Empir Econ (2018) 55:35–68
Bootstrap LM tests for higher-order spatial effects
in spatial linear regression models
Received: 15 May 2017 / Accepted: 15 January 2018 / Published online: 28 May 2018
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract This paper ﬁrst extends the methodology of Yang (J Econom 185:33–59,
2015) to allow for non-normality and/or unknown heteroskedasticity in obtaining
asymptotically reﬁned critical values for the LM-type tests through bootstrap. Boot-
strap reﬁnements in critical values require the LM test statistics to be asymptotically
pivotal under the null hypothesis, and for this we provide a set of general methods
for constructing LM and robust LM tests. We then give detailed treatments for two
general higher-order spatial linear regression models: namely the SARAR(p, q) model
and the MESS(p, q) model, by providing a complete set of non-normality robust LM
and bootstrap LM tests for higher-order spatial effects, and a complete set of LM and
bootstrap LM tests robust against both unknown heteroskedasticity and non-normality.
Monte Carlo experiments are run, and results show an excellent performance of the
bootstrap LM-type tests.
Keywords Asymptotic pivot · Bootstrap · Heteroskedasticity · LM test · Spatial lag ·
Spatial error · Matrix exponential · Wild bootstrap · Bootstrap critical values
JEL Classiﬁcations: C12 · C15 · C18 · C21
I am grateful to Singapore Management University for ﬁnancial support under Grant C244/MSS16E003. I
thank Badi Baltagi for the invitation, and Harry Kelejian and two referees for the helpful comments.
School of Economics, Singapore Management University, Singapore, Singapore