assumptions 1, 6 part (c) and the central limit theorem (30) in Pötscher and Prucha (2000), it follows that

$$\begin{aligned} (NT)^{-1/2}H^{\prime }Q_{0}R\varepsilon \overset{D}{\rightarrow }N(0,\Omega _{H^{\prime }RR^{\prime }H}) \end{aligned}$$

(36)

Therefore, by the continuous map** theorem and (31)–(35)

$$\begin{aligned} (NT)^{1/2}[\tilde{\xi }_{i}-\xi _{i}]\overset{D}{\rightarrow }N(0,\digamma ^{*}\Omega _{H^{\prime }RR^{\prime }H}\digamma ^{*\prime }) \end{aligned}$$

(37)