Model Checking in Partially Linear Spatial Autoregressive Models

Abstract

This article proposes two new classes of nonparametric tests for the correct specification of linear spatial autoregressive models based on the “integrated conditional moment” approach. Our test statistics are constructed as continuous functionals of a residual marked empirical process as well as its projected version. We derive asymptotic properties of the test statistics under the null hypothesis, the alternative hypothesis, and a sequence of local alternatives. The proposed tests do not involve the selection of tuning parameters such as bandwidths and are able to detect a broad class of local alternatives converging to the null at the parametric rate $n^{-1/2}$, with n being the sample size. We also propose a multiplier bootstrap procedure that is computationally simple to approximate the critical values. Monte Carlo simulations illustrate that our tests have a reasonable size and satisfactory power for different types of data-generating processes. Finally, an empirical analysis of growth convergence is carried out to demonstrate the usefulness of the tests.

Keywords:

• Multiplier bootstrap
• Projection
• Residual marked empirical process
• Spatial dependence

Supplementary Materials

The online Supplementary Appendix contains proofs of the main theoretical results, discussions on the assumptions in Section 3, additional simulation results, and empirical applications on Boston house prices and China air quality.

Acknowledgments

The authors thank the coeditor, associate editor, and two anonymous referees for their constructive comments.

Disclosure statement

The authors report there are no competing interests to declare.

Notes

1 There are some other commonly used weighting functions $w(z_{n,i},z)$ such that (6) is equivalent to (5), see Escanciano (2006a). Under spatial dependence, the exponential weighting family $w(z_{n,i},z)=\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\{z\prime z_{n,i}\}$ of Bierens (1990) has been studied in Lee, Phillips, and Rossi (2020). Our results based on the simple indicator weighting function can be extended to work for other weighting functions.

2 At this point, one may wonder why we choose $s_{n,i}(\theta )$ as (11) instead of ${({\overline{y}}_{n,i},x_{n,i}^{\prime },{\overline{z}}_{n,i}^{\prime })}^{\prime }$. As will be seen later, if we use the latter one, the projection-based process in (12) will diverge to infinity for every z under H₀, since the noncentralized quadratic form $(U_n^{\prime }{G}_n{U}_n)/\sqrt{n}$ will diverge when $n\to \infty$ if $\text{tr}(G_n)\ne 0$, where $G_n=W_n{(I_n-{\lambda }_0{W}_n)}^{-1}$. This implies that the resulting test will not have the correct size.

3 We maintain the linearity of x and only include z-variables in the moment condition (5) for both tests.

Additional information

Funding

Song gratefully acknowledges the financial support from the National Natural Science Foundation of China (No. 72373007) and Yu gratefully acknowledges the financial support from the National Natural Science Foundation of China (No. 71925006). Song and Yu also gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. 72333001) and support from the Center for Statistical Science of Peking University, China, and the Key Laboratory of Mathematical Economics and Quantitative Finance (Peking University) of the Ministry of Education, China.