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Testing linearity of spatial interaction functions \`a la Ramsey
\allowdisplaybreaks There has been a recent surge in the study of econometric models capturing social interactions between agents, see e.g. paula_2017 for a survey. By virtue of its close links to network and social interaction models, the spatial econometrics literature has also grown commensurately. A key component of this literature is the linear spatial autoregressive (SAR) model, introduced by Cliff1968, cliff1973spatial. The `spatial' moniker is unfortunate in some sense because it belies the generality of the SAR model. In fact the `space' in question can be any economic dimension that connects individuals, e.g. social networks and conflict alliances. In this paper we propose and theoretically justify a convenient nonparametric test for the linearity of the SAR specification, which is by far the most prevalent in the literature.
The assumption of linearity in the preponderance of studies involving SAR models is perhaps unsurprising given that even in straightforward multiple regression the linear model wins the popularity contest. Nevertheless, linearity is a strong assumption, criticized by Pinkse2010 for instance, and discussed in detail in paula_2017, Section 3.2. In the case of spatial autoregressions, nonlinearity in the spatial lag also induces immense technical difficulties due to the simultaneity inherent in the model. This has led to the seminal development of sophisticated machinery Jenish2009, Jenish2012, but the flipside is that the conditions tend to be technically challenging and restricted to geographic data.
The econometric implications of nonlinearity in spatial or network interactions are manifold. The focus of this paper is to test for potential nonlinearity in the so called `link' function, which transmits a network connection-weighted average of peer outcomes to an individual's own outcome. Tincani2018 finds evidence of nonlinearities in education peer effects in Chile, the link function here operating on the distribution of peer outcomes. Other implications of nonlinearities can include multiple equilibria, or potentially none at all. We refer the reader to detailed discussions of various forms of nonlinearity and their implications in, for example, Blume2011 and paula_2017 .
While of importance in its own econometric right, and possibly in a reduced form sense, testing for the linearity of a SAR specification can also address deeper economic considerations. Many network games yield equilibria with linear best responses and, when taken to the data, imply linear SAR specifications. Thus, a rejection of the linearity of the SAR specification can in fact be viewed as a rejection of the underlying structural economic model. Examples of such models include contest success functions (CSFs) in the Tullock1980 style konig2017networks, quadratic utility network games ballester2006s, calvo2009peer and public goods provision games in networks bramoulle2007public. These are discussed in more detail in the next section.
Nonlinearity of best responses in network games can have profound economic implications. acemoglu2015networks emphasize that convexity or concavity of nonlinear network interactions can shape how the underlying network transmits shocks. Most strikingly, they show that with concave interactions, densely connected economies outperform sparser ones in the sense of expected macroeconomic outcomes. Indeed, an economy in which interlinkages are maximally dense outperforms all other (symmetric) economies. On the contrary, with convex interactions this pattern is completely reversed, with the maximally dense economy being the worst performing. The finding that concavity of interactions acts as a `shock absorber' while convexity turns this into a `shockwave propogator' aligns with the groundbreaking work of allen2000financial on financial contagion. Thus, if a theoretical structural model begets a linear best response equilibrium, it behooves the economist to have some statistical confidence in the resulting empirical model.
We employ a residual based test statistic reminiscent of the Lagrange Multiplier (LM) test to avoid nonparametric estimation altogether. The basic idea is to approximate the potentially nonlinear part of the model with a sieve expansion of length $p$. By choosing a basis of polynomial components, a test of linearity may be constructed by setting as zero all coefficients except the one corresponding to the linear term. This is an approach in the spirit of the famous Ramsey RESET test described in Ramsey1969. Because we employ the LM principle, we need only estimate the model under the null hypothesis, which is the familiar linear SAR model.
Because $p$ is a diverging nonparametric bandwidth, the number of coefficients on nonlinear terms must grow with sample size. Therefore our test will be for an increasing number of restrictions $p\rightarrow\infty$ asymptotically, and the usual $\chi^2_p$ asymptotic distribution cannot be relied upon. To address this, we will use a centred and scaled version of the statistic. In particular our test statistic will be of the form $(\mathcal{T}_p-p)/\sqrt{2p}$, where $\mathcal{T}_p$ is computable by estimating the model under the null of linearity. We will show that this is asymptotically standard normal under the null, noting that a $\chi^2_p$ random variable has mean $p$ and variance $2p$, in the spirit of DeJong1994, Hong1995, Donald2003 and Gupta2023, to cite a few examples. We also establish the test's consistency and ability to detect local alternatives at a suitably dampened nonparametric rate.
There are several specification tests in the literature for linearity of the regression function in a linear SAR model, see for instance Su2017, Gupta2022 and Chen2025. On the other hand, the cupboard is rather bare as far as tests of linearity of the spatial lag are concerned. The most direct linearity test is provided by hoshino2022sieve and relies on the nonparametric function being estimated. This is a different approach from our Ramsey-style test, which stresses simplicity. The model considered is also somewhat different from ours: we test for linearity of the spatial lag as a whole while hoshino2022sieve focuses on testing linearity in the outcome variable. Thus, the two approaches to testing linearity complement each other. Another type of linearity test is proposed by Malikov2017, but there the spatial parameter itself is modelled as a varying coefficient while the spatial lag is linear. This is rather different from our setup. Finally, another class of tests is of the omnibus type where model misspecification can arise from a variety of sources, see Lee2024 and Yang2024 for tests of the `integrated conditional moments' type.
In a Monte Carlo simulation study, we show that critical values based on our theory deliver excellent size and power performance for a wide variety of commonly encountered social interaction networks and spatial links. These include contiguity, nearest neighbours, distance cutoff networks and more general structures. We experiment with both standard normal critical values as well as standardized $(\chi^2_p-p)/\sqrt{2p}$ critical values, and observe that the latter can do well with smaller values of $p$.
The linear SAR model is by far the most commonly employed empirical specification. In an empirical study, we study a model of tax competition and show how our test for linearity can raise new questions or indeed refine existing analysis. Our test corroborates Lyytikaeinen2012's finding of absence of tax competition between Finnish municipalities in their property tax setting behaviour, illustrating how different model specifications can lead to contrasting conclusions and how our test can offer insightful guidance on the specification choice.
The next section introduces our basic setup and a number of structural economic models that imply linear SAR specifications. In Section (ref) we define our test statistic while Section (ref) presents the key assumptions and asymptotic results. Section (ref) contains the results of our Monte Carlo experiments and Section (ref) presents our empirical study of tax competition in Finland. All proofs are collected in the appendix.
Let $W$ be a spatial weight matrix with rows $w_i^{\prime }$ and zero diagonal. Given an $n\times 1$ response vector $y$ and $k\times 1$ covariate vector $x_i$ with unity as first element, we commence from
with $f(\cdot)$ being an unknown function and $\epsilon_i$ independent disturbances. The $x_i$ can contain both exogenous and endogenous components. The first term on the RHS represents the usual linear SAR component and $f(\cdot)$ is a nonparametric function that embodies the potential non-linearity of the true data generating process.
Apart from being an often used econometric specification, the linear SAR model is implied by a number of theoretical models. Thus an empirical test of its linearity in fact serves as test of the underlying economic structure that leads to the estimation of a SAR structure. In this sense our test can be viewed as test of economic theory, rather than simply the statistical fit of an econometric model. We discuss some examples of such theories in this section, with $x_i$ always denoting a $k\times 1 $ vector of observable characteristics for the $i$-th individual, and $\beta$ a $k\times 1$ parameter vector.
We aim to develop a test of
for all $x$ and for some admissible $\lambda$ and $\beta$. Our testing strategy will involve a sieve expansion of $f(\cdot)$ but parameter estimates that can be obtained simply under the null hypothesis ((ref)).
Let $\psi_j(z)$, for $j=1,\ldots,p$, be a user-chosen set of basis functions such that
with $p=p_n$ being a divergent deterministic sequence, $\alpha= (\alpha_1,\ldots, \alpha_p)^\prime$ a vector of unknown series coefficients and $r(z)$ an approximation error. Let $\theta= (\lambda, \beta^\prime)^\prime$ denote admissible parameter values. We define our approximate null hypothesis as
where $\Theta= \Lambda \times \Re^k$, not necessarily compact but obeying Assumption (ref) below. Define the $n\times 1$ vector
Under $\mathcal{H}_{0A}$, we have $S_p(\lambda_0,0_{p\times 1}, y)= y - \lambda_0 Wy$, consistent with the linear SAR model.
Our test statistic is based on determining if the moment conditions for the instrumental variables (IV) estimate of $\theta_0$ under the null hypothesis are close enough to zero. We allow for over-identification, and thus we refer to our estimation method as Two-Stage Least Squares (2SLS) henceforth, see e.g. kelejian1998generalized. Given the expansion in ((ref)), the approximate IV objective function is
where $Z$ is a $n\times m$ matrix of valid instruments, with $m\geq p+k+1$, $m\sim p$, and $P_Z= Z(Z^\prime Z)^{-1} Z^\prime$. We will specify the components of the instrument matrix in more detail in the next section. Now, for each $j=1,\ldots,p$, define the $n\times 1$ vector $\Upsilon_j(y)= \left(\psi_j(w_1'y),\ldots,\psi_j(w_n'y)\right)'$ and the $n\times (p+ k+1) $ matrix $U=
$ with elements $u_{ij}$.
To construct our test statistic, we will now introduce the gradient of ((ref)) and its covariance matrix. Define the $(p + k+1)\times 1$ gradient vector $\tilde{d}(\lambda, \beta, y)$ under $\mathcal{H}_{0A}$ as
We denote by $\hat{\theta}=(\hat{\lambda}, \hat{\beta}^\prime)^\prime$ the 2SLS estimate of $\theta_0=(\lambda_0, \beta_0^\prime)^\prime$ under $\mathcal{H}_{0A}$. Then the gradient evaluated at the residuals corresponding to $\hat\theta$ is
Likewise, let $\hat{\Sigma}$ be the diagonal matrix with diagonal elements $\hat{\epsilon}_i^2= \left(y_i- \hat{\lambda} \sum_j w_{ij}y_j -x_i^\prime\hat{\beta}\right)^2$, for $i=1,\ldots,n$, $, \hat{J}=n^{-1}Z^\prime U$, where $\hat{J}$ is an $m \times (p+k+1) $, and define the two $m\times m$ matrices $\hat{M}= n^{-1}{Z^\prime Z}$, $\hat{\Omega}=n^{-1}{Z^\prime \hat{\Sigma} Z}.$ The covariance matrix of the gradient evaluated at the estimates is
and we thence define our test statistic as
This is a weighted measure of the distance of the gradient from zero, centred and rescaled to account for $p\rightarrow\infty$.
We commence this section by introducing some technical assumptions to establish the limiting behaviour of ((ref)) under $\mathcal{H}_{0A}$.
The possibility that $cov(\epsilon_i, x_{ij}) \neq 0$, for some $j=1,\ldots, k$, is not ruled out by Assumptions (ref) and (ref) and thus $X$ might contain some endogenous columns for which external instruments would be needed in $Z$. We denote by $X_1$ the $n \times k_1$ matrix containing the subset of exogenous columns of $X$, while $X_2$ ($n\times k_2$, with $k_2=k-k_1$) contains the endogenous ones.
Now, for a generic symmetric positive-definite matrix $A$ let $\overline{\textit{eig}}(A)$ and $\underline{\textit{eig}}(A)$ denote its largest and smallest eigenvalues, respectively. For a generic matrix $B$, denote by $\left\Vert B\right\Vert=\sqrt{\overline{\textit{eig}}(B'B)}$, i.e. the spectral norm of $B$, and by $\left\Vert B\right\Vert_\infty$ its largest absolute row sum.
Assumptions (ref) and (ref) are rather standard restrictions on the spatial weight matrix, helping to control spatial dependence, see e.g. lee2002consistency, lee2004asymptotic. Assumption (ref) imposes regularity on model primitives and the approximation error and permits conditional heteroskedasticity. In particular, ((ref)) implies that $\hat{J}$ has full rank $p+k+1$ for sufficiently large $n$, while ((ref)) is a standard asymptotic boundedness and no-collinearity condition. Our instrument set $Z$ contains, together with at least $k_2$ columns of instruments for the endogenous covariates $X_2$, the columns also of $X_1$, $WX_1$ and a set of instruments $z_{ij}$, $i=1,\ldots,n$. These would be of the form $\psi_q\left(\sum_j w_{ij}x_{1, jl} \right)$, where $q=1,\ldots,p$, and $x_{1, jl}$ denotes the $(j,l)$th element of $X_1$, with $l=1,\ldots,k_1$. Thus, uncorrelatedness between $\epsilon_i$ and $z_j$ requires that $\epsilon_i$ and $x_{1,jl}$ are uncorrelated for all $i,j=1,\ldots, n$ and $l=1,\ldots,k_1$. For more discussion on approximation error decay rates see e.g. Chen2007.
Finally, we impose
Assumption (ref) guarantees that a type of weak law of large numbers holds and enforces a suitable bound on cross-sectional dependence in the spirit of Lee2016. This can be checked under a variety of conditions such as linear process representations for the underlying random variables or with the near epoch dependence conditions of Jenish2012. Writing $J=E(\hat J)$, we can now state the following result that is analogous to but stronger than a weak law of large numbers.
In this section we develop the asymptotic theory to establish the distribution of the test statistic in ((ref)) under $\mathcal{H}_{0A}$. Our null asymptotic theory comprises of approximating the test statistic $\mathcal{T}$ with a quadratic form in $\epsilon$, weighted by population quantities. We will then show that this approximation is asymptotically standard normal. To start, we define the population quantities corresponding to ((ref)), or to its equivalent representation in ((ref)), and to ((ref)) under $\mathcal{H}_{0A}$ as
where $\mathcal{M}_{NM}= \left(I - M^{-1/2} N\left(N^\prime M^{-1} N \right)^{-1} N^\prime M^{-1/2} \right)$ is $m\times m$ and $N= \mathbb{E}(\hat{N})$, with $\hat{N}= n^{-1} Z^{\prime}(Wy, X)$, which is an $m \times (k+1)$ matrix with full rank under ((ref)) in Assumption (ref), and
with $\Omega= n^{-1}\mathbb{E}(Z^\prime \Sigma Z)$ and $\Sigma$ the $n\times n$ diagonal matrix with diagonal given by $\sigma_i^2, i=1,\ldots,n$. Under Assumptions (ref) and (ref), $H^{-1}$ exists and is non-singular for $n$ large enough, via the following lemma for the eigenvalues of $\Omega$.
Also, by construction, the $m\times m$ matrix $\mathcal{M}_{NM}$ has rank $m-k-1$, which grows like $m$ as $n \rightarrow \infty$ because $k$ is fixed. By relying on the auxiliary Theorem (ref), reported in Appendix A, we can show that $\mathcal{T}$ can be approximated by a quadratic form in $\epsilon$, as desired. Indeed, we derive
Thence, we are now ready to prove the main result of this section, which is the asymptotic standard normality of $\mathcal{T}$ under the null hypothesis.
Theorem (ref) provides asymptotic justification for using one-sided, standard normal critical values as observed also by Hong1995. However, in practice, a user of the test may be faced with moderate values of $p$. In this situation, the results of the asymptotic test can be compared with a test that employs $(\chi^2_p - p)/\sqrt{2p}$ as the distribution from which its critical values are computed. The comparison will be studied by means of a Monte Carlo experiment in Section (ref). Observe that one could equivalently just compare the $n\hat d'\hat H ^{-1}\hat d$ to critical values from a $\chi^2_p$ distribution in practice. The standardized $\chi^2_p$ version however allows us to compare the critical values to those from standard normal distribution.
In this section we establish the consistency of our test, specifically against the alternative
To examine power properties, we introduce the unrestricted quantities
where $\tilde{\Sigma}_{U}$ is a diagonal matrix with $\tilde{\Sigma}_{Uii}= \epsilon_{Ui}^2(\alpha, \lambda, \beta)$. Clearly, $\hat{\Omega}= \tilde{\Omega}_U(0_{p\times 1}, \hat{\lambda}, \hat{\beta})$.
Let $\gamma = (\alpha, \lambda, \beta)\in\Gamma= \Re^p \times \Lambda \times \Re^k$ and introduce:
This assumption places some mild regularity on behaviour under the sequence of alternatives $\mathcal{H}_{1A}$, reminiscent of standard boundedness and invertibility conditions. We can now state the main theorem of this section.
We aim to assess the local power of our test by considering the sequence of local alternatives
where $\delta_j$ is a finite constant and the $p^{1/4}$ factor accounts for the cost of the nonparametric approach; see e.g DeJong1994, Hong1995 and Gupta2023 for a similar dampening factor. We denote by $\alpha_{0n}$ the value of the $p\times 1$ vector $\alpha$ under $\mathcal{H}_{\ell}$ and $\delta= (\delta_1,\ldots., \delta_p)^\prime$ with $\Vert \delta \Vert =1$. We also define
where $\Xi= \mathbb{E}(\hat{\Xi})$, $\hat{\Xi}= n^{-1} Z^\prime
$ and $A^{11}$ denotes the top-left $p\times p$ block of ${A}^{-1}$, a notational convention that we maintain for any generic $A^{-1}$ considered in the paper.
In this section we report the results of a simulation exercise with $k=3$, $\lambda_0=0.4$, $\beta_0=(0.5, -2,1)^\prime$. The $n \times 3$ matrix $X$ has ones in its first column, while elements in the second and third columns are generated in each replication as i.i.d. random variables from $U[-2, 2]$ and $U[-2.5,2.5]$, respectively. We employ the Hermite polynomials for the basis functions $\psi_j(\cdot)$. We generate $\epsilon_i$, for $ i=1,\ldots,n$, as
with $\zeta_{i}$ generated either from standard normal distribution, or, t-distribution with 5 degrees of freedom ($t_5$), and employ three mechanisms for the scale parameter $\sigma_i$:
For a) the $\sigma_i$ are kept fixed across simulations since $W$ are also kept fixed across iterations as explained below. In b), the $\sigma_i$ are kept fixed across simulations and also across different weight matrix scenarios, while for c), they vary across iterations according to the values of the generated $x_{i2}$ and $x_{i3}$. The heteroskedasticity design in ( (ref)) is in line with the simulation work in kelejian2010specification and Arraiz2010, and is motivated by situations in which heteroskedasticity arises as units across different regions may have different numbers of neighbours. In contrast, the design b) yields error heteroskedasticity that is independent of the spatial dependence implied by the weight matrix while the design c) models conditional heteroskedasticity.
We construct the matrix $Z$ as $n\times (p+k+2)$ matrix with $i$-th row \[ \left(1,x_{i2}, x_{i3},w_i'x_2,w_i'x_3,\psi_1(w_i'x_{\ell_1}),\psi_2(w_i'x_{\ell_2}),\cdots, \psi_p(w_i'x_{\ell_p})\right)' \] with $\ell_1, \ell_4, \ell_5, \ell_8, \ell_9, \ell_{12}=2$ and $\ell_2, \ell_3, \ell_6, \ell_7, \ell_{10},\ell_{11} =3$. We alternate $x_{i2}$ and $x_{i3}$ inside $\psi_j(\cdot)$ in this way so as to ensure both variables appear in odd and even degree polynomials.
We report below empirical sizes and powers for $n=100, 200, 400, 700, 1000, 2000$ based on $1000$ Monte Carlo replications. We set $p$ as the integer part of $n^{1/3}$, leading to $p=4,5,7,8,10, 12$ for $n=100, 200, 400, 700, 1000, 2000$, respectively, and use a nominal size set at $\alpha=0.05$. Because our test statistic is a standardized chi-squared type statistic, for small $p$, critical values based on the standardized $\chi_p^2$ distribution may provide better finite sample approximation than those based on the asymptotic standard normal approximation. Hence we use two critical values, one based on $\chi_p^2$ and defined as $(\chi^2_{p, 1-\alpha}-p)/\sqrt{2p}$ where $\chi^2_{p, 1-\alpha}$ is the $1-\alpha$-th percentile of the $\chi_p^2$ distribution, and the other as the $1-\alpha$-th percentile of the standard normal distribution. For $p=4,5, 7, 8, 10, 12$, the $\chi_p^2$-based critical values are 1.9403, 1.9195, 1.8887, 1.8767, 1.8575 and 1.8424, respectively, significantly larger than the critical value based on normal distribution, 1.645. Hence we expect tests based on asymptotic normality to be oversized for small $n$, which is indeed what we observe below.
We generate $y$ according to ((ref)) under $\mathcal{H}_{0}$ using five different configurations for $W$, motivated by a range of empirical situations.
All $W$ are normalized by their spectral norms, and for specifications 1), 2) and 4), $W$ are stochastically generated and then fixed across replications.
We present Monte Carlo size results for nominal size of 5$\%$ for Gaussian and $t_5$ errors in Tables (ref) and (ref), respectively. Tests based on the standardized $\chi_p^2$ distribution lead to good empirical sizes even for small $n$ across all weight matrix designs and heteroskedasticity specifications, while relying on asymptotic normality leads to some oversizing, as anticipated. The extent of oversizing reduces with larger $n$ and $p$, as expected, but the oversizing still remains at $n=2000$. Hence, the standardized $\chi_p^2$-based critical value can provide a useful robustness check to the normal approximation in practice.
Power analysis requires data generation from a nonlinear model, which in general requires the solving of $n$ nonlinear equations at every iteration, a futile task. We resolve this issue by generating the following two nonlinear SAR models on a lattice, similar to Jenish2016. Letting $k$ and $j$ denote indices across $\Re^2$ as before and setting $y_{0, j}=0, j=1,2,\ldots, m_2$ and $y_{k,0}=0, k=1,2,\ldots, m_1$: $$ y_{k,j} = arctan (y_{k-1,j}+y_{k,j-1})+x_{k,j}' \beta +\epsilon_{k,j}, \quad k=1, \ldots, m_1, \quad j=1, \ldots, m_2 $$ and $$ y_{k,j} = log(1+0.25(y_{k-1,j}+y_{k,j-1})^2)+x_{k,j}' \beta +\epsilon_{k,j}, \quad k=1, \ldots, m_1, \quad j=1, \ldots, m_2 $$ We use the lattice weight matrix given as case 5) in the previous subsection, which correctly picks out the relevant neighbours but the misspecification lies in the linearity of SAR model being fitted. In line with the previous section, we report power results for $p=4, 5, 7, 8, 10, 12$ for $n=100, 210, 400, 702, 992, 1980$ respectively.
Tables (ref) and (ref) present empirical power of the test of $\mathcal{H}_{0A}$, employing Gaussian and $t_5$ errors, respectively. Power is best in general when the error has conditional heteroskedasticity c), followed by a) then b). This is expected given the random nature of error heteroskedasticity in b), that is independent of both the weight matrix and regressors. The empirical power improves with larger $n$ for both settings of nonlinear SAR, although the improvement is somewhat slow for $t_5$ error combined with heteroskedasticity of type b).
Numerous studies have used a linear SAR specification to test for the presence of competitive behaviour in neighbouring governments' tax-setting decisions. While many had found the presence of tax competition based on a spatial IV approach or QML methods (see an extensive list given in Allers2005), Lyytikaeinen2012 used a policy-based IV to estimate the SAR parameter and found it to be insignificant. In this section we apply our test of linearity to data from Lyytikaeinen2012 to try and shed light on this discrepancy.
We begin with some institutional background. Finland's municipalities set their own property tax rates within limits imposed by the central government. This raises the question of whether neighbouring municipalities compete on tax rates to attract investment. To study this question, Lyytikaeinen2012 used a linear SAR model with fixed effects such that
where $t_{it}$ denotes either municipality $i$'s general property tax rates or residential building tax rates in year $t$ and $\mu _{i}$ and $\tau _{t}$ are municipality and year fixed effects, respectively. The municipality controls $X_{it}$ include per capita income, per capita grants, unemployment rate, percentage of population aged 0-16, percentage population aged 61-75 and percentage of population aged 75+.
Lyytikaeinen2012 focused on one-year differenced data using the difference between 2000 and 1999 to allow for municipality fixed effects. This choice was to exploit the variation due to a policy of raising the common statutory lower limit to the property tax rates that was implemented in 2000. Using $\Delta$ to denote a difference between 2000 and 1999, this exogenous policy change was used to construct a suitable instrument and estimate the parameters of
where $P_i$ is a dummy variable indicating whether the 1998 tax rate level for municipality $i$ was below the new lower limit imposed in 2000 and $M_i$ indicates the magnitude of the imposed increase for municipality $i$, both included to ensure exogeneity of the instruments. Lyytikaeinen2012 found the spatial parameter $\lambda $ to be insignificant for both sets of regressions with either general property tax rate or residential building tax rate, and hence concluded that there is an absence of substantial tax competition between municipalities in Finland.
We apply our test of linearity in ((ref)) using $p=4,5,6$. We estimate the model with the same policy instrument and row-normalized contiguity weight matrix used in Lyytikaeinen2012. We utilize the spatial IV, constructed by premultiplying $\Delta X_{i}$ by the weight matrix, as our additional instrumental variables required to satisfy Assumption (ref), and report the results in Table (ref). Our tests do not reject the null of linearity for any choice of $p$ for both general property tax rates and residential building tax rates. This indicates that a linear SAR specification is compatible with the differenced data for Finnish municipalities. Using standardized $\chi^2_p$-based critical values evidently does not change our conclusions.
As observed above, the absence of tax competition that Lyytikaeinen2012 finds differs from earlier findings in the literature. To try and get to the bottom of this, we observe that one notable way in which Lyytikaeinen2012 differs from previous literature listed in Allers2005 is in the inclusion of municipality fixed effects. Not accounting for the time-invariant characteristics could result in spurious spatial dependence explaining the disparity of Lyytikaeinen2012's findings from the previous ones.
With this in mind, we now apply our test of linearity to the level data from year 2000, without fixed effects, given by
Interestingly, we observe in Table (ref) that the null of linearity is strongly rejected in the case of general property tax rates where the estimated $\lambda$ is positive and significant, in contrast to the case of residential building rate where linearity is not rejected and estimated $\lambda$ is insignificant. Once again, using standardized $\chi^2_p$-based critical values evidently does not change our conclusions.
Thus the finding of significant spatial competition in the general property tax when not accounting for municipality fixed effects appears to be due to an unreliable specification. This supports the conclusion of Lyytikaeinen2012 that there is no competitive behaviour in the setting of Finnish property tax rates and that the linear SAR specification is a reliable model. It also offers further explanation to Lyytikaeinen2012's insight that previous findings of the presence of tax competition need to viewed with caution and may be resulting from specification problems.