Testing spatial regression models under nonregular conditions

Abstract

In time series context, estimation and testing issues with autoregressive and moving average (ARMA) models are well understood. Similar issues in the context of spatial ARMA models for the disturbance of the regression, however, remain largely unexplored. In this paper, we discuss the problems of testing no spatial dependence in the disturbances against the alternative of spatial ARMA process incorporating the possible presence of spatial dependence in the dependent variable. The problems of conducting such a test are twofold. First, under the null hypothesis, the nuisance parameter is not identified, resulting in a singular information matrix (IM), which is a nonregular case in statistical inference. To take account of singular IM, we follow Davies (Biometrika 64(2):247–254, 1977; Biometrika 74(1):33–43, 1987) and propose a test procedure based on the supremum of the Rao score test statistic. Second, the possible presence of spatial lag dependence will have adverse effect on the performance of the test. Using the general test procedure of Bera and Yoon (Econom Theory 9:649–658, 1993) under local misspecification, we avoid the explicit estimation of the spatial autoregressive parameter. Thus our suggested tests are entirely based on ordinary least squares estimation. Tests suggested here can be viewed as a generalization of Anselin et al. (Reg Sci Urban Econ 26:77–104, 1996). We conduct Monte Carlo simulations to investigate the finite sample properties of the proposed tests. Simulation results show that our tests have good finite sample properties both in terms of size and power, compared to other tests in the literature. We also illustrate the applications of our tests through several data sets.

Appendices

Appendices

1.1 A score functions

The model we consider is

$$\begin{aligned} \begin{aligned} y&=\rho W y+X \beta +u\\ u&=\tau Wu+ \epsilon - \lambda W \epsilon , \end{aligned} \end{aligned}$$

(A.1)

and the log-likelihood function is

$$\begin{aligned} \begin{aligned} l(\theta )=&-\frac{n}{2} \mathrm{log} 2 \pi -\frac{n}{2} \mathrm{log} \sigma ^2 - \frac{1}{2 \sigma ^2} \epsilon ' \epsilon + \mathrm{log}|I-\rho W| \\&+ \mathrm{log}|I-\tau W| + log|(I-\lambda W)^{-1}|\\ =&\mathrm{Constant} -\frac{n}{2} \mathrm{log} \sigma ^2- \frac{1}{2 \sigma ^2}(AY-X \beta )' B' C^{-1'} C^{-1} B(AY-X \beta )\\&+ {log}|A| + {log}|B|+ {log}|C^{-1}|, \end{aligned} \end{aligned}$$

(A.2)

where $\theta =(\beta ',\sigma ^2,\rho ,\tau ,\lambda )$ and $A=I-\rho W$, $B=I-\tau W$, $C=I-\lambda W$.

The first derivatives are

$$\begin{aligned} \frac{\partial l}{\partial \beta }= & {} \frac{1}{\sigma ^2}X'B'C^{-1'}\epsilon \\ \frac{\partial l}{\partial \sigma ^2}= & {} -\frac{n}{2\sigma ^2}+\frac{1}{2\sigma ^4}\epsilon ' \epsilon \\ \frac{\partial l}{\partial \rho }= & {} -tr(A^{-1}W)+\frac{1}{\sigma ^2}\epsilon 'C^{-1}BWy\\ \frac{\partial l}{\partial \tau }= & {} -tr(B^{-1}W)+\frac{1}{\sigma ^2}\epsilon 'C^{-1}W(Ay-X\beta )\\ \frac{\partial l}{\partial \lambda }= & {} -tr(WC^{-1})-\frac{1}{\sigma ^2}\epsilon 'C^{-1}W\epsilon . \end{aligned}$$

Under $H_0: \tau =\lambda$, we have $B=C$ and hence

$$\begin{aligned} \frac{\partial l}{\partial \beta }|_{H_0}= & {} \frac{1}{\sigma ^2}X'\epsilon \\ \frac{\partial l}{\partial \sigma ^2}|_{H_0}= & {} -\frac{n}{2\sigma ^2}+\frac{1}{2\sigma ^4}\epsilon ' \epsilon \\ \frac{\partial l}{\partial \rho }|_{H_0}= & {} -tr(A^{-1}W)+\frac{1}{\sigma ^2}\epsilon 'Wy\\ \frac{\partial l}{\partial \tau }|_{H_0}= & {} -tr(B^{-1}W)+\frac{1}{\sigma ^2}\epsilon 'C^{-1}W(Ay-X\beta )\\ \frac{\partial l}{\partial \lambda }|_{H_0}= & {} -tr(WC^{-1})-\frac{1}{\sigma ^2}\epsilon 'C^{-1}W\epsilon . \end{aligned}$$

Moreover, if we put $\rho =0$, then we have $A=I$ and under $H_0$, the above becomes

$$\begin{aligned} \frac{\partial l}{\partial \beta }|_{H_0}= & {} \frac{1}{\sigma ^2}X'\epsilon \\ \frac{\partial l}{\partial \sigma ^2}|_{H_0}= & {} -\frac{n}{2\sigma ^2}+\frac{1}{2\sigma ^4}\epsilon ' \epsilon \\ \frac{\partial l}{\partial \rho }|_{H_0}= & {} -tr(A^{-1}W)+\frac{1}{\sigma ^2}\epsilon 'Wy\\ \frac{\partial l}{\partial \tau }|_{H_0}= & {} -tr(B^{-1}W)+\frac{1}{\sigma ^2}\epsilon 'C^{-1}W\epsilon \\ \frac{\partial l}{\partial \lambda }|_{H_0}= & {} -tr(WC^{-1})-\frac{1}{\sigma ^2}\epsilon 'C^{-1}W\epsilon . \end{aligned}$$

1.2 B information matrix

From the previous section, the second derivatives of the log-likelihood function are

$$\begin{aligned} \frac{\partial ^2 l}{\partial \beta \partial \beta '}= & {} -\frac{1}{\sigma ^2}X'B'C^{-1'}C^{-1}BX\\ \frac{\partial ^2 l}{\partial \beta \partial \sigma ^2}= & {} \frac{\partial ^2 l}{\partial \sigma ^2 \partial \beta '}=-\frac{1}{\sigma ^4}X'\epsilon \\ \frac{\partial ^2 l}{\partial \beta \partial \rho }= & {} \frac{\partial ^2 l}{\partial \rho \partial \beta '}=-\frac{1}{\sigma ^2}X'B'C^{-1'}C^{-1}BWy\\ \frac{\partial ^2 l}{\partial \beta \partial \tau }= & {} \frac{\partial ^2 l}{\partial \tau \partial \beta '}=-\frac{1}{\sigma ^2}X'W'C^{-1'}\epsilon -\frac{1}{\sigma ^2} X'B'C^{-1'}C^{-1}W(Ay-X\beta )\\ \frac{\partial ^2 l}{\partial \beta \partial \lambda }= & {} \frac{\partial ^2 l}{\partial \lambda \partial \beta '}=\frac{1}{\sigma ^2}X'B'C^{-1'}W'C^{-1'}B\epsilon +\frac{1}{\sigma ^2}X'B'C^{-1'}C^{-1}WC^{-1}B(Ay-X\beta )\\ \frac{\partial ^2 l}{\partial (\sigma ^2)^2}= & {} \frac{n}{2\sigma ^4}-\frac{1}{\sigma ^6}\epsilon '\epsilon \\ \frac{\partial ^2 l}{\partial \sigma ^2 \partial \rho }= & {} \frac{\partial ^2 l}{\partial \rho \partial \sigma ^2}=-\frac{1}{\sigma ^4}\epsilon 'C^{-1}BWy\\ \frac{\partial ^2 l}{\partial \sigma ^2 \partial \tau }= & {} \frac{\partial ^2 l}{\partial \tau \partial \sigma ^2}=-\frac{1}{\sigma ^4}\epsilon 'C^{-1}W(Ay-X\beta )\\ \frac{\partial ^2 l}{\partial \sigma ^2 \partial \lambda }= & {} \frac{\partial ^2 l}{\partial \lambda \partial \sigma ^2}=-\frac{1}{\sigma ^4}\epsilon 'C^{-1}W\epsilon \\ \frac{\partial ^2 l}{\partial \rho ^2}= & {} -tr(A^{-1}WA^{-1}W)-\frac{1}{\sigma ^2}y'W'B'C^{-1'}C^{-1}BWy\\ \frac{\partial ^2 l}{\partial \rho \partial \tau }= & {} \frac{\partial ^2 l}{\partial \tau \partial \rho }=-\frac{1}{\sigma ^2}(Ay-X\beta )'W'C^{-1'}C^{-1}BWy -\frac{1}{\sigma ^2}\epsilon 'C^{-1}WWy\\ \frac{\partial ^2 l}{\partial \rho \partial \lambda }= & {} \frac{\partial ^2 l}{\partial \lambda \partial \rho }=\frac{1}{\sigma ^2}(Ay-X\beta )'B'C^{-1'}W'C^{-1'}C^{-1}BWy +\frac{1}{\sigma ^2}\epsilon 'C^{-1}WC^{-1}BWy\\ \frac{\partial ^2 l}{\partial \tau ^2}= & {} -tr(B^{-1}WB^{-1}W)-\frac{1}{\sigma ^2}(Ay-X\beta )'W'C^{-1'} C^{-1}W(Ay-X\beta )\\ \frac{\partial ^2 l}{\partial \tau \partial \lambda }= & {} \frac{\partial ^2 l}{\partial \lambda \partial \tau }=\frac{1}{\sigma ^2}(Ay-X\beta )'B'C^{-1'}W'C^{-1'}C^{-1}W(Ay-X\beta )\\&+\frac{1}{\sigma ^2}\epsilon 'C^{-1}WC^{-1}W(Ay-X\beta )\\ \frac{\partial ^2 l}{\partial \lambda ^2}= & {} -tr(WC^{-1}WC^{-1}) -\frac{1}{\sigma ^2}(Ay-X\beta )'B'C^{-1'}W'C^{-1'}C^{-1}W\epsilon \\&-\frac{1}{\sigma ^2}\epsilon 'C^{-1}WC^{-1}W\epsilon -\frac{1}{\sigma ^2} \epsilon 'C^{-1}WC^{-1}WC^{-1}B(Ay-X\beta ).\\ \end{aligned}$$

Therefore, the information matrix can be derived as

$$\begin{aligned} I(\theta ) = \frac{1}{n \sigma ^2}\left[ \begin{array}{lllll} J_{\beta } &{} J_{\beta \sigma ^2} &{} J_{\beta \rho } &{} J_{\beta \tau } &{} J_{\beta \lambda }\\ J_{\sigma ^2 \beta } &{} J_{\sigma ^2} &{} J_{\sigma ^2 \rho } &{} J_{\sigma ^2 \tau } &{} J_{\sigma ^2 \lambda } \\ J_{\rho \beta } &{} J_{\rho \sigma ^2} &{} J_{\rho } &{} J_{\rho \tau } &{} J_{\rho \lambda } \\ J_{\tau \beta } &{} J_{\tau \sigma ^2} &{} J_{\tau \rho } &{} J_{\tau } &{} J_{\tau \lambda } \\ J_{\lambda \beta } &{} J_{\lambda \sigma ^2} &{} J_{\lambda \rho } &{} J_{\lambda \tau } &{} J_{\lambda } \\ \end{array}\right] , \end{aligned}$$

(B.1)

where

$$\begin{aligned} J_{\beta }= & {} X'B'C^{-1'}C^{-1}BX\\ J_{\beta \sigma ^2}= & {} J_{\sigma ^2 \beta }=J_{\beta \tau }=J_{\tau \beta }=J_{\beta \lambda }=J_{\lambda \beta }=0\\ J_{\beta \rho }= & {} J_{\rho \beta }=X'B'C^{-1'}C^{-1}BA^{-1}WX\beta \\ J_{\sigma ^2}= & {} \frac{n}{2 \sigma ^2}\\ J_{\sigma ^2 \rho }= & {} J_{\rho \sigma ^2}=tr(C^{-1}BWA^{-1}B^{-1}C)\\ J_{\sigma ^2 \tau }= & {} J_{\tau \sigma ^2}=tr(C^{-1}WB^{-1}C)\\ J_{\sigma ^2 \lambda }= & {} J_{\lambda \sigma ^2}=-tr(C^{-1}W)\\ J_{\rho }= & {} \sigma ^2 tr(A^{-1}WA^{-1}W)+\beta 'X'A^{-1'}W'B'C^{-1'}C^{-1}BWA^{-1}X\beta \\&+\sigma ^2 tr(C'B^{-1'}A{-1'}W'B'C^{-1'}C^{-1}BWA^{-1}B^{-1}C) \\ J_{\rho \tau }= & {} J_{\tau \rho }=\sigma ^2[tr(C'B^{-1'}W'C^{-1'}C^{-1}BWA^{-1}B^{-1}C)+tr(C^{-1}WWA^{-1}B^{-1}C)]\\ J_{\rho \lambda }= & {} J_{\lambda \rho }=-\sigma ^2[tr(C'B^{-1'}W'C^{-1'}C^{-1}BWA^{-1}B^{-1}C)+tr(C^{-1}WC^{-1}BWA^{-1}B^{-1}C)]\\ J_{\tau }= & {} \sigma ^2[tr(C^{-1}BWA^{-1}B^{-1}C)+tr(C'B^{-1'}B'C^{-1'}W'C^{-1'}C^{-1}W)]\\ J_{\tau \lambda }= & {} J_{\lambda \tau }=-\sigma ^2[tr(C'B^{-1'}W'C^{-1'}C^{-1}BWA^{-1}B^{-1}C)+tr(C^{-1}WC^{-1}WC^{-1}BB^{-1}C)]\\ J_{\lambda }= & {} -\sigma ^2[tr(C'B^{-1'}B'C^{-1'}W'C^{-1'}C^{-1}W)+tr(C^{-1}WC^{-1}WC^{-1}BB^{-1}C)]. \end{aligned}$$

Under $H_0:\tau =\lambda$, we have $B=c$, and hence (B.1) becomes

$$\begin{aligned} I(\theta ) = \frac{1}{n \sigma ^2}\left[ \begin{array}{c|c} J_{11} &{} J_{12} \\ \hline J_{21} &{} J_{22} \\ \end{array}\right] , \end{aligned}$$

(B.2)

where the partition matrices are

When $\tau =\lambda$ and $\rho =0$, we further have $A=I$ and notice that $tr(W)=0$. Therefore (B.2), reduces to

$$\begin{aligned} I(\theta )=\frac{1}{n \sigma ^2}\left[ \begin{array}{ccccc} X'X &{} 0 &{} X'WX \beta &{} 0 &{} 0 \\ 0 &{} \frac{n}{2 \sigma ^2} &{} 0 &{} tr(C^{-1}W) &{} -tr(C^{-1}W) \\ X'WX \beta &{} 0 &{} J_{\rho } &{} J_{\rho \tau } &{} J_{\rho \lambda } \\ 0 &{} tr(C^{-1}W) &{} J_{\tau \rho } &{} J_{\tau } &{} J_{\tau \lambda } \\ 0 &{} -tr(C^{-1}W) &{} J_{\lambda \rho } &{} J_{\lambda \tau } &{} J_{\lambda } \\ \end{array}\right] , \end{aligned}$$

(B.3)

where

$$\begin{aligned} \begin{aligned} J_{\rho }&= \sigma ^2 [tr(WW)+tr(W' W)]+\beta ' X W'W X \beta \\ J_{\rho \tau }&= J^J_{\tau \rho } = \sigma ^2 [tr(W'C^{-1}W +tr(C^{-1} WW)]\\ J_{\rho \lambda }&= J^J_{\lambda \rho } = -\sigma ^2 [tr(W'C^{-1}W +tr(C^{-1} WW)]\\ J_{\tau }&= \sigma ^2 [tr(W' C^{-1'} C^{-1} W)+tr(C^{-1} W C^{-1} W)]\\ J_{\tau \lambda }&= J^J_{\lambda \tau } = -\sigma ^2 [tr(W' C^{-1'} C^{-1} W)+tr(C^{-1} W C^{-1} W)]\\ J_{\lambda }&= \sigma ^2 [tr(W' C^{-1'} C^{-1} W)+tr(C^{-1} W C^{-1} W)].\\ \end{aligned} \end{aligned}$$

1.3 C derivation of test statistics

We first derive adjusted RS test statistic assuming $\lambda$ is given. Defining $\underline{\theta }=(\beta ', \rho , \tau , \sigma ^2)'$, the log-likelihood can be rewritten as

$$\begin{aligned} \begin{aligned} l(\underline{\theta }|\lambda ) =&Constant -\frac{n}{2} log \sigma ^2- \frac{1}{2 \sigma ^2}[(I-\rho W)Y-X \beta ]' (I-\tau W)' C^{-1'} \cdot \\&C^{-1} (I-\tau W)[(I-\rho W)Y-X \beta ]+log|I-\rho W|\\&+log|I-\tau W|+log|C^{-1}|. \end{aligned} \end{aligned}$$

(C.1)

And for a given value of $\lambda$, the score functions under joint null are

$$\begin{aligned} \begin{aligned} d_{\beta }(\lambda )&= \frac{1}{\sigma ^2}X'(Y-X \beta )\\ d_{\sigma ^2}(\lambda )&= -\frac{n}{2 \sigma ^2}+\frac{1}{2 \sigma ^4} (Y-X \beta )'(Y-X \beta )\\ d_{\rho }(\lambda )&= \frac{1}{\sigma ^2} (Y-X \beta )'WY\\ d_{\tau }(\lambda )&= \frac{1}{\sigma ^2} (Y-X \beta )' C^{-1} W (Y-X \beta ).\\ \end{aligned} \end{aligned}$$

The information matrix under the null when $\rho =0$ and $\lambda$ is given, denoted as $I(\underline{\theta }|\lambda )|_{H_0}$, can be derived as

$$\begin{aligned} I(\theta |\lambda )|_{H^J_0}= & {} \frac{1}{n \sigma ^2}\left[ \begin{array}{cccc} J_{\beta }(\lambda ) &{} J_{\beta \sigma ^2}(\lambda ) &{} J_{\beta \rho }(\lambda ) &{} J_{\beta \tau }(\lambda ) \\ J_{\sigma ^2 \beta }(\lambda ) &{} J_{\sigma ^2}(\lambda ) &{} J_{\sigma ^2 \rho }(\lambda ) &{} J_{\sigma ^2 \tau }(\lambda ) \\ J_{\rho \beta }(\lambda ) &{} J_{\rho \sigma ^2}(\lambda ) &{} J_{\rho }(\lambda ) &{} J_{\rho \tau }(\lambda ) \\ J_{\tau \beta }(\lambda ) &{} J_{\tau \sigma ^2}(\lambda ) &{} J_{\tau \rho }(\lambda ) &{} J_{\tau }(\lambda ) \\ \end{array}\right] \nonumber \\= & {} \frac{1}{n \sigma ^2}\left[ \begin{array}{cccc} X'X &{} 0 &{} X'WX \beta &{} 0 \\ 0 &{} \frac{n}{2 \sigma ^2} &{} 0 &{} tr(C^{-1}W) \\ X'WX \beta &{} 0 &{} \sigma ^2 [tr(WW)+tr(W' W)]+\beta ' X W'W X \beta &{} \sigma ^2 [tr(W'C^{-1}W +tr(C^{-1} WW)] \\ 0 &{} tr(C^{-1}W) &{} \sigma ^2 [tr(W'C^{-1}W +tr(C^{-1} WW)] &{} \sigma ^2 [tr(W' C^{-1'} C^{-1} W)+tr(C^{-1} W C^{-1} W)] \\ \end{array}\right] .\nonumber \\ \end{aligned}$$

(C.2)

Denoting $\gamma =(\beta ', \sigma ^2)$, the standard RS test on $(\gamma ,0,\tau )$ given $\lambda$, has the form

$$\begin{aligned} RS(\lambda ) = \frac{1}{n}d'_{\tau }(\lambda )J^{-1}_{\tau \cdot \gamma }(\lambda )d_{\tau }(\lambda ), \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} J_{\tau \cdot \gamma }(\lambda )&=J_{\tau }(\lambda )-J_{\tau \gamma }(\lambda )J^{-1}_{\gamma }(\lambda )J_{\gamma \tau }(\lambda )\\&= \frac{1}{n \sigma ^2} \{ \sigma ^2 [tr(W'C^{-1'}C^{-1}W)+tr(C^{-1}WC^{-1}W)]-\frac{2\sigma ^2}{n}[tr(C^{-1}W)]^2 \} \\ \end{aligned} \end{aligned}$$

using (C.2), and the standard RS test statistic for fixed $\lambda$ can be derived as

$$\begin{aligned} RS(\lambda )=\frac{\{ \frac{1}{\hat{\sigma ^2}}\hat{u}'C^{-1}W\hat{u}-tr[C^{-1}W] \}^2}{tr\{W'C^{-1'}C^{-1}W+[C^{-1}W]^2\}-\frac{2}{n}[tr(C^{-1}W)]^2}. \end{aligned}$$

(C.3)

Now we consider the RS test adjusted for the presence of $\rho$. Assuming $\lambda$ given, the adjusted RS test statistic, denoted by $RS^*_{\tau }(\lambda )$ has the form

$$\begin{aligned} \begin{aligned} RS^*_{\tau }(\lambda ) =&\frac{1}{n}[d_{\tau }(\lambda )-J_{\tau \rho \cdot \gamma }(\lambda )J^{-1}_{\rho \cdot \gamma }(\lambda )d_{\rho }(\lambda )]' \\&\times [J_{\tau \cdot \gamma }(\lambda )-J_{\tau \rho \cdot \gamma }(\lambda )J^{-1}_{\rho \cdot \gamma }(\lambda )J_{\rho \tau \cdot \gamma }(\lambda )]^{-1} \\&\times [d_{\tau }(\lambda )-J_{\tau \rho \cdot \gamma }(\lambda )J^{-1}_{\rho \cdot \gamma }(\lambda )d_{\rho }(\lambda )],\\ \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} J_{\tau \rho \cdot \gamma }(\lambda )&=J_{\rho \tau \cdot \gamma }(\lambda )=J_{\tau \gamma }(\lambda )J^{-1}_{\gamma }(\lambda )J_{\gamma \rho }(\lambda )\\ J_{\rho \cdot \gamma }(\lambda )&= J_{\rho }(\lambda )-J_{\rho \gamma }(\lambda )J^{-1}_{\gamma }(\lambda )J_{\gamma \rho }(\lambda ).\\ \end{aligned} \end{aligned}$$

Using (C.2), we have

$$\begin{aligned} \begin{aligned} J_{\tau \rho \cdot \gamma }(\lambda )&=\frac{1}{n}[tr(W'C^{-1'}W)+tr(C^{-1}WW)]\\ J_{\rho \cdot \gamma }(\lambda )&= \frac{1}{n \sigma ^2} \{ \sigma ^2 [tr(W'W+WW)]+\beta 'X'W'MWX\beta \},\\ \end{aligned} \end{aligned}$$

where $M=I-X(X'X)^{-1}X'$, and hence the adjusted RS statistic for fixed $\lambda$ can be derived as

$$\begin{aligned} RS^* (\lambda )=\frac{\{\hat{u'}C^{-1}W\hat{u}/\hat{\sigma }^2 -tr(C^{-1}W)-[tr(W'C^{-1'}W+C^{-1}WW)](n\hat{J_{\rho \cdot \gamma }})^{-1}\hat{u}'Wy/\hat{\sigma }^2 \}^2}{tr(W'C^{-1'}C^{-1}W+C^{-1}WC^{-1}W)-\frac{2}{n}[tr(C^{-1}W)]^2-[tr(W'C^{-1'}W+C^{-1}WW)]^2 (n\hat{J_{\rho \cdot \gamma }})^{-1}},\nonumber \\ \end{aligned}$$

(C.4)

where

$$\begin{aligned} (n\hat{J_{\rho \cdot \gamma }})^{-1} = \hat{\sigma }^2 \cdot \{ \hat{\beta } X'W'MWX\beta +\hat{\sigma }^2 [tr((W'+W)W)] \}^{-1}. \end{aligned}$$