# Consistent specification testing under spatial dependence††thanks: We thank
the editor, co-editor and three referees for insightful comments that improved
the paper. We are grateful to Swati Chandna, Miguel Delgado, Emmanuel Guerre,
Fernando López Hernandéz, Hon Ho Kwok, Arthur Lewbel, Daisuke Murakami, Ryo
Okui and Amol Sasane for helpful comments, and audiences at YEAP 2018
(Shanghai University of Finance and Economics), NYU Shanghai, Carlos III
Madrid, SEW 2018 (Dijon), Aarhus University, SEA 2018 (Vienna), EcoSta 2018
(Hong Kong), Hong Kong University, AFES 2018 (Cotonou), ESEM 2018 (Cologne),
CFE 2018 (Pisa), University of York, Penn State, Michigan State, University of
Michigan, Texas A&M, 1st Southampton Workshop on Econometrics and Statistics
and MEG 2019 (Columbus). We also thank Xifeng Wen from the Experiment and Data
Center of Antai College of Economics and Management (SJTU) for expert
computing assistance.

Abhimanyu Gupta Department of Economics, University of Essex, Wivenhoe Park,
Colchester CO4 3SQ, UK. E-mail<EMAIL_ADDRESS>supported by ESRC
grant ES/R006032/1. Xi Qu Antai College of Economics and Management, Shanghai
Jiao Tong University, Shanghai, China, 200052. E-mail:
<EMAIL_ADDRESS>supported by the National Natural Science Foundation
of China, Project Nos. 72222007, 71973097 and 72031006.

###### Abstract

We propose a series-based nonparametric specification test for a regression
function when data are spatially dependent, the ‘space’ being of a general
economic or social nature. Dependence can be parametric, parametric with
increasing dimension, semiparametric or any combination thereof, thus covering
a vast variety of settings. These include spatial error models of varying
types and levels of complexity. Under a new smooth spatial dependence
condition, our test statistic is asymptotically standard normal. To prove the
latter property, we establish a central limit theorem for quadratic forms in
linear processes in an increasing dimension setting. Finite sample performance
is investigated in a simulation study, with a bootstrap method also justified
and illustrated. Empirical examples illustrate the test with real-world data.

Keywords: Specification testing, nonparametric regression, spatial dependence,
cross-sectional dependence

JEL Classification: C21, C55

##  1 Introduction

Models for spatial dependence have recently become the subject of vigorous
research. This burgeoning interest has roots in the needs of practitioners who
frequently have access to data sets featuring inter-connected cross-sectional
units. Motivated by these practical concerns, we propose a specification test
for a regression function in a general setup that covers a vast variety of
commonly employed spatial dependence models and permits the complexity of
dependence to increase with sample size. Our test is consistent, in the sense
that a parametric specification is tested with asymptotically unit power
against a nonparametric alternative. The ‘spatial’ models that we study are
not restricted in any way to be geographic in nature, indeed ‘space’ can be a
very general economic or social space. Our empirical examples feature conflict
alliances and technology externalities as examples of ‘spatial dependence’,
for instance.

Specification testing is an important problem, and this is reflected in a huge
literature studying consistent tests. Much of this is based on independent,
and often also identically distributed, data. However data frequently exhibit
dependence and consequently a branch of the literature has also examined
specification tests under time series dependence. Our interest centers on
dependence across a ‘space’, which differs quite fundamentally from dependence
in a time series context. Time series are naturally ordered and locations of
the observations can be observed, or at least the process generating these
locations may be modelled. It can be imagined that concepts from time series
dependence be extended to settings where the data are observed on a geographic
space and dependence can be treated as a decreasing function of distance
between observations. Indeed much work has been done to extend notions of time
series dependence in this type of setting, see e.g. Jenish and Prucha (2009,
2012).

However, in a huge variety of economics and social science applications agents
influence each other in ways that do not conform to such a setting. For
example, farmers affect the demand of farmers in the same village but not in
different villages, as in Case (1991). Likewise, price competition among firms
exhibits spatial features (Pinkse et al. (2002)), input-output relations lead
to complementarities between sectors (Conley and Dupor (2003)), co-author
connections form among scientists (Oettl (2012), Mohnen (2022)), R&D
spillovers occur through technology and product market spaces (Bloom et al.
(2013)), networks form due to allegiances in conflicts (König et al. (2017))
and overlapping bank portfolios lead to correlated lending decisions (Gupta et
al. (2021)). Such examples cannot be studied by simply extending results
developed for time series and illustrate the growing need for suitable
methods.

A very popular model for general spatial dependence is the spatial
autoregressive (SAR) class, due to Cliff and Ord (1973). The key feature of
SAR models, and various generalizations such as SARMA (SAR moving average) and
matrix exponential spatial specifications (MESS, due to LeSage and Pace
(2007)), is the presence of one or more spatial weight matrices whose elements
characterize the links between agents. As noted above, these links may form
for a variety of reasons, so the ‘spatial’ terminology represents a very
general notion of space, such as social or economic space. Key papers on the
estimation of SAR models and their variants include Kelejian and Prucha (1998)
and Lee (2004), but research on various aspects of these is active and
ongoing, see e.g. Robinson and Rossi (2015); Hillier and Martellosio (2018a,
b); Kuersteiner and Prucha (2020); Han et al. (2021); Hahn et al. (2020).

Unlike work focusing on independent or time series data, a general drawback of
spatially oriented research has been the lack of general unified theory.
Typically, individual papers have studied specific special cases of various
spatial specifications. A strand of the literature has introduced the notion
of a cross-sectional linear-process to help address this problem, and we
follow this approach. This representation can accommodate SAR models in the
error term (so called spatial error models (SEM)) as a special case, as well
as variants like SARMA and MESS, whence its generality is apparent. The
linear-process structure shares some similarities with that familiar from the
time series literature (see e.g. Hannan (1970)). Indeed, time series versions
may be regarded as very special cases but, as stressed before, the features of
spatial dependence must be taken into account in the general formulation. Such
a representation was introduced by Robinson (2011) and further examined in
other situations by Robinson and Thawornkaiwong (2012) (partially linear
regression), Delgado and Robinson (2015) (non-nested correlation testing), Lee
and Robinson (2016) (series estimation of nonparametric regression) and
Hidalgo and Schafgans (2017) (cross-sectionally dependent panels).

In this paper, we propose a test statistic similar to that of Hong and White
(1995), based on estimating the nonparametric specification via series
approximations. Assuming an independent and identically distributed sample,
their statistic is based on the sample covariance between the residual from
the parametric model and the discrepancy between the parametric and
nonparametric fitted values. Allowing additionally for spatial dependence
through the form of a linear process as discussed above, our statistic is
shown to be asymptotically standard normal, consistent and possessing
nontrivial power against local alternatives of a certain type. To prove
asymptotic normality, we present a new central limit theorem (CLT) for
quadratic forms in linear processes in an increasing dimension setting that
may be of independent interest. A CLT for quadratic forms under time series
dependence in the context of series estimation can be found in Gao and Anh
(2000), and our result can be viewed as complementary to this. The setting of
Su and Qu (2017) is a very special case of our framework. There has been
recent interest in specification testing for spatial models, see for example
Sun (2020) for a kernel-based model specification test and Lee et al. (2020)
for a consistent omnibus test. We contribute to this literature by studying a
linear process based increasing parameter dimension framework.

Our linear process framework permits spatial dependence to be parametric,
parametric with increasing dimension, semiparametric or any combination
thereof, thus covering a vast variety of settings. A class of models of great
empirical interest are ‘higher-order’ SAR models in the outcome variables, but
with spatial dependence structure also in the errors. We initially present the
familiar nonparametric regression to clarify the exposition, and then cover
this class as the main model of interest. Our theory covers as special cases
SAR, SMA, SARMA, MESS models for the error term. These specifications may be
of any fixed spatial order, but our theory also covers the case where they are
of increasing order.

Thus we permit a more complex model of spatial dependence as more data become
available, which encourages a more flexible approach to modelling such
dependence as stressed by Gupta and Robinson (2015, 2018) in a higher-order
SAR context, Huber (1973), Portnoy (1984, 1985) and Anatolyev (2012) in a
regression context and Koenker and Machado (1999) for the generalized method
of moments setting, amongst others. This literature focuses on a sequence of
true models, rather than a sequence of models approximating an infinite true
model. Our paper also takes the same approach. On the other hand, in the
spatial setting, Gupta (2018a) considers increasing lag models as
approximations to an infinite lag model with lattice data and also suggests
criteria for choice of lag length.

Our framework is also extended to the situation where spatial dependence
occurs through nonparametric functions of raw distances (these may be
exogenous economic or social distances, say), as in Pinkse et al. (2002). This
allows for greater flexibility in modelling spatial weights as the
practitioner only has to choose an exogenous economic distance measure and
allow the data to determine the functional form. It also adds a degree of
robustness to the theory by avoiding potential parametric misspecification.
The case of geographical data is also covered, for example the important
classes of Matérn and Wendland (see e.g. Gneiting (2002)) covariance
functions. Finally, we introduce a new notion of smooth spatial dependence
that provides more primitive, and checkable, conditions for certain properties
than extant ones in the literature.

To illustrate the performance of the test in finite samples, we present Monte
Carlo simulations that exhibit satisfactory small sample properties. The test
is demonstrated in three empirical examples, including two based on recently
published work on social networks: Bloom et al. (2013) (R&D spillovers in
innovation), König et al. (2017) (conflict alliances during the Congolese
civil war). Another example studies cross-country spillovers in economic
growth. Our test may or may not reject the null hypothesis of a linear
regression in these examples, illustrating its ability to distinguish well
between the null and alternative models.

The next section introduces our basic setup using a nonparametric regression
with no SAR structure in responses. We treat this abstraction as a base case,
and Section 3 discusses estimation and defines the test statistic, while
Section 4 introduces assumptions and the key asymptotic results of the paper.
Section 5 examines the most commonly employed higher-order SAR models, while
Section 6 deals with nonparametric spatial error structures. Nonparametric
specification tests are often criticized for poor finite sample performance
when using the asymptotic critical values. In Section 7 we present a bootstrap
version of our testing procedure. Sections 8 and 9 contain a study of finite
sample performance and the empirical examples respectively, while Section 10
concludes. Proofs are contained in appendices, including a supplementary
online appendix which also contains additional simulation results.

For the convenience of the reader, we collect some frequently used notation
here. First, we introduce three notational conventions for any parameter $\nu$
for the rest of the paper: $\nu\in\mathbb{R}^{d_{\nu}}$, $\nu_{0}$ denotes the
true value of $\nu$ and for any scalar, vector or matrix valued function
$f(\nu)$, we denote $f\equiv f(\nu_{0})$. Let $\overline{\varphi}(\cdot)$
(respectively $\underline{\varphi}(\cdot)$) denote the largest (respectively
smallest) eigenvalue of a generic square nonnegative definite matrix argument.
For a generic matrix $A$, denote
$\left\|A\right\|=\left[\overline{\varphi}(A^{\prime}A)\right]^{1/2}$, i.e.
the spectral norm of $A$ which reduces to the Euclidean norm if $A$ is a
vector. $\left\|A\right\|_{R}$ denotes the maximum absolute row sum norm of a
generic matrix $A$ while
$\left\|A\right\|_{F}=\left[tr(AA^{\prime})\right]^{1/2}$, the Frobenius norm.
Throughout the paper $|\cdot|$ is absolute value when applied to a scalar and
determinant when applied to a matrix. Denote by $c$ ($C$) generic positive
constants, independent of any quantities that tend to infinity, and
arbitrarily small (big).

##  2 Setup

To illustrate our approach, we first consider the nonparametric regression

$y_{i}=\theta_{0}\left(x_{i}\right)+u_{i},i=1,\ldots,n,$ (2.1)

where $\theta_{0}(\cdot)$ is an unknown function and $x_{i}$ is a vector of
strictly exogenous explanatory variables with support
$\mathcal{X}\subset\mathbb{R}^{k}$. Spatial dependence is explicitly modeled
via the error term $u_{i}$, which we assume is generated by:

$u_{i}=\sum_{s=1}^{\infty}b_{is}\varepsilon_{s},$ (2.2)

where $\varepsilon_{s}$ are independent random variables, with zero mean and
identical variance $\sigma_{0}^{2}$. Further conditions on the
$\varepsilon_{s}$ will be assumed later. The linear process coefficients
$b_{is}$ can depend on $n$, as may the covariates $x_{i}$. This is generally
the case with spatial models and implies that asymptotic theory ought to be
developed for triangular arrays. There are a number of reasons to permit
dependence on sample size. The $b_{is}$ can depend on spatial weight matrices,
which are usually normalized for both stability and identification purposes.

Such normalizations, e.g. row-standardization or division by spectral norm,
may be $n$-dependent. Furthermore, $x_{i}$ often includes underlying
covariates of ‘neighbors’ defined by spatial weight matrices. For instance,
for some $n\times 1$ covariate vector $z$ and exogenous spatial weight matrix
$W\equiv W_{n}$, a component of $x_{i}$ can be $e_{i}^{\prime}Wz$, where
$e_{i}$ has unity in the $i$-th position and zeros elsewhere, which depends on
$n$. Thus, subsequently, any spatial weight matrices will also be allowed to
depend on $n$. Finally, treating triangular arrays permits re-labelling of
quantities that is often required when dealing with spatial data, due to the
lack of natural ordering, see e.g. Robinson (2011). We suppress explicit
reference to this $n$-dependence of various quantities for brevity, although
mention will be made of this at times to remind the reader of this feature.

Now, assume the existence of a $d_{\gamma}\times 1$ vector $\gamma_{0}$ such
that $b_{is}=b_{is}(\gamma_{0})$, possibly with $d_{\gamma}\rightarrow\infty$
as $n\rightarrow\infty$, for all $i=1,\ldots,n$ and $s\geq 1$. Let $u$ be the
$n\times 1$ vector with typical element $u_{i}$, $\varepsilon$ be the infinite
dimensional vector with typical element $\varepsilon_{s},$ and $B$ be an
infinite dimensional matrix (Cooke, 1950) with typical element $b_{is}.$ In
matrix form,

$u=B\varepsilon\text{ and
}\mathcal{E}\left(uu^{\prime}\right)=\sigma_{0}^{2}BB^{\prime}=\sigma_{0}^{2}\Sigma\equiv\sigma_{0}^{2}\Sigma\left(\gamma_{0}\right).$
(2.3)

We assume that $\gamma_{0}\in\Gamma$, where $\Gamma$ is a compact subset of
$\mathbb{R}^{d_{\gamma}}$. With $d_{\gamma}$ diverging, ensuring $\Gamma$ has
bounded volume requires some care, see Gupta and Robinson (2018). For a known
function $f(\cdot)$, our aim is to test

$H_{0}:P[\theta_{0}\left(x_{i}\right)=f(x_{i},\alpha_{0})]=1,\text{ for some
}\alpha_{0}\in\mathcal{A}\subset\mathbb{R}^{d_{\alpha}},$ (2.4)

against the global alternative $H_{1}:P\left[\theta_{0}\left(x_{i}\right)\neq
f(x_{i},\alpha)\right]>0,\text{ for all }\alpha\in\mathcal{A}$.

We now nest commonly used models for spatial dependence in (2.3). Introduce a
set of $n\times n$ spatial weight (equivalently network adjacency) matrices
$W_{j}$, $j=1,\ldots,m_{1}+m_{2}$. Each $W_{j}$ can be thought of as
representing dependence through a particular space. Now, consider models of
the form $\Sigma(\gamma)=A^{-1}(\gamma)A^{\prime-1}(\gamma)$. For example,
with $\xi$ denoting a vector of iid disturbances with variance
$\sigma_{0}^{2}$, the model with SARMA$(m_{1},m_{2})$ errors is
$u=\sum_{j=1}^{m_{1}}\gamma_{j}W_{j}u+\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}W_{j}\xi+\xi$,
with
$A(\gamma)=\left(I_{n}+\sum_{j=m_{1}+1}^{m_{1}+m_{2}}\gamma_{j}W_{j}\right)^{-1}\left(I_{n}-\sum_{j=1}^{m_{1}}\gamma_{j}W_{j}\right)$,
assuming conditions that guarantee the existence of the inverse. Such
conditions can be found in the literature, see e.g. Lee and Liu (2010) and
Gupta and Robinson (2018). The SEM model is obtained by setting $m_{2}=0$
while the model with SMA errors has $m_{1}=0$. The model with MESS$(m)$ errors
(LeSage and Pace (2007), Debarsy et al. (2015)) is
$u=\exp\left(\sum_{j=1}^{m}\gamma_{j}W_{j}\right)\xi,A(\gamma)=\exp\left(-\sum_{j=1}^{m}\gamma_{j}W_{j}\right).$

In some cases the space under consideration is geographic i.e. the data may be
observed at irregular points in Euclidean space. Making the identification
$u_{i}\equiv U\left(t_{i}\right)$, $t_{i}\in\mathbb{R}^{d}$ for some $d>1$,
and assuming covariance stationarity, $U(t)$ is said to follow an isotropic
model if, for some function $\delta$ on $\mathbb{R}$, the covariance at lag
$s$ is $r(s)=\mathcal{E}\left[U(t)U(t+s)\right]=\delta(\|s\|)$. An important
class of parametric isotropic models is that of Matérn (1986), which can be
parameterized in several ways, see e.g. Stein (1999). Denoting by $\Gamma_{f}$
the Gamma function and by $\mathcal{K}_{\gamma_{1}}$ the modified Bessel
function of the second kind (Gradshteyn and Ryzhik (1994)), take
$\delta(\left\|s\right\|,\gamma)=\left(2^{\gamma_{1}-1}\Gamma_{f}(\gamma_{1})\right)^{-1}\left(\gamma_{2}^{-1}\sqrt{2\gamma_{1}}\left\|s\right\|\right)^{\gamma_{1}}\mathcal{K}_{\gamma_{1}}\left(\gamma_{2}^{-1}\sqrt{2\gamma_{1}}\left\|s\right\|\right),$
with $\gamma_{1},\gamma_{2}>0$ and $d_{\gamma}=2$. With $d_{\gamma}=3$,
another model takes
$\delta(\left\|s\right\|,\gamma)=\gamma_{1}\exp\left(-\left\|s/\gamma_{2}\right\|^{\gamma_{3}}\right)$,
see e.g. De Oliveira et al. (1997), Stein (1999). Fuentes (2007) considers
this model with $\gamma_{3}=1$, as well as a specific parameterization of the
Matèrn covariance function.

##  3 Test statistic

We estimate $\theta_{0}(\cdot)$ via a series approximation. Certain technical
conditions are needed to allow for $\mathcal{X}$ to have unbounded support. To
this end, for a function $g(x)$ on $\mathcal{X}$, define a weighted sup-norm
(see e.g. Chen et al. (2005), Chen (2007), Lee and Robinson (2016)) by
$\left\|g\right\|_{w}=\sup_{x\in\mathcal{X}}\left|g(x)\right|\left(1+\left\|x\right\|^{2}\right)^{-w/2},\text{
for some }w>0$. Assume that there exists a sequence of functions
$\psi_{i}:=\psi\left(x_{i}\right):\mathbb{R}^{k}\mapsto\mathbb{R}^{p}$, where
$p\rightarrow\infty$ as $n\rightarrow\infty$, and a $p\times 1$ vector of
coefficients $\beta_{0}$ such that

$\theta_{0}\left(x_{i}\right)=\psi_{i}^{\prime}\beta_{0}+e\left(x_{i}\right),$
(3.1)

where $e(\cdot)$ satisfies:

######  Assumption R.1.

There exists a constant $\mu>0$ such that
$\left\|e\right\|_{w_{x}}=O\left(p^{-\mu}\right),$ as $p\rightarrow\infty$,
where $w_{x}\geq 0$ is the largest value such that
$\sup_{i=1,\ldots,n}\mathcal{E}\left\|x_{i}\right\|^{w_{x}}<\infty$, for all
$n$.

By Lemma 1 in Appendix B of Lee and Robinson (2016), this assumption implies
that

$\sup_{i=1,\ldots,n}\mathcal{E}\left(e^{2}\left(x_{i}\right)\right)=O\left(p^{-2\mu}\right).$
(3.2)

Due to the large number of assumptions in the paper, sometimes with changes
reflecting only the various setups we consider, we prefix assumptions with R
in this section and the next, to signify ‘regression’. In Section 5 the prefix
is SAR, for ‘spatial autoregression’, while in Section 6 we use NPN, for
‘nonparametric network’.

Let
$y=(y_{1},\ldots,y_{n})^{\prime},{\theta_{0}}=(\theta_{0}\left(x_{1}\right),\ldots,\theta_{0}\left(x_{n}\right))^{\prime},\Psi=(\psi_{1},\ldots,\psi_{n})^{\prime}$.
We will estimate $\gamma_{0}$ using a quasi maximum likelihood estimator
(QMLE) based on a Gaussian likelihood, although Gaussianity is nowhere
assumed. For any admissible values $\beta$, $\sigma^{2}$ and $\gamma$, the
(multiplied by $2/n$) negative quasi log likelihood function based on using
the approximation (3.1) is

${L}(\beta,\sigma^{2},\gamma)=\ln\left(2\pi\sigma^{2}\right)+\frac{1}{n}\ln\left|\Sigma\left(\gamma\right)\right|+\frac{1}{n\sigma^{2}}(y-\Psi\beta)^{\prime}\Sigma\left(\gamma\right)^{-1}(y-\Psi\beta),$
(3.3)

which is minimised with respect to $\beta$ and $\sigma^{2}$ by

$\displaystyle\bar{\beta}\left(\gamma\right)$ $\displaystyle=$
$\displaystyle\left(\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}y,$
(3.4) $\displaystyle\bar{\sigma}^{2}\left(\gamma\right)$ $\displaystyle=$
$\displaystyle{n^{-1}}y^{\prime}E(\gamma)^{\prime}M(\gamma)E(\gamma)y,$ (3.5)

where
$M(\gamma)=I_{n}-E(\gamma)\Psi\left(\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\Psi^{\prime}E(\gamma)^{\prime}$
and $E(\gamma)$ is the $n\times n$ symmetric matrix such that
$E(\gamma)E(\gamma)^{\prime}=\Sigma(\gamma)^{-1}$. The use of the approximate
likelihood relies on the negligibility of $e(\cdot)$, which in turn permits
the replacement of $\theta_{0}(\cdot)$ by $\psi^{\prime}\beta_{0}$ with
asymptotically negligible cost. Thus the concentrated likelihood function is

$\mathcal{L}(\gamma)=\ln(2\pi)+\ln\bar{\sigma}^{2}(\gamma)+\frac{1}{n}\ln\left|\Sigma\left(\gamma\right)\right|.$
(3.6)

We define the QMLE of $\gamma_{0}$ as $\widehat{\gamma}=\text{arg
min}_{\gamma\in\Gamma}\mathcal{L}(\gamma)$ and the QMLEs of $\beta_{0}$ and
$\sigma_{0}^{2}$ as $\widehat{\beta}=\bar{\beta}\left(\widehat{\gamma}\right)$
and $\widehat{\sigma}^{2}=\bar{\sigma}^{2}\left(\widehat{\gamma}\right)$. At a
given $x_{1},\ldots,x_{n}$, the series estimate of $\theta_{0}$ is defined as

$\widehat{\theta}=\left(\hat{\theta}(x_{1}),\ldots,\hat{\theta}(x_{n})\right)^{\prime}=\left(\psi(x_{1})^{\prime}\widehat{\beta},\ldots,\psi(x_{n})^{\prime}\widehat{\beta}\right)^{\prime}.$
(3.7)

Let $\widehat{\alpha}_{n}\equiv\widehat{\alpha}$ denote an estimator
consistent for $\alpha_{0}$ under $H_{0}$, for example the (nonlinear) least
squares estimator. Note that $\widehat{\alpha}$ is consistent only under
$H_{0}$, so we introduce a general probability limit of $\widehat{\alpha}$, as
in Hong and White (1995).

######  Assumption R.2.

There exists a deterministic sequence $\alpha_{n}^{*}\equiv\alpha^{*}$ such
that $\widehat{\alpha}-\alpha^{*}=O_{p}\left(1/\sqrt{n}\right)$.

Examples of estimators that satisfy this assumption include (nonlinear) least
squares, generalized method of moments estimators or adaptive efficient
weighted least squares (Stinchcombe and White, 1998).

Following Hong and White (1995), define the regression error $u_{i}\equiv
y_{i}-f(x_{i},\alpha^{\ast})$ and the specification error
$v_{i}\equiv\theta_{0}(x_{i})-f(x_{i},\alpha^{\ast})$. Our test statistic is
based on a scaled and centered version of
$\widehat{m}_{n}=\widehat{\sigma}^{-2}\widehat{{v}}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{{u}}/n=\widehat{\sigma}^{-2}\left(\widehat{{\theta}}-{f}\left(x,\widehat{\alpha}\right)\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(y-{f}\left(x,\widehat{\alpha}\right)\right)/n$,
where
$f(x,\alpha)=\left(f\left(x_{1},\alpha\right),\ldots,f\left(x_{n},\alpha\right)\right)^{\prime}$.
Precisely, it is defined as

$\mathscr{T}_{n}=\frac{n\widehat{m}_{n}-p}{\sqrt{2p}}.$ (3.8)

The motivation for such a centering and scaling stems from the fact that, for
fixed $p$, $n\widehat{m}_{n}$ has an asymptotic $\chi^{2}_{p}$ distribution.
Such a distribution has mean $p$ and variance $2p$, and it is a well-known
fact that
$\left(\chi^{2}_{p}-p\right)/{\sqrt{2p}}\overset{d}{\longrightarrow}N(0,1),\text{
as }p\rightarrow\infty$. This motivates our use of (3.8) and explains why we
aspire to establish a standard normal distribution under the null hypothesis.
Intuitively, the test statistic is based on the sample covariance between the
residual from the parametric model and the discrepancy between the parametric
and nonparametric fitted values, as in Hong and White (1995).

Hong and White (1995) also note that, due to the nonparametric nature of the
problem, such a statistic vanishes faster than the parametric
($n^{\frac{1}{2}}$) rate, thus a $n^{\frac{1}{2}}$-normalization leads to
degeneracy of the test. A proper normalization as in (3.8) will yield a non-
degenerate limiting distribution. As Hong and White (1995) noted, our test is
one-sided. This is because asymptotically negative values of our test
statistic can occur only under the null, while under the alternative it tends
to a positive, increasing number. Thus, we reject the null if our test
statistic is on the right tail.

##  4 Asymptotic theory

###  4.1 Consistency of $\widehat{\gamma}$

We first provide conditions under which our estimator $\widehat{\gamma}$ of
$\gamma_{0}$ is consistent. Such a property is necessary for the results that
follow. The following assumption is a rather standard type of asymptotic
boundedness and full-rank condition on $\Sigma(\gamma)$.

######  Assumption R.3.

$\varlimsup_{n\rightarrow\infty}\sup_{\gamma\in\Gamma}\bar{\varphi}\left(\Sigma(\gamma)\right)<\infty\text{
and
}\varliminf_{n\rightarrow\infty}\inf_{\gamma\in\Gamma}\underline{\varphi}\left(\Sigma(\gamma)\right)>0.$

######  Assumption R.4.

The $u_{i},i=1,\ldots,n,$ satisfy the representation (2.2). The
$\varepsilon_{s}$, $s\geq 1$, have zero mean, finite third and fourth moments
$\mu_{3}$ and $\mu_{4}$ respectively and, denoting by $\sigma_{ij}(\gamma)$
the $(i,j)$-th element of $\Sigma(\gamma)$ and defining
$b_{is}^{\ast}={b_{is}}/{\sigma_{ii}^{\frac{1}{2}}},\;i=1,\ldots,n,\;n\geq
1,s\geq 1,$ we have

$\underset{n\rightarrow\infty}{\overline{\lim}}\sup_{i=1,\ldots,n}\sum_{s=1}^{\infty}\left|b_{is}^{\ast}\right|+\sup_{s\geq
1}\underset{n\rightarrow\infty}{\overline{\lim}}\sum_{i=1}^{n}\left|b_{is}^{\ast}\right|<\infty.$
(4.1)

By Assumption R.3, $\sigma_{ii}$ is bounded and bounded away from zero, so the
normalization of the $b_{is}$ in Assumption R.4 is well defined. The
summability conditions in (4.1) are typical conditions on linear process
coefficients that are needed to control dependence; for instance in the case
of stationary time series $b^{*}_{is}=b^{*}_{i-s}$. The infinite linear
process assumed in (2.2) is further discussed by Robinson (2011), who
introduced it, and also by Delgado and Robinson (2015). These assumptions
imply an increasing-domain asymptotic setup and preclude infill asymptotics.

Because we often need to consider the difference between values of the matrix-
valued function $\Sigma(\cdot)$ at distinct points, it is useful to introduce
an appropriate concept of ‘smoothness’. This concept has been employed before
in economics, see e.g. Chen (2007), and is defined below.

######  Definition 1.

Let $\left(X,\left\|\cdot\right\|_{X}\right)$ and
$\left(Y,\left\|\cdot\right\|_{Y}\right)$ be Banach spaces, $\mathscr{L}(X,Y)$
be the Banach space of linear continuous maps from $X$ to $Y$ with norm
$\left\|T\right\|_{\mathscr{L}(X,Y)}=\sup_{\left\|x\right\|_{X}\leq
1}\left\|T(x)\right\|_{Y}$ and $U$ be an open subset of $X$. A map
$F:U\rightarrow Y$ is said to be Fréchet-differentiable at $u\in U$ if there
exists $L\in\mathscr{L}(X,Y)$ such that

$\lim_{\left\|h\right\|_{X}\rightarrow
0}\frac{F(u+h)-F(u)-L(h)}{\left\|h\right\|_{X}}=0.$ (4.2)

$L$ is called the Fréchet-derivative of $F$ at $u$. The map $F$ is said to be
Fréchet-differentiable on $U$ if it is Fréchet-differentiable for all $u\in
U$.

The above definition extends the notion of a derivative that is familiar from
real analysis to the functional spaces and allows us to check high-level
assumptions that past literature has imposed. To the best of our knowledge,
this is the first use of such a concept in the literature on spatial/network
models. Denote by $\mathcal{M}^{n\times n}$ the set of real, symmetric and
positive semi-definite $n\times n$ matrices. Let $\Gamma^{o}$ be an open
subset of $\Gamma$ and consider the Banach spaces
$\left(\Gamma,\left\|\cdot\right\|_{g}\right)$ and $\left(\mathcal{M}^{n\times
n},\left\|\cdot\right\|\right)$, where $\left\|\cdot\right\|_{g}$ is a generic
$\ell_{p}$ norm, $p\geq 1$. The following assumption ensures that
$\Sigma(\cdot)$ is a ‘smooth’ function, in the sense of Fréchet-smoothness.

######  Assumption R.5.

The map $\Sigma:\Gamma^{o}\rightarrow\mathcal{M}^{n\times n}$ is Fréchet-
differentiable on $\Gamma^{o}$ with Fréchet-derivative denoted
$D\Sigma\in\mathscr{L}\left(\Gamma^{o},\mathcal{M}^{n\times n}\right)$.
Furthermore, the map $D\Sigma$ satisfies

$\sup_{\gamma\in\Gamma^{o}}\left\|D\Sigma(\gamma)\right\|_{\mathscr{L}\left(\Gamma^{o},\mathcal{M}^{n\times
n}\right)}\leq C.$ (4.3)

Assumption R.5 is a functional smoothness condition on spatial dependence. It
has the advantage of being checkable for a variety of commonly employed
models. For example, a first-order SEM has
$\Sigma(\gamma)=A^{-1}(\gamma)A^{\prime-1}(\gamma)$ with $A=I_{n}-\gamma W$.
Corollary CS.1 in the supplementary appendix shows
$\left(D\Sigma(\gamma)\right)\left(\gamma^{\dagger}\right)=\gamma^{\dagger}A^{-1}(\gamma)\left(G^{\prime}(\gamma)+G(\gamma)\right)A^{\prime-1}(\gamma)$,
at a given point $\gamma\in\Gamma^{o}$, where $G(\gamma)=WA^{-1}(\gamma)$.
Then, taking

$\left\|W\right\|+\sup_{\gamma\in\Gamma}\left\|A^{-1}(\gamma)\right\|<C$ (4.4)

yields Assumption R.5. Condition (4.4) limits the extent of spatial dependence
and is very standard in the spatial literature; see e.g. Lee (2004) and
numerous subsequent papers employing similar conditions.

Fréchet derivatives for higher-order SAR, SMA, SARMA and MESS error structures
are computed in supplementary appendix S.D, in Lemmas LS.5-LS.6 and
Corollaries CS.1-CS.2. Strictly speaking, Gateaux differentiability might
suffice for the type of results that we target. We opt for Fréchet
differentiability because this derivative map is linear and continuous or,
equivalently, a bounded linear operator, a property that makes Assumption R.5
more reasonable.

The following proposition is very useful in ‘linearizing’ perturbations in the
$\Sigma(\cdot)$.

######  Proposition 4.1.

If Assumption R.5 holds, then for any $\gamma_{1},\gamma_{2}\in\Gamma^{o}$,

$\left\|\Sigma\left(\gamma_{1}\right)-\Sigma\left(\gamma_{2}\right)\right\|\leq
C\left\|\gamma_{1}-\gamma_{2}\right\|.$ (4.5)

To illustrate how the concept of Fréchet-differentiability allows us to check
high-level assumptions extant in the literature, a consequence of Proposition
4.1 is the following corollary, a version of which appears as an assumption in
Delgado and Robinson (2015).

######  Corollary 4.1.

For any $\gamma^{*}\in\Gamma^{o}$ and any $\eta>0$,

$\underset{n\rightarrow\infty}{\overline{\lim}}\sup_{\gamma\in\left\\{\gamma:\left\|\gamma-\gamma^{*}\right\|<\eta\right\\}\cap\Gamma^{o}}\left\|\Sigma(\gamma)-\Sigma\left(\gamma^{*}\right)\right\|<C\eta.$
(4.6)

We now introduce regularity conditions needed to establish the consistency of
$\hat{\gamma}$. Define

$\sigma^{2}\left(\gamma\right)=n^{-1}\sigma^{2}tr\left(\Sigma(\gamma)^{-1}\Sigma\right)=n^{-1}\sigma^{2}\left\|E(\gamma)E^{-1}\right\|_{F}^{2},$

which is nonnegative by definition and bounded by Assumption R.3, red with the
matrix $E(\gamma)$ defined after (3.5).

######  Assumption R.6.

$c\leq\sigma^{2}\left(\gamma\right)\leq C$ for all $\gamma\in\Gamma$.

######  Assumption R.7.

$\gamma_{0}\in\Gamma$ and, for any $\eta>0$,

$\varliminf_{n\rightarrow\infty}\inf_{\gamma\in\overline{\mathcal{N}}^{\gamma}(\eta)}\frac{n^{-1}tr\left(\Sigma(\gamma)^{-1}\Sigma\right)}{\left|\Sigma(\gamma)^{-1}\Sigma\right|^{1/n}}>1,$
(4.7)

where
$\overline{\mathcal{N}}^{\gamma}(\eta)=\Gamma\setminus\mathcal{N}^{\gamma}(\eta)$
and
$\mathcal{N}^{\gamma}(\eta)=\left\\{\gamma:\left\|\gamma-\gamma_{0}\right\|<\eta\right\\}\cap\Gamma$.

######  Assumption R.8.

$\left\\{\underline{\varphi}\left(n^{-1}\Psi^{\prime}\Psi\right)\right\\}^{-1}+\overline{\varphi}\left(n^{-1}\Psi^{\prime}\Psi\right)=O_{p}(1)$.

Assumption R.6 is a boundedness condition originally considered in Gupta and
Robinson (2018), while Assumptions R.7 and R.8 are identification conditions.
Indeed, Assumption R.7 requires that $\Sigma(\gamma)$ be identifiable in a
small neighborhood around $\gamma_{0}$. This is apparent on noticing that the
ratio in (4.7) is at least one by the inequality between arithmetic and
geometric means, and equals one when $\Sigma(\gamma)=\Sigma$. Similar
assumptions arise frequently in related literature, see e.g. Lee (2004),
Delgado and Robinson (2015). Assumption R.8 is a typical asymptotic
boundedness and non-multicollinearity condition, see e.g. Newey (1997) and
much other literature on series estimation. Primitive conditions for this
assumption to hold require the convergence (in matrix norm) of
$n^{-1}\Psi^{\prime}\Psi$ to its expectation, and this entails restrictions on
the extent of spatial dependence in the $x_{i}$. A reference is Lee and
Robinson (2016), wherein consider Assumption A.4 and the proof of Theorem 1.
By Assumption R.3, R.8 implies
$\sup_{\gamma\in\Gamma}\left\\{\underline{\varphi}\left(n^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)\right\\}^{-1}=O_{p}(1)$.

######  Theorem 4.1.

Under either $H_{0}$ or $H_{1}$, Assumptions R.1-R.8 and
$p^{-1}+\left(d_{\gamma}+p\right)/n\rightarrow 0$ as $n\rightarrow\infty$,
$\left\|\left(\widehat{\gamma},\hat{\sigma}^{2}\right)-\left(\gamma_{0},\sigma_{0}^{2}\right)\right\|\overset{p}{\longrightarrow}0.$

###  4.2 Asymptotic properties of the test statistic

Write $\Sigma_{j}(\gamma)=\partial\Sigma(\gamma)/\partial\gamma_{j}$,
$j=1,\ldots,d_{\gamma}$, the matrix differentiated element-wise. While
Assumption R.5 guarantees that these partial derivatives exist, the next
assumption imposes a uniform bound on their spectral norms.

######  Assumption R.9.

$\varlimsup_{n\rightarrow\infty}\sup_{j=1,\ldots,d_{\gamma}}\left\|\Sigma_{j}(\gamma)\right\|<C$.

We will later consider the sequence of local alternatives

$H_{\ell n}\equiv
H_{\ell}:f(x_{i},\alpha_{n}^{\ast})=\theta_{0}(x_{i})+(p^{1/4}/n^{1/2})h(x_{i}),a.s.,$
(4.8)

where $h$ is square integrable on the support $\mathcal{X}$ of the $x_{i}$.
Under the null $H_{0}$, we have $h(x_{i})=0$, a.s..

######  Assumption R.10.

For each $n\in\mathbb{N}$ and $i=1,\ldots,n$, the function
$f:\mathcal{X}\times\mathcal{A}\rightarrow\mathbb{R}$ is such that
$f\left(x_{i},\alpha\right)$ is measurable for each $\alpha\in\mathcal{A}$,
$f\left(x_{i},\cdot\right)$ is a.s. continuous on $\mathcal{A}$, with
$\sup_{\alpha\in\mathcal{A}}f^{2}\left(x_{i},\alpha\right)\leq
D_{n}\left(x_{i}\right)$, where $\sup_{n\in\mathbb{N}}D_{n}\left(x_{i}\right)$
is integrable and $\sup_{\alpha\in\mathcal{A}}\left\|\partial
f\left(x_{i},\alpha\right)/\partial\alpha\right\|^{2}\leq
D_{n}\left(x_{i}\right)$,
$\sup_{\alpha\in\mathcal{A}}\left\|\partial^{2}f\left(x_{i},\alpha\right)/\partial\alpha\partial\alpha^{\prime}\right\|\leq
D_{n}\left(x_{i}\right)$, all holding a.s..

Define the infinite-dimensional matrix
$\mathscr{V}=B^{\prime}\Sigma^{-1}\Psi\left(\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma^{-1}B$,
which is symmetric, idempotent and has rank $p$. We now show that our test
statistic is approximated by a quadratic form in $\varepsilon$, weighted by
$\mathscr{V}$.

######  Theorem 4.2.

Under Assumptions R.1-R.10,
$p^{-1}+p\left(p+d_{\gamma}^{2}\right)/n+\sqrt{n}/p^{\mu+1/4}\rightarrow 0$,
as $n\rightarrow\infty$, and $H_{0}$,
$\mathscr{T}_{n}-{\left(\sigma_{0}^{-2}\varepsilon^{\prime}\mathscr{V}\varepsilon-p\right)}/{\sqrt{2p}}=o_{p}(1).$

######  Assumption R.11.

$\underset{n\rightarrow\infty}{\overline{\lim}}\left\|\Sigma^{-1}\right\|_{R}<\infty.$

Because $\left\|\Sigma^{-1}\right\|\leq\left\|\Sigma^{-1}\right\|_{R}$, this
restriction on spatial dependence is somewhat stronger than a restriction on
spectral norm but is typically imposed for central limit theorems in this type
of setting, cf. Lee (2004), Delgado and Robinson (2015), Gupta and Robinson
(2018). The next assumption is needed in our proofs to check a Lyapunov
condition. A typical approach would be assume moments of order $4+\epsilon$,
for some $\epsilon>0$. Due to the linear process structure under
consideration, taking $\epsilon=4$ makes the proof tractable, see for example
Delgado and Robinson (2015).

######  Assumption R.12.

The $\varepsilon_{s}$, $s\geq 1$, have finite eighth moment.

The next assumption is strong if the basis functions $\psi_{ij}(\cdot)$ are
polynomials, requiring all moments to exist in that case.

######  Assumption R.13.

$\mathcal{E}\left|\psi_{ij}\left(x\right)\right|<C$, $i=1,\ldots,n$ and
$j=1,\ldots,p$.

The next theorem establishes the asymptotic normality of the approximating
quadratic form introduced above.

######  Theorem 4.3.

Under Assumptions R.3, R.4, R.8, R.11-R.13 and $p^{-1}+p^{3}/n\rightarrow 0$,
as $n\rightarrow\infty$,
${\left(\sigma_{0}^{-2}\varepsilon^{\prime}\mathscr{V}\varepsilon-p\right)}/{\sqrt{2p}}\overset{d}{\longrightarrow}N(0,1).$

This is a new type of CLT, integrating both a linear process framework as well
as an increasing dimension element. A linear-quadratic form in iid
disturbances is treated by Kelejian and Prucha (2001), while a quadratic form
in a linear process framework is treated by Delgado and Robinson (2015).
However both results are established in a parametric framework, entailing no
increasing dimension aspect of the type we face with $p\rightarrow\infty$.

Next, we summarize the properties of our test statistic in a theorem that
records its asymptotic normality under the null, consistency and ability to
detect local alternatives at $p^{1/4}/n^{1/2}$ rate. This rate has been found
also by De Jong and Bierens (1994) and Gupta (2018b). Introduce the quantity
$\varkappa=\left({\sqrt{2}\sigma_{0}^{2}}\right)^{-1}\operatorname{plim}_{n\rightarrow\infty}{n^{-1}h^{\prime}\Sigma^{-1}h}$,
where $h=\left(h\left(x_{1}\right),\ldots,h\left(x_{n}\right)\right)^{\prime}$
and $h\left(x_{i}\right)$ is from (4.8).

######  Theorem 4.4.

Under the conditions of Theorems 4.2 and 4.3, (1)
$\mathscr{T}_{n}\overset{d}{\rightarrow}N(0,1)$ under $H_{0}$, (2)
$\mathscr{T}_{n}$ is a consistent test statistic, (3)
$\mathscr{T}_{n}\overset{d}{\rightarrow}N\left(\varkappa,1\right)$ under local
alternatives $H_{\ell}$.

##  5 Models with SAR structure in responses

We now introduce the SAR model

$y_{i}=\sum_{j=1}^{d_{\lambda}}\lambda_{0j}w_{i,j}^{\prime}y+\theta_{0}\left(x_{i}\right)+u_{i},i=1,\ldots,n,$
(5.1)

where $W_{j}$, $j=1,\ldots,d_{\lambda}$, are known spatial weight matrices
with $i$-th rows denoted $w_{i,j}^{\prime}$, as discussed earlier, and
$\lambda_{0j}$ are unknown parameters measuring the strength of spatial
dependence. We take $d_{\lambda}$ to be fixed for convenience of exposition.
The error structure remains the same as in (2.2). Here spatial dependence
arises not only in errors but also responses. For example, this corresponds to
a situation where agents in a network influence each other both in their
observed and unobserved actions. Note that the error term $u_{i}$ can be
generated by the same $W_{j}$, or different ones.

While the model in (5.1) is new in the literature, some related ones are
discussed here. Models such as (5.1) but without dependence in the error
structure are considered by Su and Jin (2010) and Gupta and Robinson (2015,
2018), but the former consider only $d_{\lambda}=1$ and the latter only
parametric $\theta_{0}(\cdot)$. Linear $\theta_{0}(\cdot)$ and $d_{\lambda}>1$
are permitted by Lee and Liu (2010), but the dependence structure in errors
differs from what we allow in (5.1). Using the same setup as Su and Jin (2010)
and independent disturbances, a specification test for the linearity of
$\theta_{0}(\cdot)$ is proposed by Su and Qu (2017). In comparison, our model
is much more general and our test can handle more general parametric null
hypotheses. We thank a referee for pointing out that (5.1) is a particular
case of Sun (2016) when $u_{i}$ are iid and of Malikov and Sun (2017) when
$d_{\lambda}=1$.

Denoting $S(\lambda)=I_{n}-\sum_{j=1}^{d_{\lambda}}\lambda_{j}W_{j}$, the
quasi likelihood function based on Gaussianity and conditional on covariates
is

$\displaystyle
L(\beta,\sigma^{2},\phi)=\log{(2\pi\sigma^{2})}-\frac{2}{n}\log{\left|{S\left(\lambda\right)}\right|}+\frac{1}{n}\log{\left|{\Sigma\left(\gamma\right)}\right|}$
$\displaystyle+\frac{1}{\sigma^{2}{n}}\left(S\left(\lambda\right)y-\Psi\beta\right)^{\prime}\Sigma(\gamma)^{-1}\left(S\left(\lambda\right)y-\Psi\beta\right),$
(5.2)

at any admissible point
$\left(\beta^{\prime},\phi^{\prime},\sigma^{2}\right)^{\prime}$ with
$\phi=\left(\lambda^{\prime},\gamma^{\prime}\right)^{\prime}$, for nonsingular
$S(\lambda)$ and $\Sigma(\gamma)$. For given
$\phi=\left(\lambda^{\prime},\gamma^{\prime}\right)^{\prime}$, (5.2) is
minimised with respect to $\beta$ and $\sigma^{2}$ by

$\displaystyle\bar{\beta}\left(\phi\right)$ $\displaystyle=$
$\displaystyle\left(\Psi^{\prime}\Sigma(\gamma)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma(\gamma)^{-1}S\left(\lambda\right)y,$
(5.3) $\displaystyle\bar{\sigma}^{2}\left(\phi\right)$ $\displaystyle=$
$\displaystyle{n^{-1}}y^{\prime}S^{\prime}\left(\lambda\right)E(\gamma)^{\prime}M(\gamma)E(\gamma)S\left(\lambda\right)y.$
(5.4)

The QMLE of $\phi_{0}$ is
$\widehat{\phi}=\operatorname*{arg\,min}_{\phi\in\Phi}\mathcal{L}\left(\phi\right)$,
where

$\mathcal{L}\left(\phi\right)=\log\bar{\sigma}^{2}\left(\phi\right)+{n^{-1}}\log\left|S^{\prime-1}\left(\lambda\right)\Sigma(\gamma)S^{-1}\left(\lambda\right)\right|,$
(5.5)

and $\Phi=\Lambda\times\Gamma$ is taken to be a compact subset of
$\mathbb{R}^{d_{\lambda}+d\gamma}$. The QMLEs of $\beta_{0}$ and
$\sigma_{0}^{2}$ are defined as
$\bar{\beta}\left(\widehat{\phi}\right)\equiv\widehat{\beta}$ and
$\bar{\sigma}^{2}\left(\widehat{\phi}\right)\equiv\widehat{\sigma}^{2}$
respectively. The following assumption controls spatial dependence and is
discussed below equation (4.4).

######  Assumption SAR.1.

$\max_{j=1,\ldots,d_{\lambda}}\left\|W_{j}\right\|+\left\|S^{-1}\right\|<C$.

Writing $T(\lambda)=S(\lambda)S^{-1}$ and
$\phi=\left(\lambda^{\prime},\gamma^{\prime}\right)^{\prime}$, define the
quantity

$\sigma^{2}\left(\phi\right)=n^{-1}\sigma_{0}^{2}tr\left(T^{\prime}(\lambda)\Sigma(\gamma)^{-1}T(\lambda)\Sigma\right)=n^{-1}\sigma_{0}^{2}\left\|E(\gamma)T(\lambda)E^{-1}\right\|_{F}^{2},$

which is nonnegative by definition and bounded by Assumptions R.3 and SAR.1.
The assumptions below directly extend Assumptions R.6 and R.7 to the present
setup.

######  Assumption SAR.2.

$c\leq\sigma^{2}\left(\phi\right)\leq C$, for all $\phi\in\Phi$.

######  Assumption SAR.3.

$\phi_{0}\in\Phi$ and, for any $\eta>0$,

$\varliminf_{n\rightarrow\infty}\inf_{\phi\in\overline{\mathcal{N}}^{\phi}(\eta)}\frac{n^{-1}tr\left(T^{\prime}(\lambda)\Sigma(\gamma)^{-1}T(\lambda)\Sigma\right)}{\left|T^{\prime}(\lambda)\Sigma(\gamma)^{-1}T(\lambda)\Sigma\right|^{1/n}}>1,$
(5.6)

where
$\overline{\mathcal{N}}^{\phi}(\eta)=\Phi\setminus\mathcal{N}^{\phi}(\eta)$
and
$\mathcal{N}^{\phi}(\eta)=\left\\{\phi:\left\|\phi-\phi_{0}\right\|<\eta\right\\}\cap\Phi$.

We now introduce an identification condition that is required in the setup of
this section.

######  Assumption SAR.4.

$\beta_{0}\neq 0$ and for any $\eta>0$,

${P\left(\varliminf_{n\rightarrow\infty}\inf_{\left(\lambda^{\prime},\gamma^{\prime}\right)^{\prime}\in\Lambda\times\overline{\mathcal{N}}^{\gamma}(\eta)}n^{-1}\beta_{0}^{\prime}\Psi^{\prime}T^{\prime}(\lambda)E(\gamma)^{\prime}M\left(\gamma\right)E(\gamma)T(\lambda)\Psi\beta_{0}/\left\|\beta_{0}\right\|^{2}>0\right)=1.}$
(5.7)

Upon performing minimization with respect to $\beta$, the event inside the
probability in (5.7) is equivalent to the event

${\varliminf_{n\rightarrow\infty}\min_{\beta\in\mathbb{R}^{p}}\inf_{\left(\lambda^{\prime},\gamma^{\prime}\right)^{\prime}\in\Lambda\times\overline{\mathcal{N}}^{\gamma}(\eta)}n^{-1}\left(\Psi\beta-T(\lambda)\Psi\beta_{0}\right)^{\prime}\Sigma(\gamma)^{-1}\left(\Psi\beta-T(\lambda)\Psi\beta_{0}\right)/\left\|\beta_{0}\right\|^{2}>0,}$

which is analogous to the identification condition for the nonlinear
regression model with a parametric linear factor in Robinson (1972), weighted
by the inverse of the error covariance matrix. This reduces the condition to a
scalar form of a rank condition, making the identifying nature of the
assumption transparent. A similar identifying assumption is used by Gupta and
Robinson (2018).

######  Theorem 5.1.

Under either $H_{0}$ or $H_{1}$, Assumptions R.1-R.5, R.8, SAR.1-SAR.4 and

$p^{-1}+\left(d_{\gamma}+p\right)/n\rightarrow 0,\text{ as
}n\rightarrow\infty,$

$\left\|\left(\widehat{\phi},\widehat{\sigma}^{2}\right)-\left(\phi_{0},{\sigma_{0}}^{2}\right)\right\|\overset{p}{\longrightarrow}0$
as $n\rightarrow\infty$.

The test statistic $\mathscr{T}_{n}$ can be constructed as before but with the
null residuals redefined to incorporate the spatially lagged terms, i.e.
$\hat{u}=S(\hat{\lambda})y-f(x,\hat{\alpha})$. Then we have the following
theorem.

######  Theorem 5.2.

Under Assumptions R.1-R.5, R.8-R.10, SAR.1-SAR.4,

$p^{-1}+p\left(p+d_{\gamma}^{2}\right)/n+\sqrt{n}/p^{\mu+1/4}+d^{2}_{\gamma}/p\rightarrow
0,\text{ as }n\rightarrow\infty,$

and $H_{0}$,
$\mathscr{T}_{n}-{\left(\sigma_{0}^{-2}\varepsilon^{\prime}\mathscr{V}\varepsilon-p\right)}/{\sqrt{2p}}=o_{p}(1).$

######  Theorem 5.3.

Under the conditions of Theorems 4.3, 5.1 and 5.2, (1)
$\mathscr{T}_{n}\overset{d}{\rightarrow}N(0,1)$ under $H_{0}$, (2)
$\mathscr{T}_{n}$ is a consistent test statistic, (3)
$\mathscr{T}_{n}\overset{d}{\rightarrow}N\left(\varkappa,1\right)$ under local
alternatives $H_{\ell}$.

##  6 Nonparametric spatial weights

In this section we are motivated by settings where spatial dependence occurs
through nonparametric functions of raw distances (this may be geographic,
social, economic, or any other type of distance), as is the case in Pinkse et
al. (2002), for example. In their kind of setup, $d_{ij}$ is a raw distance
between units $i$ and $j$ and the corresponding element of the spatial weight
matrix is given by $w_{ij}=\zeta_{0}\left(d_{ij}\right)$, where
$\zeta_{0}(\cdot)$ is an unknown nonparametric function. Pinkse et al. (2002)
use such a setup in a SAR model like (5.1), but with a linear regression
function. In contrast, in keeping with the focus of this paper we instead
model dependence in the errors in this manner. Our formulation is rather
general, covering, for example, a specification like
$w_{ij}=f\left(\gamma_{0},\zeta_{0}\left(d_{ij}\right)\right)$, with
$f(\cdot)$ a _known_ function, $\gamma_{0}$ an _unknown_ parameter of possibly
increasing dimension, and $\zeta_{0}(\cdot)$ an _unknown_ nonparametric
function. For the sake of simplicity, we do not permit the $x_{i}$ in this
section to be generated by such nonparametric weight matrices although they
can be generated from other, known weight matrices.

Let $\Xi$ be a compact space of functions, on which we will specify more
conditions later. For notational simplicity we abstract away from the SAR
dependence in the responses. Thus we consider (2.1), but with

$u_{i}=\sum_{s=1}^{\infty}b_{is}\left(\gamma_{0},\zeta_{0}\left({z_{i}}\right)\right)\varepsilon_{s},$
(6.1)

where
$\zeta_{0}(\cdot)=\left(\zeta_{01}(\cdot),\ldots,\zeta_{0d_{\zeta}}(\cdot)\right)^{\prime}$
is a fixed-dimensional vector of real-valued nonparametric functions with
$\zeta_{0\ell}\in\Xi$ for each $\ell=1,\ldots,d_{\zeta}$, and ${z}_{i}$ a
fixed-dimensional vector of data, independent of the $\varepsilon_{s}$, $s\geq
1$, with support $\mathcal{Z}$. One can also take $z_{i}$ to be a fixed
distance measure. We base our estimation on approximating each
$\zeta_{0\ell}({z_{i}})$, $\ell=1,\ldots,d_{\zeta}$, with the series
representation $\delta_{0\ell}^{\prime}\varphi_{\ell}({z_{i}})$, where
$\varphi_{\ell}\left({z_{i}}\right)\equiv\varphi_{\ell}$ is an $r_{\ell}\times
1$ ($r_{\ell}\rightarrow\infty$ as $n\rightarrow\infty$) vector of basis
functions with typical function $\varphi_{\ell k}$, $k=1,\ldots,r_{\ell}$. The
set of linear combinations $\delta_{\ell}^{\prime}\varphi_{\ell}({z_{i}})$
forms the sequence of sieve spaces $\Phi_{r_{\ell}}\subset\Xi$ as
$r_{\ell}\rightarrow\infty$, for any $\ell=1,\ldots,d_{\zeta}$, and

$\zeta_{0\ell}\left({z}\right)=\delta_{0\ell}^{\prime}\varphi_{\ell}+\nu_{\ell},$
(6.2)

with the following restriction on the function space $\Xi$:

######  Assumption NPN.1.

For some scalars $\kappa_{\ell}>0$,
$\left\|\nu_{\ell}\right\|_{w_{z}}=O\left(r_{\ell}^{-\kappa_{\ell}}\right),$
as $r_{\ell}\rightarrow\infty$, $\ell=1,\ldots,d_{\zeta}$, where $w_{z}\geq 0$
is the largest value such that
$\sup_{z\in\mathcal{Z}}\mathcal{E}\left\|z\right\|^{w_{z}}<\infty$

Just as Assumption R.1 implied (3.2), by Lemma 1 of Lee and Robinson (2016),
we obtain

$\sup_{z\in\mathcal{Z}}\mathcal{E}\left(\nu^{2}_{\ell}\right)=O\left(r_{\ell}^{-2\kappa_{\ell}}\right),\ell=1,\ldots,d_{\zeta}.$
(6.3)

Thus we now have an infinite-dimensional nuisance parameter $\zeta_{0}(\cdot)$
and increasing-dimensional nuisance parameter $\gamma$. Writing
$\sum_{\ell=1}^{d_{\zeta}}r_{\ell}=r$ and
$\tau=(\gamma^{\prime},\delta^{\prime}_{1},\ldots,\delta^{\prime}_{d_{\zeta}})^{\prime}$,
which has increasing dimension $d_{\tau}=d_{\gamma}+r$, define
$\varsigma(r)=\sup_{z\in\mathcal{Z};\ell=1,\ldots,d_{\zeta}}\left\|\varphi_{\ell}\right\|.$
Write $\Sigma(\tau)$ for the covariance matrix of the $n\times 1$ vector of
$u_{i}$ in (6.1), with $\delta_{\ell}^{\prime}\varphi_{\ell}$ replacing each
admissible function $\zeta_{\ell}(\cdot)$. This is analogous to the definition
of $\Sigma(\gamma)$ in earlier sections, and indeed after conditioning on $z$
it can be treated in a similar way because $d_{\gamma}\rightarrow\infty$ was
already permitted. For example, suppose that $u=(I_{n}-W)^{-1}\varepsilon$,
where $\left\|W\right\|<1$ and the elements satisfy
$w_{ij}=\zeta_{0}\left(d_{ij}\right)$, $i,j=1,\ldots,n$, for some fixed
distances $d_{ij}$ and unknown function $\zeta_{0}(\cdot)$, see e.g. Pinkse
(1999). Approximating $\zeta_{0}(z)=\tau_{0}^{\prime}\varphi(z)+\nu$, for some
$r\times 1$ basis function vector $\varphi(z)$ and approximation error $\nu$,
we define $W(\tau)$ as the $n\times n$ matrix with elements
$w_{ij}(\tau)=\tau_{0}^{\prime}\varphi\left(d_{ij}\right)$, and set
$\Sigma(\tau)=\text{var}\left((I_{n}-W(\tau))^{-1}\varepsilon\right)=\sigma_{0}^{2}(I_{n}-W(\tau))^{-1}(I_{n}-W^{\prime}(\tau))^{-1}$.

For any admissible values $\beta$, $\sigma^{2}$ and $\tau$, the redefined
(multiplied by $2/n$) negative quasi log likelihood function based on using
the approximations (3.1) and (6.2) is

${L}(\beta,\sigma^{2},\tau)=\ln\left(2\pi\sigma^{2}\right)+\frac{1}{n}\ln\left|\Sigma\left(\tau\right)\right|+\frac{1}{n\sigma^{2}}(y-\Psi\beta)^{\prime}\Sigma\left(\tau\right)^{-1}(y-\Psi\beta),$
(6.4)

which is minimised with respect to $\beta$ and $\sigma^{2}$ by

$\displaystyle\bar{\beta}\left(\tau\right)$ $\displaystyle=$
$\displaystyle\left(\Psi^{\prime}\Sigma\left(\tau\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\tau\right)^{-1}y,$
(6.5) $\displaystyle\bar{\sigma}^{2}\left(\tau\right)$ $\displaystyle=$
$\displaystyle{n^{-1}}y^{\prime}E(\tau)^{\prime}M(\tau)E(\tau)y,$ (6.6)

where
$M(\tau)=I_{n}-E(\tau)\Psi\left(\Psi^{\prime}\Sigma(\tau)^{-1}\Psi\right)^{-1}\Psi^{\prime}E(\tau)^{\prime}$
and $E(\tau)$ is the $n\times n$ symmetric matrix such that
$E(\tau)E(\tau)^{\prime}=\Sigma(\tau)^{-1}$. Thus the concentrated likelihood
function is

$\mathcal{L}(\tau)=\ln(2\pi)+\ln\bar{\sigma}^{2}(\tau)+\frac{1}{n}\ln\left|\Sigma\left(\tau\right)\right|.$
(6.7)

Again, for compact $\Gamma$ and sieve coefficient space $\Delta$, the QMLE of
$\tau_{0}$ is $\widehat{\tau}=\text{arg
min}_{\tau\in\Gamma\times\Delta}\mathcal{L}(\tau)$ and the QMLEs of $\beta$
and $\sigma^{2}$ are $\widehat{\beta}=\bar{\beta}\left(\widehat{\tau}\right)$
and $\widehat{\sigma}^{2}=\bar{\sigma}^{2}\left(\widehat{\tau}\right)$. The
series estimate of $\theta_{0}$ is defined as in (3.7). Define also the
product Banach space $\mathcal{T}=\Gamma\times\Xi^{d_{\zeta}}$ with norm
$\left\|\left(\gamma^{\prime},\zeta^{\prime}\right)^{\prime}\right\|_{\mathcal{T}_{w}}=\left\|\gamma\right\|+\sum_{\ell=1}^{d_{\zeta}}\left\|\zeta_{\ell}\right\|_{w}$,
and consider the map $\Sigma:\mathcal{T}^{o}\rightarrow\mathcal{M}^{n\times
n}$, where $\mathcal{T}^{o}$ is an open subset of $\mathcal{T}$.

######  Assumption NPN.2.

The map $\Sigma:\mathcal{T}^{o}\rightarrow\mathcal{M}^{n\times n}$ is Fréchet-
differentiable on $\mathcal{T}^{o}$ with Fréchet-derivative denoted
$D\Sigma\in\mathscr{L}\left(\mathcal{T}^{o},\mathcal{M}^{n\times n}\right)$.
Furthermore, conditional on ${z}$, the map $D\Sigma$ satisfies

$\sup_{t\in\mathcal{T}^{o}}\left\|D\Sigma(t)\right\|_{\mathscr{L}\left(\mathcal{T}^{o},\mathcal{M}^{n\times
n}\right)}\leq C,$ (6.8)

on its domain $\mathcal{T}^{o}$.

This assumption can be checked in a similar way to how we checked Assumption
R.5, where a diverging dimension for the argument was already permitted.

######  Proposition 6.1.

If Assumption NPN.2 holds, then for any $t_{1},t_{2}\in\mathcal{T}^{o}$,
conditional on $z$,

$\left\|\Sigma\left(t_{1}\right)-\Sigma\left(t_{2}\right)\right\|\leq
C\varsigma(r)\left\|t_{1}-t_{2}\right\|.$ (6.9)

######  Corollary 6.1.

For any $t^{*}\in\mathcal{T}^{o}$ and any $\eta>0$, conditional on $z$,

$\underset{n\rightarrow\infty}{\overline{\lim}}\sup_{t\in\left\\{t:\left\|t-t^{*}\right\|<\eta\right\\}\cap\mathcal{T}^{o}}\left\|\Sigma(t)-\Sigma\left(t^{*}\right)\right\|<C\varsigma(r)\eta.$
(6.10)

######  Assumption NPN.3.

$c\leq\sigma^{2}\left(\tau\right)\leq C$ for $\tau\in\Gamma\times\Delta$,
conditional on $z$.

Denote $\Sigma\left(\tau_{0}\right)=\Sigma_{0}$. Note that this is not the
true covariance matrix, which is
$\Sigma\equiv\Sigma\left(\gamma_{0},\zeta_{0}\right)$.

######  Assumption NPN.4.

$\tau_{0}\in\Gamma\times\Delta$ and, for any $\eta>0$, conditional on $z$,

$\varliminf_{n\rightarrow\infty}\inf_{\tau\in\overline{\mathcal{N}}^{\tau}(\eta)}\frac{n^{-1}tr\left(\Sigma(\tau)^{-1}\Sigma_{0}\right)}{\left|\Sigma(\tau)^{-1}\Sigma_{0}\right|^{1/n}}>1,$
(6.11)

where
$\overline{\mathcal{N}}^{\tau}(\eta)=(\Gamma\times\Delta)\setminus\mathcal{N}^{\tau}(\eta)$
and
$\mathcal{N}^{\tau}(\eta)=\left\\{\tau:\left\|\tau-\tau_{0}\right\|<\eta\right\\}\cap(\Gamma\times\Delta)$.

######  Remark 1.

Expressing the identification condition in Assumption NPN.4 in terms of $\tau$
implies that identification is guaranteed via the sieve spaces
$\Phi_{r_{\ell}}$, $\ell=1,\ldots,d_{\zeta}$. This approach is common in the
sieve estimation literature, see e.g. Chen (2007), p. 5589, Condition 3.1.

######  Theorem 6.1.

Under either $H_{0}$ or $H_{1}$, Assumptions R.1-R.4 (with R.3 and R.4 holding
for $t\in\mathcal{T}$ rather than $\gamma\in\Gamma$), R.8, NPN.1-NPN.4 and
$p^{-1}+\left(\min_{\ell=1,\ldots,d_{\zeta}}r_{\ell}\right)^{-1}+\left(d_{\gamma}+p+\max_{\ell=1,\ldots,d_{\zeta}}r_{\ell}\right)/n\rightarrow
0$ as $n\rightarrow\infty$,
$\left\|\left(\widehat{\tau},\hat{\sigma}^{2}\right)-\left(\tau_{0},\sigma^{2}_{0}\right)\right\|\overset{p}{\longrightarrow}0.$

######  Theorem 6.2.

Under the conditions of Theorems 4.2 and 6.1, but with $\tau$ and
$\mathcal{T}$ replacing $\gamma$ and $\Gamma$ in assumptions prefixed with R
and $p\rightarrow\infty$,

$\left(\min_{\ell=1,\ldots,d_{\zeta}}r_{\ell}\right)^{-1}+\frac{p^{2}}{n}+\frac{\sqrt{n}}{p^{\mu+1/4}}+p^{1/2}\varsigma(r)\left(\frac{d_{\gamma}+\displaystyle\max_{\ell=1,\ldots,d_{\zeta}}r_{\ell}}{\sqrt{n}}+\sqrt{\sum_{\ell=1}^{d_{\zeta}}r_{\ell}^{-2\kappa_{\ell}}}\right)\rightarrow
0,$

as $n\rightarrow\infty$, and $H_{0}$,
$\mathscr{T}_{n}-\left({\sigma_{0}^{-2}\varepsilon^{\prime}\mathscr{V}\varepsilon-p}\right)/{\sqrt{2p}}=o_{p}(1).$

######  Theorem 6.3.

Let the conditions of Theorems 4.3 and 6.2 hold, but with $\tau$ and
$\mathcal{T}$ replacing $\gamma$ and $\Gamma$ in assumptions prefixed with R.
Then (1) $\mathscr{T}_{n}\overset{d}{\rightarrow}N(0,1)$ under $H_{0}$, (2)
$\mathscr{T}_{n}$ is a consistent test statistic, (3)
$\mathscr{T}_{n}\overset{d}{\rightarrow}N\left(\varkappa,1\right)$ under local
alternatives $H_{\ell}$.

##  7 Fixed-regressor residual-based bootstrap test

The performance of nonparametric tests based on asymptotic distributions often
leaves something to be desired in finite samples. An alternative approach is
to use the bootstrap approximation. In this section, we propose a bootstrap
version of our test, focusing on the setting of Section 5. In our simulations
and empirical studies, we consider test statistics based on both
$\widehat{m}_{n}=\widehat{\sigma}^{-2}\widehat{v}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{u}/n$
and
$\widetilde{m}_{n}=\widehat{\sigma}^{-2}(\widehat{u}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{u}-\widehat{\eta}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\widehat{\eta})/n$,
where $\widehat{\eta}=S(\hat{\lambda})y-\widehat{\theta}$, i.e., the residual
from nonparametric estimation, $\hat{u}=S(\hat{\lambda})y-f(x,\hat{\alpha})$,
and $\hat{v}=\hat{\theta}-f(x,\hat{\alpha})$. Analogous to the definition of
$\mathscr{T}_{n}$, define the statistic
$\mathscr{T}_{n}^{a}={\left(n\widetilde{m}_{n}-p\right)}/{\sqrt{2p}}.$ In the
case of no spatial autoregressive term, and under the power series,
$\mathscr{T}_{n}^{a}$ and $\mathscr{T}_{n}$ are numerically identical, as was
observed by Hong and White (1995). However, in the SARSE setting a difference
arises due to the spatial structure in the response $y$. We show that
$\mathscr{T}_{n}^{a}-\mathscr{T}_{n}=o_{p}(1)$ under the null or local
alternatives in Theorem TS.1 in the online supplementary appendix.

The bootstrap versions of the test statistics $\mathscr{T}_{n}$ and
$\mathscr{T}_{n}^{a}$ are

$\displaystyle\mathscr{T}_{n}^{*}$ $\displaystyle=$
$\displaystyle\frac{n\widehat{m}_{n}^{\ast}-p}{\sqrt{2p}}=\frac{\widehat{\sigma}^{\ast-2}\widehat{{v}}^{\ast\prime}\Sigma\left(\widehat{\gamma}^{\ast}\right)^{-1}\widehat{{u}}^{\ast}-p}{\sqrt{2p}}$
$\displaystyle\mathscr{T}_{n}^{a\ast}$ $\displaystyle=$
$\displaystyle\frac{n\widetilde{m}_{n}^{\ast}-p}{\sqrt{2p}}=\frac{\widehat{\sigma}^{\ast-2}(\widehat{{u}}^{\ast\prime}\Sigma\left(\widehat{\gamma}^{\ast}\right)^{-1}\widehat{{u}}^{\ast}-\widehat{\eta}^{\ast\prime}\Sigma\left(\widehat{\gamma}^{\ast}\right)^{-1}\widehat{\eta}^{\ast})-p}{\sqrt{2p}},$

respectively, where $\widehat{{u}}^{\ast}$ is the bootstrap residual vector
under the null, $\widehat{\eta}^{*}$ is the bootstrap residual vector under
the alternative,
$\widehat{{v}}^{\ast}=\widehat{\theta}^{\ast}(x)-f(x,\widehat{\alpha}^{\ast})$,
and $\left(\widehat{\gamma}^{\ast},\lambda^{*},\widehat{\sigma}^{\ast
2},\widehat{\theta}^{\ast},\widehat{\alpha}^{\ast}\right)$ is the estimator
using the bootstrap sample. We elaborate on the bootstrap statistics using the
SARARMA($m_{1}$,$m_{2},m_{3}$) model as an example:

$y=\sum_{k=1}^{m_{1}}\lambda_{k}W_{1k}y+\theta(x)+u\text{,
}u=\sum_{l=1}^{m_{2}}\gamma_{2l}W_{2l}u+\sum_{l=1}^{m_{3}}\gamma_{3l}W_{3l}\xi+\xi.$

Following Jin and Lee (2015), we first deduct the empirical mean of the
residual vector from

$\widehat{{\xi}}=\left(\sum_{l=1}^{m_{3}}\widehat{\gamma}_{3l}W_{3l}+I_{n}\right)^{-1}\left(I_{n}-\sum_{l=1}^{m_{2}}\widehat{\gamma}_{2l}W_{2l}\right)\left(y-\sum_{k=1}^{m_{1}}\widehat{\lambda}_{k}W_{1k}y-\widehat{{\theta}}_{n}\right)$

to obtain
$\widetilde{{\xi}}=(I_{n}-\frac{1}{n}l_{n}l_{n}^{\prime})\widehat{{\xi}}$.
Next, we sample randomly with replacement $n$ times from elements of
$\widetilde{{\xi}}$ to obtain a vector of $\mathbf{\xi}^{\ast}.$ After this,
we generate the bootstrap sample $y^{\ast}$ by treating
$\widehat{{f}}=f(x,\widehat{\alpha})$, $\hat{\lambda}$ and $\widehat{\gamma}$
as the true parameter:

$y^{\ast}=\left(I_{n}-\sum_{k=1}^{m_{1}}\widehat{\lambda}_{k}W_{1k}\right)^{-1}\left(\widehat{{f}}+\left(I_{n}-\sum_{l=1}^{m_{2}}\widehat{\gamma}_{2l}W_{2l}\right)^{-1}\left(\sum_{l=1}^{m_{3}}\widehat{\gamma}_{3l}W_{3l}+I_{n}\right){\xi}^{\ast}\right).$

We estimate the model based on the bootstrap sample $y^{\ast}$ using QMLE to
obtain the estimator
$\widehat{{\theta}}^{\ast}=\psi^{\prime}\widehat{\beta}^{\ast},$
$\widehat{\lambda}^{\ast}$, and $\widehat{\gamma}^{\ast}$ under the
alternative hypothesis and $\widehat{\alpha}^{\ast}$ under the null hypothesis
of $\theta(x)=f(x,\alpha_{0}).$ Then,
$\widehat{{\eta}}^{\ast}=y^{\ast}-\sum_{k=1}^{m_{1}}\widehat{\lambda}_{k}^{\ast}W_{1k}y^{\ast}-\widehat{{\theta}}^{\ast}$,
$\widehat{{u}}^{\ast}=y^{\ast}-\sum_{k=1}^{m_{1}}\widehat{\lambda}_{k}^{\ast}W_{1k}y^{\ast}-f(x,\widehat{\alpha}^{\ast}).$

This procedure is repeated $B$ times to obtain the sequence
$\left\\{\mathscr{T}_{nj}^{\ast}\right\\}_{j=1}^{B}$. We reject the null when
$p^{\ast}=B^{-1}\sum_{j=1}^{B}\mathbf{1(}\mathscr{T}_{n}<\mathscr{T}_{nj}^{\ast})$
is smaller than the given level of significance. An identical procedure holds
for the test based on $\mathscr{T}_{n}^{a\ast}.$ The asymptotic validity of
the bootstrap method can be shown as in Theorem 4 of Su and Qu (2017) and
Lemma 2 in Jin and Lee (2015), and detailed analysis can be found in the
supplementary appendix, see proof of Theorem TS.1.

##  8 Finite sample performance

###  8.1 Parametric error spatial structure

Taking $n=60,100,200$, we choose two specifications to generate $y$ from the
SARARMA($m_{1}$,$m_{2},m_{3}$) models:

$\displaystyle\text{SARARMA(}0\text{,1,0): }$ $\displaystyle
y=\theta(x)+u,\text{ }u=\gamma_{2}W_{2}u+\xi$
$\displaystyle\text{SARARMA(}1\text{,0,1): }$ $\displaystyle
y=\lambda_{1}W_{1}y+\theta(x)+u,\text{ }u=\gamma_{3}W_{3}\xi+\xi,$

where $\xi$ is $N(0,I_{n})$. The DGP of $\theta(x)$ is

$\theta(x_{i})=x_{i}^{\prime}\alpha+cp^{1/4}n^{-1/2}\sin(x_{i}^{\prime}\alpha),$

where $x_{i}^{\prime}\alpha=1+x_{1i}+x_{2i}$, with $x_{1i}=(z_{i}+z_{1i})/2$,
$x_{2i}=(z_{i}+z_{2i})/2$. We choose two settings: compactly supported
regressors where $z_{i},z_{1i}$ and $z_{2i}$ are i.i.d., $U[0,2\pi]$ and
unboundedly supported regressors where $z_{i},z_{1i}$ and $z_{2i}$ are i.i.d.
$N(0,1).$ We report the compact support setting in the main text, while the
results for unbounded support are reported in the online supplement.

We use three series bases for our experiments: power (polynomial) series of
third and fourth order ($p=10,p=15$), trigonometric series
$trig_{1}=\left(1,\sin\left(x_{1}\right),\sin\left(x_{1}/2\right),\sin\left(x_{2}\right),\sin\left(x_{2}/2\right),\cos\left(x_{1}\right),\cos\left(x_{1}/2\right),\cos\left(x_{2}\right),\cos\left(x_{2}/2\right)\right)^{\prime}$
and
$trig_{2}=\left(trig_{1}^{\prime},\sin\left(x_{1}^{2}\right),\cos\left(x_{1}^{2}\right),\sin\left(x_{2}^{2}\right),\cos\left(x_{2}^{2}\right)\right)^{\prime}$,
and the B-spline bases of fourth and seventh order ($p=9,p=14$), We also set
$\gamma_{2}=0.3$, $\lambda_{1}=0.3$ and $\gamma_{3}=0.4$; the value $c=0,3,6$
indicates the null hypothesis and the local alternatives. The spatial weight
matrices are generated using LeSage’s code make_neighborsw from
http://www.spatial-econometrics.com/, where the row-normalized sparse matrices
are generated by choosing a specific number of the closest locations from
randomly generated coordinates and we set the number of neighbors to be
$n/20$. We employ 100 bootstrap replications in each of 500 Monte Carlo
replications except for the SARARMA(1,0,1) design with $n=200$, where we set
50 bootstrap replications in view of the computation time. We report the
rejection frequencies of tests based on bootstrap critical values in the main
text, while tests based on asymptotic critical values are reported in the
online supplement.

Tables 1-4 report the empirical rejection frequencies using the bootstrap test
statistics $\mathscr{T}_{n}^{\ast}$ (Tables 1, 3) and
$\mathscr{T}_{n}^{a\ast}$ (Tables 2, 4), when nominal levels are given by 1%,
5% and 10%. To see how the choice of $p$ and the basis functions affect small
sample outcomes, we report two sets of results for each basis function family:
the first row for each value of $c$ is from the smaller $p$ ($p=9$ or $10$),
while the second row is from the larger $p$ ($p=14$ or $15$). We summarize
some important findings. First, we see that for most DGPs, our bootstrap test
is closer to the nominal level than the asymptotic test (reported in the
online supplement) although the sizes of both types of tests improve generally
as the sample size increases. Second, both bootstrap and asymptotic tests are
powerful in detecting any deviations from linearity in the local alternatives.
The patterns are similar across all cases: the bootstrap generally affords
better size control, albeit not always.

All three types of bases give qualitatively similar results, but we note that
$\mathscr{T}_{n}^{\ast}=\mathscr{T}_{n}^{\ast a}$ when using polynomial series
under the SARARMA(0,1,0) model, as observed in Hong and White (1995). When
using trigonometric and B-spline series, tests based on these two statistics
give slightly different rejection rates. However, under the SARARMA(1,0,1)
model, all series give quantitatively different results, as illustrated in
Tables 3 and 4. When using B-spline bases, $p=14$ does not perform well
compared to $p=9$. In the other cases, both choices of $p$ work well.

###  8.2 Nonparametric error spatial structure

Now we examine finite sample performance in the setting of Section 6. The
three DGPs of $\theta(x)$ are the same as the parametric setting but we
generate the $n\times n$ matrix $W^{*}$ as
$w^{*}_{ij}=\Phi(-d_{ij})I(c_{ij}<0.05)$ if $i\neq j$, and $w^{*}_{ii}=0$,
where $\Phi(\cdot)$ is the standard normal cdf, $d_{ij}\sim$iid $U[-3,3]$, and
$c_{ij}\sim$iid $U[0,1]$. From this construction, we ensure that $W^{*}$ is
sparse with no more than $5\%$ elements being nonzero. Then, $y$ is generated
from $y=\theta(x)+u,\text{ }u=Wu+\xi,$ where $\xi\sim N(0,I_{n})$ and
$W=W^{*}/{1.2\overline{\varphi}\left(W^{*}\right)}$, ensuring the existence of
$(I-W)^{-1}$. In estimation, we know the distance $d_{ij}$ and the indicator
$I(c_{ij}<0.05)$, but we do not know the functional form of $w_{ij}$, so we
approximate elements in $W$ by
$\widehat{w}_{ij}=\sum_{l=0}^{r}a_{l}d_{ij}^{l}I(c_{ij}<0.05)\text{ if }i\neq
j\text{; }\widehat{w}_{ii}=0.$

Table 5 reports the rejection rates using 500 Monte Carlo simulation at the 5%
asymptotic level 1.645 using polynomial bases with $r=2,3,4,5$ and
$p=10,15,20$. We take $n=150,300,500,600,700$, larger sample sizes than
earlier because two nonparametric functions must be estimated in this spatial
setting. The two largest bandwidths ($r=5,p=20$) are only employed for the
largest sample size $n=700$. We observe a clear pattern of rejection rates
approaching the theoretical level as sample size increases. Power improves as
$c$ increases for all designs and is non-trivial in all cases even for $c=3$.
Sizes are acceptable for $n=500$, particularly when $p=15$. Size performance
improves further as $n=600$, indicating asymptotic stability. Note that with
two diverging bandwidths ($p$ and $r$), we expect sizes to improve in a
diagonal pattern going from top left corner to bottom right corner in Table 5.
This is indeed the case. For $n=700$, we observe that the pairs
$(r,p)=(5,15),(5,20)$ deliver acceptable sizes.

##  9 Empirical applications

In this section, we illustrate the specification test presented in previous
sections using several empirical examples.

###  9.1 Conflict alliances

This example is based on a study of how a network of military alliances and
enmities affects the intensity of a conflict, conducted by König et al.
(2017). They stress that understanding the role of informal networks of
military alliances and enmities is important not only for predicting outcomes,
but also for designing and implementing policies to contain or put an end to
violence. König et al. (2017) obtain a closed-form characterization of the
Nash equilibrium and perform an empirical analysis using data on the Second
Congo War, a conflict that involves many groups in a complex network of
informal alliances and rivalries.

To study the fighting effort of each group the authors use a panel data model
with individual fixed effects, where key regressors include total fighting
effort of allies and enemies. They further correct the potential spatial
correlation in the error term by using a spatial heteroskedasticity and
autocorrelation robust standard error. We use their data and the main
structure of the specification and build a cross-sectional SAR(2) model with
two weight matrices, $W^{A}$ ($W^{A}_{ij}=1$ if group $i$ and $j$ are allies,
and $W^{A}_{ij}=0$ otherwise) and $W^{E}$ ($W^{E}_{ij}=1$ if group $i$ and $j$
are enemies, and $W^{E}_{ij}=0$ otherwise):

$y=\lambda_{1}W^{A}y+\lambda_{2}W^{E}y+\mathbf{1}_{n}\beta_{0}+X\beta+u,$

where $y$ is a vector of fighting efforts of each group and $X$ includes the
current rainfall, rainfall from the last year, and their squares.111We follow
the analysis in the original paper and do not row normalize. This is because
the economic content of the weight matrices is defined by total fights of
allies or enemies. To consider the spatial correlation in the error term, we
consider both the Error SARMA(1,0) and Error SARMA(0,1) structures. For these,
we employ a spatial weight matrix $W^{d}$, based on the inverse distance
between group locations and set to be 0 after 150 km, following König et al.
(2017). The idea is that geographical spatial correlation dies out as groups
become further apart. We also report results using a nonparametric estimator
of the spatial weights, as described in Section 6 and studied in simulations
in Section 8. For the nonparametric estimator we take $r=2$.

In the original dataset, there are 80 groups, but groups 62 and 63 have the
same variables and the same locations, so we drop one group and end up with a
sample of 79 groups. We use data from 1998 as an example and further use the
pooled data from all years as a robustness check. $H_{0}$ stands for
restricted model where the linear functional form of the regression is
imposed, while $H_{1}$ stands for the unrestricted model where we use basis
functions comprising of power series with $p=10$. In all our specifications,
the test statistics are negative, so we cannot reject the null hypothesis that
the model is correctly specified. As Table 6 indicates, this failure to reject
the null persists when we use pooled data from 13 years, yielding 1027
observations. Thus we conclude that a linear specification is not
inappropriate for this setting. One possible reason is that the original
regression, though linear, has already included the squared terms of the
rainfall as regressors. This finding is robust to using the bootstrap tests of
Section 7, which generally yield smaller p-values but unchanged conclusions.

###  9.2 Innovation spillovers

This example is based on the study of the impact of R&D on growth from Bloom
et al. (2013). They develop a general framework incorporating two types of
spillovers: a positive effect from technology (knowledge) spillovers and a
negative ‘business stealing’ effect from product market rivals. They implement
this model using panel data on U.S. firms.

We consider the Productivity Equation in Bloom et al. (2013):

$\ln
y=\varphi_{1}\ln(R\&D)+\varphi_{2}\ln(Sptec)+\varphi_{3}\ln(Spsic)+\varphi_{4}X+error,$
(9.1)

where $y$ is a vector of sales, $R\&D$ is a vector of R&D stocks, and
regressors in $X$ include the log of capital ($Capital$), log of labor
($Labor$), $R\&D$, a dummy for missing values in $R\&D$, a price index, and
two spillover terms constructed as the log of $W_{SIC}R\&D$ ($Spsic$) and the
log of $W_{TEC}R\&D$ ($Sptec$), where $W_{SIC}$ measures the product market
proximity and $W_{TEC}$ measures the technological proximity. Specifically,
they define

$W_{SIC,ij}={S_{i}S_{j}^{\prime}}/{(S_{i}S_{i}^{\prime})^{1/2}(S_{j}S_{j}^{\prime})^{1/2}},W_{TEC,ij}={T_{i}T_{j}^{\prime}}/{(T_{i}T_{i}^{\prime})^{1/2}(T_{j}T_{j}^{\prime})^{1/2}},$

where $S_{i}=(S_{i1},S_{i2},\ldots,S_{i597})^{\prime}$, with $S_{ik}$ being
the share of patents of firm $i$ in the four digit industry $k$ and
$T_{i}=(T_{i1},T_{i2},\ldots,T_{i426})^{\prime}$, with $T_{i\tau}$ being the
share of patents of firm $i$ in technology class $\tau$. Focusing on a cross-
sectional analysis, we use observations from the year 2000 and obtain a sample
size of 577\. Both weight matrices are row normalized.

The column FE of Table 7 is from Table 5 of Bloom et al. (2013) based on their
panel fixed effects estimation and we use it as a baseline for comparison.
This table reports results for SARARMA(0,1,0) models using $W_{SIC}$ and
$W_{TEC}$ separately. We use both $W_{SIC}$ and $W_{TEC}$ simultaneously in
SARARMA(0,2,0), SARARMA(0,2,0), and Error MESS(2) models, reported in Table 8.
In all of these specifications, the test statistics are larger than 1.645, so
we reject the null hypothesis of the linear specification. This rejection also
persists with the bootstrap tests, albeit the p-values go up compared to the
asymptotic ones. However, we can say even more as our estimation also sheds
light on spatial effects in the disturbances in (9.1). As before $H_{0}$
imposes linear functional form of the regressors, while $H_{1}$ uses the
nonparametric series estimate employing power series with $p=10$. Regardless
of the specification of the regression function, the disturbances suggest a
strong spatial effect as the coefficients on $W_{TEC}$ and $W_{SIC}$ are large
in magnitude.

###  9.3 Economic growth

The final example is based on the study of economic growth rate in Ertur and
Koch (2007). Knowledge accumulated in one area might depend on knowledge
accumulated in other areas, especially in its neighborhoods, implying the
possible existence of spatial spillover effects. These questions are of
interest to both economists as well as regional scientists. For example,
Autant-Bernard and LeSage (2011) examine spatial spillovers associated with
research expenditures for French regions, while Ho et al. (2013) examine the
international spillover of economic growth through bilateral trade amongst
OECD countries, Cuaresma and Feldkircher (2013) study spatially correlated
growth spillovers in the income convergence process of Europe, and Evans and
Kim (2014) study the spatial dynamics of growth and convergence in Korean
regional incomes.

In this section, we want to test the linear SAR model specification in Ertur
and Koch (2007). Their dataset covers a sample of 91 countries over the period
1960-1995, originally from Heston et al. (2002), obtained from the Penn World
Tables (PWT version 6.1). The variables in use include per worker income in
1960 ($y60$) and 1995 ($y95$), average rate of growth between 1960 and 1995
$(gy)$, average investment rate of this period ($s$), and average rate of
growth of working-age population ($n_{p}$).

Ertur and Koch (2007) consider the model

$y=\lambda Wy+X\beta+WX\theta+\varepsilon,$ (9.2)

where the dependent variable is log real income per worker $\ln(y95)$,
elements of the explanatory variable $X=(x_{1}^{\prime},x_{2}^{\prime})$
include log investment rate $\ln(s)=x_{1}$ and log physical capital effective
rate of depreciation $\ln(n_{p}+0.05)=x_{2}$, with corresponding subscripted
coefficients $\beta_{1},\beta_{2},\theta_{1},\theta_{2}$. A restricted
regression based on the joint constraints $\beta_{1}=-\beta_{2}$ and
$\theta_{1}=-\theta_{2}$ (these constraints are implied by economic theory) is
also considered in Ertur and Koch (2007). The model (9.2) has regressors
$(X,WX)$ and iid errors, so the test derived in Section 5 can be directly
applied here. Denoting by $d_{ij}$ the great-circle distance between the
capital cities of countries $i$ and $j$, one construction of $W$ takes
$w_{ij}=d_{ij}^{-2}$ while the other takes $w_{ij}=e^{-2d_{ij}}$, following
Ertur and Koch (2007).

Table 9 presents the estimation and testing results based on using linear and
quadratic power series basis functions with $p=10$ and a sample size of
$n=91$. We impose additive structure in our estimation to at least alleviate
the curse of dimensionality, always a concern in nonparametric estimation. We
also use only linear and quadratic basis functions to reduce the number of
terms for series estimation.

We cannot reject linearity of the regression function for the unrestricted
model. On the other hand, linearity is rejected for the restricted model,
which is the preferred specification of Ertur and Koch (2007), with
$w_{ij}=e^{-2d_{ij}}$. Thus, not only can we conclude that the specification
of the model is under suspicion we can also infer this is due to constraints
from economic theory. The findings are supported by the bootstrap tests of
Section 7.

##  10 Conclusion

This paper justifies a specification test for the regression function in a
model where data are spatially dependent. The test is based on a nonparametric
series approximation and is consistent. The paper also allows for some
robustness in error spatial dependence by permitting this to be a
nonparametric function of an underlying economic distance. On the other hand,
our Section 5 imposes correct specification of the spatial weight matrices
$W_{j}$ in the SAR context, while Sun (2020) allows these to be nonparametric
functions as well. Thus our work acts as a complement to existing results in
the literature and future work might combine both aspects.

$\mathscr{T}_{n}^{\ast}$ | SARARMA(0,1,0)
---|---
|  | PS |  |  |  | Trig |  |  |  | B-s |
$n=60$ | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.1
${\small c=0}$ | ${\small 0.008}$ | ${\small 0.032}$ | ${\small 0.084}$ |  | ${\small 0.004}$ | ${\small 0.04}$ | ${\small 0.092}$ |  | ${\small 0.006}$ | ${\small 0.048}$ | ${\small 0.104}$
| ${\small 0.004}$ | ${\small 0.038}$ | ${\small 0.096}$ |  | ${\small 0.004}$ | ${\small 0.038}$ | ${\small 0.094}$ |  | ${\small 0.006}$ | ${\small 0.034}$ | ${\small 0.098}$
${\small c=3}$ | ${\small 0.036}$ | ${\small 0.154}$ | ${\small 0.296}$ |  | ${\small 0.092}$ | ${\small 0.276}$ | ${\small 0.39}$ |  | ${\small 0.098}$ | ${\small 0.292}$ | ${\small 0.470}$
| ${\small 0.154}$ | ${\small 0.414}$ | ${\small 0.62}$ |  | ${\small 0.056}$ | ${\small 0.22}$ | ${\small 0.374}$ |  | ${\small 0.036}$ | ${\small 0.150}$ | ${\small 0.292}$
${\small c=6}$ | ${\small 0.22}$ | ${\small 0.544}$ | ${\small 0.748}$ |  | ${\small 0.454}$ | ${\small 0.794}$ | ${\small 0.908}$ |  | ${\small 0.432}$ | ${\small 0.814}$ | ${\small 0.938}$
| ${\small 0.844}$ | ${\small 0.992}$ | ${\small 1}$ |  | ${\small 0.314}$ | ${\small 0.714}$ | ${\small 0.872}$ |  | ${\small 0.174}$ | ${\small 0.542}$ | ${\small 0.732}$
${\small n=100}$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.006}$ | ${\small 0.044}$ | ${\small 0.098}$ |  | ${\small 0.002}$ | ${\small 0.04}$ | ${\small 0.09}$ |  | ${\small 0.008}$ | ${\small 0.038}$ | ${\small 0.110}$
| ${\small 0.012}$ | ${\small 0.046}$ | ${\small 0.096}$ |  | ${\small 0.006}$ | ${\small 0.036}$ | ${\small 0.102}$ |  | ${\small 0.01}$ | ${\small 0.056}$ | ${\small 0.108}$
${\small c=3}$ | ${\small 0.294}$ | ${\small 0.578}$ | ${\small 0.72}$ |  | ${\small 0.214}$ | ${\small 0.508}$ | ${\small 0.626}$ |  | ${\small 0.272}$ | ${\small 0.572}$ | ${\small 0.712}$
| ${\small 0.37}$ | ${\small 0.662}$ | ${\small 0.824}$ |  | ${\small 0.194}$ | ${\small 0.45}$ | ${\small 0.632}$ |  | ${\small 0.188}$ | ${\small 0.46}$ | ${\small 0.63}$
${\small c=6}$ | ${\small 0.95}$ | ${\small 0.99}$ | ${\small 0.996}$ |  | ${\small 0.902}$ | ${\small 0.99}$ | ${\small 0.998}$ |  | ${\small 0.922}$ | ${\small 0.994}$ | ${\small 1}$
| ${\small 0.992}$ | ${\small 0.998}$ | ${\small 1}$ |  | ${\small 0.856}$ | ${\small 0.988}$ | ${\small 1}$ |  | ${\small 0.852}$ | ${\small 0.98}$ | ${\small 0.998}$
$\small n=200$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.006}$ | ${\small 0.038}$ | ${\small 0.104}$ |  | ${\small 0.008}$ | ${\small 0.042}$ | ${\small 0.112}$ |  | ${\small 0.024}$ | ${\small 0.074}$ | ${\small 0.132}$
| ${\small 0.006}$ | ${\small 0.048}$ | ${\small 0.088}$ |  | ${\small 0.016}$ | ${\small 0.038}$ | ${\small 0.082}$ |  | ${\small 0.022}$ | ${\small 0.074}$ | ${\small 0.144}$
${\small c=3}$ | ${\small 0.178}$ | ${\small 0.402}$ | ${\small 0.55}$ |  | ${\small 0.162}$ | ${\small 0.374}$ | ${\small 0.532}$ |  | ${\small 0.314}$ | ${\small 0.516}$ | ${\small 0.654}$
| ${\small 0.282}$ | ${\small 0.56}$ | ${\small 0.694}$ |  | ${\small 0.136}$ | ${\small 0.346}$ | ${\small 0.468}$ |  | ${\small 0.19}$ | ${\small 0.37}$ | ${\small 0.542}$
${\small c=6}$ | ${\small 0.846}$ | ${\small 0.968}$ | ${\small 0.984}$ |  | ${\small 0.796}$ | ${\small 0.95}$ | ${\small 0.98}$ |  | ${\small 0.89}$ | ${\small 0.976}$ | ${\small 0.986}$
| ${\small 0.982}$ | ${\small 0.998}$ | ${\small 1}$ |  | ${\small 0.776}$ | ${\small 0.934}$ | ${\small 0.974}$ |  | ${\small 0.852}$ | ${\small 0.946}$ | ${\small 0.982}$
Table 1: Rejection probabilities of SARARMA(0,1,0) using bootstrap test $\mathscr{T}_{n}^{\ast}$ at 1, 5, 10% levels, power series (PS), trigonometric (Trig) and B-spline (B-s) bases. $\mathscr{T}_{n}^{a\ast}$ | SARARMA(0,1,0)
---|---
|  | PS |  |  |  | Trig |  |  |  | B-s |
$n=60$ | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.1
${\small c=0}$ | ${\small 0.008}$ | ${\small 0.032}$ | ${\small 0.084}$ |  | ${\small 0.004}$ | ${\small 0.04}$ | ${\small 0.092}$ |  | ${\small 0.01}$ | ${\small 0.07}$ | ${\small 0.132}$
| ${\small 0.004}$ | ${\small 0.038}$ | ${\small 0.096}$ |  | ${\small 0.004}$ | ${\small 0.038}$ | ${\small 0.094}$ |  | ${\small 0.004}$ | ${\small 0.038}$ | ${\small 0.096}$
${\small c=3}$ | ${\small 0.036}$ | ${\small 0.154}$ | ${\small 0.296}$ |  | ${\small 0.09}$ | ${\small 0.274}$ | ${\small 0.384}$ |  | ${\small 0.164}$ | ${\small 0.376}$ | ${\small 0.558}$
| ${\small 0.154}$ | ${\small 0.414}$ | ${\small 0.62}$ |  | ${\small 0.056}$ | ${\small 0.22}$ | ${\small 0.376}$ |  | ${\small 0.036}$ | ${\small 0.152}$ | ${\small 0.288}$
${\small c=6}$ | ${\small 0.22}$ | ${\small 0.544}$ | ${\small 0.748}$ |  | ${\small 0.444}$ | ${\small 0.794}$ | ${\small 0.906}$ |  | ${\small 0.56}$ | ${\small 0.892}$ | ${\small 0.956}$
| ${\small 0.844}$ | ${\small 0.992}$ | ${\small 1}$ |  | ${\small 0.312}$ | ${\small 0.714}$ | ${\small 0.87}$ |  | ${\small 0.174}$ | ${\small 0.532}$ | ${\small 0.732}$
${\small n=100}$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.006}$ | ${\small 0.044}$ | ${\small 0.098}$ |  | ${\small 0.004}$ | ${\small 0.038}$ | ${\small 0.092}$ |  | ${\small 0.012}$ | ${\small 0.048}$ | ${\small 0.112}$
| ${\small 0.012}$ | ${\small 0.046}$ | ${\small 0.096}$ |  | ${\small 0.006}$ | ${\small 0.036}$ | ${\small 0.106}$ |  | ${\small 0.01}$ | ${\small 0.056}$ | ${\small 0.106}$
${\small c=3}$ | ${\small 0.294}$ | ${\small 0.578}$ | ${\small 0.72}$ |  | ${\small 0.214}$ | ${\small 0.504}$ | ${\small 0.63}$ |  | ${\small 0.28}$ | ${\small 0.564}$ | ${\small 0.72}$
| ${\small 0.37}$ | ${\small 0.662}$ | ${\small 0.824}$ |  | ${\small 0.194}$ | ${\small 0.45}$ | ${\small 0.632}$ |  | ${\small 0.196}$ | ${\small 0.466}$ | ${\small 0.64}$
${\small c=6}$ | ${\small 0.95}$ | ${\small 0.99}$ | ${\small 0.996}$ |  | ${\small 0.900}$ | ${\small 0.99}$ | ${\small 0.998}$ |  | ${\small 0.932}$ | ${\small 0.992}$ | ${\small 1}$
| ${\small 0.992}$ | ${\small 0.998}$ | ${\small 1}$ |  | ${\small 0.856}$ | ${\small 0.988}$ | ${\small 1}$ |  | ${\small 0.86}$ | ${\small 0.984}$ | ${\small 0.998}$
$\small n=200$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.006}$ | ${\small 0.038}$ | ${\small 0.104}$ |  | ${\small 0.012}$ | ${\small 0.046}$ | ${\small 0.114}$ |  | ${\small 0.014}$ | ${\small 0.048}$ | ${\small 0.132}$
| ${\small 0.006}$ | ${\small 0.048}$ | ${\small 0.088}$ |  | ${\small 0.016}$ | ${\small 0.042}$ | ${\small 0.08}$ |  | ${\small 0.022}$ | ${\small 0.07}$ | ${\small 0.14}$
${\small c=3}$ | ${\small 0.178}$ | ${\small 0.402}$ | ${\small 0.55}$ |  | ${\small 0.162}$ | ${\small 0.38}$ | ${\small 0.524}$ |  | ${\small 0.282}$ | ${\small 0.476}$ | ${\small 0.608}$
| ${\small 0.282}$ | ${\small 0.56}$ | ${\small 0.694}$ |  | ${\small 0.134}$ | ${\small 0.35}$ | ${\small 0.466}$ |  | ${\small 0.198}$ | ${\small 0.37}$ | ${\small 0.514}$
${\small c=6}$ | ${\small 0.846}$ | ${\small 0.968}$ | ${\small 0.984}$ |  | ${\small 0.802}$ | ${\small 0.952}$ | ${\small 0.978}$ |  | ${\small 0.848}$ | ${\small 0.95}$ | ${\small 0.982}$
| ${\small 0.982}$ | ${\small 0.998}$ | ${\small 1}$ |  | ${\small 0.774}$ | ${\small 0.934}$ | ${\small 0.972}$ |  | ${\small 0.84}$ | ${\small 0.932}$ | ${\small 0.97}$
Table 2: Rejection probabilities of SARARMA(0,1,0) using bootstrap test $\mathscr{T}_{n}^{a\ast}$ at 1, 5, 10% levels, power series (PS), trigonometric (Trig) and B-spline (B-s) bases. $\mathscr{T}_{n}^{\ast}$ | SARARMA(1,0,1)
---|---
|  | PS |  |  |  | Trig |  |  |  | B-s |
$n=60$ | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.1
${\small c=0}$ | ${\small 0.006}$ | ${\small 0.054}$ | ${\small 0.08}$ |  | ${\small 0.012}$ | ${\small 0.062}$ | ${\small 0.106}$ |  | ${\small 0.016}$ | ${\small 0.044}$ | ${\small 0.086}$
| ${\small 0.016}$ | ${\small 0.062}$ | ${\small 0.118}$ |  | ${\small 0.026}$ | ${\small 0.09}$ | ${\small 0.138}$ |  | ${\small 0.016}$ | ${\small 0.048}$ | ${\small 0.088}$
${\small c=3}$ | ${\small 0.08}$ | ${\small 0.264}$ | ${\small 0.402}$ |  | ${\small 0.082}$ | ${\small 0.256}$ | ${\small 0.406}$ |  | ${\small 0.08}$ | ${\small 0.288}$ | ${\small 0.475}$
| ${\small 0.132}$ | ${\small 0.41}$ | ${\small 0.578}$ |  | ${\small 0.096}$ | ${\small 0.222}$ | ${\small 0.354}$ |  | ${\small 0.048}$ | ${\small 0.192}$ | ${\small 0.282}$
${\small c=6}$ | ${\small 0.266}$ | ${\small 0.588}$ | ${\small 0.748}$ |  | ${\small 0.266}$ | ${\small 0.616}$ | ${\small 0.782}$ |  | ${\small 0.218}$ | ${\small 0.604}$ | ${\small 0.772}$
| ${\small 0.444}$ | ${\small 0.804}$ | ${\small 0.894}$ |  | ${\small 0.204}$ | ${\small 0.474}$ | ${\small 0.658}$ |  | ${\small 0.198}$ | ${\small 0.496}$ | ${\small 0.612}$
${\small n=100}$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.006}$ | ${\small 0.054}$ | ${\small 0.116}$ |  | ${\small 0.012}$ | ${\small 0.046}$ | ${\small 0.114}$ |  | ${\small 0.014}$ | ${\small 0.042}$ | ${\small 0.09}$
| ${\small 0.02}$ | ${\small 0.056}$ | ${\small 0.112}$ |  | ${\small 0.012}$ | ${\small 0.044}$ | ${\small 0.088}$ |  | ${\small 0.034}$ | ${\small 0.058}$ | ${\small 0.118}$
${\small c=3}$ | ${\small 0.134}$ | ${\small 0.366}$ | ${\small 0.496}$ |  | ${\small 0.132}$ | ${\small 0.346}$ | ${\small 0.514}$ |  | ${\small 0.162}$ | ${\small 0.46}$ | ${\small 0.59}$
| ${\small 0.222}$ | ${\small 0.556}$ | ${\small 0.732}$ |  | ${\small 0.242}$ | ${\small 0.542}$ | ${\small 0.698}$ |  | ${\small 0.08}$ | ${\small 0.234}$ | ${\small 0.372}$
${\small c=6}$ | ${\small 0.566}$ | ${\small 0.832}$ | ${\small 0.916}$ |  | ${\small 0.59}$ | ${\small 0.888}$ | ${\small 0.96}$ |  | ${\small 0.548}$ | ${\small 0.898}$ | ${\small 0.952}$
| ${\small 0.732}$ | ${\small 0.964}$ | ${\small 0.986}$ |  | ${\small 0.476}$ | ${\small 0.846}$ | ${\small 0.918}$ |  | ${\small 0.432}$ | ${\small 0.796}$ | ${\small 0.874}$
${\small n=200}$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.04}$ | ${\small 0.086}$ | ${\small 0.11}$ |  | ${\small 0.026}$ | ${\small 0.076}$ | ${\small 0.108}$ |  | ${\small 0.02}$ | ${\small 0.06}$ | ${\small 0.09}$
| ${\small 0.03}$ | ${\small 0.078}$ | ${\small 0.114}$ |  | ${\small 0.032}$ | ${\small 0.074}$ | ${\small 0.118}$ |  | ${\small 0.038}$ | ${\small 0.086}$ | ${\small 0.112}$
${\small c=3}$ | ${\small 0.186}$ | ${\small 0.4}$ | ${\small 0.524}$ |  | ${\small 0.242}$ | ${\small 0.432}$ | ${\small 0.526}$ |  | ${\small 0.29}$ | ${\small 0.516}$ | ${\small 0.626}$
| ${\small 0.402}$ | ${\small 0.636}$ | ${\small 0.754}$ |  | ${\small 0.244}$ | ${\small 0.42}$ | ${\small 0.542}$ |  | ${\small 0.184}$ | ${\small 0.36}$ | ${\small 0.458}$
${\small c=6}$ | ${\small 0.718}$ | ${\small 0.904}$ | ${\small 0.962}$ |  | ${\small 0.78}$ | ${\small 0.942}$ | ${\small 0.982}$ |  | ${\small 0.73}$ | ${\small 0.948}$ | ${\small 0.978}$
| ${\small 0.872}$ | ${\small 0.98}$ | ${\small 0.998}$ |  | ${\small 0.794}$ | ${\small 0.948}$ | ${\small 0.98}$ |  | ${\small 0.772}$ | ${\small 0.914}$ | ${\small 0.94}$
Table 3: Rejection probabilities of SARARMA(1,0,1) using bootstrap test $\mathscr{T}_{n}^{\ast}$ at 1, 5, 10% levels, power series (PS), trigonometric (Trig) and B-spline (B-s) bases. $\mathscr{T}_{n}^{a\ast}$ | SARARMA(1,0,1)
---|---
|  | PS |  |  |  | Trig |  |  |  | B-s |
$n=60$ | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.10 |  | 0.01 | 0.05 | 0.1
${\small c=0}$ | ${\small 0.006}$ | ${\small 0.052}$ | ${\small 0.084}$ |  | ${\small 0.014}$ | ${\small 0.064}$ | ${\small 0.096}$ |  | ${\small 0.012}$ | ${\small 0.044}$ | ${\small 0.104}$
| ${\small 0.012}$ | ${\small 0.068}$ | ${\small 0.114}$ |  | ${\small 0.024}$ | ${\small 0.088}$ | ${\small 0.13}$ |  | ${\small 0.018}$ | ${\small 0.038}$ | ${\small 0.068}$
${\small c=3}$ | ${\small 0.092}$ | ${\small 0.27}$ | ${\small 0.396}$ |  | ${\small 0.08}$ | ${\small 0.25}$ | ${\small 0.406}$ |  | ${\small 0.118}$ | ${\small 0.382}$ | ${\small 0.56}$
| ${\small 0.164}$ | ${\small 0.408}$ | ${\small 0.596}$ |  | ${\small 0.102}$ | ${\small 0.242}$ | ${\small 0.37}$ |  | ${\small 0.046}$ | ${\small 0.15}$ | ${\small 0.23}$
${\small c=6}$ | ${\small 0.268}$ | ${\small 0.596}$ | ${\small 0.752}$ |  | ${\small 0.248}$ | ${\small 0.61}$ | ${\small 0.792}$ |  | ${\small 0.23}$ | ${\small 0.56}$ | ${\small 0.808}$
| ${\small 0.518}$ | ${\small 0.824}$ | ${\small 0.9}$ |  | ${\small 0.206}$ | ${\small 0.484}$ | ${\small 0.658}$ |  | ${\small 0.176}$ | ${\small 0.43}$ | ${\small 0.56}$
${\small n=100}$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.008}$ | ${\small 0.058}$ | ${\small 0.122}$ |  | ${\small 0.01}$ | ${\small 0.046}$ | ${\small 0.116}$ |  | ${\small 0.004}$ | ${\small 0.04}$ | ${\small 0.82}$
| ${\small 0.024}$ | ${\small 0.062}$ | ${\small 0.118}$ |  | ${\small 0.014}$ | ${\small 0.044}$ | ${\small 0.096}$ |  | ${\small 0.028}$ | ${\small 0.056}$ | ${\small 0.074}$
${\small c=3}$ | ${\small 0.14}$ | ${\small 0.36}$ | ${\small 0.494}$ |  | ${\small 0.122}$ | ${\small 0.354}$ | ${\small 0.52}$ |  | ${\small 0.186}$ | ${\small 0.4}$ | ${\small 0.524}$
| ${\small 0.252}$ | ${\small 0.566}$ | ${\small 0.73}$ |  | ${\small 0.272}$ | ${\small 0.568}$ | ${\small 0.696}$ |  | ${\small 0.04}$ | ${\small 0.148}$ | ${\small 0.214}$
${\small c=6}$ | ${\small 0.536}$ | ${\small 0.818}$ | ${\small 0.914}$ |  | ${\small 0.554}$ | ${\small 0.884}$ | ${\small 0.948}$ |  | ${\small 0.58}$ | ${\small 0.914}$ | ${\small 0.95}$
| ${\small 0.786}$ | ${\small 0.958}$ | ${\small 0.974}$ |  | ${\small 0.478}$ | ${\small 0.834}$ | ${\small 0.916}$ |  | ${\small 0.328}$ | ${\small 0.586}$ | ${\small 0.678}$
${\small n=200}$ |  |  |  |  |  |  |  |  |  |  |
${\small c=0}$ | ${\small 0.04}$ | ${\small 0.08}$ | ${\small 0.116}$ |  | ${\small 0.03}$ | ${\small 0.076}$ | ${\small 0.102}$ |  | ${\small 0.016}$ | ${\small 0.036}$ | ${\small 0.072}$
| ${\small 0.026}$ | ${\small 0.064}$ | ${\small 0.108}$ |  | ${\small 0.028}$ | ${\small 0.06}$ | ${\small 0.122}$ |  | ${\small 0.008}$ | ${\small 0.014}$ | ${\small 0.02}$
${\small c=3}$ | ${\small 0.176}$ | ${\small 0.382}$ | ${\small 0.516}$ |  | ${\small 0.22}$ | ${\small 0.438}$ | ${\small 0.526}$ |  | ${\small 0.262}$ | ${\small 0.45}$ | ${\small 0.55}$
| ${\small 0.41}$ | ${\small 0.632}$ | ${\small 0.738}$ |  | ${\small 0.256}$ | ${\small 0.428}$ | ${\small 0.538}$ |  | ${\small 0.06}$ | ${\small 0.124}$ | ${\small 0.164}$
${\small c=6}$ | ${\small 0.704}$ | ${\small 0.894}$ | ${\small 0.948}$ |  | ${\small 0.746}$ | ${\small 0.934}$ | ${\small 0.976}$ |  | ${\small 0.69}$ | ${\small 0.916}$ | ${\small 0.974}$
| ${\small 0.914}$ | ${\small 0.986}$ | ${\small 0.996}$ |  | ${\small 0.776}$ | ${\small 0.93}$ | ${\small 0.976}$ |  | ${\small 0.482}$ | ${\small 0.612}$ | ${\small 0.66}$
Table 4: Rejection probabilities of SARARMA(1,0,1) using bootstrap test $\mathscr{T}_{n}^{a\ast}$ at 1, 5, 10% levels, power series (PS), trigonometric (Trig) and B-spline (B-s) bases. | $r=2$ | $r=3$ | $r=4$ | $r=5$
---|---|---|---|---
$n=150$ | $p=10$ | $p=15$ | $p=10$ | $p=15$ | $p=10$ | $p=15$ | $p=10$ | $p=15$ | $p=20$
$c=0$ | 0.0860 | 0.2020 | 0.1180 | 0.2060 | 0.1420 | 0.2240 |  |  |
$c=3$ | 0.3320 | 0.6340 | 0.3700 | 0.6380 | 0.3760 | 0.6700 |  |  |
$c=6$ | 0.9060 | 0.9920 | 0.9180 | 0.9940 | 0.9220 | 0.9960 |  |  |
$n=300$ |  |  |  |  |  |  |  |  |
$c=0$ | 0.0820 | 0.0960 | 0.0880 | 0.1080 | 0.1060 | 0.1100 |  |  |
$c=3$ | 0.2680 | 0.5980 | 0.2600 | 0.6120 | 0.2780 | 0.6220 |  |  |
$c=6$ | 0.8140 | 0.9980 | 0.8160 | 0.9980 | 0.8220 | 0.9980 |  |  |
$n=500$ |  |  |  |  |  |  |  |  |
$c=0$ | 0.0280 | 0.0420 | 0.0260 | 0.0400 | 0.0360 | 0.0480 |  |  |
$c=3$ | 0.2320 | 0.6660 | 0.2400 | 0.6620 | 0.2460 | 0.6680 |  |  |
$c=6$ | 0.8920 | 1 | 0.9040 | 1 | 0.9000 | 1 |  |  |
$n=600$ |  |  |  |  |  |  |  |  |
$c=0$ | 0.0320 | 0.0500 | 0.0340 | 0.0540 | 0.0360 | 0.0540 |  |  |
$c=3$ | 0.3140 | 0.6480 | 0.3080 | 0.6280 | 0.3120 | 0.6460 |  |  |
$c=6$ | 0.9220 | 1 | 0.9180 | 1 | 0.9180 | 1 |  |  |
$n=700$ |  |  |  |  |  |  |  |  |
$c=0$ | 0.0260 | 0.0300 | 0.0280 | 0.0380 | 0.0280 | 0.0380 | 0.0280 | 0.0420 | 0.0580
$c=3$ | 0.2420 | 0.5540 | 0.2400 | 0.5480 | 0.2520 | 0.5500 | 0.2420 | 0.5600 | 0.6920
$c=6$ | 0.9580 | 0.9980 | 0.9560 | 0.9980 | 0.9600 | 0.9980 | 0.9500 | 0.9980 | 1
Table 5: Rejection probabilities of $\mathscr{T}_{n}$ at 5% asymptotic level, nonparametric spatial error structure. | 1998 | Pooled
---|---|---
| $H_{0}$ | p-value | $H_{1}$ | p-value | $H_{0}$ | p-value | $H_{1}$ |
| SARARMA(2,1,0)
$W^{A}y$ | -0.005 |  $<$0.001 | -0.003 |  $<$0.001 | 0.013 |  $<$0.001 | 0.013 |  $<$0.001
$W^{E}y$ | 0.130 |  $<$0.001 | 0.129 |  $<$0.001 | 0.121 |  $<$0.001 | 0.121 |  $<$0.001
$W^{d}$ | -0.159 | 0.281 | -0.225 |  $<$0.001 | -0.086 | 0.033 | -0.086 | 0.033
$\mathscr{T}_{n}$ |  |  | -1.921 | 0.973 |  |  | -2.531 | 0.994
$\mathscr{T}_{n}^{\ast}$ |  |  |  | 0.840 |  |  |  | 0.940
$\mathscr{T}_{n}^{a}$ |  |  | -1.918 | 0.972 |  |  | -2.547 | 0.995
$\mathscr{T}_{n}^{a\ast}$ |  |  |  | 0.870 |  |  |  | 0.730
| SARARMA(2,0,1)
$W^{A}y$ | 0.001 |  $<$0.01 | 0.011 |  $<$0.01 | 0.013 |  $<$0.01 | 0.013 |  $<$0.01
$W^{E}y$ | 0.127 |  $<$0.01 | 0.122 |  $<$0.01 | 0.121 |  $<$0.01 | 0.121 |  $<$0.01
$W^{d}$ | -0.153 |  $<$0.01 | -0.050 |  $<$0.01 | -0.086 |  $<$0.01 | -0.086 | 0.025
$\mathscr{T}_{n}$ |  |  | -1.763 | 0.961 |  |  | -2.421 | 0.992
$\mathscr{T}_{n}^{\ast}$ |  |  |  | 0.900 |  |  |  | 0.990
$\mathscr{T}_{n}^{a}$ |  |  | -2.349 | 0.991 |  |  | -2.423 | 0.992
$\mathscr{T}_{n}^{a\ast}$ |  |  |  | 0.850 |  |  |  | 0.790
| Nonparametric
$W^{A}y$ | -0.052 |  $<$0.001 | -0.011 |  $<$0.001 | 0.033 |  $<$0.001 | 0.033 |  $<$0.001
$W^{E}y$ | 0.149 |  $<$0.001 | 0.133 |  $<$0.001 | 0.110 |  $<$0.001 | 0.109 |  $<$0.001
$W^{d}$ |  |  |  |  |  |  |  |
$\mathscr{T}_{n}$ |  |  | -1.294 | 0.902 |  |  | -2.314 | 0.990
$\mathscr{T}_{n}^{\ast}$ |  |  |  | 0.830 |  |  |  | 0.850
$\mathscr{T}_{n}^{a}$ |  |  | -1.898 | 0.971 |  |  | -2.530 | 0.994
$\mathscr{T}_{n}^{a\ast}$ |  |  |  | 0.660 |  |  |  | 0.910
Table 6: The estimates and test statistics for the conflict data. ∗ denotes
the bootstrap p-value.

Variables | FE |  SARARMA(0,1,0), $W_{TEC}$
---|---|---
|  | p-value | $H_{0}$ | p-value | $H_{1}$ | p-value
$\ln(Spsic)$ | -0.005 | 0.649 |  0.007 | 0.574 | 0.015 | 0.166
$\ln(Sptec)$ | 0.191 |  $<$0.001 | 0.006 | 0.850 | -0.001 | 0.998
$\ln(Lab.)$ | 0.636 |  $<$0.001 | 0.572 |  $<$0.001 |  |
$\ln(Cap.)$ | 0.154 |  $<$0.001 | 0.336 |  $<$0.001 |  |
$\ln(R\&D)$ | 0.043 |  $<$0.001 | 0.081 |  $<$0.001 |  |
$W_{TEC}$ |  |  | 0.835 |  $<$0.001 | 0.829 |  $<$0.001
$\mathscr{T}_{n}$ |  |  |  |  | 15.528 |  $<$0.001
$\mathscr{T}_{n}^{*}$ |  |  |  |  |  | 0.050

Variables |  SARARMA(0,1,0), $W_{SIC}$
---|---
| $H_{0}$ | p-value | $H_{1}$ | p-value
$\ln(Spsic)$ | 0.008 | 0.620 | 0.017 | 0.193
$\ln(Sptec)$ | 0.039 | 0.157 | 0.020 | 0.336
$\ln(Lab.)$ | 0.571 |  $<$0.001 |  |
$\ln(Cap.)$ | 0.318 |  $<$0.001 |  |
$\ln(R\&D)$ | 0.082 |  $<$0.001 |  |
$W_{SIC}$ | 0.722 |  $<$0.001 | 0.724 |  $<$0.001
$\mathscr{T}_{n}$ |  |  | 10.451 |  $<$0.001
$\mathscr{T}_{n}^{*}$ |  |  |  |  $<$0.001

Table 7: The estimates and test statistics for the R&D data, SARARMA(0,1,0). ∗
denotes the bootstrap p-value. The price index as well as a dummy variable for
missing value in R&D are included, but we only report the coefficients
reported in Bloom et al. (2013).

Variables | SARARMA(0,2,0)
---|---
| ${\small H}_{0}$ | p-value | ${\small H}_{1}$ | p-value
$\ln{\small(Spsic)}$ | 0.009 | 0.587 | 0.018 | 0.170
$\ln{\small(Sptec)}$ | 0.044 | 0.112 | 0.026 | 0.236
$\ln{\small(Lab.)}$ | 0.573 |  $<$0.001 |  |
$\ln{\small(Cap.)}$ | 0.315 |  $<$0.001 |  |
$\ln{\small(R\&D)}$ | 0.082 |  $<$0.001 |  |
${\small W}_{SIC}$ | 0.696 |  $<$0.001 | 0.693 |  $<$0.001
${\small W}_{TEC}$ | 0.157 | 0.092 | 0.164 | 0.079
$\mathscr{T}_{n}$ |  |  | 10.485 |  $<$0.001
$\mathscr{T}_{n}^{*}$ |  |  |  | 0.060

Variables | SARARMA(0,0,2)
---|---
| ${\small H}_{0}$ | p-value | ${\small H}_{1}$ | p-value
$\ln{\small(Spsic)}$ | -0.0002 | 0.991 | 0.013 | 0.266
$\ln{\small(Sptec)}$ | 0.033 | 0.200 | 0.017 | 0.434
$\ln{\small(Lab.)}$ | 0.565 |  $<$0.01 |  |
$\ln{\small(Cap.)}$ | 0.334 |  $<$0.01 |  |
$\ln{\small(R\&D)}$ | 0.076 |  $<$0.01 |  |
${\small W}_{SIC}$ | 0.624 |  $<$0.01 | 0.728 |  $<$0.001
${\small W}_{TEC}$ | 0.312 | 0.123 | 0.321 | 0.112
$\mathscr{T}_{n}$ |  |  | 15.144 |  $<$0.001
$\mathscr{T}_{n}^{*}$ |  |  |  | 0.020

Variables | Error MESS(2)
---|---
| ${\small H}_{0}$ | p-value | ${\small H}_{1}$ | p-value
$\ln{\small(Spsic)}$ | 0.002 | 0.788 | 0.014 | 0.040
$\ln{\small(Sptec)}$ | 0.045 | 0.025 | 0.027 | 0.088
$\ln{\small(Lab.)}$ | 0.569 |  $<$0.001 |  |
$\ln{\small(Cap.)}$ | 0.323 |  $<$0.001 |  |
$\ln{\small(R\&D)}$ | 0.077 |  $<$0.001 |  |
${\small W}_{SIC}$ | 0.775 |  $<$0.001 | 0.836 |  $<$0.001
${\small W}_{TEC}$ | 0.338 | 0.010 | 0.380 | 0.004
$\mathscr{T}_{n}$ |  |  | 12.776 |  $<$0.001
$\mathscr{T}_{n}^{*}$ |  |  |  | 0.050

Table 8: The estimates and test statistics for the R&D data, SARARMA(0,2,0) and Error MESS(2). ∗ denotes the bootstrap p-value. The price index as well as a dummy variable for missing value in R&D are included, but we only report the coefficients reported in Bloom et al. (2013). Variable |  $w_{ij}^{\ast}=d_{ij}^{-2}$ for $i\neq j$ |  $w_{ij}^{\ast}=e^{-2d_{ij}}$ for $i\neq j$
---|---|---
| estimate | p-value | estimate | p-value
Constant | $1.0711$ | $0.608$ | $0.5989$ | $0.798$
$\ln(s)$ | $0.8256$ | $<0.001$ | $0.7938$ | $<0.001$
$\ln(n_{p}+0.05)$ | $-1.4984$ | $0.008$ | $-1.4512$ | $0.009$
$W\ln(s)$ | $-0.3159$ | $0.075$ | $-0.3595$ | $0.020$
$W\ln(n_{p}+0.05)$ | $0.5633$ | $0.498$ | $0.1283$ | $0.856$
$Wy$ | $0.7360$ | $<0.001$ | $0.6510$ | $<0.001$
$\mathscr{T}_{n}$ | $-1.88$ | 0.970 | $-2.08$ | 0.981
$\mathscr{T}_{n}^{*}$ |  | 0.850 |  | 0.900
$\mathscr{T}_{n}^{a}$ | $-1.90$ | 0.971 | $-2.05$ | 0.980
$\mathscr{T}_{n}^{a*}$ |  | 0.820 |  | 0.810
Restricted regression |  |  |  |
Constant | $2.1411$ | $<0.001$ | $2.9890$ | $<0.001$
$\ln(s)-\ln(n+0.05)$ | $0.8426$ | $<0.001$ | $0.8195$ | $<0.001$
$W[\ln(s)-\ln(n_{p}+0.05)]$ | $-0.2675$ | $0.122$ | $-0.2589$ | $0.098$
$W\ln(y)$ | $0.7320$ | $<0.001$ | $0.6380$ | $<0.001$
$\mathscr{T}_{n}$ | $0.30$ | 0.382 | $4.04$ | $<0.001$
$\mathscr{T}_{n}^{*}$ |  | 0.500 |  | $<0.001$
$\mathscr{T}_{n}^{a}$ | $0.10$ | 0.460 | $4.50$ | $<0.001$
$\mathscr{T}_{n}^{a*}$ |  | 0.560 |  | $0.040$
Table 9: The estimates and test statistics of the linear SAR model for the
growth data. ∗ denotes the bootstrap p-value.

Appendix

##  Appendix A Proofs of theorems and propositions

###### Proof of Proposition 4.1:.

Because the map $\Sigma:\Gamma^{o}\rightarrow\mathcal{M}^{n\times n}$ is
Fréchet-differentiable on $\Gamma^{o}$, it is also Gâteaux-differentiable and
the two derivative maps coincide. Thus by Theorem 1.8 of Ambrosetti and Prodi
(1995),
$\left\|\Sigma\left(\gamma_{1}\right)-\Sigma\left(\gamma_{2}\right)\right\|\leq\sup_{\gamma\in\ell\left[\gamma_{1},\gamma_{2}\right]}\left\|D\Sigma(\gamma)\right\|\left\|\gamma_{1}-\gamma_{2}\right\|,$
where
$\ell\left[\gamma_{1},\gamma_{2}\right]=\left\\{t\gamma_{1}+(1-t)\gamma_{2}:t\in[0,1]\right\\}$.
The claim now follows by (4.3) in Assumption 8. ∎

###### Proof of Theorem 4.1.

This a particular case of the proof of Theorem 5.1 with $\lambda=0$, and so
$S(\lambda)=I_{n}$. ∎

###### Proof of Theorem 4.2.

In the supplementary appendix. ∎

###### Proof of Theorem 4.3.

We would like to establish the asymptotic unit normality of

$\frac{\sigma_{0}^{-2}\varepsilon^{\prime}\mathscr{V}\varepsilon-p}{\sqrt{2p}}.$
(A.1)

Writing $q=\sqrt{2p}$, the ratio in (A.1) has zero mean and variance equal to
one, and may be written as $\sum_{s=1}^{\infty}w_{s}$, where
$w_{s}=\sigma_{0}^{-2}q^{-1}v_{ss}\left(\varepsilon_{s}^{2}-\sigma_{0}^{2}\right)+2\sigma_{0}^{-2}q^{-1}\mathbf{1}(s\geq
2)\varepsilon_{s}\sum_{t<s}v_{st}\varepsilon_{t},$ with $v_{st}$ the typical
element of $\mathscr{V}$, with $s,t=1,2,\ldots,$. We first show that

$w_{\ast}\overset{p}{\longrightarrow}0,$ (A.2)

where $w_{\ast}=w-w_{S}$, $w_{S}=\sum_{s=1}^{S}w_{s}$ and $S=S_{n}$ is a
positive integer sequence that is increasing in $n$. All expectations in the
sequel are taken conditional on $X$. By Chebyshev’s inequality proving

$\mathcal{E}w_{\ast}^{2}\overset{p}{\rightarrow}0$ (A.3)

is sufficient to establish (A.2). Notice that $\mathcal{E}w_{s}^{2}\leq
Cq^{-2}v_{ss}^{2}+Cq^{-2}\mathbf{1}(s\geq 2)\sum_{t<s}v_{st}^{2}\leq
Cq^{-2}\sum_{t\leq s}v_{st}^{2},$ so that, writing
$\mathscr{M}=\Sigma^{-{1}}\Psi[\Psi^{\prime}\Sigma^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma^{-{1}}$,

$\displaystyle\sum_{s=S+1}^{\infty}\mathcal{E}w_{s}^{2}\leq
Cq^{-2}\sum_{s=S+1}^{\infty}\sum_{t\leq s}v_{st}^{2}\leq
Cq^{-2}\sum_{s=S+1}^{\infty}b_{s}^{\prime}M\sum_{t\leq
s}b_{t}b_{t}^{\prime}\mathscr{M}b_{s}$ (A.4) $\displaystyle\leq$
$\displaystyle
Cq^{-2}\left\|\Sigma\right\|\sum_{s=S+1}^{\infty}b_{s}^{\prime}\mathscr{M}^{2}b_{s}\leq
Cq^{-2}\sum_{s=S+1}^{\infty}\sum_{i,j,k=1}^{n}b_{is}b_{kt}m_{ij}m_{kj}$
$\displaystyle\leq$ $\displaystyle
Cq^{-2}\sum_{s=S+1}^{\infty}\sum_{i,k=1}^{n}\left|b_{is}^{\ast}\right|\left|b_{ks}^{\ast}\right|\sum_{j=1}^{n}\left(m_{kj}^{2}+m_{ij}^{2}\right),$

where $m_{ij}$ is the $(i,j)$-th element of $\mathscr{M}$ and we have used the
inequality $|ab|\leq\left(a^{2}+b^{2}\right)/2$ in the last step. Now, denote
by $h_{i}^{\prime}$ the $i$-th row of the $n\times p$ matrix
$\Sigma^{-1}\Psi$. Denoting the elements of $\Sigma^{-1}$ by
$\Sigma^{-1}_{ij}$ and $\psi_{jl}=\psi\left(x_{jl}\right)$, $h_{i}$ has
entries $h_{il}=\sum_{j=1}^{n}\Sigma^{-1}_{ij}\psi_{jl}$, $l=1,\ldots,p$. We
have
$\left|h_{il}\right|=O_{p}\left(\sum_{j=1}^{n}\left|\Sigma^{-1}_{ij}\right|\right)=O_{p}\left(\left\|\Sigma^{-1}\right\|_{R}\right)=O_{p}(1)$,
uniformly, by Assumptions R.11 and R.13. Thus, we have
$\left\|h_{i}\right\|=O_{p}\left(\sqrt{p}\right)$, uniformly in $i$. As a
result,

$\left|m_{ij}\right|=n^{-1}\left|h_{i}^{\prime}\left(n^{-1}\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}h_{j}\right|=O_{p}\left(n^{-1}\left\|h_{i}\right\|\left\|h_{j}\right\|\right)=O_{p}\left(pn^{-1}\right),$
(A.5)

uniformly in $i,j$, by Assumption R.11. Similarly, note that

$\displaystyle\sum_{j=1}^{n}m_{ij}^{2}$ $\displaystyle=$ $\displaystyle
n^{-1}h_{i}^{\prime}\left(n^{-1}\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}\left(n^{-1}\Psi^{\prime}\Sigma^{-2}\Psi\right)\left(n^{-1}\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}h_{i}$
(A.6) $\displaystyle\leq$ $\displaystyle
n^{-1}\left\|h_{i}\right\|^{2}\left\|\left(n^{-1}\Psi^{\prime}\Sigma^{-1}\Psi\right)^{-1}\right\|^{2}\left\|n^{-1}\Psi^{\prime}\Sigma^{-2}\Psi\right\|$
$\displaystyle=$ $\displaystyle
O_{p}\left(pn^{-2}\left\|\Psi\right\|^{2}\left\|\Sigma^{-1}\right\|^{2}\right)=O_{p}\left(pn^{-1}\right),$

uniformly in $i$. Thus (A.4) is

$O_{p}\left(q^{-2}pn^{-1}\sum_{i=1}^{n}\sum_{s=S+1}^{\infty}\left|b_{is}^{\ast}\right|\sum_{t=1}^{n}\left|b_{ks}^{\ast}\right|\right)=O_{p}\left(q^{-2}p\sup_{i=1,\ldots,n}\sum_{s=S+1}^{\infty}\left|b_{is}^{\ast}\right|\right),$
(A.7)

by Assumption R.4. By the same assumption, there exists $S_{in}$ such that
$\sum_{s=S_{in}+1}^{\infty}\left|b_{is}^{\ast}\right|\leq\epsilon_{n}$ for any
decreasing sequence $\epsilon_{n}\rightarrow 0$ as $n\rightarrow\infty$.
Choosing $S=\max_{i=1,\ldots,n}S_{in}$ in $w_{S}$, we deduce that (A.7) is
$O_{p}\left(q^{-2}p\epsilon_{n}\right)=O_{p}\left(\epsilon_{n}\right)=o_{p}(1)$,
proving (A.3). Thus we need only focus on $w_{S}$, and seek to establish that

$w_{S}\longrightarrow_{d}N(0,1),\text{ as }n\rightarrow\infty.$ (A.8)

From Scott (1973), (A.8) follows if

$\sum_{s=1}^{S}\mathcal{E}w_{s}^{4}\overset{p}{\longrightarrow}0,\text{ as
}n\rightarrow\infty,$ (A.9)

and

$\sum_{s=1}^{S}\left[\mathcal{E}\left(w_{s}^{2}\left.{}\right|\varepsilon_{t},t<s\right)-\mathcal{E}\left(w_{s}^{2}\right)\right]\overset{p}{\longrightarrow}0,\text{
as }n\rightarrow\infty.$ (A.10)

We show (A.9) first. Evaluating the expectation and using (A.6) yields

$\displaystyle\mathcal{E}w_{s}^{4}$ $\displaystyle\leq$ $\displaystyle
Cq^{-4}v_{ss}^{4}+Cq^{-4}\sum_{t<s}v_{st}^{4}\leq Cq^{-4}\left(\sum_{t\leq
s}v_{st}^{2}\right)^{2}\leq Cq^{-4}\left(b_{s}^{\prime}\mathscr{M}\sum_{t\leq
s}b_{t}b_{t}^{\prime}\mathscr{M}b_{s}\right)^{2}$ $\displaystyle\leq$
$\displaystyle
Cq^{-4}\left(b_{s}^{\prime}\mathscr{M}^{2}b_{s}\right)^{2}=Cq^{-4}\sum_{i,j,k=1}^{n}b_{is}b_{ks}m_{ij}m_{kj}\leq
Cq^{-4}\sum_{i,k=1}^{n}\left|b^{*}_{is}\right|\left|b^{*}_{ks}\right|\sum_{j=1}^{n}\left(m^{2}_{ij}+m^{2}_{kj}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}pn^{-1}\left(\sum_{i=1}^{n}\left|b^{*}_{is}\right|\right)^{2}\right),$

whence

$\displaystyle\sum_{s=1}^{S}\mathcal{E}w_{s}^{4}$ $\displaystyle=$
$\displaystyle
O_{p}\left(q^{-4}pn^{-1}\sum_{s=1}^{S}\left(\sum_{i=1}^{n}\left|b^{*}_{is}\right|\right)^{2}\right)=O_{p}\left(q^{-4}pn^{-1}\sum_{s=1}^{S}\left(\sum_{i=1}^{n}\left|b_{is}^{\ast}\right|\right)\right)=O_{p}\left(q^{-4}p\right),$

by Assumption R.4. Thus (A.9) is established. Notice that
$\mathcal{E}\left(\left.w_{s}^{2}\right|\epsilon_{t},t<s\right)$ equals

$4q^{-2}\sigma_{0}^{-4}\left\\{\left(\mu_{4}-\sigma_{0}^{4}\right)v_{ss}^{2}+2\mu_{3}\mathbf{1}(s\geq
2)\sum_{t<s}v_{st}v_{ss}\varepsilon_{t}\right\\}+4q^{-2}\sigma_{0}^{-2}\mathbf{1}(s\geq
2)\left(\sum_{t<s}v_{st}\varepsilon_{t}\right)^{2},$

and
$\mathcal{E}w_{s}^{2}=4q^{-2}\sigma_{0}^{-4}\left(\mu_{4}-\sigma_{0}^{4}\right)v_{ss}^{2}+4q^{-2}\mathbf{1}(s\geq
2)\sum_{t<s}v_{st}^{2},$ so that (A.10) is bounded by a constant times

$q^{-2}\sum_{s=2}^{S}\sum_{t<s}v_{st}v_{ss}\varepsilon_{t}+\left\\{\sum_{s=2}^{S}\left(\sum_{t<s}v_{st}\varepsilon_{t}\right)^{2}-\sigma_{0}^{2}\sum_{t<s}v_{st}^{2}\right\\}.$
(A.11)

By transforming the range of summation, the square of the first term in (A.11)
has expectation bounded by

$Cq^{-4}\mathcal{E}\left(\sum_{t=1}^{S-1}\sum_{s=t+1}^{S}v_{st}v_{ss}\varepsilon_{t}\right)^{2}\leq
Cq^{-4}\sum_{t=1}^{S-1}\left(\sum_{s=t+1}^{S}v_{st}v_{ss}\right)^{2},$ (A.12)

where the factor in parentheses on the RHS of (A.12) is

$\displaystyle\sum_{s,r=t+1}^{S}b_{s}^{\prime}\mathscr{M}b_{s}b_{s}^{\prime}\mathscr{M}b_{t}b_{r}^{\prime}\mathscr{M}b_{r}b_{r}^{\prime}\mathscr{M}b_{t}\leq\sum_{s,r=t+1}^{S}\left|b_{s}^{\prime}\mathscr{M}b_{s}b_{r}^{\prime}\mathscr{M}b_{r}\right|\left|b_{s}^{\prime}\mathscr{M}b_{t}\right|\left|b_{r}^{\prime}\mathscr{M}b_{t}\right|$
$\displaystyle\leq$ $\displaystyle
C\sum_{s,r=t+1}^{S}\sum_{i,j,k,l=1}^{n}\left|b_{is}\right|\left|m_{ij}\right|\left|b_{jr}\right|\left|b_{ks}\right|\left|m_{lk}\right|\left|b_{kr}\right|\left|b_{s}^{\prime}\mathscr{M}b_{t}\right|\left|b_{r}^{\prime}\mathscr{M}b_{t}\right|$
$\displaystyle\leq$ $\displaystyle
C\left(\sup_{i,j}\left|m_{ij}\right|\right)^{2}\left(\sup_{s\geq
1}\sum_{i=1}^{n}\left|b_{is}^{\ast}\right|\right)^{4}\sum_{s,r=t+1}^{S}\left|b_{s}^{\prime}\mathscr{M}b_{t}\right|\left|b_{r}^{\prime}\mathscr{M}b_{t}\right|$
$\displaystyle=$ $\displaystyle
O_{p}\left(p^{2}n^{-2}\left(\sum_{s=t+1}^{S}\left|b_{t}^{\prime}\mathscr{M}b_{s}\right|\right)^{2}\right)=O_{p}\left(p^{2}n^{-2}\left(\sum_{s=t+1}^{S}\sum_{i,j=1}^{n}\left|b_{it}^{\ast}\right|\left|m_{ij}\right|\left|b_{js}^{\ast}\right|\right)^{2}\right),$

where we used Assumptions R.4 and (A.5). Now Assumptions R.4, R.11 and (A.5)
imply that

$\displaystyle\sum_{s=t+1}^{S}\sum_{i,j=1}^{n}\left|b_{it}^{\ast}\right|\left|m_{ij}\right|\left|b_{js}^{\ast}\right|$
$\displaystyle=$ $\displaystyle
O_{p}\left(\sup_{i,j}\left|m_{ij}\right|\sup_{t}\sum_{i=1}^{n}\left|b_{it}^{\ast}\right|\sum_{j=1}^{n}\sum_{s=t+1}^{S}\left|b_{js}^{\ast}\right|\right)=O_{p}\left(p\sup_{t}\sum_{i=1}^{n}\left|b_{it}^{\ast}\right|\right),$

so (A.12) is
$O_{p}\left(q^{-4}p^{4}n^{-2}\sup_{t}\left(\sum_{i=1}^{n}\left|b_{it}^{\ast}\right|\right)\left(\sum_{i=1}^{n}\left(\sum_{t=1}^{S-1}\left|b_{it}^{\ast}\right|\right)\right)\right)$.
By Assumption R.4 the latter is $O_{p}\left(q^{-4}p^{4}n^{-1}\right)$ and
therefore the first term in (A.11) is $O_{p}\left(p^{2}n^{-1}\right)$, which
is negligible.

Once again transforming the summation range and using the inequality
$|a+b|^{2}\leq C\left(a^{2}+b^{2}\right)$, we can bound the square of the
second term in (A.11) by a constant times

$\left(\sum_{t=1}^{S-1}\sum_{s=t+1}^{S}v_{st}^{2}\left(\varepsilon_{t}^{2}-\sigma_{0}^{2}\right)\right)^{2}+\left(2\sum_{t=1}^{S-1}\sum_{r=1}^{t-1}\sum_{s=t+1}^{S}v_{st}v_{sr}\varepsilon_{t}\varepsilon_{r}\right)^{2}.$
(A.13)

Using Assumption R.4, the expectations of the two terms in (A.13) are bounded
by a constant times $\alpha_{1}$ and a constant times $\alpha_{2}$,
respectively, where
$\alpha_{1}=\sum_{t=1}^{S-1}\left(\sum_{s=t+1}^{S}v_{st}^{2}\right)^{2},\alpha_{2}=\sum_{t=1}^{S-1}\sum_{r=1}^{t-1}\left(\sum_{s=t+1}^{S}v_{st}v_{sr}\right)^{2}.$
Thus (A.13) is $O_{p}\left(\alpha_{1}+\alpha_{2}\right)$. Now by (A.5),
Assumptions R.4, R.11 and elementary inequalities $\alpha_{2}$ is bounded by

$\displaystyle\sum_{t=1}^{S-1}\sum_{r=1}^{t-1}\sum_{s=t+1}^{S}\sum_{u=t+1}^{S}b_{s}^{\prime}\mathscr{M}b_{t}b_{s}^{\prime}\mathscr{M}b_{r}b_{u}^{\prime}\mathscr{M}b_{t}b_{u}^{\prime}\mathscr{M}b_{r}$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}\sum_{s,r,t,u=1}^{S}\sum_{i,j=1}^{n}\left|b_{ir}^{\ast}\right|\left|m_{ij}\right|\left|b_{js}^{\ast}\right|\sum_{i,j=1}^{n}\left|b_{ir}^{\ast}\right|\left|m_{ij}\right|\left|b_{ju}^{\ast}\right|\sum_{i,j=1}^{n}\left|b_{it}^{\ast}\right|\left|m_{ij}\right|\left|b_{js}^{\ast}\right|\sum_{i,j=1}^{n}\left|b_{it}^{\ast}\right|\left|m_{ij}\right|\left|b_{ju}^{\ast}\right|\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}pn^{-1}\sum_{s,r,t=1}^{S}\left(\sum_{i,j=1}^{n}\left|b_{ir}^{\ast}\right|\left|m_{ij}\right|\left|b_{js}^{\ast}\right|\right)\left(\sum_{i,j=1}^{n}\left|b_{ir}^{\ast}\right|\left|m_{ij}\right|\sum_{u=1}^{S}\left|b_{ju}^{\ast}\right|\right)\right.$
$\displaystyle\times$
$\displaystyle\left.\sum_{i,j=1}^{n}\left|b_{it}^{\ast}\right|\left|m_{ij}\right|\left|b_{js}^{\ast}\right|\sum_{i=1}^{n}\left|b_{it}^{\ast}\right|\sup_{u}\sum_{j=1}^{n}\left|b_{ju}^{\ast}\right|\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}p^{2}n^{-2}\sum_{s,r=1}^{S}\left(\sum_{i,j=1}^{n}\left|b_{ir}^{\ast}\right|\left|m_{ij}\right|\left|b_{js}^{\ast}\right|\right)\sum_{i=1}^{n}\left|b_{ir}^{\ast}\right|\sum_{j=1}^{n}\left(\sum_{u=1}^{S}\left|b_{ju}^{\ast}\right|\right)\left(\sum_{i,j=1}^{n}\sum_{t=1}^{S}\left|b_{it}^{\ast}\right|\left|m_{ij}\right|\left|b_{js}^{\ast}\right|\right)\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}p^{2}n^{-1}\sum_{i,j=1}^{n}\left(\sum_{r=1}^{S}\left|b_{ir}^{\ast}\right|\right)\left|m_{ij}\right|\left(\sum_{s=1}^{S}\left|b_{js}^{\ast}\right|\right)\left(\sup_{j}\sum_{i=1}^{n}\left|m_{ij}\right|\right)\sum_{j=1}^{n}\left|b_{js}^{\ast}\right|\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}p^{2}n^{-1}\sup_{k}\sum_{i,j=1}^{n}\left|m_{ij}\right|\sum_{i=1}^{n}\left|m_{ik}\right|\right)=O_{p}\left(q^{-4}p^{2}n^{-1}\sup_{k}\sum_{i,j,\ell=1}^{n}\left|m_{ij}\right|\left|m_{\ell
k}\right|\right)$ $\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}p^{2}n^{-1}\sup_{k}\sum_{i,j,\ell=1}^{n}\left(m_{ij}^{2}+m_{\ell
k}^{2}\right)\right)=O_{p}\left(q^{-4}p^{2}n^{-1}\sum_{i,j,\ell=1}^{n}\left(m_{ij}^{2}+m_{\ell
j}^{2}\right)\right)$ $\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}p^{2}n^{-1}\sum_{i,j=1}^{n}m_{ij}^{2}\right)=O_{p}\left(q^{-4}p^{2}\sup_{j}\sum_{i=1}^{n}m_{ij}^{2}\right)=O_{p}\left(pn^{-1}\right),$

where we used (A.6) in the last step. A similar use of the conditions of the
theorem and (A.5) implies that $\alpha_{1}$ is

$\displaystyle
O_{p}\left(q^{-4}\sum_{t=1}^{S-1}\left\\{\sum_{s=t+1}^{S}\left(\sum_{i,j=1}^{n}\left|m_{ij}\right|\left|b_{jt}^{\ast}\right|\left|b_{is}^{\ast}\right|\right)^{2}\right\\}^{2}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}\left(\sup_{i,j}\left|m_{ij}\right|\right)^{4}\sum_{t=1}^{S-1}\left\\{\sum_{s=t+1}^{S}\left(\sum_{i=1}^{n}\left|b_{is}^{\ast}\right|\sum_{j=1}^{n}\left|b_{jt}^{\ast}\right|\right)^{2}\right\\}^{2}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}p^{4}n^{-4}\sum_{t=1}^{S-1}\left\\{\sum_{s=t+1}^{S}\left(\sum_{i=1}^{n}\left|b_{is}^{\ast}\right|\right)^{2}\left(\sum_{j=1}^{n}\left|b_{jt}^{\ast}\right|\right)^{2}\right\\}^{2}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}p^{4}n^{-4}\sum_{t=1}^{S-1}\left(\sum_{s=t+1}^{S}\left(\sum_{i=1}^{n}\left|b_{is}^{\ast}\right|\right)^{2}\right)^{2}\left(\sum_{j=1}^{n}\left|b_{jt}^{\ast}\right|\right)^{4}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}p^{4}n^{-4}\left(\sum_{t=1}^{S-1}\sum_{j=1}^{n}\left|b_{jt}^{\ast}\right|\right)\left(\sum_{s=t+1}^{S}\sum_{i=1}^{n}\left|b_{is}^{\ast}\right|\right)^{2}\sup_{s}\left(\sum_{i=1}^{n}\left|b_{is}^{\ast}\right|\right)^{2}\sup_{t}\left(\sum_{j=1}^{n}\left|b_{jt}^{\ast}\right|\right)^{3}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(q^{-4}p^{4}n^{-1}\right)=O_{p}\left(p^{2}n^{-1}\right)$

proving (A.10), as $p^{2}/n\rightarrow 0$ by the conditions of the theorem. ∎

###### Proof of Theorem 4.4.

In supplementary appendix. ∎

###### Proof of Theorem 5.1.

Due to the similarity with proofs in Delgado and Robinson (2015) and Gupta and
Robinson (2018), the details are in the supplementary appendix. ∎

###### Proof of Theorem 5.2.

Denote $\theta^{\ast}$ as the solution of
$\min_{\theta}\mathcal{E}\left(y_{i}-\sum_{j=1}^{d_{\lambda}}\lambda_{j}w_{i,j}^{\prime}y-\theta(x_{i})\right)^{2}$.
Put $\theta_{i}^{\ast}=\theta^{\ast}(x_{i})$, $\theta_{0i}=\theta_{0}(x_{i})$,
$\widehat{\theta}_{i}=\psi_{i}^{\prime}\widehat{\beta}$ ,
$\widehat{f}_{i}=f(x_{i},\widehat{\alpha})$,
$f_{i}^{\ast}=f(x_{i},\alpha^{\ast})$. Then
$\widehat{u}_{i}=y_{i}-\sum_{j=1}^{d_{\lambda}}\widehat{\lambda}_{j}w_{i,j}^{\prime}y-f(x_{i},\widehat{\alpha})=u_{i}+\theta_{0i}+\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})w_{i,j}^{\prime}y-\widehat{f}_{i}$.
Proceeding as in the proof of Theorem 4.2, we obtain
$n\widehat{m}_{n}=\widehat{\sigma}^{-2}u^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}u+\widehat{\sigma}^{-2}\sum_{j=1}^{7}A_{j}$.
Thus, compared to the test statistic with no spatial lag, cf. the proof of
Theorem 4.2, we have the additional terms

$\displaystyle A_{5}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})y^{\prime}W_{j}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})W_{j}y,$
$\displaystyle A_{6}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})y^{\prime}W_{j}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}(u+\theta_{0}-\widehat{f}),$
$\displaystyle A_{7}$ $\displaystyle=$
$\displaystyle\left(\Psi\left(\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\left(u\mathbf{+}e\right)-e+\theta_{0}-\widehat{f}\right)^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})W_{j}y.$

We now show that $A_{\ell}=o_{p}(\sqrt{p}),\ell>4$, so the leading term in
$n\widehat{m}_{n}$ is the same as before. First
$\left\|y\right\|=O_{p}(\sqrt{n})$ from
$y=(I_{n}-\sum_{j=1}^{d_{\lambda}}\lambda_{j_{0}}W_{j})^{-1}\left(\theta_{0}+u\right)$.
Then, with
$\left\|\lambda_{{}_{0}}-\widehat{\lambda}\right\|=O_{p}\left(\sqrt{d_{\gamma}/n}\right)$
by Lemma LS.2, we have

$\displaystyle\left|A_{5}\right|$ $\displaystyle\leq$
$\displaystyle\left\|\lambda_{{}_{0}}-\widehat{\lambda}\right\|^{2}\sum_{j=1}^{d_{\lambda}}\left\|W_{j}\right\|^{2}\sup_{\gamma,j}\left\|\Sigma\left(\gamma\right)^{-1}\frac{1}{n}\Psi\left(\frac{1}{n}\Psi^{\prime}\Sigma\left({\gamma}\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\right\|\left\|y\right\|^{2}$
$\displaystyle=$ $\displaystyle
O_{p}\left(d_{\gamma}/n\right)O_{p}(1)O_{p}(n)=O_{p}\left(d_{\gamma}\right)=o_{p}(\sqrt{p}).$

Uniformly in $\gamma$ and $j$,

$\displaystyle\mathcal{E}\left(u^{\prime}S^{-1\prime}W_{j}^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}u\right)$
$\displaystyle=$
$\displaystyle\mathcal{E}tr\left(\left(\frac{1}{n}\Psi^{\prime}\Sigma\left({\gamma}\right)^{-1}\Psi\right)^{-1}\frac{1}{n}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Sigma
S^{-1\prime}W_{j}^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right)=O_{p}(p)$

and

$\displaystyle\mathcal{E}\left(\theta_{0}^{\prime}S^{-1\prime}W_{j}^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}u\right)^{2}$
$\displaystyle=$ $\displaystyle
O_{p}\left(\left\|S^{-1}\right\|^{2}\sup_{\gamma}\left\|\Sigma\left(\gamma\right)^{-1}\right\|^{4}\left\|\frac{1}{n}\Psi\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right)^{-1}\Psi^{\prime}\right\|^{2}\sup_{j}\left\|W_{j}\right\|^{2}\left\|\Sigma\right\|\left\|\theta_{0}\right\|^{2}\right)=O_{p}(n).$

Similarly,
$\theta_{0}^{\prime}S^{-1\prime}W_{j}^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}W_{j}\theta_{0}=O_{p}(n),$
uniformly. Therefore,

$\displaystyle\left|\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})y^{\prime}W_{j}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}u\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})\left(\theta_{0}+u\right)^{\prime}S^{-1\prime}W_{j}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}u\right|$
$\displaystyle\leq$ $\displaystyle
d_{\lambda}\left\|\lambda_{{}_{0}}-\widehat{\lambda}\right\|\sup_{\gamma,j}\left|\theta_{0}^{\prime}S^{-1\prime}W_{j}^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}u\right|$
$\displaystyle+d_{\lambda}\left\|\lambda_{{}_{0}}-\widehat{\lambda}\right\|\sup_{\gamma,j}\left|u^{\prime}S^{-1\prime}W_{j}^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}u\right|$
$\displaystyle=$ $\displaystyle
O_{p}\left(\sqrt{d_{\gamma}/n}\right)O_{p}(\sqrt{n})+O_{p}\left(\sqrt{d_{\gamma}/n}\right)O_{p}(p)=O_{p}\left(\sqrt{d_{\gamma}}\right)=o_{p}\left(\sqrt{p}\right),$

and

$\displaystyle\left|\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})y^{\prime}W_{j}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}(\theta_{0}-\widehat{f})\right|$
$\displaystyle\leq$ $\displaystyle
d_{\lambda}\left\|\lambda_{{}_{0}}-\widehat{\lambda}\right\|\left\|y\right\|\sup_{j}\left\|W_{j}\right\|\sup_{\gamma}\left\|\frac{1}{n}\Psi\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right)^{-1}\Psi\right\|\sup_{\gamma}\left\|\Sigma\left(\gamma\right)^{-1}\right\|^{2}\left\|\theta_{0}-\widehat{f}\right\|$
$\displaystyle=$ $\displaystyle
O_{p}\left(\sqrt{d_{\gamma}/n}\right)O_{p}\left(\sqrt{n}\right)O_{p}\left(p^{1/4}\right)=O_{p}\left(\sqrt{d_{\gamma}}p^{1/4}\right)=o_{p}(\sqrt{p}),$

so that $A_{6}=o_{p}(\sqrt{p})$. Finally,

$\displaystyle\left|\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})y^{\prime}W_{j}^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi[\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\Psi]^{-1}\Psi^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}e\right|$
$\displaystyle\leq$ $\displaystyle
d_{\lambda}\left\|\lambda_{{}_{0}}-\widehat{\lambda}\right\|\left\|y\right\|\sup_{j}\left\|W_{j}\right\|\sup_{\gamma}\left\|\frac{1}{n}\Psi\left(\frac{1}{n}\Psi^{\prime}\Sigma\left(\gamma\right)^{-1}\Psi\right)^{-1}\Psi\right\|\sup_{\gamma}\left\|\Sigma\left(\gamma\right)^{-1}\right\|^{2}\left\|e\right\|$
$\displaystyle=$ $\displaystyle
O_{p}\left(\sqrt{d_{\gamma}/n}\right)O_{p}\left(\sqrt{n}\right)O_{p}\left(p^{-\mu}\sqrt{n}\right)=O_{p}\left(\sqrt{d_{\gamma}}p^{-\mu}\sqrt{n}\right)=o_{p}(\sqrt{p}),$

and

$\displaystyle\left|(e+\theta_{0}-\widehat{f})^{\prime}\Sigma\left(\widehat{\gamma}\right)^{-1}\sum_{j=1}^{d_{\lambda}}(\lambda_{j_{0}}-\widehat{\lambda}_{j})W_{j}y\right|$
$\displaystyle\leq$ $\displaystyle
d_{\lambda}\left\|\lambda_{{}_{0}}-\widehat{\lambda}\right\|\left(\left\|e\right\|+\left\|\theta_{0}-\widehat{f}\right\|\right)\sup_{\gamma}\left\|\Sigma\left(\gamma\right)^{-1}\right\|\sup_{j}\left\|W_{j}\right\|\left\|y\right\|$
$\displaystyle=$ $\displaystyle
O_{p}\left(\sqrt{d_{\gamma}/n}\right)O_{p}\left(p^{-\mu}\sqrt{n}+p^{1/4}\right)O_{p}\left(\sqrt{n}\right)=O_{p}\left(\sqrt{d_{\gamma}}p^{-\mu}\sqrt{n}+\sqrt{d_{\gamma}}p^{1/4}\right)=o_{p}(\sqrt{p}),$

implying that $A_{7}=o_{p}(\sqrt{p}).$ ∎

###### Proof of Theorem 5.3.

Omitted as it is similar to the proof of Theorem 4.4. ∎

###### Proof of Proposition 6.1:.

Because the map $\Sigma:\mathcal{T}^{o}\rightarrow\mathcal{M}^{n\times n}$ is
Fréchet-differentiable on $\mathcal{T}^{o}$, it is also Gâteaux-differentiable
and the two derivative maps coincide. Thus by Theorem 1.8 of Ambrosetti and
Prodi (1995),

$\left\|\Sigma(t_{1})-\Sigma(t_{1})\right\|\leq\sup_{t\in\mathcal{T}^{o}}\left\|D\Sigma(t)\right\|_{\mathscr{L}\left(\mathcal{T}^{o},\mathcal{M}^{n\times
n}\right)}\left(\left\|\gamma_{1}-\gamma_{2}\right\|+\sum_{\ell=1}^{d_{\zeta}}\left\|\left(\delta_{\ell
1}-\delta_{\ell 2}\right)^{\prime}\varphi_{\ell}\right\|_{w}\right),$ (A.14)

where

$\displaystyle\sum_{\ell=1}^{d_{\zeta}}\left\|\left(\delta_{\ell
1}-\delta_{\ell 2}\right)^{\prime}\varphi_{\ell}\right\|_{w}$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{d_{\zeta}}\sup_{z\in\mathcal{Z}}\left|\left(\delta_{\ell
1}-\delta_{\ell
2}\right)^{\prime}\varphi_{\ell}\right|\left(1+\left\|z\right\|^{2}\right)^{-w/2}$
$\displaystyle\leq$ $\displaystyle\sum_{\ell=1}^{d_{\zeta}}\left\|\delta_{\ell
1}-\delta_{\ell
2}\right\|\sup_{z\in\mathcal{Z}}\left\|\varphi_{\ell}\right\|\left(1+\left\|z\right\|^{2}\right)^{-w/2}$
$\displaystyle\leq$ $\displaystyle
C\varsigma(r)\sum_{\ell=1}^{d_{\zeta}}\left\|\delta_{\ell 1}-\delta_{\ell
2}\right\|\leq C\varsigma(r)\left\|t_{1}-t_{2}\right\|.$

The claim now follows by (6.8) in Assumption NPN.2, because
$\left\|\gamma_{1}-\gamma_{2}\right\|\leq
C\varsigma(r)\left\|t_{1}-t_{2}\right\|$ for some suitably chosen $C$. ∎

###### Proof of Theorem 6.1.

The proof is omitted as it is entirely analogous to that of Theorem 5.1, with
the exception of one difference when proving equicontinuity. In the setting of
Section 6, we obtain via Proposition 6.1 that
$\left\|\Sigma(\tau)-\Sigma\left(\tau^{*}\right)\right\|=O_{p}\left(\varepsilon\right)$,
the $\varsigma(r)$ factor being omitted because only finitely many
neighborhoods contribute due to compactness of $\mathcal{T}$. ∎

###### Proof of Theorem 6.2.

Writing,
$\delta(z)=\left(\widehat{\delta}_{1}^{\prime}\varphi_{1}(z),\ldots,\widehat{\delta}_{d_{\zeta}}^{\prime}\varphi_{d_{\zeta}}(z)\right)^{\prime}$
and taking
$t_{1}=\left(\widehat{\gamma}^{\prime},\hat{\delta}(z)^{\prime}\right)^{\prime}$
and $t_{2}=\left(\gamma_{0}^{\prime},\zeta_{0}(z)^{\prime}\right)^{\prime}$ in
Proposition 6.1 implies (we suppress the argument $z$)

$\displaystyle\left\|\Sigma\left(\widehat{\tau}\right)-\Sigma\right\|=O_{p}\left(\varsigma(r)\left(\left\|\widehat{\gamma}-\gamma_{0}\right\|+\left\|\widehat{\delta}-\zeta_{0}\right\|\right)\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(\varsigma(r)\left(\left\|\widehat{\tau}-\tau_{0}\right\|+\left\|\nu\right\|\right)\right)$
$\displaystyle=$ $\displaystyle