Non-Identifiability in Network Autoregressions
Federico Martellosio***School of Economics, University of Surrey,
<EMAIL_ADDRESS>

November 20, 2020

Abstract

We study identification in autoregressions defined on a general network. Most
identification conditions that are available for these models either rely on
repeated observations, are only sufficient, or require strong distributional
assumptions. We derive conditions that apply even if only one observation of a
network is available, are necessary and sufficient for identification, and
require weak distributional assumptions. We find that the models are
generically identified even without repeated observations, and analyze the
combinations of the interaction matrix and the regressor matrix for which
identification fails. This is done both in the original model and after
certain transformations in the sample space, the latter case being important
for some fixed effects specifications.

Keywords: fixed effects, invariance, networks, quasi maximum likelihood
estimation.
JEL Classification: C12, C21.

##  1 Introduction

A simple way to model interaction on a general network is to use an
autoregressive process for an outcome variable, usually conditional on
covariates. Models of this type can be traced back at least to Whittle (1954),
and have since proved useful in many applications, across many scientific
fields. In economics, and the social sciences more generally, they are
currently particularly popular in the analysis of peer effects and social
networks. The models are known as simultaneous autoregressions in the
statistics literature (e.g., Cressie, 1993), spatial autoregressions in the
econometrics literature (e.g., LeSage and Pace, 2009), are closely related to
linear-in-means models (e.g., Manski, 1993a), and have important connections
to linear structural equation models (e.g., Drton et al., 2011). To emphasize
their wide applicability, we refer to them as network autoregressions.

This paper is concerned with identifiability of the parameters in a network
autoregression. Not surprisingly, this is a topic that has been considered
several times in the literature, more or less explicitly, and notably in two
very influential papers: Lee (2004) and Bramoullé et al. (2009). Lee (2004)
studies asymptotic properties of the quasi maximum likelihood estimator based
on the Gaussian distribution. His consistency conditions are sufficient for
identification. Bramoullé et al. (2009) analyzes identifiability by looking at
the mapping from the reduced form parameters to the structural parameters, an
approach that has become standard in the social network literature. The
present paper departs from previous studies in two main ways. First, we look
at identifiability directly from the first two moments of the outcome
variable. Compared to the approach via reduced form parameters, identification
from moments does not require repeated observations of the cross section.
Second, we aim to understand what combinations of the interaction matrix $W$
and the regressor matrix $X$ lead to a failure of identification. To this end,
we restrict attention to the case where both $W$ and $X$ are nonstochastic and
known, as in Lee (2004).111It would be possible, alternatively, to study
identifiability conditional on $W$ and/or $X$, under suitable exogeneity
assumptions (see, e.g., Bramoullé et al., 2009; Gupta, 2019), at the cost of
some notational complexity. Allowing for endogeneity of $W$ and/or $X$ would
instead require different methods; see Section 6.

We show that identification from the first moment is generally possible, and
characterize the cases when it is impossible. We focus on one class of such
cases, which is particularly relevant in fixed effects models (for example,
the classical linear-in-means model with group fixed effects belongs to this
class of cases). In this class, non-identifiability from the first moment is
linked to the impossibility of invariant inference; that is, the parameters
cannot be identified from any statistic that is invariant with respect to a
certain group of transformations under which the model itself is invariant.
This fundamental type of non-identifiability occurs despite the fact that the
parameters may be identifiable from the second moment of the outcome variable.

Section 2 sets out the framework. Section 3 studies identifiability from the
first and second moments of the outcome variable. Identifiability after
imposition of invariance is discussed in Section 4, and implications for
likelihood inference in Section 5. Section 6 briefly concludes. The appendices
contain additional material and all proofs.

Notation. Throughout the paper, $\iota_{n}$ denotes the $n\times 1$ vector of
ones, $M_{A}$ denotes the orthogonal projector onto
$\operatorname{col}^{\perp}(A)$ ($M_{A}\coloneqq
I_{n}-A(A^{\prime}A)^{-1}A^{\prime}$ if $A$ has full column rank),
$\mu_{\mathbb{R}^{n}}$ denotes the Lebesgue measure on $\mathbb{R}^{n}$,
“a.s.” stands for almost surely, with respect to $\mu_{\mathbb{R}^{n}}$, and
$A\oplus B$ denotes the direct sum of the matrices $A$ and $B$ (that is, if
$A$ is $n\times m$ and $B$ is $p\times q$, $A\oplus B$ is the $np\times mq$
block diagonal matrix with $A$ as top diagonal block and $B$ as bottom
diagonal block).

##  2 The model

The model of interest is the network autoregression

$y=\lambda Wy+X\beta+\sigma\varepsilon,$ (2.1)

where $y$ is the $n\times 1$ vector of outcomes, $\lambda$ is a scalar
parameter, $W$ is an interaction matrix, $X$ is an $n\times k$ matrix of
regressors with full column rank and with $k\leq n-2$,
$\beta\in\mathbb{R}^{k}$, $\sigma$ is a positive scale parameter, and
$\varepsilon$ is an unobserved zero mean $n\times 1$ random vector. As
mentioned earlier, both $W$ and $X$ are taken to be nonstochastic and known.
The entries of $W$ are supposed to reflect the pairwise interaction between
the observational units; in particular, the $(i,j)$-th entry of $W$ is zero if
unit $j$ is not deemed to be a neighbor of unit $i$. Some of the columns of
$X$ may be spatial lags of some other columns (the spatial lag of a vector $x$
being the vector $Wx$). That is, in the terminology of social networks, we
allow for “contextual effects” or “exogenous spillovers”.

When the index set of $y$ has more than one dimension (e.g., individuals and
time, or individuals and networks), it is often useful to include in the error
term additive unobserved components relative to those dimensions. In that
case, we take a fixed effects approach and treat the unobserved effects as
parameters to be estimated. Accordingly, for inferential purposes, we
incorporate the fixed effects into $\beta$ and the corresponding dummy
variables into $X$. Two examples of fixed effects specifications that can be
nested into the general model (2.1) are given next.

######  Example 1 (Panel data model).

There are $N$ individuals, followed over $T$ time periods. Let $\widetilde{W}$
be an $N\times N$ matrix describing the interaction between individuals, and
$\widetilde{X}$ an $NT\times\tilde{k}$ regressor matrix. The interaction
matrix $\widetilde{W}$ is assumed to be constant over time for simplicity. A
panel data version of the network autoregression (2.1) is given by
$y_{it}=\lambda\sum_{ij}\widetilde{W}_{ij}y_{jt}+\widetilde{x}_{it}^{\prime}\tilde{\beta}+u_{it}$,
where $\widetilde{W}_{ij}$ are the entries of $\widetilde{W}$, and
$\widetilde{x}_{it}^{\prime}$ are the $\tilde{k}\times 1$ rows of
$\widetilde{X}$, for $i=1,\ldots,N$ and $t=1,\ldots,T.$ The error $u_{it}$ is
decomposed into $c_{i}+\sigma\varepsilon_{it}$ (one-way model) or
$c_{i}+\alpha_{t}+\sigma\varepsilon_{it}$ (two-way model), where $c_{i}$ and
$\alpha_{t}$ are, respectively, individual and time fixed effects, and
$\varepsilon_{it}$ is an idiosyncratic error. Following a fixed effects
approach (i.e., treating the random components $c_{i}$ and $\alpha_{t}$ as
parameters to be estimated), the model can be written in the notation of
equation (2.1), with $W=I_{T}\otimes\widetilde{W}$, and, for the two-way
model, $X=(\widetilde{X},\iota_{T}\otimes I_{N},I_{T}\otimes\iota_{N})$ and
$\beta=(\tilde{\beta}^{\prime},c^{\prime},\alpha^{\prime})^{\prime}$, where
$c$ and $\alpha$ are the vectors with entries $c_{i}$ and $\alpha_{t}$,
respectively.222Obviously for identification of $\beta$ one column of the
matrix $(\iota_{T}\otimes I_{N},I_{T}\otimes\iota_{N})$ should be omitted from
$X$, or some normalization should be imposed on the fixed effects, and no
regressor should be constant over time or over individuals. ∎

######  Example 2 (Network fixed effects).

There are $R$ networks, with network $r$ having $m_{r}$ individuals. The model
is

$y_{r}=\lambda
W_{r}y_{r}+\widetilde{X}_{r}\gamma+\alpha_{r}\iota_{m_{r}}+\sigma\varepsilon_{r},\text{
}r=1,\ldots,R,$ (2.2)

where $W_{r}$ is the $m_{r}\times m_{r}$ interaction matrix of network $r$,
$\alpha_{r}$ is a network fixed effect, $\widetilde{X}_{r}$ is an
$m_{r}\times\tilde{k}$ matrix of regressors, and $\gamma$ is a
$\tilde{k}\times 1$ parameter. In the notation of equation (2.1),
$y=(y_{1}^{\prime},\ldots,y_{R}^{\prime})^{\prime}$,
$W=\bigoplus_{r=1}^{R}W_{r}$,
$\beta=(\gamma^{\prime},\alpha^{\prime})^{\prime}$,
$\varepsilon=(\varepsilon_{1}^{\prime},\ldots,\varepsilon_{R}^{\prime})^{\prime}$,
and $X=(\widetilde{X},\bigoplus_{r=1}^{R}\iota_{m_{r}})$, with
$\widetilde{X}\coloneqq(\widetilde{X}_{1}^{\prime},\ldots,\widetilde{X}_{R}^{\prime})^{\prime}$.
∎

We now present an assumption that plays a crucial role throughout the paper.

######  Assumption 1.

There is no real eigenvalue $\omega$ of $W$ for which $M_{X}(\omega
I_{n}-W)=0.$

Assumption 1 is required to rule out some pathological combinations of $W$ and
$X$. More precisely, we shall see in Section 4 that a failure of Assumption 1
implies a particular type of non-identifiability. A condition equivalent to
$M_{X}(\omega I_{n}-W)=0$ is $\operatorname{col}(\omega
I_{n}-W)\subseteq\operatorname{col}(X)$. That is, a pair $(X,W)$ causes
Assumption 1 to fail if and only if $\operatorname{col}(X)$ contains the
subspace $\operatorname{col}(\omega I_{n}-W)$, for some real eigenvalue
$\omega$ of $W$. Also, note that if, for a given $W$, Assumption 1 is violated
for some $X=X_{1}$, then it is also violated for $X=(X_{1},X_{2})$, for any
$X_{2}$ (such that $X$ is full rank). It is helpful to look at two examples in
which Assumption 1 fails (further examples are given in Appendix A).

######  Example 3 (Balanced Group Interaction model).

A particular case of model (2.2), which we refer to as the Group Interaction
model, is when all members of a group interact homogeneously, that is,
$W_{r}=\frac{1}{m_{r}-1}(\iota_{m_{r}}\iota_{m_{r}}^{\prime}-I_{m_{r}})\eqqcolon
B_{m_{r}}$, for $r=1,\ldots,R$. Following Manski (1993b), this specific
structure has played a central role in the literature on peer effects. We say
that the Group Interaction model is balanced if all group sizes $m_{r}$ are
the same. In that case, letting $m$ denote the common group size,
$W=I_{R}\otimes B_{m}$. It is easily verified that, for the matrix
$W=I_{R}\otimes B_{m}$, $\omega_{\min}=-\frac{1}{m-1}$ and
$\operatorname{col}(\omega_{\min}I_{n}-W)=\operatorname{col}(I_{R}\otimes\iota_{m})$.
Since $I_{R}\otimes\iota_{m}$ is the design matrix of the group fixed effects,
it follows that the balanced group interaction model violates Assumption 1
whenever it includes group fixed effects. ∎

######  Example 4 (Complete Bipartite model).

In a complete bipartite graph the $n$ observational units are partitioned into
two groups of sizes $p$ and $q,$ say, with all units within a group
interacting with all in the other group, but with none in their own group. For
$p=1$ or $q=1$ this corresponds to the graph known as a star. The adjacency
matrix of a complete bipartite graph is

$A\coloneqq\mathopen{}\mathclose{{}\left(\begin{array}[c]{cc}0_{pp}&\iota_{p}\iota_{q}^{\prime}\\\
\iota_{q}\iota_{p}^{\prime}&0_{qq}\end{array}}\right).$

The associated row-normalized interaction matrix is333A row-normalized matrix
is obtained by dividing each entry of a matrix by the corresponding row-sum,
and is therefore a row-stochastic matrix.

$W=\mathopen{}\mathclose{{}\left(\begin{array}[c]{cc}0_{pp}&\frac{1}{q}\iota_{p}\iota_{q}^{\prime}\\\
\frac{1}{p}\iota_{q}\iota_{p}^{\prime}&0_{qq}\end{array}}\right).$ (2.3)

Alternatively, $A$ can be rescaled by its largest eigenvalue, yielding the
symmetric interaction matrix

$W=\frac{1}{\sqrt{pq}}A.$ (2.4)

We refer to the network autoregression with interaction matrix (2.3) or (2.4),
as, respectively, the row-normalized Complete Bipartite model and the
symmetric Complete Bipartite model. It is easily verified that, for both (2.3)
and (2.4), $\operatorname{col}(W)$ is spanned by the vectors
$(\iota_{p}^{\prime},0_{q}^{\prime})^{\prime}$ and
$(0_{p}^{\prime},\iota_{q}^{\prime})^{\prime}$. Hence, for both the row-
normalized Complete Bipartite model and the symmetric Complete Bipartite
model, Assumption 1 is violated (for $\omega=0$) if $\operatorname{col}(X)$
contains $(\iota_{p}^{\prime},0_{q}^{\prime})^{\prime}$ and
$(0_{p}^{\prime},\iota_{q}^{\prime})^{\prime}$. This is the case whenever $X$
contains an intercept for each of the two groups, and also in the two
following circumstances: (i) $X$ contains an intercept and a contextual effect
term $Wx$, for some $x\in\mathbb{R}^{n}$;444This is because, when $W$ is the
interaction matrix of a complete bipartite model, $Wx$ is in the span of
$(\iota_{p}^{\prime},0_{q}^{\prime})^{\prime}$ and
$(0_{p}^{\prime},\iota_{q}^{\prime})^{\prime}$, for any $x\in\mathbb{R}^{n}$.
(ii) $X$ contains two contextual effect terms $Wx_{1}$ and $Wx_{2}$, for some
$x_{1},x_{2}\in\mathbb{R}^{n}$. ∎

##  3 Identifiability

This section studies identifiability of $(\lambda,\beta)$ from the first two
moments of $y$. Note that in models containing fixed effects one would often
consider a transformation of $y$ that removes the fixed effects. We do not
discuss, at this stage, identifiability after removal of the fixed effects,
which, depending on the specific model and the specific transformation, may be
a different question—see Section 4. Instead, this section asks the more
primitive question of whether all model parameters, including the fixed
effects, are identifiable.

We shall use the following definitions. Consider an observable random vector
$z\in\mathbb{R}^{n}$ with cumulative distribution function $F(z;\theta)$
depending on a parameter $\theta\in\Theta\subseteq\mathbb{R}^{p}$. A
particular value
$\overset{\sim}{\smash{\theta}\rule{0.0pt}{4.86528pt}}\in\Theta_{I}\subseteq\Theta$
of $\theta$ is said to be identified (from the distribution of $z$) on a set
$\Theta_{I}$ if there is no other
$\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\theta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\in\Theta_{I}$
such that
$F(z;\overset{\sim}{\smash{\theta}\rule{0.0pt}{4.86528pt}})=F(z;\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\theta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})$
for all $z\in\mathbb{R}^{n}$. If all values
$\overset{\sim}{\smash{\theta}\rule{0.0pt}{4.86528pt}}\in\Theta_{I}$ are
identified on $\Theta_{I}$, we say that the parameter $\theta$ is identified
on $\Theta_{I}$. If all values
$\overset{\sim}{\smash{\theta}\rule{0.0pt}{4.86528pt}}\in\Theta_{I}$ except
for those in a $\mu_{\mathbb{R}^{n}}$-null set are identified on $\Theta_{I}$,
we say that the parameter $\theta$ is generically identified on $\Theta_{I}$.
Next, the value
$\overset{\sim}{\smash{\theta}\rule{0.0pt}{4.86528pt}}\in\Theta_{I}$ is said
to be identified from a moment $m(\theta)$ of $z$ on a set $\Theta_{I}$ if
there is no other
$\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\theta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\in\Theta_{I}$
such that
$m(\overset{\sim}{\smash{\theta}\rule{0.0pt}{4.86528pt}})=m(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\theta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})$.
Clearly, identification from a moment of $z$ is sufficient but not necessary
for identification from the distribution of $z$.

###  3.1 Identifiability from first moment

When no distributional assumption other than $\mathrm{E}(\varepsilon)=0$ is
imposed on model (2.1), identification can only occur via the first moment of
$Y$. To explore this case, we need to be clear about the set over which we
wish to identify $\lambda$. Letting $S(\lambda)\coloneqq I_{n}-\lambda W$,
rewrite equation (2.1) as $S(\lambda)y=X\beta+\sigma\varepsilon$. In order for
$y$ to be uniquely determined, given $X$ and $\varepsilon$, it is necessary
that $\det(S(\lambda))\neq 0$, which requires $\lambda\neq\omega^{-1}$, for
any nonzero real eigenvalue $\omega$ of $W$. We refer to the set
$\Lambda_{\mathrm{u}}\coloneqq\\{\lambda\in\mathbb{R}:\det(S(\lambda))\neq
0\\}$ as the unrestricted parameter space for $\lambda$. In practice, the
parameter space for $\lambda$ is usually restricted much further, but, for
now, it is convenient to focus on $\Lambda_{\mathrm{u}}$. Of course, if
$\lambda$ is identified on $\Lambda_{\mathrm{u}}$ it is also identified on any
subset of $\Lambda_{\mathrm{u}}$.

######  Lemma 3.1 (Identifiability from first moment).

In the network autoregression (2.1),

  1. (i)

if $\mathrm{rank}(X,WX)>k$, the parameter $(\lambda,\beta)$ is generically
identified on $\Lambda_{\mathrm{u}}\times\mathbb{R}^{k}$;

  2. (ii)

if $\mathrm{rank}(X,WX)=k$, no value of the parameter $(\lambda,\beta)$ is
identified on $\Lambda_{\mathrm{u}}\times\mathbb{R}^{k}$ from $\mathrm{E}(Y)$.

Lemma 3.1 says that the parameters $\lambda$ and $\beta$ are generically
identified (from the first moment of $y$) if the matrices $X$ and $W$ are such
that $\mathrm{rank}(X,WX)>k$. Conversely, if $\mathrm{rank}(X,WX)=k$,
$\lambda$ and $\beta$ cannot be identified, and hence consistently estimated,
without distributional assumptions beyond $\mathrm{E}(\varepsilon)=0$. For
example, the 2SLS estimators of Kelejian and Prucha (1998) and Lee (2003),
which are based on the specification of the first moment only of $y$, are not
defined if $\mathrm{rank}(X,WX)=k$, because in that case no internal
instruments are available for the endogenous variable $Wy$.

The condition $\mathrm{rank}(X,WX)=k$ is trivially satisfied when $k=0$ (pure
model); otherwise, it is typically very strong. Indeed, for any given $W$, the
set of (full rank) $n\times k$ matrices $X$ such that $\mathrm{rank}(X,WX)=k$
is a $\mu_{\mathbb{R}^{n\times k}}$-null set. Accordingly, Lemma 3.1 could be
stated by saying that identification from the first moment of $y$ is possible
for generic parameter values $(\lambda,\beta)$ and for generic regressor
matrices $X$. Nevertheless, specific combinations of $W$ and $X$ such that
$\mathrm{rank}(X,WX)=k$ may arise in some cases of interest, particularly in
fixed effects models. Some examples worth mentioning where it is easily
verified that $\mathrm{rank}(X,WX)=k$ are as follows:

  1. (a)

Any network autoregression such that Assumption 1 is violated (because
$M_{X}(\omega I_{n}-W)=0$ implies $M_{X}WX=0$, which is equivalent to
$\mathrm{rank}(X,WX)=k$).

  2. (b)

Some network fixed effects models of the type in Example 2:555In case (b.i),
Assumption 1 is satisfied for generic matrices
$\widetilde{X}_{1},\ldots,\widetilde{X}_{r}$ if the model is unbalanced, and
is violated if the model is balanced (see Example 3). In cases (b.ii) and
(b.iii), Assumption 1 is satisfied for generic $\widetilde{X}$.

    1. (b.i)

A Group Interaction model with group specific slope coefficients, group fixed
effects, and with at least two groups ($R>1$). In this model,
$X=\bigoplus_{r=1}^{R}(\widetilde{X}_{r},\iota_{m_{r}})$, where the matrix
$\widetilde{X}_{r}$ of regressors is $m_{r}\times k_{r}$, with $0\leq
k_{r}<m_{r}$, so that $k=R+\sum_{r=1}^{R}k_{r}$.

    2. (b.ii)

A Balanced Group Interaction model with contextual effects, and with at least
two groups ($R>1$). In this model, $X=(\widetilde{X},W\widetilde{X})$ for some
$n\times\tilde{k}$ matrix $\widetilde{X}$ of regressors, so that
$k=2\tilde{k}$.666The condition $\mathrm{rank}(X,WX)=k$ is also satisfied if,
when $\tilde{k}=1$, an intercept is added to $X$, i.e.,
$X=(\iota_{n},\widetilde{X},W\widetilde{X})$. When $\tilde{k}>1$,
$\iota_{n}\in$ $\operatorname{col}(\widetilde{X},W\widetilde{X})$, and
therefore an intercept cannot be added to $(\widetilde{X},W\widetilde{X})$
(one could, of course, replace one of the columns of
$(\widetilde{X},W\widetilde{X})$ with an intercept, and this would still give
$\mathrm{rank}(X,WX)=k$).

    3. (b.iii)

The network fixed effects model (2.2) with each $W_{r}$ being the symmetric or
row-normalized adjacency matrix of a complete bipartite graph, with contextual
effects, and with at least two groups ($R>1$). In this model,
$X=(\widetilde{X},W\widetilde{X},\bigoplus_{r=1}^{R}\iota_{m_{r}})$ for some
$n\times\tilde{k}$ matrix $\widetilde{X}$ of regressors, with $\tilde{k}\geq
0$, so that $k=R+2\tilde{k}$.

  3. (c)

Some models with fixed effects and no regressors (i.e., $X$ contains only the
dummies corresponding to the fixed effects):

    1. (c.i)

The one-way model of Example 1 with no regressors (i.e., $X=\iota_{T}\otimes
I_{N}$), as, for instance, in Robinson and Rossi (2015).

    2. (c.ii)

The two-way model of Example 1 with no regressors (i.e., $X=(\iota_{T}\otimes
I_{N},I_{T}\otimes\iota_{N})$) and row-stochastic $\widetilde{W}$ (a matrix is
said to be row-stochastic if all its row sums are 1).

    3. (c.iii)

The network fixed effects model (2.2) with no regressors (i.e.,
$X=\bigoplus_{r=1}^{R}\iota_{m_{r}}$) and all matrices $W_{r}$’s being row-
stochastic. Note that, when $R=1$, this reduces to an intercept-only network
autoregression (2.1) with row-stochastic interaction matrix.

In cases such as those just listed, $\mathrm{rank}(X,WX)=k$ and therefore
$\lambda$ and $\beta$ cannot be identified from $\mathrm{E}(Y)$. As noted
earlier, however, the condition $\mathrm{rank}(X,WX)=k$ is very strong in
general. What might be more relevant in applications is that the condition is
close, in some sense, to being satisfied. In such a situation, it is natural
to expect that identification from the first moment will be weak. We confirm
this with a small simulation experiment. We generate $10,000$ replications
from model (2.1) with $W$ a row-normalized 2-ahead 2-behind interaction matrix
(before row-standardization, this is a matrix with all entries in the two
diagonals above and the two diagonals below the main diagonal equal to one,
and zero everywhere else), and a single regressor equal to $\iota_{n}+bz$,
where $b\in\mathbb{R}$ and $z\thicksim\mathrm{N}(0,I_{n})$, with $z$ being
generated once and then kept fixed across replications. We set $\beta=1$,
$\sigma=1$, and draw the errors independently from either a standard normal
distribution or a gamma distribution with shape parameter 1 and scale
parameter 1, demeaned by the population mean. Mean, variance, skewness, and
kurtosis are 0, 1, 0, and 3 for the former distribution and 0, 1, 2, and 9 for
the latter. Note that $\lambda$ and $\beta$ cannot be identified from the
first moment if $b=0$, because in that case $\mathrm{rank}(X,WX)=k=1$. Thus,
we expect any estimator of $\lambda$ and $\beta$ that relies entirely on the
specification of the first moment of $y$ to perform poorly if $b$ is close to
$0$. For illustration, we consider the 2SLS estimator with instruments $WX$
and $W^{2}X$ for $Wy$ (Kelejian and Prucha, 1998), and we compare it with the
quasi maximum likelihood estimator (QMLE), which also uses the second moment
(the QMLE is the MLE that maximizes the likelihood obtained when
$\varepsilon\thicksim\mathrm{N}(0,I_{n})$; see Section 5). Table 1 displays
the root median square error of the 2SLS and (Q)ML estimators of $\lambda$ and
$\beta$. The root median square error is reported rather than the more usual
root mean square error because, in the setting we are considering, the
variance of the 2SLS estimator does not exist (see Roberts, 1995, Section
7.2.2). For both $\lambda$ and $\beta$, and for both the normal and the gamma
distributions, the performance of the 2SLS estimator is good, compared to the
(Q)MLE benchmark, when $b=1$, but deteriorates rapidly as $b$ gets smaller.
Such a deterioration is due to both the bias and the dispersion of the 2SLS
estimator growing large as $b$ decreases, for any $n$.

Table 1: Root median square error of the 2SLS and (Q)ML estimator of $\lambda$
and $\beta$.

$\begin{array}[c]{llllm{1cm}llcllcllc}\hline\cr&&&\lx@intercol\text{Normal}\hfil\lx@intercol&&&&&\lx@intercol\text{Gamma}\hfil\lx@intercol&&&\\\
\cline{4-8}\cr\cline{10-14}\cr&&&\lambda&&&\beta&&&\lambda&&&\beta\bigstrut[t]&\\\
\cline{4-5}\cr\cline{7-8}\cr\cline{10-11}\cr\cline{13-14}\cr
n&b&&\text{2SLS}&\text{MLE}&&\text{2SLS}&\text{MLE}&&\text{2SLS}&\text{QMLE}&&\text{2SLS}&\text{QMLE}\bigstrut[t]\\\
\hline\cr
100&1&&0.080&0.061&&0.067&0.059&&0.081&0.062&&0.066&0.058\bigstrut[t]\\\
&0.1&&0.598&0.095&&0.593&0.116&&0.593&0.095&&0.582&0.114\\\
&0.01&&1.658&0.096&&1.585&0.118&&1.618&0.096&&1.598&0.115\\\
1000&1&&0.024&0.019&&0.020&0.018&&0.024&0.019&&0.020&0.018\\\
&0.1&&0.189&0.030&&0.188&0.036&&0.190&0.030&&0.189&0.036\\\
&0.01&&1.194&0.031&&1.191&0.037&&1.189&0.030&&1.189&0.037\\\
\hline\cr\end{array}$

In the simulation experiment, $b$ can be interpreted as a measure of the
distance from non-identifiability via the first moment. In more complex
situations, one could construct a measure of distance by observing that, since
$\mathrm{rank}(X)=k$, $\mathrm{rank}(X,WX)=k$ is equivalent to
$\operatorname{col}(WX)\subseteq\operatorname{col}(X)$ (i.e., in matrix
theoretic language, to $\operatorname{col}(X)$ being an invariant subspace of
$W$) or, which is the same, to $M_{X}WX=0$. A distance from the condition
$\mathrm{rank}(X,WX)=k$ could then be provided by some norm of the matrix
$M_{X}WX$. We do not intend to study this rigorously here, but such a measure
should help model users to avoid not only the cases in which inference based
on the first moment is impossible (the norm of $M_{X}WX$ is zero), but also
the cases close to these (the norm of $M_{X}WX$ is close to zero), in which
inference is likely to be very challenging without additional distributional
assumptions.

###  3.2 Connections to the literature

It is useful to briefly compare Lemma 3.1 with some related results available
in the literature, obtained by two different approaches. First, Lee (2004)
studies asymptotic properties of the quasi maximum likelihood estimator based
on the Gaussian distribution. The condition $\mathrm{rank}(X,WX)>k$ appearing
in Lemma 3.1 can be interpreted as the finite sample equivalent of Assumption
8 in Lee (2004). Indeed, under the latter assumption (and other regularity
assumptions) the limit of the Gaussian quasi-likelihood has a unique maximum
at the true value of the parameters, which is sufficient (and necessary under
correct specification of the likelihood) for identification; see Newey and
McFadden (1994). Second, in the social network literature, identification of
the structural parameters in model (2.1) is typically established by checking
that those parameters can be uniquely recovered from the reduced form
parameters (e.g., Bramoullé et al., 2009; Blume et al., 2011; Kwok, 2019).
Such a strategy obviously relies on the reduced form parameters being
identified, which, in the case of a fixed $W$, would typically require
repeated observations of the cross-section, over time or some other dimension.
Because of this, identification via reduced form parameters may not be
appropriate in applications where a single observation of a network is
available. Lemma 3.1 can establish identifiability not only when repeated
observations are available (in which case $W$ is block diagonal with identical
blocks, as in Example 1), but also when a single observation of the network is
available. The following example shows that it is possible that parameters are
identified with repeated observations, but not with a single observation.

######  Example 5.

Consider a row-normalized or symmetric Complete Bipartite model with
$X=\mathopen{}\mathclose{{}\left(\iota_{n},x,Wx}\right)$, for some
$x\in\mathbb{R}^{n}$ (such that $X$ is full rank). Since the matrices $I_{n}$,
$W$, $W^{2}$ are linearly independent, Proposition 1 in Bramoullé et al.
(2009) implies that $\lambda$ and $\beta$ are identified from an i.i.d. sample
of observations from the model. However, as noted in Example 4, Assumption 1
fails, and therefore $\mathrm{rank}(X,WX)=k$. Thus, according to Lemma 3.1,
$\lambda$ and $\beta$ cannot be identified from a single observation of the
model, whatever the value of $x$. ∎

The applicability to the case of a single observation of a network is the most
important difference between Lemma 3.1 and the approach in Bramoullé et al.
(2009). With repeated observations, Lemma 3.1 yields results that are similar
to those in Bramoullé et al. (2009),777Indeed, identification under repeated
observations for the Complete Bipartite model, which is established via
Proposition 1 in Bramoullé et al. (2009) in Example 5, can also be established
by Lemma 3.1. To see this, note that $R$ observations of the row-normalized or
symmetric Complete Bipartite model with
$X=\mathopen{}\mathclose{{}\left(\iota_{n},x,Wx}\right)$ correspond to a
network autoregression with interaction matrix $W^{\ast}=I_{R}\otimes W$ and
regressor matrix
$X^{\ast}=\mathopen{}\mathclose{{}\left(\iota_{nR},x^{\ast},W^{\ast}x^{\ast}}\right)$
for some $x^{\ast}\in\mathbb{R}^{nR}$. Then one can see that
$\mathrm{rank}(X^{\ast},W^{\ast}X^{\ast})>k$ if and only if $R>1$. That is,
Lemma 3.1 establishes that identification is achieved if and only if $R>1$.
but with two less important differences. Firstly, Lemma 3.1 does not restrict
attention to the case when $X$ contains contextual effects; our results can be
used for that case, but also for the case when no contextual effects are
included, or only some contextual effects are included. Secondly, Bramoullé et
al. (2009) assume that $X$ is random with $\mathrm{E}(\varepsilon|X)=0$,
whereas, for the reasons mentioned in the Introduction, $X$ is nonrandom in
Lemma 3.1.

###  3.3 Identifiability from second moment

So far, we have considered identifiability from the first moment of $y$, under
the restriction $\mathrm{E}(\varepsilon)=0$. When identification from the
first moment fails, identification may be achieved by imposing further
restrictions on the model. The simplest of such restrictions is
$\mathrm{var}(\varepsilon)=I_{n}$, in which case identification can occur via
the second moment of $y$.888If $X$ and $W$ were random, the restriction would
be imposed on $\mathrm{var}(\varepsilon|W,X)$, rather than on
$\mathrm{var}(\varepsilon)$. To see this, it is convenient to focus on a
parameter space for $\lambda$ that is smaller than $\Lambda_{\mathrm{u}}$.
Consider the case when $W$ has at least one (real) negative eigenvalue and at
least one (real) positive eigenvalue.999While not needed for Lemma 3.1, this
restriction rules out the case when $W$ is a scalar multiple of $I_{n}$, which
trivially leads to non-identification in Lemma 3.1. This is typically
satisfied in both applications and theoretical studies. Denote the smallest
real eigenvalue of $W$ by $\omega_{\min}$, and, without loss of generality,
normalize the largest real eigenvalue to 1. The parameter space for $\lambda$
is often restricted to the largest interval containing the origin in which
$S(\lambda)$ is nonsingular, that is,

$\Lambda\coloneqq(\omega_{\min}^{-1},1),$

or a subset thereof (possibly independent of $n$) such as $(-1,1)$. Without
such restrictions, the models are believed to be too erratic to be useful in
practice, and $\lambda$ is difficult to interpret.

######  Lemma 3.2 (Identifiability from second moment).

Consider a network autoregression (2.1) with
$\mathrm{var}(\varepsilon)=I_{n}$, and assume that $W$ has at least one
negative eigenvalue and at least one positive eigenvalue. The parameter
$(\lambda,\sigma^{2})$ is identified on $\Lambda\times(0,\infty)$.

Of course, once $\lambda\ $is identified, $\beta$ can be identified from the
first moment $\mathrm{E}(y)=(I_{n}-\lambda W)^{-1}X\beta$, for any $W$ and any
(full rank)$\ X$. Lemma 3.2 complements two results available in the
literature that are concerned with identifiability from $\mathrm{var}(y)$ on a
different parameter space for $\lambda$. Firstly, Lemma 3.2 is an extension of
Lemma 4.2 in Preinerstorfer and Pötscher (2017), which establishes
identification of $(\lambda,\sigma^{2})$ on $(0,1)\times(0,\infty)$. Secondly,
Lemma 4 in Lee and Yu (2016) says that a sufficient condition for
$(\lambda,\sigma^{2})$ to be identified from $\mathrm{var}(y)$ on
$\Lambda_{\mathrm{u}}\times(0,\infty)$ is that the matrices $I_{n}$,
$W+W^{\prime}$ and $W^{\prime}W$ are linearly independent.101010See also
Theorem 3.2 in Davezies et al. (2009). Conditions for $(\lambda,\sigma^{2})$
to be identified from $\mathrm{var}(y)$ can be seen as finite sample
counterparts of Assumption 9 in Lee (2004) (cf. Section 3.2). The following
example considers a case when identification cannot be established by Lemma 4
in Lee and Yu (2016), but can be by Lemma 3.2.

######  Example 6.

Consider a balanced group interaction model (see Example 3) with
$\mathrm{var}(\varepsilon)=I_{n}$. According to Lemma 4 in Lee and Yu (2016),
$(\lambda,\sigma^{2})$ is not identified from $\mathrm{var}(y)$ on
$\Lambda_{\mathrm{u}}\times(0,\infty)$, because the matrices $I_{n}$,
$W+W^{\prime}$ and $W^{\prime}W$ are linearly dependent when $W=I_{R}\otimes
B_{m}$.111111More precisely, for the variance matrix
$\sigma^{2}\mathopen{}\mathclose{{}\left(S^{\prime}(\lambda)S(\lambda)}\right)^{-1}$
of the balanced group interaction model we have
$\sigma_{1}^{2}\mathopen{}\mathclose{{}\left(S^{\prime}(\lambda_{1})S(\lambda_{1})}\right)^{-1}=\sigma_{2}^{2}\mathopen{}\mathclose{{}\left(S^{\prime}(\lambda_{2})S(\lambda_{2})}\right)^{-1}$
if and only if $\sigma_{2}^{2}=m^{2}\sigma_{1}^{2}/(2\lambda_{1}+m-2)^{2}$ and
$\lambda_{2}=-((m-2)\lambda_{1}+2(1-m))/(2\lambda_{1}+m-2)$. Note that
$\lambda_{2}\notin\Lambda$ if $\lambda_{1}\in\Lambda$. However, Lemma 3.2
asserts that $(\lambda,\sigma^{2})$ is identified (from $\mathrm{var}(y)$) on
$\Lambda\times(0,\infty)$ (and hence on any subset thereof). ∎

It should be noted that the restriction $\mathrm{var}(\varepsilon)=I_{n}$ is
imposed only for simplicity, and is by no means crucial for identification
from $\mathrm{var}(y)$. Indeed, one could assume some parametric structure for
$\mathrm{var}(\varepsilon)$, say $\mathrm{var}(\varepsilon)=\Sigma(\eta)$, and
study identifiability of the parameter $(\lambda,\sigma^{2},\eta)$ from
$\mathrm{var}(y)=\sigma^{2}(I_{n}-\lambda W)^{-1}\Sigma(\eta)(I_{n}-\lambda
W^{\prime})^{-1}$, but we refrain from doing this here.

At this point, it is worth considering the network (or spatial) error model

$y=X\beta+u,\text{ }u=\lambda Wu+\sigma\varepsilon,$ (3.1)

even though this specification is considerably less popular than model (2.1)
in economic applications. The same set of assumptions as in the paragraph
after equation (2.1) will be maintained for model (3.1). Lemma 3.2 also
applies to the network error model, because equations (2.1) and (3.1) imply
the same variance structure for $y$. On the other hand, in the network error
mode $\lambda$ cannot obviously be identified from the first moment $X\beta$
of $y$. In fact, the result in Lemma 3.1 can be interpreted as saying that
$\lambda$ and $\beta$ cannot be identified from $\mathrm{E}(y)$ in a network
autoregression that behaves like a network error model. This point is made
precise by the following argument. If $\mathrm{rank}(X,WX)=k$, there exists a
unique $k\times k$ matrix $A$ such that $WX=XA$, and hence
$S^{-1}(\lambda)X=X(I_{k}-\lambda A)^{-1},$ for any $\lambda$ such that
$S(\lambda)$ is invertible.121212It is easily seen that the eigenvalues of $A$
are eigenvalues of $W$. Hence, $I_{k}-\lambda A$ is invertible if $S(\lambda)$
is. It follows that, when $\mathrm{rank}(X,WX)=k$, the network autoregression
$y=S^{-1}(\lambda)X\beta+\sigma S^{-1}(\lambda)\varepsilon$ can be written as
$y=X(I_{k}-\lambda A)^{-1}\beta+\sigma S^{-1}(\lambda)\varepsilon$, which is a
network error model with regression coefficients $(I_{k}-\lambda
A)^{-1}\beta$.131313According to Lemma C.1 in Appendix C, the model
$y=X(I_{k}-\lambda A)^{-1}\beta+\sigma S^{-1}(\lambda)\varepsilon$ has the
same profile quasi log-likelihood $l(\lambda,\sigma^{2})$ as model (3.1), even
though, clearly, the MLE of $\beta$ will be different in the two models.

##  4 Invariance

This section discusses the full identifiability content of Assumption 1. We
already know from Section 3.1 that, in the network autoregression, a failure
of Assumption 1 precludes identification from the first moment of $y$, but not
from the higher order moments of $y$. We are now going to show that Assumption
1 is necessary for identification from statistics that are invariant under
certain transformations. Similar results to those in this section are obtained
in Preinerstorfer and Pötscher (2017) for a general regression model with
correlated errors and for the particular case of a network error model with
arbitrary $W$. We will need some notions of group invariance (e.g., Lehmann
and Romano, 2005, Chapter 6). Let $\mathcal{G}$ be a group of transformations
from the sample space into itself. A statistic is said to be invariant under
$\mathcal{G}$ (or $\mathcal{G}$-invariant) if it is constant on the orbits of
$\mathcal{G}$. It is said to be a maximal invariant under $\mathcal{G}$ if it
is invariant and takes different values on each orbit. A necessary and
sufficient condition for a statistic to be invariant under $\mathcal{G}$ is
that it depends on the data only through a maximal invariant under
$\mathcal{G}$. Lastly, a family of distributions
$\mathopen{}\mathclose{{}\left\\{P_{\theta},\theta\in\Theta}\right\\}$, where
$\Theta$ is the parameter space is said to be invariant under $\mathcal{G}$ if
every pair $g\in\mathcal{G}$, $\theta\in\Theta$ determine a unique element in
$\Theta$ denoted by $\bar{g}\theta$, such that when $y$ has distribution
$P_{\theta}$, $gy$ has distribution $P_{\bar{g}\theta}$.

In order to apply the theory of invariance, in this section the network
autoregression (2.1) and the network error model (3.1) are regarded as
families of distributions
$\mathopen{}\mathclose{{}\left\\{P_{\theta},\theta\in\Theta}\right\\}$ for
$y$, where $\theta\coloneqq(\lambda,\beta,\sigma^{2},\eta)$, with $\eta$ being
a parameter indexing the distribution of $\varepsilon$, and $\theta$ is
assumed to be identified (from the distribution of $y$). For a given regressor
matrix $X,$ we will consider the group
$\mathcal{G}_{X}\coloneqq\\{g_{\kappa,\delta}:\kappa>0,\delta\in\mathbb{R}^{k}\\}$,
where $g_{\kappa,\delta}$ denotes the transformation $y\rightarrow\kappa
y+X\delta$, and its subgroup
$\mathcal{G}_{X}^{1}\coloneqq\\{g_{1,\delta}:\delta\in\mathbb{R}^{k}\\}$. A
maximal invariant under $\mathcal{G}_{X}^{1}$ is $C_{X}y$, where $C_{X}$ is an
$\mathopen{}\mathclose{{}\left(n-k}\right)\times n$ matrix such that
$C_{X}C_{X}^{\prime}=I_{n-k}$ and $C_{X}^{\prime}C_{X}=M_{X}$, and a maximal
invariant under $\mathcal{G}_{X}$ is $v\coloneqq
C_{X}y/\mathopen{}\mathclose{{}\left\|C_{X}y}\right\|$ (with the convention
that $v=0$ if $C_{X}y=0$). We also say that an expectation $\mu_{y}(\theta)$
is $\mathcal{G}$-invariant if every pair $g\in\mathcal{G}$, $\theta\in\Theta$
determine a unique $\bar{g}\theta$ such that
$\mu_{gy}(\theta)=\mu_{y}(\bar{g}\theta)$. The non-identifiability result in
Lemma 3.1(ii) can be seen as a consequence of the fact that, when
$\mathrm{rank}(X,WX)=k$, the mean $\mu_{y}(\lambda,\beta)\coloneqq
S^{-1}(\lambda)X\beta$ is $\mathcal{G}_{X}^{1}$-invariant.141414If
$\mathrm{rank}(X,WX)=k$, there exists a unique $k\times k$ matrix $A$ such
that $WX=XA$, and hence $S^{-1}(\lambda)X=X(I_{k}-\lambda A)^{-1},$ for any
$\lambda$ such that $S(\lambda)$ is invertible (note that $I_{k}-\lambda A$ is
invertible if $S(\lambda)$ is, because the eigenvalues of $A$ must be
eigenvalues of $W$). Hence
$\mu_{g_{1,\delta}y}(\lambda,\beta)=\mu_{y}(\bar{g}\mathopen{}\mathclose{{}\left(\lambda,\beta}\right))$,
with
$\bar{g}\mathopen{}\mathclose{{}\left(\lambda,\beta}\right)=(I_{k}-\lambda
A)^{-1}\beta+\delta.$ This type of invariance implies that, when
$\mathrm{rank}(X,WX)=k$, the mean can only identify a $k$-dimensional
parameter, not the $(k+1)$-dimensional parameter $(\lambda,\beta)$. Under the
same rank restriction and with an additional assumption that we now state, the
network autoregression (not its mean only) is invariant under
$\mathcal{G}_{X}^{1}$, in fact under $\mathcal{G}_{X}$.

######  Assumption 2.

The distribution of $\varepsilon$ does not depend on the parameters $\lambda$,
$\beta$, and $\sigma^{2}$.

Let $P_{\theta}$ denote the distribution for $y$ in the network error model,
with $\theta\coloneqq(\lambda,\beta,\sigma^{2},\eta)$. Under Assumption 2, the
network error model is $\mathcal{G}_{X}$-invariant, because $gy$ has
distribution $P_{\bar{g}\theta},$ with
$\bar{g}\theta=(\lambda,\kappa\beta+\delta,\kappa^{2}\sigma^{2},\eta)$, for
any $g\in\mathcal{G}_{X}$. For the network autoregression we have the
following result.

######  Lemma 4.1.

Suppose Assumption 2 holds. The network autoregression (2.1) is
$\mathcal{G}_{X}$-invariant if and only if $\mathrm{rank}(X,WX)=k$.

We are now in a position to discuss the implications of Assumption 1. The
“principle of invariance” asserts that inference in a model should be
invariant under any group of transformations under which the model is
invariant. Accordingly, under Assumption 2, inference in a network error model
should be based on $\mathcal{G}_{X}$-invariant procedures, whatever $X$ and
$W$ are, and inference in a network autoregression should be based on
$\mathcal{G}_{X}$-invariant procedures whenever $\mathrm{rank}(X,WX)=k$,
which, as we have seen in Section 3.1, is the case if Assumption 1 is
violated. However, the imposition of $\mathcal{G}_{X}$-invariance causes an
identifiability issue when Assumption 1 fails. To see this, observe that if
Assumption 1 fails then $C_{X}S(\lambda)=(1-\lambda\omega)C_{X}$, and
therefore premultiplying both sides of the network autoregression equation
$S(\lambda)y=X\beta+\sigma\varepsilon$ by $C_{X}$ yields

$C_{X}y=\frac{\sigma}{1-\lambda\omega}C_{X}\varepsilon.$ (4.1)

Equation (4.1) shows that, when Assumption 2 is satisfied but Assumption 1 is
not, $(\lambda,\beta,\sigma^{2})$ cannot be identified from the distribution
of $C_{X}y$ and hence, since $C_{X}y$ is a maximal invariant under
$\mathcal{G}_{X}^{1}$, cannot be identified from the distribution of any
$\mathcal{G}_{X}^{1}$-invariant statistic. Exactly the same conclusion obtains
starting from the network error model $y=X\beta+\sigma
S^{-1}(\lambda)\varepsilon$. The result is particularly perverse for the
network autoregression: when Assumption 1 fails, and under Assumption 2, the
model is $\mathcal{G}_{X}^{1}$-invariant, and yet its parameters cannot be
identified from any $\mathcal{G}_{X}^{1}$-invariant statistic.

It is possible to be more precise about the cause of non-identification.
Suppose Assumption 1 is violated for some eigenvalue $\omega$ of $W$, and let
$g_{\omega}$ be the geometric multiplicity of $\omega$.151515Note that, for
fixed $W$ and $X$, the condition $M_{X}(\omega I_{n}-W)=0$ that leads to a
violation of Assumption 1 can be satisfied at most by one eigenvalue $\omega$.
This is because $M_{X}(\omega_{1}I_{n}-W)=M_{X}(\omega_{2}I_{n}-W)$ implies
$\omega_{1}=\omega_{2}$. Also, note that $M_{X}(\omega I_{n}-W)=0$ implies
that $\omega$ is real. Recall from Section 2 that a pair $(X,W)$ causes
Assumption 1 to fail if and only if some of the columns of $X$ span the
subspace $\operatorname{col}(\omega I_{n}-W)$. Observe that this requires
$k\geq n-g_{\omega}$, because the dimension of $\operatorname{col}(\omega
I_{n}-W)$ is $\mathrm{rank}(\omega I_{n}-W)=n-\mathrm{nullity}(\omega
I_{n}-W)=n-g_{\omega}$. Let $X_{\omega}$ be the $n\times(n-g_{\omega})$ matrix
containing the columns of $X$ that span $\operatorname{col}(\omega I_{n}-W)$,
and reorder the columns of $X$ as in $X=(X_{\omega},X^{\ast})$, where
$X^{\ast}$ is $n\times(k-(n-g_{\omega}))$, with $k-(n-g_{\omega})\geq 0$.
Generalizing the argument leading to equation (4.1), if Assumption 1 fails
then $C_{X_{\omega}}S(\lambda)=(1-\lambda\omega)C_{X_{\omega}}$, and therefore

$C_{X_{\omega}}y=\frac{1}{1-\lambda\omega}C_{X_{\omega}}X^{\ast}\beta^{\ast}+\frac{\sigma}{1-\lambda\omega}C_{X_{\omega}}\varepsilon,$
(4.2)

where $\beta^{\ast}$ is the component of $\beta$ corresponding to $X^{\ast}$.
This shows that, under Assumption 2, $(\lambda,\beta,\sigma^{2})$ cannot be
identified from the distribution of $C_{X_{\omega}}y$ if Assumption 1 fails.
That is, what really causes non-identification when Assumption 1 fails is the
imposition of invariance with respect to the subgroup
$\mathcal{G}_{X_{\omega}}^{1}$ of $\mathcal{G}_{X}$, and what we said above
about $\mathcal{G}_{X}^{1}$-invariant statistics applies to the (larger) set
of $\mathcal{G}_{X_{\omega}}^{1}$-invariant statistics. We summarize this
result in the following theorem, and then provide an example.

######  Theorem 1.

Suppose that, in the network autoregression (2.1) or in the network error
model (3.1), Assumption 2 is satisfied, but Assumption 1 fails for some
eigenvalue $\omega$ of $W$. Then $(\lambda,\beta,\sigma^{2})$ cannot be
identified from the distribution of any
$\mathcal{G}_{X_{\omega}}^{1}$-invariant statistic.

Theorem 1 says that, for any $W$, there are matrices of regressors that make
invariant inference impossible—these are the matrices leading to a violation
of Assumption 1, that is, the matrices whose column space contains one of the
subspaces $\operatorname{col}(\omega I_{n}-W)$, where $\omega$ is an
eigenvalue of $W$. It is worth emphasizing that this result does not require
any distributional assumption other than Assumption 2.

######  Example 7.

Consider a balanced group interaction model with group fixed effects. We have
seen in Example 3 that in this model Assumption 1 fails, because the columns
of the fixed effects matrix $I_{R}\otimes\iota_{m}$ span
$\operatorname{col}(\omega_{\min}I_{n}-W)$ (i.e., in the notation introduced
just before equation (4.2), $X_{\omega_{\min}}=I_{R}\otimes\iota_{m}$).
Theorem 1 therefore implies that, under Assumption 2,
$(\lambda,\beta,\sigma^{2})$ cannot be identified from any statistic that is
invariant under $\mathcal{G}_{I_{R}\otimes\iota_{m}}^{1}$, even though the
model is invariant under that group. ∎

Since $C_{X}X=0$, imposing invariance with respect to
$\mathcal{G}_{I_{R}\otimes\iota_{m}}^{1}$ removes the group fixed
effects.161616For the use of invariance arguments to solve incidental
parameter problems, see also Chamberlain and Moreira (2009). Thus, Example 7
can be seen as a revisitation of the well-known identification failure that
occurs in a balanced group interaction model upon removal of the group fixed
effects (Lee, 2007).

To conclude this section, it is useful to make a connection with the results
obtained in Section 3. Recall that the parameters of a network autoregression
can generally be identified by specifying the variance structure of
$\varepsilon$, regardless of whether Assumption 1 holds; for example, this is
certainly the case if $\mathrm{var}(\varepsilon)=I_{n}$, by Lemma 3.2.
According to Theorem 1, however, any result establishing identification from
the distribution of $y$ cannot be helpful for invariant inference if
Assumption 1 is not satisfied, because in that case identification is lost
after imposition of invariance with respect to the group
$\mathcal{G}_{X_{\omega}}^{1}$.171717Consider the model in Example 7. Due to
the failure of Assumption 1, any result establishing identification from the
distribution of $y$ cannot help to achieve reasonable inference in that model.
This is so, for example, for Proposition 2 in de Paula (2017), which
establishes identification from the variance of $y$ for the particular case
$R=1$, when $\mathopen{}\mathclose{{}\left|\lambda}\right|<1$. Inference based
on such a result cannot respect the invariance properties of the model,
because the model is invariant under the group $\mathcal{G}_{\iota_{n}}^{1}$
of transformations $y\rightarrow y+\alpha\iota_{n}$, $\alpha\in\mathbb{R}$,
but identification is lost on imposition of the group
$\mathcal{G}_{\iota_{n}}^{1}$. The next section considers this point from a
likelihood perspective.

##  5 Likelihood

We now study the consequences of Theorem 1 for likelihood estimation of the
network autoregression. The MLE that is typically used for a network
autoregression is the one based on the likelihood that would obtain if
$\varepsilon$ were distributed as $\mathrm{N}(0,I_{n})$, often referred simply
as the QMLE (quasi MLE); see, e.g., Lee (2004). It will also be useful to
consider the adjusted QMLE, which is obtained from the QMLE by centering the
profile score for $(\lambda,\sigma^{2})$ (see Yu et al., 2015). For estimation
of $(\lambda,\sigma^{2})$, the adjusted QMLE usually performs better than the
QMLE when the dimension of $\beta$ is large with respect to the sample size
$n$ (including in fixed effects models, in which case the dimension of $\beta$
is increasing with $n$).

Let $l(\lambda,\beta,\sigma^{2})$ denote the Gaussian quasi log-likelihood for
$(\lambda,\beta,\sigma^{2})$ in a network autoregression or in a network error
model, $l(\lambda)$ the corresponding profile likelihood for $\lambda,$ and
$l_{\mathrm{a}}(\lambda)$ the adjusted profile likelihood for $\lambda$. The
precise definitions of these likelihoods are given in Appendix B. We say that
a parameter $\theta$ is identified on a set $\Theta_{I}$ from a quasi
likelihood $L(\theta)$ if it is identified on $\Theta_{I}$ from the
distribution underlying $L(\theta)$ (so, if $\theta$ is identified on
$\Theta_{I}$ from the quasi likelihood $L(\theta)$, then
$L(\overset{\sim}{\smash{\theta}\rule{0.0pt}{4.86528pt}})=L(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\theta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})$
for almost all $y\in\mathbb{R}^{n}$ implies
$\overset{\sim}{\smash{\theta}\rule{0.0pt}{4.86528pt}}=\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\theta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}$,
for any
$\overset{\sim}{\smash{\theta}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\theta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\in\Theta_{I}$).
Clearly, Lemma 3.2 is sufficient to guarantee identification of
$(\lambda,\beta,\sigma^{2})$ on $\Lambda\times\mathbb{R}^{k}\times(0,\infty)$
from $l(\lambda,\beta,\sigma^{2})$ for any pair $X,W$, including those pairs
such that Assumption 1 is violated. However, a violation of Assumption 1 makes
inference based on $l(\lambda,\beta,\sigma^{2})$ pointless, in the following
sense.

######  Proposition 5.1.

Consider the network autoregression (2.1) or the network error model (3.1). If
Assumption 1 is violated, then, for any $\lambda$ such that
$\det(S(\lambda))\neq 0$, and for any $y\notin\operatorname{col}(X)$,

  1. (i)

the profile score associated with the profile log-likelihood $l(\lambda)$ does
not depend on $y$;

  2. (ii)

the adjusted profile log-likelihood function $l_{\mathrm{a}}(\lambda)$ is
flat.

In other words, when Assumption 1 fails, a maximizer of $l(\lambda)$ (over
$\Lambda$ or any other subset of $\mathbb{R}$), if it exists, is non-random,
and $l_{\mathrm{a}}(\lambda)$ does not contain any identifying information
about $\lambda$.

Part (ii) of Proposition 5.1 can be linked back to the invariance results of
Section 4. By standard arguments (available for instance in Rahman and King,
1997), $l_{\mathrm{a}}(\lambda)$ corresponds to the density of the maximal
invariant $v\coloneqq C_{X}y/\mathopen{}\mathclose{{}\left\|C_{X}y}\right\|$
under $\mathcal{G}_{X}$, for any network autoregression model violating
Assumption 1 and for any network error model. Then, the flatness of
$l_{\mathrm{a}}(\lambda)$ can be understood in terms of the distribution of
$v$ being free of $\lambda$ if the distribution of $\varepsilon$ is free of
$\lambda$, which follows from equation (4.1).181818It is easily verified that
the maximal invariant induced by $\mathcal{G}_{X}$ on the parameter space is
$\lambda$ (it would be $\mathopen{}\mathclose{{}\left(\lambda,\eta}\right)$ in
the presence of a parameter $\eta$ in the distribution of $\varepsilon$). This
may seem to contradict one of the fundamental results on invariance, which is
usually stated by saying that the distribution of an invariant statistic
depends only on a maximal invariant induced on the parameter space (e.g.,
Lehmann and Romano, 2005, Theorem 6.3.2). The apparent contradiction is due to
the non-identification caused by the violation of Assumption 1.

##  6 Conclusion

We have studied identification of an autoregression defined on a general
network, under weak distributional assumptions and without requiring repeated
observations of the network. In this context, identification is possible for
generic parameter values and for generic regressor matrices, whatever the
network. However, important cases do exist when identification fails, either
in the original sample space or after some transformation (this could be, for
instance, a transformation aimed at removing fixed effects). We have shown
that in the latter case it is impossible to conduct inference that respects
the invariance properties of the model, regardless of whether the parameters
are identified from the second moment of the outcome variable.

It should be emphasized that our results have been derived under the
assumption that the network is fully known and exogenous, which may be
unrealistic in many applications. The study of identification when the network
is (partially) unknown and/or endogenous remains a key challenge in the
literature (e.g., Blume et al., 2015; de Paula et al., 2020; Lewbel et al.,
2019), and we hope that the results obtained in this paper can prove useful in
that setting too.

##  Appendix A Further examples when Assumption 1 fails

Further to Examples 3 and 4, other two simple models in which Assumption 1
fails are as follows.

######  Example 8.

Consider the modification of Example 3 in which exclusive averaging is
replaced by inclusive averaging, meaning that each unit interacts not only
with all other units in a group but also with itself. If there are $R$ groups,
each of size $m_{r}$, the interaction matrix is
$W=\bigoplus_{r=1}^{R}\frac{1}{m_{r}}\iota_{m_{r}}\iota_{m_{r}}^{\prime}$.
Since
$\operatorname{col}(\bigoplus_{r=1}^{R}\frac{1}{m_{r}}\iota_{m_{r}}\iota_{m_{r}}^{\prime})=\operatorname{col}(\bigoplus_{r=1}^{R}\iota_{m_{r}})$,
Assumption 1 is violated (at $\omega=0$) whenever $X$ contains group
intercepts. Note that in this case, contrary to the case of exclusive
averaging, Assumption 1 fails regardless of whether the model is balanced or
not. ∎

######  Example 9.

Example 4 generalizes immediately to complete $R$-partite graphs, with $R\geq
2$ (a complete $R$-partite graph is a graph in which the $n$ observational
units can be divided into $R$ partitions, with all units in a partition
interacting with all in other partitions, but with none in their own
partition). In that case, Assumption 1 is violated (at $\omega=0$) whenever
$X$ contains an intercept for each of the $R$ partitions. ∎

Examples 8 and 9 share important similarities, due to the fact that the graphs
underlying the two models are complements of each other, in the graph
theoretic sense. Indeed, for both models, the condition
$\operatorname{col}(\bigoplus_{r=1}^{R}\iota_{m_{r}})\subseteq\operatorname{col}(X)$
leading to a failure of Assumption 1 is also satisfied if: (i) $X$ contains an
intercept and $R-1$ contextual effect terms $Wx_{i}$, for some
$x_{1},\ldots,x_{R-1}\in\mathbb{R}^{n};$ (ii) $X$ contains $R$ contextual
effect terms $Wx_{1},\ldots,Wx_{R}$, for some
$x_{1},\ldots,x_{R}\in\mathbb{R}^{n}$.191919In order to be full rank, $X$ can
contain at most $R-1$ contextual effects if it contains an intercept, $R$
contextual effect terms otherwise.

##  Appendix B The QMLE and the adjusted QMLE

Omitting additive constants, the quasi log-likelihood for
$\varepsilon\sim\mathrm{N}(0,I_{n})$ in the network autoregression (2.1) is

$l(\lambda,\beta,\sigma^{2})\coloneqq-\frac{n}{2}\log(\sigma^{2})+\log\mathopen{}\mathclose{{}\left|\det\mathopen{}\mathclose{{}\left(S(\lambda)}\right)}\right|-\frac{1}{2\sigma^{2}}(S(\lambda)y-X\beta)^{\prime}(S(\lambda)y-X\beta),$
(B.1)

for any $\lambda$ such that $S(\lambda)$ is nonsingular. To avoid tedious
repetitions, we often omit the “quasi-” in front of “log-likelihood”. The QMLE
in most common use maximizes $l(\lambda,\beta,\sigma^{2})$ under the condition
that $\lambda$ is in $\Lambda$ (or in a subset thereof).202020This assumes
that $\Lambda$ is well defined. If $W$ did not have a negative (resp.,
positive) eigenvalue, then the left (resp., right) extreme of $\Lambda$ could
be taken to be $-\infty$ (resp., $+\infty$). That is, the QMLE of
$(\lambda,\beta,\sigma^{2})$ is

$(\hat{\lambda}_{\mathrm{ML}},\hat{\beta}_{\mathrm{ML}},\hat{\sigma}_{\mathrm{ML}}^{2})\coloneqq\underset{\beta\in\mathbb{R}^{k},\hskip
1.66702pt\sigma^{2}>0,\hskip
1.66702pt\lambda\in\Lambda}{\operatorname*{argmax}}l(\lambda,\beta,\sigma^{2}).$

Maximization with respect to $\beta$ and $\sigma^{2}$ gives
$\hat{\beta}_{\mathrm{ML}}(\lambda)\coloneqq(X^{\prime}X)^{-1}X^{\prime}S(\lambda)y$
and
$\hat{\sigma}_{\mathrm{ML}}^{2}(\lambda)\coloneqq\frac{1}{n}y^{\prime}S^{\prime}(\lambda)M_{X}S(\lambda)y$.
Thus, $\hat{\lambda}_{\mathrm{ML}}$ can be conveniently computed by maximizing
over $\Lambda$ the profile likelihood for $\lambda$,

$l(\lambda)\coloneqq
l(\lambda,\hat{\beta}_{\mathrm{ML}}(\lambda),\hat{\sigma}_{\mathrm{ML}}^{2}(\lambda))=-\frac{n}{2}\log\mathopen{}\mathclose{{}\left(\hat{\sigma}_{\mathrm{ML}}^{2}(\lambda)}\right)+\log\mathopen{}\mathclose{{}\left|\det\mathopen{}\mathclose{{}\left(S(\lambda)}\right)}\right|,$
(B.2)

where additive constants have again been omitted.

When the dimension of $\beta$ is large compared to the sample size, the QMLE
of $(\lambda,\sigma^{2})$ may perform poorly. To tackle this problem, the QMLE
of $(\lambda,\sigma^{2})$ can be adjusted by recentering the profile score
$s(\lambda,\sigma^{2})$ associated to the profile log-likelihood for
$(\lambda,\sigma^{2})$, $l(\lambda,\sigma^{2})\coloneqq
l(\hat{\beta}_{\mathrm{ML}}(\lambda),\sigma^{2},\lambda)$. Under the
assumptions $\mathrm{E}(\varepsilon)=0$ and $\mathrm{var}(\varepsilon)=I_{n}$,
$\mathrm{E}(s(\lambda,\sigma^{2}))$ is available analytically and does not
depend on the nuisance parameter $\beta$. Thus, calculation of the adjusted
profile score $s_{\mathrm{a}}(\lambda,\sigma^{2})\coloneqq
s(\sigma^{2},\lambda)-\mathrm{E}(s(\lambda,\sigma^{2}))$ is straightforward.
Given $s_{\mathrm{a}}(\lambda,\sigma^{2})$, one can define the adjusted
likelihood $l_{\mathrm{a}}(\lambda,\sigma^{2})$ as the function with gradient
equal to $s_{\mathrm{a}}(\lambda,\sigma^{2})$, and hence the adjusted QMLE
$(\hat{\lambda}_{\mathrm{aML}},\hat{\sigma}_{\mathrm{aML}}^{2})$ as the
maximizer of $l_{\mathrm{a}}(\lambda,\sigma^{2})$. Also, letting
$\hat{\sigma}_{\mathrm{aML}}^{2}(\lambda)$ be the adjusted QMLE of
$\sigma^{2}$ for given $\lambda$, we define the adjusted likelihood for
$\lambda$ only as $l_{\mathrm{a}}(\lambda)\coloneqq
l_{\mathrm{a}}(\lambda,\hat{\sigma}_{\mathrm{aML}}^{2}(\lambda))$. See Yu et
al. (2015) for details on these constructions.

##  Appendix C Proofs

######  Lemma C.1.

The network autoregression (2.1) and the network error model (3.1) imply the
same profile quasi log-likelihood function for $(\lambda,\sigma^{2})$ if and
only if $\mathrm{rank}(X,WX)=k$.

Proof of Lemma C.1. On concentrating the nuisance parameter $\beta$ out of the
likelihood (B.1), the profile quasi log-likelihood for $(\lambda,\sigma^{2})$
in a network autoregression is, up to an additive constant,

$l(\lambda,\sigma^{2})\coloneqq
l(\hat{\beta}_{\mathrm{ML}}(\lambda),\sigma^{2},\lambda)=-\frac{n}{2}\log(\sigma^{2})+\log\mathopen{}\mathclose{{}\left|\det\mathopen{}\mathclose{{}\left(S(\lambda)}\right)}\right|-\frac{1}{2\sigma^{2}}y^{\prime}S^{\prime}(\lambda)M_{X}S(\lambda)y.$
(C.1)

Similarly, the profile quasi log-likelihood function for
$(\lambda,\sigma^{2})$ in a network error model, based again on the assumption
$\varepsilon\sim\mathrm{N}(0,I_{n})$, is

$l(\lambda,\sigma^{2})\coloneqq-\frac{n}{2}\log(\sigma^{2})+\log\mathopen{}\mathclose{{}\left|\det\mathopen{}\mathclose{{}\left(S(\lambda)}\right)}\right|-\frac{1}{2\sigma^{2}}y^{\prime}S^{\prime}(\lambda)M_{S(\lambda)X}S(\lambda)y.$
(C.2)

The two log-likelihood functions are the same if and only if
$M_{S(\lambda)X}=M_{X}$ for any $\lambda$ such that $S(\lambda)$ is
invertible. But, for any $\lambda$ such that $S(\lambda)$ is invertible, the
condition $M_{S(\lambda)X}=M_{X}$ is equivalent to
$\operatorname{col}(S(\lambda)X)=\operatorname{col}(X)$, and hence to
$\operatorname{col}(WX)\subseteq\operatorname{col}(X)$, which in turn is the
same as $\mathrm{rank}(X,WX)=k.$ ∎

Proof of Lemma 3.1. The parameter $(\lambda,\beta)$ is identified on
$\Lambda_{\mathrm{u}}\times\mathbb{R}^{k}$ from
$\mathrm{E}(Y)=S^{-1}(\lambda)X\beta$ if
$S^{-1}(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}})X\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}=S^{-1}(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})X\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}$
implies
$(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})=(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}},\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})$
for any two values
$(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}),(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}},\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})$
of $(\lambda,\beta)$ in $\Lambda_{\mathrm{u}}\times\mathbb{R}^{k}.$ One
immediately has that
$S^{-1}(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}})X\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}=S^{-1}(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})X\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}$
if and only if

$X(\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})+WX(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})=0.$
(C.3)

We analyze separately three (exhaustive) cases, depending on the rank of the
$n\times 2k$ matrix $(X,WX)$. Recall that $X$ is assumed to be of full column
rank.

  1. (i)

$\mathrm{rank}(X,WX)=2k$. In this case equation (C.3) is equivalent to
$\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}=\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}$
and
$\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}=\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}},$
from which
$(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})=(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}},\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})$
if and only if
$\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}=\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\neq
0.$ That is, $\mathopen{}\mathclose{{}\left(\lambda,\beta}\right)$ is
identified on $\Lambda_{\mathrm{u}}\times\mathbb{R}^{k}\backslash\\{0\\}$ from
$\mathrm{E}(Y).$

  2. (ii)

$k<\mathrm{rank}(X,WX)<2k$. Partition $X$ as $(X_{1},X_{2})$ where $X_{1}$ is
$n\times k_{1}$ and $X_{2}$ is $n\times k_{2}$, with $0<k_{1}<k$. The case
$k<\mathrm{rank}(X,WX)<2k$ may be characterized by assuming
$\mathrm{rank}(X,WX_{1})=k+k_{1}$ and $WX_{2}=XB+WX_{1}C,$ for some $k\times
k_{2}$ matrix $B$ and some $k_{1}\times k_{2}$ matrix $C$, so that
$\mathrm{rank}(X,WX)=k+k_{1}$. Replacing $WX$ with
$\mathopen{}\mathclose{{}\left(WX_{1},XB+WX_{1}C}\right)$ in (C.3), and
letting
$\mathopen{}\mathclose{{}\left(\beta_{1}^{\prime},\beta_{2}^{\prime}}\right)$
be the partition of $\beta^{\prime}$ conformable with that of $X$, we obtain

$X(\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}+B(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}_{2}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}_{2}))+WX_{1}(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}_{1}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}_{1}+C(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}_{2}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}_{2}))=0,$

which is satisfied if and only if
$\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}+B(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}_{2}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}_{2})=0$
and
$\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}_{1}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}_{1}+C(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}_{2}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}_{2})=0$.
As a linear system in the unknowns
$\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}$
and
$\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}$,
these two equations are

$M(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})\begin{pmatrix}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\\\
\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\end{pmatrix}=\begin{pmatrix}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}\\\
0_{k_{1}}\end{pmatrix},$ (C.4)

where the matrix

$M(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})\coloneqq\mathopen{}\mathclose{{}\left(\begin{array}[c]{cc}B\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}_{2}&I_{k}-\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}(0_{k,k_{1}},B)\\\
\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}_{1}+C\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}_{2}&-\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}(I_{k_{1}},C)\end{array}}\right)$

is of dimension $\mathopen{}\mathclose{{}\left(k+k_{1}}\right)\times(1+k)$.
Now, identification of $(\lambda,\beta)$ from $\mathrm{E}(Y)$ is equivalent to
$(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})$
being the unique solution to system (C.4), and this occurs if and only if
${\mathrm{rank}(M(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}))}=1+k$,
or, equivalently,
$\det(M(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})^{\prime}M(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}))\neq
0.$ But
${\det(M(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})^{\prime}M(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}))}$
is a polynomial in
$(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})$
and hence the set of its zeros is either the whole $\mathbb{R}^{k+1}$ or has
zero measure with respect to $\mu_{\mathbb{R}^{k+1}}$. The former case is
easily ruled out (e.g.,
$M(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})$
has rank $k+1$ for
$(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})=(0,(1_{k_{1}}^{\prime},0_{k_{2}}^{\prime})^{\prime})$),
which means that $(\lambda,\beta)$ is generically identified from
$\mathrm{E}(Y)$.

  3. (iii)

$\mathrm{rank}(X,WX)=k$. This happens if and only if there is a $k\times k$
matrix $A$ such that $WX=XA$. In that case, equation (C.3) becomes
$X(\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}+A(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}))=0$,
which, since $\mathrm{rank}(X)=k$, is equivalent to
$\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}+A(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})=0$.
Rewrite the last equality as
$(I_{k}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}A)\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}-(I_{k}-\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}A)\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}=0$.
Since the eigenvalues of $A$ are eigenvalues of $W,$ $I_{k}-\lambda A$ is
invertible for any $\lambda\in\Lambda_{\mathrm{u}}$, and therefore
$\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}=(I_{k}-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}A)^{-1}(I_{k}-\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}A)\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}$.
This shows that for any
$(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}},\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})\in\Lambda_{\mathrm{u}}\times\mathbb{R}^{k}$,
it is possible to find
$(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}})\neq(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}},\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})$
such that
$S^{-1}(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}})X\overset{\sim}{\smash{\beta}\rule{0.0pt}{4.86528pt}}=S^{-1}(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})X\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\beta}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}$.

Summarizing, $\mathopen{}\mathclose{{}\left(\lambda,\beta}\right)$ is
generically identified from $\mathrm{E}(Y)$, and hence generically identified,
on $\Lambda_{\mathrm{u}}\times\mathbb{R}^{k}$ in cases (i) and (ii), and not
identified from $\mathrm{E}(Y)$ on $\Lambda_{\mathrm{u}}\times\mathbb{R}^{k}$
in case (iii). ∎

Proof of Lemma 3.2. This proof is similar to the proof of Lemma 4.2 in
Preinerstorfer and Pötscher (2017). Under the assumption that
$\mathrm{var}(\varepsilon)=I_{n}$,
$\mathrm{var}(y)=\sigma^{2}(S^{\prime}(\lambda)S(\lambda))^{-1}$. We show
that, if
$\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\sigma}\rule{0.0pt}{2.71251pt}}}}\rule{0.0pt}{4.30554pt}}\rule{0.0pt}{4.86528pt}^{2}S^{\prime}(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}})S(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}})=\overset{\sim}{\smash{\sigma}\rule{0.0pt}{2.71251pt}}\rule{0.0pt}{4.86528pt}^{2}S^{\prime}(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})S(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})$
for any two parameter values
${(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\sigma}\rule{0.0pt}{2.71251pt}}\rule{0.0pt}{4.86528pt}^{2}),(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}},\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\sigma}\rule{0.0pt}{2.71251pt}}}}\rule{0.0pt}{4.30554pt}}\rule{0.0pt}{4.86528pt}^{2})}\in\Lambda\times(0,\infty)$,
then
$(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}},\overset{\sim}{\smash{\sigma}\rule{0.0pt}{2.71251pt}}\rule{0.0pt}{4.86528pt}^{2})=(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}},\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\sigma}\rule{0.0pt}{2.71251pt}}}}\rule{0.0pt}{4.30554pt}}\rule{0.0pt}{4.86528pt}^{2})$.
The maintained assumption that $W$ has at least one negative eigenvalue and at
least one positive eigenvalue guarantees the existence of a nonzero vector
$f\in\mathrm{null}(W-I_{n})$ and a nonzero vector
$g\in\mathrm{null}(W-\omega_{\min}I_{n})$. Multiplying both sides of the
equality
$\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\sigma}\rule{0.0pt}{2.71251pt}}}}\rule{0.0pt}{4.30554pt}}\rule{0.0pt}{4.86528pt}^{2}S^{\prime}(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}})S(\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}})=\overset{\sim}{\smash{\sigma}\rule{0.0pt}{2.71251pt}}\rule{0.0pt}{4.86528pt}^{2}S^{\prime}(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})S(\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})$
by $f^{\prime}$ on the left and $f$ on the right gives
$\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\sigma}\rule{0.0pt}{2.71251pt}}}}\rule{0.0pt}{4.30554pt}}\rule{0.0pt}{4.86528pt}^{2}(1-\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}})^{2}f^{\prime}f=\overset{\sim}{\smash{\sigma}\rule{0.0pt}{2.71251pt}}\rule{0.0pt}{4.86528pt}^{2}(1-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})^{2}f^{\prime}f$.
Since $1-\lambda>0$ for any $\lambda\in\Lambda$ , and $f^{\prime}f\neq 0,$ the
last equality is equivalent to
$\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\sigma}\rule{0.0pt}{2.71251pt}}}}\rule{0.0pt}{4.30554pt}}/\overset{\sim}{\smash{\sigma}\rule{0.0pt}{2.71251pt}}=(1-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})/(1-\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}})$.
Repeating with $g$ in place of $f$ gives
$\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\sigma}\rule{0.0pt}{2.71251pt}}}}\rule{0.0pt}{4.30554pt}}/\overset{\sim}{\smash{\sigma}\rule{0.0pt}{2.71251pt}}={(1-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\omega_{\min})}/(1-\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\omega_{\min})$.
Thus, we must have
$(1-\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}\omega_{\min})/(1-\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}})=(1-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}\omega_{\min})/(1-\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}})$.
Since the function
$\lambda\mapsto\mathopen{}\mathclose{{}\left(1-\lambda\omega_{\min}}\right)/\mathopen{}\mathclose{{}\left(1-\lambda}\right)$
is strictly increasing on $\Lambda$, we have
$\overset{\sim}{\smash{\lambda}\rule{0.0pt}{4.86528pt}}=\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\lambda}\rule{0.0pt}{4.86528pt}}}}\rule{0.0pt}{6.45831pt}}$,
and hence
$\overset{\sim}{\smash{\sigma}\rule{0.0pt}{2.71251pt}}\rule{0.0pt}{4.86528pt}^{2}=\overset{\sim}{\smash{\stackrel{{\scriptstyle\sim}}{{\smash{\sigma}\rule{0.0pt}{2.71251pt}}}}\rule{0.0pt}{4.30554pt}}\rule{0.0pt}{4.86528pt}^{2}$.
∎

Proof of Lemma 4.1. For any $\lambda$ such that $S(\lambda)$ is nonsingular
and under Assumption 2, it is clear from the reduced form
$y=S^{-1}(\lambda)X\beta+\sigma S^{-1}(\lambda)\varepsilon$ that a network
autoregression is invariant under $\mathcal{G}_{X}$ if and only if
$\operatorname{col}(S^{-1}(\lambda)X)=\operatorname{col}(X)$, or, which is the
same, $\operatorname{col}(S(\lambda)X)=\operatorname{col}(X)$. But this is all
that is required, because, as noted in the proof of Lemma C.1, the condition
$\operatorname{col}(S(\lambda)X)=\operatorname{col}(X)$ for any $\lambda$ such
that $S(\lambda)$ is invertible is equivalent to $\mathrm{rank}(X,WX)=k$. ∎

Proof of Proposition 5.1. For any $\lambda$ such that
$\mathrm{rank}\mathopen{}\mathclose{{}\left(S(\lambda)}\right)=n$, and for any
$y\notin\mathrm{null}(M_{X}S(\lambda))$, the profile log-likelihood
$l(\lambda)$ for a network autoregression is given by equation (B.2). Note
that equation (B.2) holds a.s. for any fixed $\lambda$ such that
rank$\mathopen{}\mathclose{{}\left(S(\lambda)}\right)=n$, because
$\mathrm{null}(M_{X}S(\lambda))$ is a $\mu_{\mathbb{R}^{n}}$-null set when
rank$\mathopen{}\mathclose{{}\left(S(\lambda)}\right)=n$ (since $k<n$). If
Assumption 1 is violated for an eigenvalue $\omega$ of $W$, then $M_{X}(\omega
I_{n}-W)=0$ and hence $M_{X}S(\lambda)=(1-\lambda\omega)M_{X}$, which
substituted into (B.2) gives

$l(\lambda)=\log\mathopen{}\mathclose{{}\left|\det\mathopen{}\mathclose{{}\left(S(\lambda)}\right)}\right|-n\log\mathopen{}\mathclose{{}\left|1-\lambda\omega}\right|-\frac{n}{2}\log(y^{\prime}M_{X}y),$
(C.5)

for any $y\notin\operatorname{col}(X)$. Since a violation of Assumption 1
implies $\mathrm{rank}(X,WX)=k$, equation (C.5) also applies to a network
error model, by Lemma C.1. Part (i) of the proposition follows on noting that
the terms in (C.5) that contain $\lambda$ do not contain $y$. Next, let
$s(\lambda)$ be the profile score associated with $l(\lambda)$, let
$s_{\mathrm{a}}(\lambda)\coloneqq s(\lambda)-\mathrm{E}(s(\lambda))$ be its
adjusted counterpart, and let $l_{\mathrm{a}}^{\ast}(\lambda)\coloneqq\int
s_{\mathrm{a}}(\lambda)\mathop{}\\!\mathrm{d}\lambda$ be the likelihood
corresponding to $s_{\mathrm{a}}(\lambda)$. It can be easily verified that
$l_{\mathrm{a}}(\lambda)=\frac{n-k}{n}l_{\mathrm{a}}^{\ast}(\lambda)$ (the
adjusted profile likelihood $l_{\mathrm{a}}$ being defined in Appendix B). If
Assumption 1 is violated, then, from part (i),
$\mathrm{E}\mathopen{}\mathclose{{}\left(s(\lambda)}\right)=s(\lambda)$, and
hence $s_{\mathrm{a}}(\lambda)=0$, which in turn implies that
$l_{\mathrm{a}}^{\ast}(\lambda)$, and hence $l_{\mathrm{a}}(\lambda)$, is
constant. This completes the proof. ∎

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