Looking for the Causes of Instability in Spatial Econometric Models

Abstract

It is relatively easy to find symptoms of misspecification in a model that suffers from instability. Problems arise in relation to their interpretation: They may be due to a wrong selection of the functional form. There may be a problem of omitted variables or the parameters of the model may not be homogeneous across space. The authors focus their attention on the last aspect. Specifically, the intention is to develop a strategy to analyze the hypothesis of stability in the parameters of a spatial econometric model. It is important that this procedure should provide the user with information about the cause of instability. The content of the article is mixed. First, the authors develop a collection of robust Multipliers to test the hypothesis of stability in the main elements of a spatial econometric model. Then, they solve a Monte Carlo to analyze the behavior of the Multipliers. Finally, they present an application to the case of the Spanish elections by municipalities.

Keywords

spatial dependence spatial instability Monte Carlo

Introduction

Model specification issues are of great interest in spatial data analysis (see, e.g., Anselin 1988a, 1988b, 1992; Blommestein 1983; Florax and Folmer 1992; Florax, Folmer, and Rey 2003; Mur and Angulo 2009). Two main topics are usually studied, heterogeneity and spatial dependence, and in most cases they are taken separately. Anselin (1988c) was one of the first to combine the two topics, giving rise to the well-known battery of tests of autocorrelation and heteroskedasticity; Anselin (1990) and Páez, Uchida, and Miyamoto (2001) continue along the same lines, adapting the Chow test to models with spatial dependence. Later, Brunsdon, Fotheringham, and Charlton (1998a), Pace and Lesage (2004), and López, Mur, and Angulo (2009) develop models where the mechanism of spatial dependence is specific for each point in space. These last three articles may be seen as a natural extension of the Locally Weighted Regressions algorithm, introduced by Cleveland (1979) and Cleveland and Devlin (1988) and subsequently developed by, among others, McMillen (1996), McMillen and McDonald (1997), Brunsdon, Fotheringham, and Charlton (1998b), and Páez, Uchida, and Miyamoto (2002a, 2002b).

Clearly, then, previous literature in spatial modeling has dealt with the problem of heterogeneous models, through the analysis of instabilities in the three elements of an equation: (a) in the coefficients of the regression, (b) in the variance of the random term, and (c) in the mechanisms of spatial dependence. However, usually, these different forms of instabilities have been treated separately. As far as we know, this is the first time that a simultaneous and systematic approach to all of them is proposed. Specifically, in this article, we present results on two main issues. On one hand, we obtain a collection of Lagrange Multipliers for testing different sources of instability in a spatial econometric model, which may act individually or combined. The tests are robust to the presence of nuisance parameters (which, in turn, may cause instability in other aspects of the model). On the other hand, we combine this battery of diagnostics into a model strategy process for identifying the sources of instability that are affecting a given spatial econometric model. We concentrate the discussion on the cases of the Spatial Lag Model (SLM) and the Spatial Error Model (SEM), though the discussion can be generalized.

The structure of the article is as follows. In the section on A General Model of Spatial Dependence with Heterogeneity, we present a general model in which various mechanisms of instability are introduced simultaneously. In the section on Testing for the Sources of Instability, we obtain a collection of robust Multipliers to test for different sources of instability. Moreover, a specification strategy, directed at identifying the sources of instability, is presented and justified. In the section on A Monte Carlo Experiment, we solve a Monte Carlo experiment to analyze the behavior of the Multipliers under different situations: (a) instability in the coefficient of spatial autocorrelation; (b) instability in the coefficients of regression; and (c) instability in the variance of the error terms. Some results on the probability of selecting the true Data Generating Process (DGP) using the suggested specification strategy complete the content of the section. The section on An Application to the Spatial Analysis of the 2008 Spanish General Elections includes an application of the procedure to the results of the 2008 Spanish General Election by municipalities. Finally, the main conclusions are discussed in the section on Main Conclusions.

A General Model of Spatial Dependence with Heterogeneity

In this section we analyze a model with spatial dependence of an autoregressive type and different forms of heterogeneity. Our intention is to develop a technique to detect instabilities in these specifications. Instability might be present in the mechanism of spatial autocorrelation, in the coefficients of regression and/or in the dispersion parameter. To cope with this objective, we start with a general model proposed by Mur, López, and Angulo (2009), which includes instabilities in all the three elements:

\begin{matrix} y = ρ H W y + X m + ϵ \Rightarrow 0.8 p c A y = X m + ϵ \\ ϵ ~ N (0, Ω); Ω = σ^{2} D; A = I - ρ H W \end{matrix}\},

where

y = \underset{R \times 1}{[\begin{matrix} y_{1} \\ y_{2} \\ \dots \\ y_{R} \end{matrix}]}; X = \underset{R \times R k}{[\begin{matrix} x_{1} & 0 & \dots & 0 \\ 0 & x_{2} & \dots & 0 \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & x_{R} \end{matrix}]}; x_{i} = \underset{_{1 \times k}}{[\begin{matrix} x_{i 1} & x_{i 2} & \dots & x_{i k} \end{matrix}]} .

m = \underset{_{R k \times 1}}{[\begin{matrix} m_{1} \\ m_{2} \\ \dots \\ m_{R} \end{matrix}]}; m_{i} = \underset{_{k \times 1}}{[\begin{matrix} m_{i 1} \\ m_{i 2} \\ \dots \\ m_{i k} \end{matrix}]}; m_{i j} = β_{j} p_{j} [{g^{'}}_{i} μ_{j}] \to \{\begin{matrix} \begin{matrix} g_{i}^{^{'}} = [g_{i 1}; g_{i 2;} \dots; g_{i p}] \end{matrix} \\ μ_{j}^{^{'}} = [μ_{j 1}; μ_{j 2;} \dots; μ_{j p}] \end{matrix} .

H = d i a g \{h (z_{i} α); i = 1, 2, . . . ., R\} .

\begin{matrix} ϵ ~ N (0, Ω); Ω = σ^{2} D \\ D_{i i} = d [n_{i}^{^{'}} δ] \\ D_{i j} = 0; i \neq j \end{matrix}\} .

p_{j} (0) = κ_{j 2} > 0; h (0) = κ_{1} > 0; d (0) = κ_{3} > 0.

There are R spatial units in the sample. As defined in equation 1a, y is the (R × 1) vector of the endogenous variable. There are k exogenous variables whose observations are arranged in the block diagonal

X

matrix. The terms x_i (i = 1,2,…,R) that appear in the diagonal of this matrix are (1 × k) vectors of observations corresponding to the ith spatial unit and the 0s distributed throughout the matrix are (1 × k) vectors of zeroes.

The purpose of this arrangement of the X data is to allow for the existence of a vector of regression coefficients in each spatial unit, m_i , of order (k × 1), as in equation 1b, similarly as the complex heterogeneous formulation derived in the spatial expansion method proposed by Casetti (1972), “terminal model” in Casetti’s formulation. Hence, vector m contains Rk parameters and, consequently, taking into account that we have only R observations, equation 1 suffers from an incidental parameter problem. Our rationale is that these parameters are not constant but evolve over space according to a regular process. In this way, parameter j corresponding to the ith spatial unit, m_ij , is the product of two terms: β_j, which is constant across space, and a function, p_j , of a group of p variables (g ₁, g ₂,…, g_p ) and a set of parameters (μ_j1, μ_j2,…, μ_jp). We assume that function p_j and parameters (μ_j1, μ_j2, …, μ_jp) are specific to the regression coefficient j, m_j , and that they are the same across space. Note that, if vector μ_j is zero, according to the identification restriction of equation 1e, there is a single and unique vector of coefficients in model 1: $m = l_{R} \otimes β$ , where l_R is an (R × 1) vector of ones, $\otimes$ denotes the Kronecker product and $β = {[\begin{matrix} β_{1} & β_{2} & \dots & β_{k} \end{matrix}]}^{^{'}}$ , the original simple homogeneous model, or “initial model” in Casetti’s (1972) formulation.

In model 1, we allow the mechanism of spatial dependence to adapt to the peculiarities of each spatial unit. ρ denotes an overall level of common spatial dependence (let us say, associated with the model), which is flexible to the local conditions, as in equation 1c. Matrix H is an (R × R) diagonal matrix that reflects these local peculiarities; they are measured through a function h of a certain variable (or group of variables), z_i , and a parameter α. As before, we assume that the function h and the parameter are constant across space (the assumption can be relaxed). Note that, if α is zero, according to equation 1e, the mechanism of spatial dependence can be simplified as a SLM. W is the usual weighting matrix.

Finally, we also allow for heteroskedasticity in the error term of equation 1. As stated in equation 1d, the covariance matrix of this vector is the product of a scalar, σ², a common term of variance, and an (R × R) diagonal D matrix. The terms of the main diagonal of the latter matrix are a function, d, of a variable (or group of variables), n, and a common parameter, δ. The same comments apply in relation to the constancy of both d and δ. Note also that the error term is homoskedastic if δ = 0.

The estimation of the model of equation 1, as it has been specified, is a difficult task. However, in our case, we are concerned only with the question of testing for instabilities, which simplifies matters considerably. It is clear that our approach is similar to that of Breusch and Pagan when they developed their well-known test of heteroskedasticity (Breusch and Pagan 1979). This approach does not require a specific form of the p_j (−), h(−), and d(−) functions. In short, these functions may remain unknown if our objective is just to test for breaks. The basic information that we need is the indicators of heterogeneity, the variables z, g_j , and n. We only need to assume that these functions, at the origin, verify that $p_{j} (0) = κ_{j 2} > 0; h (0) = κ_{1} > 0; d (0) = κ_{3} > 0$ with $κ_{j 2}$ , $κ_{1}$ , and $κ_{3}$ are finite constants.

The log likelihood function of model 1, assuming normality, is as usual:

l (y; γ) = - \frac{R}{2} ln (2 π) - \frac{R}{2} ln σ^{2} + ln |A| - \frac{1}{2} ln |D| - \frac{1}{2 σ^{2}} {(A y - X m)}^{'} D^{- 1} (A y - X m),

where γ=[β, ρ, σ², μ, α, δ]’ is the (K × 1) vector of parameters, being K = (p + 1) k + 4. The score vector is the following:

g (γ) = [\begin{array}{l} l_{β} \\ l_{ρ} \\ l_{σ^{2}} \\ l_{μ} \\ l_{α} \\ l_{δ} \end{array}] = - \frac{1}{σ^{2}} [\begin{matrix} m_{β}^{1'} X' D^{- 1} ε \\ σ^{2} t r A^{- 1} H W - ε' D^{- 1} H W y \\ \frac{R}{2} (1 - \frac{ε' D^{- 1} ε}{R σ^{2}}) \\ m_{μ}^{1'} X' D^{- 1} ε \\ ρ (σ^{2} t r A^{- 1} H_{1} \hat{Z} W - ε' D^{- 1} H_{1} \hat{Z} W y) \\ \frac{1}{2} (σ^{2} t r D^{- 1} D_{1} \hat{N} - ε' D^{- 1} D_{1} \hat{N} D^{- 1} ε) \end{matrix}],

where

l_{γ_{i}} = (\partial l / \partial γ_{i}); m_{β}^{1} = (\partial m / \partial β); m_{μ}^{1} = (\partial m / \partial μ)

. Moreover,

\begin{aligned} \begin{matrix} H_{1} = d i a g [\frac{\partial h (z_{i}^{^{'}} α)}{\partial (z_{i}^{^{'}} α)}; i = 1, 2, \dots, R] & \hat{Z} = d i a g [z_{i}; i = 1, 2, \dots, R] \end{matrix} \\ \begin{matrix} D_{1} = d i a g [\frac{\partial d (n_{i}^{^{'}} δ)}{\partial (n_{i}^{^{'}} δ)}; i = 1, 2, \dots, R] & \hat{N} = d i a g [n_{i}; i = 1, 2, \dots, R] \end{matrix} . \end{aligned}

Global stability of the elements of model 1 leads to the following composite hypothesis:

\begin{matrix} H_{0} : μ = α = δ = 0 \\ H_{A} : a t l e a s t o n e o f μ, α o r δ \neq 0 \end{matrix}\} .

Under these restrictions, the complex model of equation 1 simplifies as a simple SLM (y = ρWy + xβ + ϵ, ϵ~N(0;σ²I)), which facilitates the use of the Lagrange Multipliers to test for the hypothesis of equation 5. As is well known, these Multiplier are the quadratic form of the score vector,

{g (γ)}_{H_{0}}

, on the inverse of the information matrix,

{I M (γ)}_{H_{0}}^{- 1}

, both terms evaluated under the null hypothesis:

{L M}_{β ρ σ^{2}}^{S L M} = {g (γ)}_{H_{0}}^{^{'}} {I M (γ)}_{H_{0}}^{- 1} {g (γ)}_{H_{0}} ~ χ^{2} (p + 2) .

Asymptotically, the distribution of the Multiplier under the null is a chi-square with (p + 2) degrees of freedom. The expression of the score under the null hypothesis is very simple:

g {(γ)}_{| H_{0}} = \frac{1}{σ^{2}} [\begin{matrix} 0 \\ 0 \\ 0 \\ G' \hat{Y} ε \\ ρ ({ε' \hat{Z} W y - σ}^{2} t r A^{- 1} \hat{Z} W) \\ 0.5 (ε' \hat{N} ε - σ^{2} t r \hat{N}) \end{matrix}] = [\begin{matrix} \begin{array}{l} 0 \end{array} \\ q \end{matrix}] \to {\begin{matrix} \begin{array}{l} 0 = [\begin{matrix} 0 & 0 & 0 \end{matrix}]' \end{array} \\ q = [\begin{matrix} G' \hat{Y} ε / σ^{2} \\ ρ ((ε' \hat{Z} W y / σ^{2}) - t r A^{- 1} \hat{Z} W) \\ 0.5 ((ε' \hat{N} ε / σ^{2}) - t r \hat{N}) \end{matrix}], \end{matrix}

where G is an (R × p) matrix with the observations of the p variables associated with the break in vector β:

G = [\begin{matrix} g_{11} & g_{12} & g_{13} & \dots & g_{1 p} \\ g_{21} & g_{22} & g_{23} & \dots & g_{2 p} \\ g_{31} & g_{32} & g_{33} & \dots & g_{3 p} \\ \dots & \dots & \dots & \dots & \dots \\ g_{R 1} & g_{R 2} & g_{R 3} & \dots & g_{R p} \end{matrix}],

\hat{Y}

is a diagonal matrix

\hat{Y} = d i a g \{x_{i}^{^{'}} \hat{β}; i = 1, 2, . . ., R\}

, where

\hat{β}

is the restricted ML estimation of vector β under the null hypothesis;

\hat{ϵ}

is the vector of error terms of the corresponding SLM.

The obtaining of the information matrix is a more laborious and tedious task that requires the use of standard statistical techniques. The main results appear in Appendix A. Finally, we obtain a compact expression for the Lagrange Multiplier of equation 6:

{L M}_{β ρ σ^{2}}^{S L M} = {q^{'} I M}^{22} q \underset{a s}{~} χ^{2} (p + 2)

q has been defined in equation 7 whereas IM²² is a (p + 2) × (p + 2) covariance matrix defined in expression A5 of Appendix A. The SEM case is completely analogous to the SLM discussed so far and it is solved in Appendix B.

Testing for the Sources of Instability

The Multipliers of the A General Model of Spatial Dependence with Heterogeneity section are useful to test for the assumption of stability in the elements of a spatial econometric model. The problem that we now propose is to identify the specific sources of instability of the model. Note that the null hypothesis of equation 5 is composite. The rejection of this null does not necessarily imply that all the three elements (regression coefficients, spatial dependence parameter, and variance term) are unstable across space but that some (all) of them are. Furthermore, there is a great distance between the models of the null and the alternative hypotheses. The first is a simple SLM, or SEM, model that is perfectly homogenous across space whereas that of the alternative includes instabilities in all its elements. For these reasons, we think that it would be interesting to identify which elements are unstable across space in order to amend the estimation.

On the basis of the framework presented in the A General Model of Spatial Dependence with Heterogeneity section, we may think of, at least, three possibilities for dealing with this problem. The first is the development of more specific Lagrange Multipliers, directed at testing a single null hypothesis. This is the approach of Mur, López, and Angulo (2009) who obtain a collection of raw specialized Lagrange Multipliers for different null hypotheses of interest. It is shown that these Multipliers are very sensitive to any symptom of instability but, also, that they are not a good technique for identifying the causes of this instability. The Lagrange Multipliers, as is well known (Davidson 2000), are not robust to specification errors in the alternative or, equivalently, the null of equation 5 is compatible with different alternative hypotheses. Accordingly, the second possibility involves a nonambiguous definition of the null and the alternative hypotheses of the test (i.e., we want to test for the stability of the coefficients of regression in a model where the parameters of spatial dependence and the variance of the error term are allowed to vary across space). In this case, we may use the marginal Multipliers. These tests are, in general, more specific and powerful (Anselin and Bera 1998); however, the estimation of the model of the alternative hypothesis is needed. As noted previously, this may be a complex task in some circumstances. The third alternative implies using the robust Multipliers as an improved and corrected version of the raw Multipliers after correcting for some biases that appear in the ML algorithm (Bera and Yoon 1993).

Considering the pros and cons, the robust Multipliers are, at the very least, a satisfactory alternative: computationally, the solution is simple (we only need the estimation of the model of the null hypothesis) and offers a great flexibility (they may be adapted, without additional efforts, to test different cases of interest). There are also drawbacks: The robust Multipliers have less power than the raw or the marginal Multipliers, the adjustment of the raw Multipliers needs a larger sample size in order to perform correctly, and they tend to be undersized because the correction is not always effective, especially when the competing models of the alternative are similar (Anselin et al. 1996; Florax, Folmer, and Rey 2003; Mur and Angulo 2009). Other restrictions, such as normality, functional form, non-omitted variables, and so on, also apply as for every ML technique.

The rationale of the robust Multipliers is as follows. The likelihood function of the general model of equation 1, $L [γ]$ , depends on two groups of parameters: $γ_{1}^{^{'}} = [β^{'}; ρ; σ^{2}]$ , and those that introduce heterogeneity, $γ_{2}^{^{'}} = [μ^{'}; α; δ]$ . Let us suppose that the error term is homoskedastic and that the regression coefficients are the same across space (δ = 0 and μ = 0). We partition the vector γ₂ as $γ_{2}^{^{'}} = [γ_{21}^{^{'}}; γ_{22}^{^{'}}]$ where $γ_{21}^{^{'}} = [μ^{'}; δ]$ and $γ_{22}^{^{'}} = [α]$ . Given that vector γ₂₁ is zero, the likelihood function collapses in $L_{1} [γ_{1}; γ_{22}]$ . The unidirectional test in relation to the stability of ρ, H₀: α=0, leads to a raw Lagrange Multiplier distributed as a centered chi-square $χ^{2} (1, 0)$ . Under alternative hypotheses of the type $H_{A} : γ_{22} = ξ / \sqrt{R}$ , with $ξ \neq 0$ , there is a noncentrality parameter, $χ^{2} (1, π_{1})$ , equal to $π_{1} = ξ^{'} I_{γ_{22} / γ_{1}} ξ$ with ${I M}_{γ_{22} / γ_{1}} = {I M}_{γ_{22} γ_{22}} - {I M}_{γ_{22} γ_{1}} {I M}_{γ_{1} γ_{1}}^{- 1} {I M}_{γ_{1} γ_{22}}$ , where “a/b” means “a conditioned to b.” This noncentrality parameter gives power to the test.

If we now assume that the correct likelihood function is $L_{2} [γ_{1}; γ_{21}]$ , where γ₂₂ is zero but γ₂₁ is different from zero, the previous (raw) Lagrange Multiplier is biased even if the null hypothesis, H₀: γ₂₂=0, holds. In particular, for alternatives of the type $H_{A^{'}} : γ_{21} = ζ / \sqrt{R}; ζ \neq 0$ , the test has a $χ^{2} (1, π_{2})$ distribution where $π_{2} = ζ^{'} {I M}_{\underset{22}{γ} \underset{21}{γ} / γ_{1}} {I M}_{γ_{22} / γ_{1}}^{- 1} {I M}_{γ_{22} γ_{21} / γ_{1}} ζ$ with ${I M}_{γ_{22} γ_{21} / γ_{1}} = {I M}_{γ_{22} γ_{21}} - {I M}_{γ_{22} γ_{1}} {I M}_{γ_{1} γ_{1}}^{- 1} {I M}_{γ_{1} γ_{21}}$ and ${I M}_{γ_{22} / γ_{1}} = {I M}_{γ_{22} γ_{22}} - {I M}_{γ_{22} γ_{1}} {I M}_{γ_{1} γ_{1}}^{- 1} {I M}_{γ_{1} γ_{22}}$ . The factor of noncentrality, Π₂, will be zero only if the terms of the score associated with $γ_{22}$ and $γ_{21}$ ( $g_{(γ_{22})_{|H_{0}}}$ and $g_{(γ_{21})_{|H_{0}}}$ , respectively) are independent. In the case of model 1, it can be shown that, whichever the partition chosen, $g_{(γ_{22})_{|H_{0}}}$ and $g_{(γ_{21})_{|H_{0}}}$ will not be orthogonal and the corresponding raw Lagrange Multiplier for the hypothesis H₀: γ₂₂ = 0 will tend to, unduly, reject the null.

The proposal of Bera and Yoon (1993, 652) “is to construct a size-resistant test … where we adapt the statistic for the nuisance parameter.” The correction consists of adjusting the bias that appears both in the score and in the covariance matrix. In the preceding example:

\begin{aligned} \begin{matrix} H_{0} : γ_{22} = 0 \\ H_{A} : N o_{} H_{0} \end{matrix}\} \\ {L M}_{γ_{22} / γ_{21}}^{*} = {[g_{(γ_{22})_{|H_{0}}} - b (g_{(γ_{22})_{|H_{0}}})]}^{'} {[I_{γ_{22} / γ_{1}} - b (I_{γ_{22} / γ_{1}})]}^{- 1} \\ [g_{(γ_{22})_{|H_{0}}} - b (g_{(γ_{22})_{|H_{0}}})] \underset{a s}{~} χ^{2} (1), \end{aligned}

where the biases are, respectively (Anselin et al. 1996):

\begin{aligned} b (g_{(γ_{22})_{|H_{0}}}) = {I M}_{γ_{22} γ_{21} / γ_{1}} {I M}_{γ_{21} / γ_{1}}^{- 1} g_{(γ_{21})_{|H_{0}}} \\ b (I_{γ_{22} / γ_{1}}) = I_{γ_{22} γ_{21} / γ_{1}} I_{γ_{21} / γ_{1}}^{- 1} I_{γ_{22} γ_{21} / γ_{1}} . \end{aligned}

The results needed in order to obtain the value of the test, for either partition desired of vector γ₂, appear in expressions 7, A1, and A2 of Appendix A, for the SLM case, and in expressions B4 and B5 of Appendix B, for the SEM case. Table 1 shows the complete relation of robust Multipliers that extend the global Multiplier of equation 7.

Table 1.

Robust Multipliers for Testing the Hypothesis of Stability

Hypotheses	Robust to	Unrestricted Parameters	Statistics	Asymptotic Distribution
Instability in σ² H₀:δ=0 γ₂₂=[δ]	Instability in ρ and β γ₂₁=[α,μ]	ρ, β, σ² γ₁=[ρ, β, σ²]	${L M}_{σ^{2} / β ρ}^{* S L M}$ ${L M}_{σ^{2} / β ρ}^{* S L M}$	$χ^{2} (1)$
Instability in β H₀:μ=0 γ₂₂=[μ]	Instability in ρ and σ² γ₂₁=[α,δ]	ρ, β, σ² γ₁=[ρ, β, σ²]	${L M}_{β / ρ σ^{2}}^{* S L M}$ ${L M}_{β / ρ σ^{2}}^{* S L M}$	$χ^{2} (p + 1)$
Instability in ρ H₀:α=0 γ₂₂=[α]	Instability in σ² and β γ₂₁=[μ,δ]	ρ, β, σ² γ₁=[ρ, β, σ²]	${L M}_{ρ / {β σ}^{2}}^{* S L M}$ ${L M}_{ρ / {β σ}^{2}}^{* S L M}$	$χ^{2} (1)$
Instability in σ² and β H₀:δ=μ=0 γ₂₂=[δ,μ]	Instability in ρ γ₂₁=[α]	ρ, β, σ² γ₁=[ρ, β, σ²]	${L M}_{{β σ}^{2} / ρ}^{* S L M}$ ${L M}_{{β σ}^{2} / ρ}^{* S L M}$	$χ^{2} (p + 1)$
Instability in σ² and ρ H₀:δ=α=0 γ₂₂=[δ,α]	Instability in β γ₂₁=[μ]	ρ, β, σ² γ₁=[ρ, β, σ²]	${L M}_{ρ σ^{2} / β}^{* S L M}$ ${L M}_{ρ σ^{2} / β}^{* S L M}$	$χ^{2} (2)$
Instability in β and ρ H₀: μ=α=0 γ₂₂=[α,μ]	Instability in σ² γ₂₁=[δ]	ρ, β, σ² γ₁=[ρ, β, σ²]	${L M}_{β ρ / σ^{2}}^{* S L M}$ ${L M}_{β ρ / σ^{2}}^{* S L M}$	$χ^{2} (p + 1)$

Besides the collection of Multipliers, we also need a guide in order to read the information. The question, detecting the source of instability may be seen as a problem of model selection, which can be solved following different strategies. In general terms, we may think in two main approaches, the General-to-Specific, Gets, or the Specific-to-General, Stge. The two are procedures are well-known in spatial econometrics, whose properties have been discussed recently by Florax, Folmer, and Rey (2003, 2006), Hendry (2006), and Mur and Angulo (2009).

In our case, the Gets approach implies to begin with the most general model of equation 1, compatible with the data, which we will try to simplify afterwards. Obviously, we would need very precise information about the nature of the instabilities just to specify the nesting general model; then adequate algorithms will give us the corresponding estimates. We may think in a maximum-likelihood algorithm and in a sequence of, say, Wald tests where the null hypotheses will introduce more regularity into the model. The feasibility of this procedure depends, as said, on the availability of information in relation to the nature of the breaks. The Stge approach begins with the simplest model that is checked for misspecification. The initial equation is amended according to the errors detected in the process. This framework is the most popular among the practitioners and adapts better to our situation: We do not need full information in relation to the characteristics of the breaks; it is enough with partial information about the (suspected) causes. That is, our motivation is mainly testing the starting model and, if necessary, introducing adequate corrections. In this sense, we think that an Stge approach might be preferable. Our proposal, in the form of a flow chart, appears in Figure 1 .

Figure 1.

A strategy for interpreting the symptoms of instability in a spatial model (SLM or SEM) using robust Multipliers.

The first step consists of testing the composite null hypothesis of homogeneity in all the parameters (expression 5). If this hypothesis is not rejected, the process ends at this point with the specification of a homogeneous model. On the other hand, if the hypothesis is rejected, the procedure must continue by testing three single hypotheses: (i) homogeneity in the coefficients of regression; (ii) homogeneity in the spatial dependence parameter; and (iii) homogeneity in the variance of the error term. Only if one of these hypotheses is rejected and the other two are not, will the conclusion be unambiguous. When two of them are rejected, the next step should consist of testing the implied joint hypothesis: If the corresponding composite null hypothesis is rejected, we can corroborate that there are two simultaneous sources of heterogeneity. Finally, when the three individual tests all reject the null hypothesis, we conclude that the spatial model presents instability in the three groups of parameters.

A Monte Carlo Experiment

In this section, we evaluate the performance of the robust Multipliers derived in the Testing for the Sources of Instability section as well as the effectiveness of the model selection strategy outlined in Figure 1. The next subsection describes the characteristics of the experiment and the second focuses on the results.

Design of the Monte Carlo

We focus on a SLM model, specified under different degrees of instability. We leave aside the case of the SEM model, which produced very similar results.¹ The question is to detect the existence of instability in the components of a SLM model and to identify the origins of the instability, be they in the spatial dependence parameter, in the coefficients of regression and/or in the variance of the error term. It is clear that there may exist other forms of instability (in the functional form, in the exogenous variables, in the spatial interaction mechanisms, and so on); we concentrate on the cases, which, in our opinion, may be of greater interest for the applied literature.

As regards instability in the mechanism of spatial dependence, $H = d i a g \{h (z_{i}^{} α); i = 1, 2, . . . ., R\}$ , only one variable has been introduced into the break, z_i , whose values have been obtained from a uniform distribution; h(−) has been specified as an exponential function together with several values for $α$ (0, 0.50, and 1). The value zero in α corresponds to the control case, no break.

The same exponential function has been introduced into the function d(−) that operates in the heteroskedasticity process, $D = d i a g \{d (n_{i}^{} δ); i = 1, 2, . . . ., R\}$ . We have repeated the range of values of the parameter δ (0, 0.50, and 1), although the values of n_i proceed from an N(0,1) distribution (in order to intensify the heteroskedastic signal).

Finally, we have created instability in the coefficients of regression following a quasi-linear pattern:

m_{i j} = β_{j} [1 + μ g_{i}^{2}] .

Once again, only one variable has been introduced, g_i , whose values have been obtained from a uniform distribution. We have maintained the set of values in relation to parameter

μ

(0, 0.50, and 1).

The remaining characteristics of the exercise are as follows:

Only one regressor, plus the autoregressive term, has been used in the model. The coefficient associated with this regressor has been fixed to 3 and 2 for the intercept. Both magnitudes guarantee that, in the absence of spatial effects, the R ² of the model will be close to 0.8.

The observations of the regressor and of the random terms ϵ have been obtained from univariate normal distributions with zero mean and unit variance. That is, $σ_{ϵ}^{2} = 1$ in all cases.

We have used hexagon-based regular tessellation, of orders (7 × 7) and (20 × 20), which means that the sample size is 49 or 400 observations, respectively. The use of hexagons responds to the fact that they more closely resemble the properties of irregular systems.² For reasons of space, the results for the 400 sample size appear, in a more compact form, in Appendix C.

The weighting matrices have been specified under the assumption of contiguity. The resulting matrices are far from being densely connected weight matrices, in the terminology of authors such as Smith (2009) or Farber, Páez, and Volz (2009). More precisely, respectively for (7 × 7) and (20 × 20) lattices, the average number of connections per unit are 4.898 and 5.605 and the percentages of nonzeros values are 9.996 and 1.401. Afterwards, the resulting matrices have been row-standardized.

In each case, two values of parameter ρ have been simulated: 0.5 and 0.9.

Each combination has been repeated 1000 times.

Results of the Experiment

First we present the results corresponding to each Multiplier developed in Section 3, then the efectiveness of the strategy outlined in Figure 1 is checked.

Tables 2-9 show the percentage of rejection of the respective null hypothesis attained by each of the Multipliers. The composition of Tables 10-17 is similar but here the data refer to the effectiveness of the selection strategy shown in Figure 1. We highlight the DGP that should have been selected in each simulation.

Table 2.

Percentage of Rejection of the Null Hypothesis (R = 49). No Break

ρ	α	μ	δ	${L M}_{β ρ σ^{2}}^{S L M}$	${L M}_{ρ / {β σ}^{2}}^{* S L M}$	${L M}_{β / ρ σ^{2}}^{* S L M}$	${L M}_{σ^{2} / β ρ}^{* S L M}$	${L M}_{β ρ / σ^{2}}^{* S L M}$	${L M}_{ρ σ^{2} / β}^{* S L M}$	${L M}_{{β σ}^{2} / ρ}^{* S L M}$
0.5	0	0	0	3.3	7.4	7.3	3.8	6.8	5.6	6.1
0.9	0	0	0	4.4	9.7	9.3	3.9	8.6	7.4	8.5

Table 3.

Percentage of Rejection of the Null Hypothesis (R = 49). Break in ρ

ρ	α	${L M}_{β ρ σ^{2}}^{S L M}$	${L M}_{ρ / {β σ}^{2}}^{* S L M}$	${L M}_{β / ρ σ^{2}}^{* S L M}$	${L M}_{σ^{2} / β ρ}^{* S L M}$	${L M}_{β ρ / σ^{2}}^{* S L M}$	${L M}_{ρ σ^{2} / β}^{* S L M}$	${L M}_{{β σ}^{2} / ρ}^{* S L M}$
0.5	0.5	11.4	21.5	6.5	4.1	23.8	16.4	5.6
0.9	0.5	91.6	89.6	6.5	5.0	96.0	84.3	6.7
0.5	1	24.8	39.0	6.2	4.2	41.8	27.5	5.6
0.9	1	94.4	94.2	5.5	5.1	97.2	91.6	6.2

Table 4.

Percentage of Rejection of the Null Hypothesis (R = 49). Break in β

ρ	μ	${L M}_{β ρ σ^{2}}^{S L M}$	${L M}_{ρ / {β σ}^{2}}^{* S L M}$	${L M}_{β / ρ σ^{2}}^{* S L M}$	${L M}_{σ^{2} / β ρ}^{* S L M}$	${L M}_{β ρ / σ^{2}}^{* S L M}$	${L M}_{ρ σ^{2} / β}^{* S L M}$	${L M}_{{β σ}^{2} / ρ}^{* S L M}$
0.5	0.5	73.8	8.0	77.0	20.3	87.4	19.3	71.1
0.9	0.5	72.4	14.3	74.5	20.4	83.8	22.8	68.2
0.5	1	99.4	7.6	98.3	60.0	99.5	51.8	98.2
0.9	1	95.3	16.0	92.9	58.9	95.0	54.1	93.3

Table 5.

Percentage of Rejection of the Null Hypothesis (R = 49). Break in σ2

ρ	δ	${L M}_{β ρ σ^{2}}^{S L M}$	${L M}_{ρ / {β σ}^{2}}^{* S L M}$	${L M}_{β / ρ σ^{2}}^{* S L M}$	${L M}_{σ^{2} / β ρ}^{* S L M}$	${L M}_{β ρ / σ^{2}}^{* S L M}$	${L M}_{ρ σ^{2} / β}^{* S L M}$	${L M}_{{β σ}^{2} / ρ}^{* S L M}$
0.5	0.5	29.7	5.8	7.5	53.9	5.9	46.7	47.1
0.9	0.5	30.2	6.7	8.9	55.3	8.1	45.6	46.9
0.5	1	84.8	6.0	7.3	96.1	6.1	92.7	93.1
0.9	1	85.2	7.5	8.9	95.8	7.7	91.5	91.8

Table 6.

Percentage of Rejection of the Null Hypothesis (R = 49). Break in β and ρ

ρ	α	μ	${L M}_{β ρ σ^{2}}^{S L M}$	${L M}_{ρ / {β σ}^{2}}^{* S L M}$	${L M}_{β / ρ σ^{2}}^{* S L M}$	${L M}_{σ^{2} / β ρ}^{* S L M}$	${L M}_{β ρ / σ^{2}}^{* S L M}$	${L M}_{ρ σ^{2} / β}^{* S L M}$	${L M}_{{β σ}^{2} / ρ}^{* S L M}$
0.5	0.5	0.5	57.9	28.8	81.4	12.5	73.7	28.4	73.4
0.9	0.5	0.5	91.4	96.1	68.5	17.9	95.9	94.0	64.0
0.5	1	0.5	58.6	45.8	81.4	11.7	74.3	41.8	73.1
0.9	1	0.5	94.4	98.4	65.2	17.7	97.3	96.3	60.9
0.5	0.5	1	98.2	25.1	99.4	42.2	99.6	48.7	99.1
0.9	0.5	1	97.3	96.8	98.3	27.4	98.8	95.1	97.5
0.5	1	1	98.0	41.9	99.7	33.4	99.4	55.5	99.1
0.9	1	1	98.3	98.3	98.0	27.7	99.4	97.3	96.4

Table 7.

Percentage of Rejection of the Null Hypothesis (R = 49). Break in σ2 and ρ

ρ	α	δ	${L M}_{β ρ σ^{2}}^{S L M}$	${L M}_{ρ / {β σ}^{2}}^{* S L M}$	${L M}_{β / ρ σ^{2}}^{* S L M}$	${L M}_{σ^{2} / β ρ}^{* S L M}$	${L M}_{β ρ / σ^{2}}^{* S L M}$	${L M}_{ρ σ^{2} / β}^{* S L M}$	${L M}_{{β σ}^{2} / ρ}^{* S L M}$
0.5	0.5	0.5	40.8	23.0	7.3	51.5	24.3	54.7	44.4
0.9	0.5	0.5	95.2	86.3	6.0	28.1	96.2	86.9	23.9
0.5	1	0.5	51.9	37.9	6.8	48.6	41.7	63.5	40.0
0.9	1	0.5	97.0	93.0	5.6	26.3	98.2	92.1	21.3
0.5	0.5	1	87.4	17.6	6.9	94.5	19.2	93.4	91.0
0.9	0.5	1	97.6	77.3	6.0	77.6	90.1	95.2	71.6
0.5	1	1	89.2	30.5	6.8	93.3	32.1	93.5	89.0
0.9	1	1	98.5	86.0	5.4	73.6	93.4	96.1	67.3

Table 8.

Percentage of Rejection of the Null Hypothesis (R = 49). Break in σ2 and β

ρ	μ	δ	${L M}_{β ρ σ^{2}}^{S L M}$	${L M}_{ρ / {β σ}^{2}}^{* S L M}$	${L M}_{β / ρ σ^{2}}^{* S L M}$	${L M}_{σ^{2} / β ρ}^{* S L M}$	${L M}_{β ρ / σ^{2}}^{* S L M}$	${L M}_{ρ σ^{2} / β}^{* S L M}$	${L M}_{{β σ}^{2} / ρ}^{* S L M}$
0.5	0.5	0.5	79.5	7.8	71.5	38.8	83.0	33.3	79.5
0.9	0.5	0.5	77.9	14.4	68.6	38.9	80.7	35.0	75.3
0.5	1	0.5	99.5	8.0	98.5	20.5	99.5	19.1	97.9
0.9	1	0.5	95.4	19.5	93.6	20.8	96.1	28.9	92.2
0.5	0.5	1	93.5	7.4	59.7	87.4	67.4	80.6	95.6
0.9	0.5	1	90.6	12.4	57.6	87.4	66.4	79.7	92.9
0.5	1	1	99.4	7.8	95.7	58.3	98.8	51.6	98.0
0.9	1	1	95.7	17.8	90.6	58.3	94.9	55.9	93.1

Table 9.

Percentage of Rejection of the Null Hypothesis (R = 49). Break in β, ρ and σ2

ρ	α	μ	δ	${L M}_{β ρ σ^{2}}^{S L M}$	${L M}_{ρ / {β σ}^{2}}^{* S L M}$	${L M}_{β / ρ σ^{2}}^{* S L M}$	${L M}_{σ^{2} / β ρ}^{* S L M}$	${L M}_{β ρ / σ^{2}}^{* S L M}$	${L M}_{ρ σ^{2} / β}^{* S L M}$	${L M}_{{β σ}^{2} / ρ}^{* S L M}$
0.5	0.5	0.5	0.5	69.7	28.1	75.2	42.0	66.0	51.3	83.8
0.9	0.5	0.5	0.5	91.7	96.0	63.1	29.9	93.2	95.7	70.3
0.5	1	0.5	0.5	71.3	47.3	75.8	41.3	67.6	62.2	83.9
0.9	1	0.5	0.5	94.6	97.7	60.9	26.2	96.7	97.6	65.7
0.5	0.5	1	0.5	99.0	26.6	99.2	21.9	99.5	34.5	98.9
0.9	0.5	1	0.5	97.2	96.3	97.2	21.9	98.0	94.6	97.4
0.5	1	1	0.5	98.2	43.7	99.3	22.0	99.4	46.8	99.2
0.9	1	1	0.5	97.9	97.6	96.6	20.5	98.5	96.9	96.8
0.5	0.5	0.5	1	90.9	22.6	61.6	90.7	52.6	89.4	97.6
0.9	0.5	0.5	1	96.2	90.9	52.8	79.2	86.9	98.6	89.5
0.5	1	0.5	1	91.2	38.6	63.1	90.1	54.9	91.6	97.0
0.9	1	0.5	1	97.3	94.4	51.7	73.5	91.3	98.7	86.3
0.5	0.5	1	1	98.9	23.3	97.9	66.4	97.3	67.9	99.4
0.9	0.5	1	1	97.9	92.6	94.5	65.9	96.2	97.0	97.9
0.5	1	1	1	98.3	37.4	98.1	67.1	96.7	74.6	99.5
0.9	1	1	1	98.3	94.9	94.3	61.6	96.8	97.8	97.9

Table 2 corresponds to the SLM model with stability in all its parameters. As expected, the percentage of rejection of the composite null hypothesis of homogeneity (the empirical size of the Multiplier $L M_{β ρ σ^{2}}^{S L M}$ ) is low. The model selection strategy works quite well, since the percentage of times that we select the correct model is around 97 percent (see Table 10), even for a sample size of forty-nine observations.

Table 10.

Percentage that Each DGP is Selected^a (R = 49). No Break

ρ	α	μ	δ	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
0.5	0	0	0	96.7	0.5	0.6	0.3	0.9	0.2	0.3	0.2
0.9	0	0	0	95.6	0.3	0.8	0.2	2.2	0.2	0.0	0.3

Note: ^a(1) SLM; (2) SLM, break in ρ; (3) SLM, break in β (4) SLM break in σ²; (5) SLM, break in β and ρ (6) SLM, break in σ² and ρ; (7) SLM, break in σ² and β; (8) SLM, break in β, ρ and σ².

Table 3 corresponds to cases in which the data are generated with instability in the spatial dependence parameter. In this case, both the global Multiplier, $L M_{β ρ σ^{2}}^{S L M}$ , and the specific Multiplier, $L M_{ρ / β σ^{2}}^{* S L M}$ tend to reject the null hypothesis. Table C1 of Appendix C confirms this tendency for the case of R = 400. As a consequence, the probability of making the correct decision, Table 11 , is around 80 percent (90 percent when R = 400).

Table 11.

Percentage that Each DGP is Selected^a (R = 49). Break in ρ

ρ	α	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
0.5	0.5	88.6	5.3	1.1	0.5	1.7	0.9	0.3	0.2
0.9	0.5	8.4	77.3	0.9	0.7	4.3	3.2	0.3	0.8
0.5	1	75.2	15.4	1.1	0.5	2.5	1.5	0.3	0.3
0.9	1	5.6	82.7	0.3	0.3	4.1	3.8	0.2	0.7

Note: ^a(1) SLM; (2) SLM, break in ρ; (3) SLM, break in β; (4) SLM break in σ²; (5) SLM, break in β and ρ (6) SLM, break in σ² and ρ; (7) SLM, break in σ² and β; SLM, break in β, ρ and σ².

Tables 4 and 12 correspond to situations where the heterogeneity comes from the coefficient of regression. In this case, the global and the specific Multipliers, $L M_{β ρ σ^{2}}^{S L M}$ and $L M_{β / ρ σ^{2}}^{* S L M}$ , are very sensitive, but the results are not as good as before because the heteroskedasticity Multiplier also tends to reject its null. This unexpected reaction of the $L M_{σ^{2} / β ρ}^{* S L M}$ statistic introduces confusion into the functioning of the specification strategy; the result is that the probability of getting a correct decision is very low, 40 percent, even in the large sample case.

Table 12.

Percentage that Each DGP is Selected^a (R = 49). Break in β

ρ	μ	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
0.5	0.5	26.2	1.1	41.7	1.7	5.3	0.0	16.9	1.3
0.9	0.5	27.6	0.9	38.5	1.5	8.8	0.3	14.0	3.6
0.5	1	0.6	0.1	36.5	0.5	2.5	0.0	54.2	5.0
0.9	1	4.7	0.2	31.7	1.4	6.1	0.2	45.6	9.4

Tables 5 and 13 refer to the purely heteroskedastic processes, well interpreted by our Multipliers. In most cases, only the relevant statistics, $L M_{β ρ σ^{2}}^{S L M}$ and $L M_{σ^{2} / β ρ}^{* S L M}$ , react with very high percentages. This consistent behavior assures a good performance of the discrimination strategy with ratios of correct identification well above 70 percent (90 percent in the case of R = 400).

Table 13.

Percentage that Each DGP is Selected^a (R = 49). Break in σ2

ρ	δ	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
0.5	0.5	70.3	0.3	0.3	22.5	0.8	1.7	2.7	1.2
0.9	0.5	69.8	0.2	0.5	22.6	1.6	1.3	2.3	1.6
0.5	1	15.2	0.0	0.2	74.8	0.0	2.9	4.1	2.8
0.9	1	14.8	0.1	0.2	73.5	0.3	2.8	4.1	4.2

Note: ^a(1) SLM; (2) SLM, break in ρ; (3) SLM, break in β; (4) SLM break in σ²; (5) SLM, break in β and ρ; (6) SLM, break in σ² and ρ; (7) SLM, break in σ² and β; (8) SLM, break in β, ρ and σ².

The other cases included in Tables 6-9 and 14-17 are related to situations in which the data are obtained by using, simultaneously, several sources of instability. The combination of instability in the spatial dependence parameter and in the coefficient of regression is often confused with instability in all the parameters. This is the most problematic case, because, the other combinations (the last three tables) are correctly interpreted. As before, the probability of selecting the right model is much higher for the larger size sample and reaches values of 96.6 percent (for the combinations of instability in the spatial dependence parameter and in the variance of the error term), 97.0 percent (for instability in the coefficient of regression and in the variance), and 100 percent (for instability in all the parameters).

Table 14.

Percentage that Each DGP is Selected^a (R = 49). Break in β and ρ

ρ	α	μ	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
0.5	0.5	0.5	42.1	0.5	25.4	0.7	19.6	0.0	7.2	4.0
0.9	0.5	0.5	8.6	22.6	0.2	0.0	50.9	1.8	0.0	15.7
0.5	1	0.5	41.4	0.8	16.5	0.4	30.5	0.1	4.6	5.6
0.9	1	0.5	5.6	27.4	0.0	0.0	49.5	2.9	0.0	14.6
0.5	0.5	1	1.8	0.0	40.3	0.0	15.6	0.0	32.5	9.5
0.9	0.5	1	2.7	0.4	1.5	0.0	68.4	0.1	0.5	26.4
0.5	1	1	2.0	0.0	34.7	0.0	29.9	0.0	21.5	11.8
0.9	1	1	1.7	1.1	0.6	0.0	69.2	0.3	0.2	26.9

Note: ^a(1) SLM; (2) SLM, break in ρ; (3) SLM, break in β; (4) SLM break in σ²; (5) SLM, break in β and ρ; (6) SLM, break in σ² and ρ; (7) SLM, break in σ² and β; SLM, break in β, ρ and σ².

Table 15.

Percentage that Each DGP is Selected^a (R = 49). Break in σ2 and ρ

ρ	α	δ	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
0.5	0.5	0.5	59.2	4.4	0.5	20.7	1.3	8.3	1.8	1.9
0.9	0.5	0.5	4.8	56.5	0.6	4.4	4.0	22.2	0.0	1.3
0.5	1	0.5	48.1	11.4	0.9	18.1	1.4	14.2	1.1	2.2
0.9	1	0.5	3.0	64.1	0.4	2.2	3.3	22.1	0.0	1.9
0.5	0.5	1	12.6	0.4	0.0	66.7	0.1	13.6	3.5	2.8
0.9	0.5	1	2.4	17.5	0.7	17.7	0.9	54.7	0.5	3.9
0.5	1	1	10.8	1.2	0.0	56.3	0.3	24.8	3.3	2.9
0.9	1	1	1.5	22.1	0.5	10.5	1.1	58.8	0.2	3.6

Table 16.

Percentage that Each DGP is Selected^a (R = 49). Break in σ2 and β

ρ	μ	δ	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
0.5	0.5	0.5	20.5	0.9	33.6	10.0	3.8	0.6	24.1	2.3
0.9	0.5	0.5	22.1	0.6	30.1	10.1	7.8	0.4	19.8	5.0
0.5	1	0.5	0.5	0.3	71.8	0.1	6.1	0.0	18.8	1.6
0.9	1	0.5	4.6	0.2	58.3	0.4	15.4	0.0	16.0	3.8
0.5	0.5	1	6.5	0.1	7.1	32.3	0.9	2.0	45.8	4.4
0.9	0.5	1	9.4	0.1	5.6	31.6	1.7	1.7	39.9	8.9
0.5	1	1	0.6	0.2	36.8	2.4	3.2	0.3	51.4	4.1
0.9	1	1	4.3	0.3	30.1	3.0	7.2	0.5	43.6	9.7

Note: ^a(1) SLM; (2) SLM, break in ρ; (3) SLM, break in β; (4) SLM break in σ²; (5) SLM, break in β and ρ (6) SLM, break in σ² and ρ; (7) SLM, break in σ² and β; (8) SLM, break in β, ρ and σ².

Table 17.

Percentage that Each DGP is Selected^a (R = 49). Break in β, ρ and σ2

ρ	α	μ	δ	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
0.5	0.5	0.5	0.5	30.3	0.0	17.3	7.9	14.8	0.6	18.2	10.5
0.9	0.5	0.5	0.5	8.3	18.5	0.1	0.9	44.0	10.8	0.0	17.4
0.5	1	0.5	0.5	28.7	0.1	10.5	5.5	23.3	2.8	13.6	15.4
0.9	1	0.5	0.5	5.4	23.6	0.1	0.6	45.0	10.2	0.0	15.1
0.5	0.5	1	0.5	1.0	0.1	56.0	0.1	20.9	0.0	16.2	5.6
0.9	0.5	1	0.5	2.8	0.8	1.7	0.1	73.3	0.5	0.4	20.4
0.5	1	1	0.5	1.8	0.0	42.3	0.1	33.9	0.0	12.0	9.8
0.9	1	1	0.5	2.1	1.2	1.0	0.1	75.5	0.8	0.2	19.1
0.5	0.5	0.5	1	9.1	0.0	3.0	29.4	2.5	2.1	36.2	17.7
0.9	0.5	0.5	1	3.8	6.4	0.0	6.1	11.8	31.3	0.6	39.9
0.5	1	0.5	1	8.8	0.1	1.9	24.2	4.2	6.0	27.3	27.5
0.9	1	0.5	1	2.7	9.2	0.0	3.8	15.0	32.8	0.2	36.2
0.5	0.5	1	1	1.1	0.1	24.4	1.4	8.4	0.0	49.8	14.8
0.9	0.5	1	1	2.1	0.6	1.5	1.0	30.7	2.4	3.4	58.3
0.5	1	1	1	1.7	0.1	19.3	1.0	12.5	0.2	40.6	24.6
0.9	1	1	1	1.7	0.8	0.8	1.0	35.8	2.7	2.1	55.1

To conclude, we can say that these results are quite interesting. According to our data, the most difficult situation occurs when the variance of the error term is constant but there is a break in the coefficient of regression. In this case, the strategy outlined in Figure 1 points, wrongly, toward heteroskedastic patterns in the error term. In the other cases, the specification strategy appears to be highly efficient. The results are especially satisfactory when the sample size increases.

An Application to the Spatial Analysis of the 2008 Spanish General Elections

Explaining party choice is one of the key challenges facing analysts of political behavior. There are three main approaches to the problem: the sociological, the psychological, and the rational (Criado 2008). The sociological approach is centered on voters' sociodemographic characteristics, since it argues that people vote for the party closest to their own ideological position or for the party that best represents the interest of their social class or ethnic background. For supporters of the psychological approach, variables such as voters' psychological emotions and attitudes toward parties are the variables that best predict their voting choices (Campbell et al. 1960). Finally, the rational approach maintains that voters behave rationally in the sense that they vote for the party that maximizes their expected utility.

In spite of the huge theoretical differences that exist between these three approaches, they coincide in their preference for citizens' personal characteristics and their attitudes or preferences in order to explain electoral behavior. Variables such as social class, ethnicity, or religion are factors of great importance for explaining citizens' voting decisions (Lipset and Rokkan 1967; Wolfinger and Rosenstone 1980; Van der Eijk, Franklin, and Oppenhuis 1996; Nieuwbeerta and Ultee 1999; Laitin 1998). Of them, social class is especially important because, in almost all Western countries, blue-collar working class are more likely to vote for left-wing parties than people from other social classes (Nieuwbeerta and Ultee 1999). Furthermore, it is also very important to analyze the context in which citizens are located (Anduiza 2002; Van Egmond, De Graaf, and Van der Eijk 1998). This implies paying attention to the contextual features of the municipalities in which voters live, such as level of employment, earnings, wealth, and so on (social welfare, in sum, Ward and O’Loughlin 2002).

However, consideration of the previous two types of social and geographical variables may not suffice, because space also matters (O’Loughlin 2002; O’Loughlin, Flint, and Anselin 1994). This issue has usually been omitted in the previous literature. One exception is the work by Kim, Elliott, and Wang (2003) who identify spatial patterns for the case of county-level U.S. presidential election outcomes from 1988 to 2000. In line with their work, in this article, we analyze the choice of the socialist party in the 2008 Spanish general elections at municipality level. Specifically, our explained variable will be the ratio of voting for the socialist party, the Partido Socialista Obrero Español (PSOE) and winner of the elections, through 3,234 Spanish municipalities with more than one thousand inhabitants accounting for 96 percent of the Spanish population. The source of data is the Ministerio del Interior (Gobierno de España 2008). The map of the voting ratio for the PSOE party is shown in Figure 2 . As can be seen, the socialist party received the highest percentage of votes in the south-east and the north-west of Spain.

Figure 2.

Map of ratio of voting for the PSOE party in 2008 elections, in percentages (WPSOE08).

To explain voting behavior, we examined several economic variables referring to the population in terms of gender, nationality, unemployment rates, population density, consumption capacity, and activity level. This information, also at municipality level, is obtained from the Anuario Económico de España (La Caixa, 2008).

In our application, we take into account not only the expected spatial autocorrelation of the specification but also any other symptoms of instability in the spatial econometric model. On the basis of the discussion in the Testing for the Sources of Instability section, we will consider the unemployment rate as the candidate for generating instability due to its potential disturbing effects on social welfare.

Following Kim, Elliott, and Wang (2003), we formulate a basic model in which the voting ratio for the socialist party is explained by a set of variables related to the sociodemographic characteristics of the voters and also by other contextual variables, referring to the economic situation of the municipalities in which the voters live, as follows:

W P S O E 08 = ι β_{0} + X β_{1} + Z β_{2} + ϵ,

where

W P S O E 08

is the (R × 1) vector of votes, in percentages, for the PSOE in the elections of 2008 in each municipality;

ι

represents a vector of ones; X is an (R × k) matrix of explanatory variables referring to the citizens' personal characteristics; and Z is an (R × s) matrix of explanatory variables referring to relevant aspects that characterize the municipality. As candidates for X, we consider the following variables: sex, measured as the percentage of men in each municipality (denoted by PERMEN); nationality, measured as the percentage of Spaniards in the total population (PERSPANISH); and, finally, a variable that reflects the consumption capacity of each municipality (CONSCAP), elaborated as a synthetic index of six other variables (see the Anuario for the details). As candidates for Z, the following variables are considered: the population density of each municipality (POPDEN); the annual average increase of population between 2004 and 2008 (INCPOP); and two factors (FACT1 and FACT2) that approximate the economic situation of each municipality.³ The specification is completed with a variable of inertia (WPSOE04) that represents the percentage of votes for the PSOE party in the preceding 2004 elections. We have also introduced a spatial tendency through the coordinates of each municipality (XC and YC). Finally, a collection of dummy variables is included in the model with the purpose of diminishing the impact of different anomalous observations (they account for 3 percent of the sample).

As can be seen in Figure 2, the data are extracted from a very heterogeneous collection of spatial units. There are some municipalities with many neighbors (in the centre or in the East of Spain), whereas others are almost isolated (in the West of the peninsula and in the islands). Because of this heterogeneity, we decided to specify a weighted matrix that combines distance (two municipalities are considered neighbors if they are less than 200 km apart) with the nearest neighbors criteria (assuring, at least, 4 neighbors for the most isolated municipalities). The resulting binary matrix is also far from being a densely connected weight matrix: the average number of connections is 4.944 and the percentage of nonzeros values is 0.153. Afterwards, the binary matrix has been row-standardized.

Results obtained for different models estimated are gathered in Table 18. First, the static model defined in equation 13 is estimated. Then, the null of no spatial autocorrelation is tested through the usual standard specification statistics. The static model is clearly misspecified and the evidence points to the existence of a SEM structure in the errors. Consequently, a SEM model has been estimated under the assumption of the homogeneity of all the parameters (second block of Table 18). Then, we apply the battery of robust Multipliers derived in Testing for the Sources of Instability section to decide whether there is heterogeneity in the data that it is not captured by the model. For all the cases, we have considered that differences in the municipalities' unemployment rate are the cause of heterogeneity. As can be seen in Table 18, all parameters, the coefficient of regression, the variance of the error term and the spatial dependence parameter suffer from instability.

Table 18.

Estimated Models for the Case of the 2008 Spanish General Elections (PSOE vote)^a,b

	STATIC		SEM with stability in all parameters
	Coefficient	t Ratio	Coefficient	t Ratio
c	−15.722*	−4.717	−13.2815*	−3.61
PERMEN	0.026	0.496	−0.0534	−1.21
PERSPANISH	0.097*	9.544	0.0427*	3.97
CONSCAP	−0.003*	−2.384	−0.0023*	−2.31
POPDEN	0.000	−0.347	0.0001	1.23
INCPOP	−0.228*	−8.089	−0.0929*	−3.68
FACT1	−0.115	−1.425	−0.0457	−0.68
FACT2	−0.563*	−6.121	−0.2215*	−3.06
WPSOE04	0.829*	126.811	0.8517*	121.82
XC	0.004*	14.339	0.0031*	6.44
YC	0.003*	8.040	0.0042*	7.27
ρ			0.581*	4.89
Diagnostic measures
MORAN	33.824*
LMEL	835.024*
LMLE	0.001
SARMA	1129.0*
${L M}_{β ρ σ^{2}}^{S E M}$			37.457*
${L M}_{ρ \| β σ^{2}}^{* S E M}$			10.112*
${L M}_{β \| ρ σ^{2}}^{* S E M}$			26.312*
${L M}_{σ^{2} \| β ρ}^{* S E M}$			4.179*
${L M}_{β ρ \| σ^{2}}^{* S E M}$			36.425*
${L M}_{ρ σ^{2} \| β}^{* S E M}$			11.144*
${L M}_{β σ^{2} \| ρ}^{* S E M}$			30.492*
R ²	0.885		0.929

Note: ^aAn asterisk means that the corresponding null hypothesis is rejected at the 5% level of significance. ^bMORAN means Moran’s I statistics of spatial dependence, distributed as an N(0;1) after standardizing (Anselin 1988a); LMEL means Lagrange Multiplier for spatial correlation in the errors of the equation robust to the omission of lags in the main equation (Anselin et al. 1996); LMLE means Lagrange Multiplier for omitted lags in the main equation robust to correlation in the errors (Anselin et al. 1996); SARMA means Lagrange Multiplier for omitted lags in the main equation and spatial correlation in the errors of the equation (Anselin 1988a). The first two Multipliers, LMEL and LMLE, are a chi-square with 1 degree of freedom under the null and the SARMA is a chi-square with 2 degrees of freedom.

In order to continue with the discussion, we would need to reestimate the SEM introducing mechanisms of instability in the coefficients of regression and of spatial dependence, depending on the unemployment rate of each municipality. However, there is the problem of the unknown functional forms associated with these elements of instability. At the moment, it is quite difficult even to suggest an acceptable alternative. As an intermediate solution, we are going to obtain the local estimation of the SEM model as a case of complete heterogeneity. Following Pace and Lesage (2004) and Mur, López, and Angulo (2010), we take as our reference model:

\begin{matrix} {WPSOE08}_{r}^{(m_{r})} = ι β_{0}_{, r}^{(m_{r})} + X_{r}^{(m_{r})} β_{1,}_{r}^{(m_{r})} + Z_{r}^{(m_{r})} β_{2,}_{r}^{(m_{r})} + ε_{r}^{(m_{r})} \\ ε_{r}^{(m_{r})} = ρ_{r}^{(m_{r})} W_{r}^{(m_{r})} ε_{r}^{(m_{r})} + η_{r}^{(m_{r})} \\ η_{r}^{(m_{r})} ~ N [0, σ_{r, m_{r}}^{2} I_{m_{r}}] \end{matrix}} .

The indexes r and m mean that the data correspond to the local system defined by the m_r nearest neighbors of the index r. For example, where

i_{k} \in N (r)

, being N(r) the bundle of indexes that are the nearest neighbors to point r. The same criterion is used for the other variables of the model,

X_{r}^{(m_{r})}

and

Z_{r}^{(m_{r})}

. Matrix

{W_{r}^{(m_{r})}}^{}

refers to the weighting matrix obtained for the respective local system, defined with the same connectivity criteria used to specify the W matrix. Finally,

ρ_{r}^{(m_{r})}

β_{r}^{(m_{r})}

, and

σ_{r, m_{r}}^{2}

are the local parameters of interest. We refer to m_r as the zoom size (equivalent to window size in nonparametric literature). In our case, we use a flexible zoom size in the sense that the set N(r) adapts to the point r. To be more exact, we use all the other municipalities that belong to the same Autonomous Community as neighbors of municipality r (Eurostat NUTS II spatial units). This decision implies that the zoom size varies for each municipality ranging from a size of 29 to 580.

The first consequence of the local estimation is that the symptoms of instability are greatly reduced. In Figure 3 , we depict the spatial distribution of the global Multiplier ${L M}_{β ρ σ^{2}}^{S E M}$ corresponding to the local estimates. The null hypothesis of stability in the coefficients of the associated local estimation (using the unemployment rate as the variable that generates instability) is rejected in only 15.8 percent of the municipalities.

Figure 3.

$L M_{β ρ σ^{2}}^{S E M}$ of each estimation of the zoom estimation technique.

Figure 4 shows the local estimation of the spatial dependence parameter. As can be observed, there are important differences among municipalities ranging from patterns of negative to positive autocorrelation. The parameter of spatial dependence is significant for 45.4 percent of cases but negative in only 2.8 percent of these cases. Figure 5 shows the results corresponding to the effect of the voting ratio for the socialist party in the 2004 general elections (WPSOE04). This variable is significant in 99.9 percent of cases and takes values ranging from 0.72 to 1.20. From these results, we can conclude that inertia is a very important factor for explaining the results of the 2008 Spanish general elections, although its impact is not homogeneous among the different municipalities. The increase of population between 2004 and 2008 is also significant for 53 percent of cases, with negative estimates for 97 percent of cases (Figure 6). The gender variable is also significant for 52 percent of the municipalities and nationality in 43 percent of the local estimates.

Figure 4.

Spatial dependence estimated parameter with the zoom estimation technique.

Figure 5.

Estimation for effect of voting ratio for PSOE party in 2004 elections (WPSOE04) with the zoom estimation.

Figure 6.

Estimation for increase of population between 2004 and 2008 (INCPOP) with the zoom estimation.

Figure 7.

Estimation for percentage of men (PERMEN) variable with the zoom estimation.

Figure 8.

Estimation for percentage of Spaniards (PERSPANISH) variable with the zoom estimation.

Main Conclusions

In this article, we tried to relax the basic assumption of homogeneity in the parameters of a spatial dependence model. The purpose is to allow these specifications to better capture the symptoms of heterogeneity present in a cross-sectional sample (be they in the coefficients of regression, in the variance of the error term or in the mechanisms of spatial dependence). We propose a battery of robust Multipliers to test simple and composite hypotheses in a well-defined strategy to tackle the objective of selecting the correct model that underlies the data.

The Monte Carlo experiment carried out has shown that the results are very promising. The battery of single and composite Multipliers are well behaved, both in power and in size, which assures a correct functioning of the strategy developed in order to identify the causes of instability in the model; in most cases, we identify the right source of instability. The outcome of this battery may be to corroborate the assumption of stability or to confirm the suspicious of structural breaks. In the last case, our strategy identifies the causes of the instabilities but other aspects need to be discussed. Perhaps, one of the most important issues refers to the functional forms associated to the breaks. This is an open question, in relation to which the local estimation algorithms have shown to be helpful.

The utility of this methodology is tested with an application to the case of the 2008 Spanish general elections. A SEM model with heterogeneity in all parameters has been clearly identified. The resulting heterogeneous model has been estimated using a Zoom algorithm, as a case of complete heterogeneity.

Footnotes

Appendix A

Appendix B

Appendix C

More results of the Monte Carlo. The case of R = 400.

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors have received financial support from the project ECO-2009-10534-ECON of the Ministerio de Ciencia y Tecnología del Reino de España. Fernando A López like to thank the project 11897/PHCS/09 of the Fundación Séneca de la Región de Murcia.

Notes

References

Anduiza

2002. “Individual Characteristics, Institutional Incentives and Electoral Abstention in Western Europe.” European Journal of Political Research 41:643–73.

Anselin

1988a. Spatial Econometrics. Methods and Models. Dordrecht: Kluwer.

Anselin

1988b. “Model Validation in a Spatial Econometrics: A Review and Evaluation of Alternative Approaches.” International Regional Science Review 11:279–316.

Anselin

1988c. “Lagrange Multiplier Test Diagnostics for Spatial Dependence and Spatial Heterogeneity.” Geographical Analysis 20:1–17.

Anselin

1990. “Spatial Dependence and Spatial Structural Instability in Applied Regression Analysis.” Journal of Regional Science 30:185–207.

Anselin

Bera

1998. “Spatial Dependence In Linear Regression Models With An Introduction To Spatial Econometrics.” In Handbook of Applied Economic Statistics, edited by Ullah

Giles

, 237–90. New York: Marcel Dekker.

Anselin

Bera

Florax

Yoon

1996. “Simple Diagnostic Tests for Spatial Dependence.” Regional Science and Urban Economics 26:77–104.

Bera

Yoon

1993. “Specification Testing with Locally Misspecified Alternatives.” Econometric Theory 9:649–58.

Blommestein

1983. “Specification and Estimation of Spatial Econometric Models. A Discussion of Alternative Strategies for Spatial Economic Modelling.” Regional Science and Urban Economics 13:251–70.

10.

Breusch

Pagan

1979. “A Simple Test for Heteroscedasticity and Random Coefficient Variation.” Econometrica 47:1287–294.

11.

Brunsdon

Fotheringham

Charlton

1998a. “Spatial Nonstationarity and Autoreggresive Models.” Environment and Planning A 30:957–73.

12.

Brunsdon

Fotheringham

Charlton

1998b. “Geographically Weighted Regression-Modelling Spatial Non-Stationarity.” The Statistician 47:431–43.

13.

La Caixa. 2008. Anuario Económico de España 2008. Barcelona, Spain: Caja de Ahorros y Pensiones de Barcelona.

14.

Campbell

Converse

Miller

Stokes

1960. The American Voter. New York: John Wiley & Sons.

15.

Casetti

1972. “Generating Models by the Expansion Method. Application to Geographical Research.” Geographical Analysis 4:81–91.

16.

Cleveland

1979. “Robust Locally Weighted Regression and Smoothing Scatterplots.” Journal of the American Statistical Association 74:829–36.

17.

Cleveland

Devlin

1988. “Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting.” Journal of the American Statistical Association 83:596–610.

18.

Criado

2008. “The Effect of Party Mobilisation Strategies on the Vote: The PSOE and the PP in the 1996 Spanish Election.” European Journal of Political Research 47:80–100.

19.

Davidson

2000. Econometric Theory. New York: Blackwell.

20.

Farber

Páez

Volz

2009. “Topology and Dependency Tests in Spatial and Network Autoregressive Models.” Geographical Analysis 41:158–80.

21.

Florax

Folmer

1992. “Specification and Estimation of Spatial Linear Regression Models: Monte Carlo Evaluation of Pre-Test Estimators.” Regional Science and Urban Economics 22:405–32.

22.

Florax

Folmer

Rey

2003. “Specification Searches in Spatial Econometrics: The Relevance of Hendry’s Methodology.” Regional Science and Urban Economics 33:557–79.

23.

Florax

Folmer

Rey

2006. “A Comment on Specification Searches in Spatial Econometrics: The Relevance of Hendry’s Methodology: A Reply.” Regional Science and Urban Economics 36:300–308.

24.

Gobierno de España (Ministerio del Interior). 2008. “Elecciones a Cortes Generales 2008.” http://www.generales2008.mir.es. Accessed July 29, 2011.

25.

Hendry

2006. “A Comment on “Specification Searches in Spatial Econometrics: The Relevance of Hendry’s Methodology.”” Regional Science and Urban Economics 36:309–12.

26.

Kim

Elliott

Wang

2003. “A Spatial Analysis of County-Level Outcomes in US Presidential Elections: 1988-2000.” Electoral Studies 22:741–61.

27.

Laitin

1998. Identify in Formation: The Russian-Speaking Population in the Near-Abroad. Ithaca, NY: Cornell University Press.

28.

Lipset

S. M.

Rokkan

1967. “Cleavage Structures, Party Systems and Voter Alignments: An Introduction.” In Party Systems and Voter Alignments: Cross-National Perspectives, edited by Lipset

S. M.

Rokkan

, 1–64. New York: The Free Press.

29.

López

Mur

Angulo

2009. “Local Estimation of Spatial Autocorrelation Processes.” In Progress in Spatial Analysis: Theory, Computation and Thematic Applications, edited by Paez

Le Gallo

Buliung

Dall’Erba

, 111–40. Berlin: Springer.

30.

McMillen

1996. “One Hundred Fifty Years of Land Values in Chicago: A Nonparametric Approach.” Journal of Urban Economics 40:100–24.

31.

McMillen

McDonald

1997. “A Nonparametric Analysis of Employment Density in a Policentric City.” Journal of Regional Science 37:591–612.

32.

Mur

Angulo

2009. “Model Selection Strategies in a Spatial Setting: Some Additional Results.” Regional Science and Urban Economics 39:200–13.

33.

Mur

López

Angulo

2008. “Symptoms of Instability in Models of Spatial Dependence.” Geographical Analysis 40:189–211.

34.

Mur

López

Angulo

2009. “Testing the Hypothesis of Stability in Spatial Econometric Models.” Papers in Regional Science 88:409–44.

35.

Mur

López

Angulo

2010. “Instability in Spatial Error Models. An Application to the Hypothesis of Convergence in the European Case.” Journal of Geographical Systems 12:259–80.

36.

Nieuwbeerta

Ultee

1999. “Class Voting in Western Industrialized Countries, 1945-1990: Systematizing and Testing Explanations.” European Journal of Political Research 35:123–60.

37.

O’Loughlin

2002. “The Electoral Geography of Weimar Germany: Exploratory Spatial Data Analysis (ESDA) of Protestant Support for the Nazi Party.” Political Analysis 11:217–43.

38.

O’Loughlin

Flint

Anselin

1994. “The Geography of Nazi Vote: Context, Confession and Class in the Reichstag Election of 1930.” Annals, Association of American Geographers 84:351–80.

39.

Pace

Lesage

2004. “Spatial Autoregressive Local Estimation.” In Spatial Econometrics and Spatial Statistics, edited by Getis

Mur

Zoller

, 31–51. London: Palgrave.

40.

Páez

Uchida

Miyamoto

2001. “Spatial Association and Heterogeneity Issues in Land Price Models.” Urban Studies 38:1493–508.

41.

Páez

Uchida

Miyamoto

2002a. “A General Framework for Estimation and Inference of Geographically Weighted Regression Models: 1: Location-Specific Kernel Bandwidth and a Test for Locational Heterogeneity.” Environment and Planning A 34:733–54.

42.

Páez

Uchida

Miyamoto

2002b. “A General Framework for Estimation and Inference of Geographically Weighted Regression Models: 2: Spatial Association and Model Specification Tests.” Environment and Planning A 34:883–904.

43.

Smith

T. E.

2009. “Estimation Bias in Spatial Models with Strongly Connected Weight Matrices.” Geographical Analysis 41:307–32.

44.

Van der Eijk

Franklin

Oppenhuis

1996. “The Strategic Context: Party Choice.” In Choosing Europe: The European Electorate and National Politics in the Face of Union, edited by Van der Eijk

Kranklin

, 332–65. Ann Arbor, MI: Michigan University Press.

45.

Van Egmond

De Graaf

N. D.

Van der Eijk

1998. “Electoral Participation in the Netherlands: Individual and Contextual Influences.” European Journal of Political Research 342:281–300.

46.

Ward

O’Loughlin

2002. “Spatial Processes and Political Methodology.” Political Analysis 11:211–17.

47.

Wolfinger

Rosenstone

1980. Who Votes? New Haven: Yale University Press.