Modeling Spatial Regimes With Smooth Transitions

Abstract

In this paper, we introduce a new way of modeling spatial regimes using smooth transitions. We propose an autoregressive spatial lag model where a logistic function captures structural variation in the spatial lag parameter. In the regime-switching spatial lag model with smooth transitions, the effect of the spatial neighbors depends on the transition variable that governs the regime switch. An LM test for detecting nonlinearity is derived, and a simulation study, where the properties of the test are investigated, is conducted. The test shows good power in relatively small samples with moderate deviation from linearity. An empirical application is included, where data on median house prices from the Boston area is used to explore the spatial dependence between census tracts. A smooth shift between regimes governed by the variable property tax rate is found. The smooth spatial lag model performs slightly better than the regular spatial lag model in terms of model fit and capturing spatial dependence.

Introduction

Geographical data generally possess spatial effects, meaning that observations in neighboring locations influence each other more than those farther apart. Taking these effects into account is of great importance in applied statistical work, as it will influence results used for predictions and the implementation and evaluation of policies. Spatial dependence is one of two types of spatial effects found in cross-sectional data. It can be described as a functional relationship between what happens at different points in space and is typically modeled using spatial linear regression models (Anselin 1988b). The other type of effect is spatial heterogeneity, which occurs if the effects of space are structurally different for different locations. Spatial heterogeneity can manifest through problems such as heteroscedasticity, random coefficient variation, and switching regressions. The latter is used when the heterogeneity is of a form that categorizes the observations into different regimes. The regimes may possess different regression coefficients, weight matrices, or explanatory variables. In addition to spatial heterogeneity, switching regressions can also account for spatial dependence (Anselin 1988b). In this paper, we introduce a model that deals with systematic variation in the parameter measuring the average level of influence spatial neighbors have on the dependent variable. By letting one of the exogenous variables determine the switch, we create a new type of spatial regimes model in which the effects of neighbors differ and where observations are allowed to exist between regimes.

The most common choices for modeling spatial dependence are the spatial lag and spatial error models. The former includes a spatially lagged dependent variable in the model, while the latter accounts for spatial dependence in the error terms. The theory of spatial linear regression models is well developed in the literature, where maximum likelihood is most frequently used for testing and estimating the parameters of the models (Anselin 1988a, Anselin and Rey 1991; LeSage and Pace 2009). Alternative approaches include GMM and IV estimation, which have been put forth and developed by Kelejian and Prucha (1999), Kelejian, Prucha, and Yuzefovich (2004), Kelejian and Prucha (2010), Lin and Lee (2010), and Saavedra (2003), among others.

Spatial regimes refer to distinct patterns or characteristics observed within specific geographic areas. Examples of spatial regimes include rural and urban areas, which generally exhibit substantial differences in population density, infrastructure, and socioeconomic factors, leading to spatial heterogeneity. Correctly identifying regimes and getting a deeper understanding of the effect of spatial neighbors is crucial in many applications. For example, it can help in developing targeted policies to promote economic growth in certain areas (see Andreano, Benedetti, and Postiglione 2017) or to find crime patterns that can guide crime prevention initiatives. It is further important in health geography, where a proper understanding of the effect of spatial neighbors can help with resource allocation and healthcare planning, as well as in climate studies when there is a need to develop region-specific adaptation strategies to climate change. Some recent applications covering these, and similar scenarios include Graif and Sampson (2009), Auteri et al. (2019), Vandenbulcke et al. (2011), and Billé, Salvioni, and Benedetti (2018).

The traditional way of modeling spatial regimes is to identify the different regimes a priori and then estimate the model parameters (Anselin 1988b). Structural instability in a standard spatial lag model can be detected with a Chow test, and regimes can subsequently be identified from exploratory analyses. Although spatial regimes models account for structural instability in the regression parameters, they typically assume constant spatial lag parameters across regimes. Different weight matrices can be estimated, although it requires knowledge of how to allocate observations into regimes. In many applications, the observations cannot be easily categorized into regimes such as “south/north,” “east/west,” or “urban/rural.” Furthermore, the influence of neighboring locations may be structurally different across space, something that can be caused by phenomena such as localized effects or contextual factors. For example, in certain ecological studies, the influence of neighboring habitats on species' abundance might vary depending on the specific characteristics of each location. In social science studies, the impact of neighboring districts might differ depending on local socioeconomic conditions or cultural factors. In these situations, we propose to use a spatial lag model with smoothly varying lag parameters. The proposed model drives the transition between regimes by a theoretically justified transition variable likely to influence the regime switch. Our main contribution is to present a test for detecting smooth transition type nonlinearity in the spatial lag parameter in a spatial regression model. Suppose the null hypothesis of linearity is rejected for a certain variable. In that case, that variable can be used in estimating a spatial regimes model with systematic variation in the influence of neighbors. Choosing the appropriate transition variable can be done in several ways and depends on the objective of the analysis. In the smooth transition literature, it is common practice to choose one of the explanatory variables in the model, based on theory, and/or test several variables and choose the one with the lowest p-value (Teräsvirta 1994; Teräsvirta, van Dijk, and Franses 2002). One benefit of the approach we suggest is that it does not require a priori categorization into regimes, as this is done in estimating the model. The switching point and speed at which the observations move between regimes are also estimated, which makes the model more flexible than a traditional spatial two-regime model.

The standard smooth transition autoregressive (STAR) model developed for time series data extends the threshold model that separates a process into two regimes to one where the switch between regimes is smooth. The STAR framework has received much attention and is well established in the non-linear time series literature (Luukkonen, Saikkonen, and Teräsvirta 1988; Teräsvirta 1994; Teräsvirta, van Dijk, and Franses 2002). Previous work that combines spatial analysis and STAR models include that of Pede, Florax, and Lambert (2014), which introduces a spatial STAR model that accounts for STAR-type nonlinearity in the exogenous variables but assumes a non-varying spatial lag parameter. Like these authors, we include a logistic transition function in the model but place it with the spatial lag parameter. This approach enables the interpretation that the effect of spatial neighbors depends on the transition variable that governs the regime switch. The resulting smooth spatial lag model can be seen as having two extreme regimes that occur when the transition function is zero or one. Observations with values in-between are seen as being in transition between regimes. For example, if we consider modeling crime rate in a certain area where there is no obvious classification of regimes, a socioeconomic variable such as education level could be a candidate transition variable. The average influence neighboring locations have on crime levels would be structurally different in the two regimes, and which regime a given location belongs to would depend on the education level. If the transition variable is estimated to be a low number, many locations will be in transition between regimes. Although this model focuses specifically on the structural difference in the lag parameter, it could be extended to include varying coefficients for the exogenous regressors. It could further be extended to include spatially lagged exogenous variables, thus encompassing a spatial Durbin model.

To test for linearity, we derive an LM-type test based on that of Luukkonen, Saikkonen, and Teräsvirta (1988). A Monte Carlo study shows that the small sample properties for the test are good, in terms of size and power, for moderate sample sizes with non-negligible differences between regimes.We further demonstrate the practical use of the model in an empirical application, estimating median house prices using the Boston house price data available from the R package SpData. We find a smooth shift from a regime with a weak average influence of neighbors on house prices to one with a strong influence governed by property tax rates.

The remainder of the paper is organized as follows. In the next section, we present the smooth spatial lag model, how to estimate it, and derive a nonlinearity test. Following this, a Monte Carlo experiment investigating the small sample properties of the test, is conducted. Subsequently, we demonstrate the use of the model in a empirical application. The paper concludes with a summary of the findings.

The Smooth Spatial Lag Model

We start this section by introducing and defining the smooth spatial lag model, followed by a description of how to estimate the model and how to test for nonlinearities.

One of the most common ways of modeling spatial dependence in cross-sectional data sampled at one time point is through the mixed regressive-spatial autoregressive model. This model is represented as $\begin{array}{ll}{y}_i=\rho Wy+X_i\beta +{ϵ}_i,& {ϵ}_i\sim N\left(0,{\sigma }^2\right),\end{array}$ (1)where X is a n × k matrix of non-stochastic exogenous variables, and W is an n × n spatial weight matrix specifying the dependence structure between observations. What constitutes spatial neighbors and the strength of the interaction between them is thus determined by the matrix W. Equation (1) is also known as the spatial lag model. The scalar spatial lag parameter ρ measures the average level of influence neighboring observations have on y. The weight matrix W is typically considered known and non-stochastic and is row-standardized so that each row sums to one. Correctly specifying the structure of W is of importance and can be done based on a distance or contiguity criterion (Anselin 1988b). In the presence of spatial heterogeneity of the form that can be described by a small number of regimes, each characterized by different parameter values, Anselin (1988b) suggests using spatially switching regressions. The estimation of such models can be complicated when the number of regimes and switching points are unknown. The first to introduce smooth transitions in time series threshold models were Chan and Tong (1986), whose work has later been extended and further developed by Luukkonen, Saikkonen, and Teräsvirta (1988) and Teräsvirta (1994), amongst others. By including a transition function that is bounded between 0 and 1, we allow the parameters to vary smoothly and the observations to exist in transition between regimes. Another interpretation, perhaps more appropriate when applied to spatial data, is that of a continuum of regimes where each unique value represents a distinct regime (Teräsvirta, van Dijk, and Franses (2002)). The resulting model is $y_i={\alpha }_{1}Wy\times \left(1-G\left(s_i;\gamma ,c\right)\right)+{\alpha }_{2}Wy\times G\left(s_i;\gamma ,c\right)+X_i\beta +{ϵ}_i$ (2) $G\left(s_i;\gamma ,c\right)={\left(1+\exp \left\{-\gamma \left(s_i-c\right)\right\}\right)}^{-1},$ (3)where W is a spatial weight matrix, α₁ and α₂ are spatial lag parameters, X _i is a matrix containing the exogenous explanatory variables, and β is a conforming parameter vector. The transition speed is determined by γ, the switching point by c, and s _i is the transition variable that governs the switch from one regime to another. The parameters α₁ and α₂ in (2) will be interpreted as the spatial lag coefficients of the first and second regimes, respectively. In equation (3), G(s _i ; γ, c) is a logistic function that can take on values between 0 and 1. When γ = 0, G(s _i ; γ, c) is constant at 0.5 and the model in equation (2) reduces to a regular spatial lag model. When γ → ∞, all values of G(s _i ; γ, c) will approach zero or one, with an abrupt shift at the c, making the model in equation (2) a strict two-regimes model. When 0 < γ < ∞, G _i will fall below 0.5 if s _i < c, and more weight will be given to α₁ in the low regime. Conversely, when s _i exceeds the threshold c, G _i > 0.5, giving more weight to α₂ in the high regime. The effect of changing the scale parameter γ is visualized in Figure A1 in the Appendix. Equation (2) can be rewritten as $\begin{array}{l}{y}_i={\alpha }_{1}Wy+\left({\alpha }_2-{\alpha }_1\right)Wy\times G\left(s_i;\gamma ,c\right)+X_i\beta +{ϵ}_i\\ ={\rho }_{1}Wy+{\rho }_{2}Wy\times G\left(s_i;\gamma ,c\right)+X_i\beta +{ϵ}_i,\end{array}$ (4)where ρ₁ and ρ₂ denote the spatial lag parameters for the first regime and the difference between the second and first regime, respectively. The spatial heterogeneity is thus manifested through smoothly varying effects of spatial neighbors.

Estimation

Estimation of the spatial lag model can be done in several ways. As the OLS estimator is inconsistent in the presence of a spatial weight matrix, much of the early work has focused on a maximum likelihood approach. Maximum likelihood estimators for the spatial lag model have been derived and applied by Anselin (1988a), amongst others. Their results can easily be extended to the smooth spatial lag model. The analytical maximum likelihood estimators for β and σ² in (4) are ${\widehat{\beta }}_{ML}={\left(X^{T}X\right)}^{-1}{X}^T\left(I-{\rho }_1{W}_1-{\rho }_2{W}_2\right)y$ (5)and ${\widehat{\sigma }}_{ML}^2={\left(y-{\rho }_1{W}_{1}y-{\rho }_2{W}_{2}y-X{\widehat{\beta }}_{ML}\right)}^T\left(y-{\rho }_1{W}_{1}y-{\rho }_2{W}_{2}y-X{\widehat{\beta }}_{ML}\right)/n,$ (6)where $W_1=W$ (7) $W_2=W\times G\left(s_i,\gamma ,c\right).$ (8)

The concentrated log-likelihood function follows as $L=\ln \left|I-{\rho }_{1}W-{\rho }_{2}W\times G\left(s_i,\gamma ,c\right)\right|-\frac{N}{2}\ln \left({\widehat{\sigma }}_{ML}^2\right).$ (9)

Using the simplification proposed by Ord (1975), the concentrated log-likelihood function can be expressed as $L=\ln ({{\displaystyle \prod }}_{i=1}^n\left(1-{\rho }_1{w}_i-{\rho }_2{w}_i\times G\left(s_i,\gamma ,c\right)\right)-\frac{N}{2}\ln \left({\widehat{\sigma }}_{ML}^2\right)$ (10)where w _i , …, w _n are the n eigenvalues of W. In order to find starting values for estimating the parameters in the model, a two-dimensional grid search is performed over a set of values for c and log(γ). This is done by evaluating the log-likelihood for all possible combinations of fixed values of γ and c compiled in a grid. As suggested by Schleer (2015), we redefine the smoothing parameter as γ = exp(v), where v is the parameter to be estimated. This is done to avoid taking the log of a negative value in the numerical optimization. Creating an equidistant grid in the parameter space of v thus gives a search space for γ that is more dense for lower values, which is a reasonable choice in many applications. It further removes the need to make the restriction γ > 0. The parameter space of γ denoted Γ, is restricted to values that identify the model as a smooth transition rather than a linear- or threshold model. A redefined parameter space ln(Γ) is used for v, while sample percentiles of the transition variable s _i constitute an appropriate set of grid values for c. The starting values are then obtained from the evaluation of the log-likelihood in (9) at each combination of γ and c in the grid. The pair for which the log-likelihood is maximized are chosen as starting values.

LM Test for Nonlinearity

We use a Lagrange multiplier test to test the smooth spatial lag against a regular spatial lag. When testing for smooth transitions in a time series setting, the standard procedure is to use a first-order Taylor approximation of the transition function, around γ = 0. This is done to avoid the problem of unidentified nuisance parameters under the null hypothesis (Teräsvirta, Tjøstheim, and Granger (2010)). A Taylor approximation of the transition function around γ = 0 is given by $G\approx \left[\frac{1}{2}-\frac{c\gamma }{4}+\frac{\gamma }{4}s\right].$ (11)

Substituting (11) for (4) yields the auxiliary regression model $\begin{array}{l}y=\left[{\rho }_1+{\rho }_2\frac{1}{2}-{\rho }_2\frac{c\gamma }{4}\right]Wy+\frac{\gamma }{4}{\rho }_{2}Wy\times s+X\beta +ϵ\\ ={\eta }_{0}Wy+{\eta }_{1}Wy\times s+X\beta +ϵ,\end{array}$ (12)and testing η₁ = 0 in (12) is therefore equal to testing ρ₂ = 0 in (4). The log likelihood function of (12) is $L=\ln \left|I-{\eta }_{0}W-{\eta }_{1}W\times s\right|-\frac{N}{2}\ln \left(2\pi \right)-\frac{N}{2}\ln \left({\sigma }^2\right)$ (13) $-{\left(y-{\eta }_{0}Wy-{\eta }_{1}Wys-X\beta \right)}^{\prime }\left(y-{\eta }_{0}Wy-{\eta }_{1}Wys-X\beta \right)/2{\sigma }^2.$ (14)

The LM test for η₁ = 0 is a straightforward extension of the test for ρ = 0 in (1), derived by Anselin (1988 b), and closely follows the tests presented by Pede, Florax, and Lambert (2014). The derivations of the test are presented in the Appendix.

Monte Carlo Simulations

To investigate the small sample properties of the derived LM test, we perform a Monte Carlo experiment where size and power simulations are conducted.

The simulation design is set up to resemble that of Pede, Florax, and Lambert (2014) as closely as possible. As such, it follows standard Monte Carlo practices for evaluating tests in spatial econometrics. By their design, three types of weight matrices are used in the experiments. Firstly, we consider a row standardized weight matrix based on the k nearest neighbors of each spatial unit. The number of neighbors k is defined as the square root of the sample size n. Secondly, we consider a weight matrix based on the inverse of the Euclidean distance of the k nearest neighbors. These matrices are based on x, y coordinates simulated from a U(0, 1) distribution. Finally, a queens contiguity weight matrix is considered, set up as a regular n × n grid. The three matrices are row-standardized to have the sum of each row equal one. For the distance-based weight matrices, we thus have a random placement based on simulated coordinates, and for the queens matrix, we have a regular lattice, meaning that each location has three, five, or eight neighbors. These weight matrices are considered known in the experiment. The spatial lag model with smooth transitions is defined as $y_i={\rho }_{1}Wy+{\rho }_{2}Wy\times G\left(s_i,\gamma ,c\right)+X_i\beta +{ϵ}_i,$ (15)where X is an n × (p + 1) matrix with a first column of ones corresponding to the intercept, and where the remaining p columns are covariates generated from a multivariate standard normal distribution. The covariates are held fixed throughout the experiment. The (p + 1) × 1 parameter vector β is fixed to unity and the random errors ϵ are considered i.i.d ∼ N(0, 1). Following Pede, Florax, and Lambert (2014), we set p = 2, which means we have two exogenous covariates, and use a spatially lagged version of the first covariate in X, WX₁, as the transition variable s _i . The location parameter c is set to the mean of s, one of the values considered in Pede, Florax, and Lambert (2014), and a common choice in the Smooth transition literature (Teräsvirta, Tjøstheim, and Granger (2010)). The spatial lag parameter ρ₁ is fixed to 0.1 while the values of ρ₂ considered are 0.3, 0.5, 0.8. As the difference between the two extreme regimes is given by ρ₂, these values cover situations with a small and large difference between states. The smoothing parameter γ is also free to vary, and the values considered are γ = 0, 1, 3, 5, 10, where γ = 0 corresponds to the regular spatial lag model. The simulations are performed for sample sizes N = 25, 49, 100, 400, and the number of replicates is 2000.

The results of the size simulations are reported in Table 1, for all sample sizes and weight matrices, and ρ₁ = 0.8. ¹ Notably, the size is reasonably good for all specified weight matrices. The only case where the test size falls outside the 95% confidence interval is for the KNN weight matrix and sample size 100. Figure 1 displays the power of the test, adjusted for empirical size, for each sample size and weight matrix. The power curves are based on the simulations where ρ₁ = 0.1 and ρ₂ = 0.8. The test is shown to have the strongest power, in general, when W is based on the queen’s criterion with a regular grid. Furthermore, the power increases quickly with γ, even in the moderate sample sizes of 49 and 100, when there is a non-negligible difference between the two regimes. This implies that in any empirical application with two distinct spatial regimes, the test will do a very good job detecting even slight nonlinearities if we have a sample of approximately n = 100 or more. When the sample size is smaller than that and/or the difference between regimes is very small, the power is greatly weakened, which can be seen from Table 2.

Table 1.

Empirical Size of the LM Test for Linearity, for Three Types of Weight Matrices.

N	Parameter Value			Weight Matrix
	γ	ρ 1	ρ 2	KNN (1/N)	KNN (1/d ij )	queen
25	0	0.1	0.8	0.0450*	0.0440*	0.0515*
49				0.0505*	0.0575*	0.0505*
100				0.0690	0.0580*	0.0555*
400				0.0475*	0.0450*	0.0560*

Notes: A 95% confidence interval for the rejection proportion p is given by 0.0425 < p < 0.05975. Values that fall within the interval are marked with an asterisk.

Figure 1.

Size adjusted power when ρ₁ = 0.1 and ρ₂ = 0.8.

Table 2.

Size Adjusted Power of the LM Test for Linearity, for Three Types of Weight Matrices.

		KNN (1/d ij )				Queen				KNN (1/N)
ρ 2	γ	N = 25	N = 49	N = 100	N = 400	N = 25	N = 49	N = 100	N = 400	N = 25	N = 49	N = 100	N = 400
0.8	1	0.0885	0.1091	0.3282	0.4315	0.1339	0.2450	0.2052	0.5637	0.0850	0.0778	0.2573	0.3035
	5	0.3725	0.4686	0.9903	1.0000	0.5298	0.8338	0.9516	1.0000	0.4815	0.5446	0.9899	1.0000
	10	0.4670	0.4591	0.9785	1.0000	0.5353	0.8871	0.9272	1.0000	0.7220	0.7439	0.9592	0.9980
0.5	1	0.0505	0.0760	0.1204	0.1980	0.0720	0.1005	0.0810	0.3051	0.0530	0.0533	0.0922	0.1090
	5	0.1240	0.2046	0.6192	0.9515	0.1941	0.3740	0.4500	0.9931	0.1650	0.1761	0.4830	0.8315
	10	0.1525	0.2737	0.7257	0.9935	0.2317	0.4610	0.6205	0.9995	0.2655	0.2807	0.5994	0.9880
0.3	1	0.0440	0.0585	0.0661	0.0995	0.0578	0.0689	0.0592	0.1327	0.0405	0.0510	0.0637	0.0680
	5	0.0650	0.1004	0.2065	0.4785	0.1009	0.1558	0.1640	0.6830	0.0770	0.0887	0.1523	0.3310
	10	0.0735	0.1174	0.2583	0.6645	0.1127	0.1897	0.2114	0.8214	0.1050	0.1126	0.1888	0.5940

Boston House Prices

To demonstrate the practical application of the smooth spatial lag model, we utilize the Boston house price data from the open-source R package SpData. Our primary focus lies in estimating the spatial lag parameters, aiming to gain deeper insights into the influence of neighboring areas. Additionally, we intend to evaluate the model’s performance in capturing spatial dependencies and assess whether it outperforms a conventional linear spatial lag model.

Data

The data set consists of 506 census tracts within the Boston Standard Metropolitan Statistical Area (SMSA) in 1970. Initially analyzed by Harrison and Rubinfeld (1978) to study median house values and their relation to willingness to pay for clean air, the data has undergone minor corrections by Gilley and Pace (1996). In our application, we employ the same housing value equation as Harrison and Rubinfeld (1978) but introduce a spatial lag with smooth transitions. The dependent variable, denoted as y, represents the natural logarithm of median housing values in USD 1000 (ln(CMEDV)) and is depicted in Figure 2. The independent variables within X consist of two accessibility metrics, eight neighborhood variables, two structural attributes, and one air pollution variable, mirroring the variables used in Harrison and Rubinfeld (1978) (details provided in Table 3).

Figure 2.

Distribution of median house prices in the Boston area in 1970, measured in USD 1000. Natural breaks (Jenks) are used for classification.

Table 3.

Description of Variables Included in the Housing Value Equation.

Variable key	Description	Transformation
CMEDV	Median values of housing in USD 1000	Natural logarithm
CRIM	Per capita crime
ZN	Proportion of residential land zoned for lots over 25,000 sq. ft per town
INDUS	Proportion of non-retail business acres per town
CHAS	Dummy: takes value 1 if a tract borders Charles River, 0 otherwise
NOX	Nitric oxides concentration (parts per 10 million) per town	Squared
RM	Average number of rooms per dwelling	Squared
AGE	Proportion of owner-occupied units built prior to 1940
DIS	Weighted distances to five Boston employment centers	Natural logarithm
RAD	Index of accessibility to radial highways per town	Natural logarithm
TAX	Full-value property-tax rate per USD 10,000 per town
PTRATIO	Pupil-teacher ratio per town
BLACK	1000 × (Bk − 0.63)2 where Bk is the proportion of blacks
LSTAT	Percentage values of lower status population	Natural logarithm

Testing

When estimating the smooth spatial lag model, the initial step involves defining the weight matrix W and identifying a suitable transition variable s. To ensure our findings are robust to different choices of weight matrices, we conduct the analysis using two different weight matrices. Firstly, we use a weight matrix based on k ² nearest neighbors (KNN). Secondly, we use one based on the queen’s criterion.

In line with the modeling procedure outlined by Teräsvirta, Tjøstheim, and Granger (2010) for estimating smooth transition models, the first stage involves testing for linearity. To do this, we select a range of transition variables and evaluate the suitability of a spatial lag model versus a smooth spatial lag model using the LM-test previously derived. In cases where linearity is rejected for multiple variables, the practice is to choose the one with the smallest p-value. If the rejection of linearity persists across various transition variable choices, Teräsvirta, Tjøstheim, and Granger (2010) advises estimating the model using several of these and deferring the final selection to the evaluation stage. For our analysis, we consider each variable within the set X as a potential transition variable s. To facilitate interpretation and make γ scale-free, we standardize these variables to have mean zero and variance one. The results of these linearity tests are presented in Table 4. Notably, linearity is rejected for several transition variable choices. Among the potential transition variables yielding a test statistic with a p-value below 0.01, selecting s = TAX provides the best model fit for both weight matrix options. This variable measures the full-value property tax rate per USD 10,000 and remains constant for census tracts within the same town. This implies that a distinct regime will be encountered once the property tax surpasses a certain threshold. Consequently, different underlying structures come into play, influencing how house prices are affected by neighboring districts.

Table 4.

Results of the LM Test for Linearity for all Potential Transition Variables.

Transition Variable	KNN		Queen
	LM	p-Value	LM	p-Value
CRIM	42.6146	<0.0001	6.4277	0.0112
ZN	0.5435	0.4610	1.1038	0.2934
INDUS	6.7380	0.0094	6.7441	0.0094
CHAS	4.9088	0.0267	0.4465	0.5040
NOX2	13.1738	0.0003	5.2142	0.0224
RM2	0.0330	0.8559	0.0271	0.8692
AGE	7.7987	0.0052	5.1562	0.0232
ln(DIS)	12.5942	0.0004	11.8398	0.0006
ln(RAD)	5.2495	0.0220	8.3309	0.0039
TAX	19.7757	<0.0001	14.5593	0.0001
PTRATIO	4.8629	0.0274	6.8833	0.0087
BLACK	3.3557	0.0670	1.2857	0.2568
ln(LSTAT)	38.0195	<0.0001	11.2011	0.0008

Weight matrices are based on KNN and queen neighbors.

Estimation

Model estimation becomes the next step once a suitable transition variable is identified. We first focus on the scenario where the weight matrix is constructed using KNN. Table 5 reports the results of model estimations in this context. Here, the threshold parameter c is estimated at 0.5686 (equivalent to 504.0638 for the non-standardized variable), while the spatial lag parameters in the two extreme regimes are estimated as ${\widehat{\alpha }}_1=-0.0366$ and ${\widehat{\alpha }}_2=0.8992$ , respectively. The smoothing parameter γ is estimated at $\widehat{\gamma }=e^{\widehat{\eta }}=1.0445$ , indicating a smooth transition. Notably, 137 census tracts belong to towns with property-tax rates exceeding the threshold value. This implies that approximately 73% of the values of the transition function $G\left(s_i;\widehat{\gamma },\widehat{c}\right)<0.5$ , while around 27% exceed this threshold. The low value of $\widehat{\gamma }$ signifies that the extreme points of the function G are not reached; all observations fall within the range of 0.1229-0.7829. Essentially, this means that all observations are in a transitional phase between the two extremes. Alternatively, a continuum of regimes is observed: one for each value of G. In one extreme regime, a notably positive effect exists for census tracts in towns with high property tax rates. In contrast, a small negative spill-over effect is observed in the regime characterized by lower property taxes, although this negative estimate is not significantly different from zero. The value of the spatial lag parameter in each area is determined by the value of ${\widehat{G}}_i$ . When s _i is below the threshold $\widehat{c}$ , ${\widehat{G}}_i<0.5$ and more weight is given to the parameter ${\widehat{\alpha }}_1$ . Conversely, when s _i is above $\widehat{c}$ more weight is given to ${\widehat{\alpha }}_2$ . The estimated spatial lag coefficient value in each census tract is visualized in Figure 3(a). Comparing model fits, the smooth spatial lag model surpasses the spatial lag model in terms of maximizing likelihood (changing from 199.3684 to 228.6419) and minimizing the sum of squared errors (SSE, decreasing from 13.3718 to 11.9184). The spatial lag model is rejected in favor of the smooth spatial lag model in a Likelihood ratio test.

Table 5.

Estimation Results - KNN.

Parameter	Spatial Lag	Smooth Spatial Lag
α 1	0.4161 (0.0394)	−0.0366 (0.0716)
α 2		0.8992 (0.1496)
c		0.5686 (0.2180)
log(γ)		0.0436 (0.2582)
Constant	284.7790 (21.0951)	492.6235 (20.1840)
CRIM	−0.9795 (0.1121)	−0.9556 (0.1062)
ZN	0.0450 (0.0452)	0.0409 (0.0436)
INDUS	−0.1142 (0.2112)	−0.3507 (0.2026)
CHAS	1.1203 (3.0450)	2.3058 (2.9132)
NOX2	−45.4841 (10.2503)	−60.8126 (9.7344)
RM2	0.6508 (0.1170)	0.7281 (0.1081)
AGE	0.0266 (0.0471)	−0.0080 (0.0382)
ln(DIS)	−22.9424 (2.9819)	−24.6346 (2.7326)
ln(RAD)	6.6892 (1.7078)	6.5500 (1.6419)
TAX	−0.0265 (0.0110)	−0.3812 (0.0063)
PTRATIO	−1.2216 (0.4729)	−2.5399 (0.4614)
BLACK	0.0272 (0.0093)	0.0227 (0.0068)
ln(LSTAT)	−32.3970 (2.2671)	−29.6412 (2.0713)
SSE	13.3718	11.9184
Log.likelihood	199.3684	228.6419
LR test value:	58.54699	p-value: <0.0001

Notes: Parameter estimates with standard errors in parentheses. The estimates of the parameters and standard errors corresponding to the exogenous variables are given as β _i × 100, and should be interpreted as percentage change in median house price, conditional on X.

The results for models employing a weight matrix based on the queen criterion are presented in Table 6. Notably, model fit improves for both the smooth spatial lag and spatial lag models when using this weight matrix choice. The improvements in model fit observed when adding smooth transitions mirror those observed when KNN weights were used. The estimated parameters for this weight matrix choice remain similar to those obtained with KNN weights. The smooth transition parameters are estimated as $\widehat{\gamma }=e^{\widehat{\eta }}=1.07262$ and $\widehat{c}=0.5118$ (503.4612 when not standardized), resulting in the same proportions of G below and above 0.5 as observed with KNN weights. Figure 4 visualizes the estimated transition function. In this case, the spatial lag parameters in the two extreme regimes are estimated as ${\widehat{\alpha }}_1=0.0431$ and ${\widehat{\alpha }}_2=0.8358$ , both positive, with the effect being notably stronger in the high regime. The estimated spatial lag parameter value in each area is depicted in Figure 3(b).

Table 6.

Estimation Results - Queen.

Parameter	Spatial Lag	Smooth Spatial Lag
α 1	0.4874 (0.0311)	0.0431 (0.0571)
α 2		0.8358 (0.1140)
c		0.5118 (0.1896)
log(γ)		0.0701 (0.2505)
Constant	226.7616 (18.2749)	428.7918 (16.8668)
CRIM	−0.7564 (0.0992)	−0.6784 (0.0938)
ZN	0.0403 (0.0392)	0.0333 (0.0380)
INDUS	0.0872 (0.1833)	−0.2084 (0.1765)
CHAS	2.6537 (2.6012)	2.7787 (2.4458)
NOX2	−30.0536 (8.9607)	−41.7531 (8.4641)
RM2	0.7219 (0.1020)	0.9120 (0.0907)
AGE	−0.0090 (0.0408)	−0.0567 (0.0315)
ln(DIS)	−16.6634 (2.5990)	−19.5370 (2.3189)
ln(RAD)	6.2042 (1.4941)	5.5087 (1.4238)
TAX	−0.0321 (0.0096)	−0.3468 (0.0058)
PTRATIO	−0.9444 (0.4079)	−2.1780 (0.3909)
BLACK	0.0233 (0.0081)	0.0202 (0.0063)
ln(LSTAT)	−24.7399 (2.0728)	−23.0209 (1.8512)
SSE	10.1234	8.9218
Log.likelihood	259.1874	295.3958
LR test value:	72.41681	p-value: <0.0001

Notes: Parameter estimates with standard errors in parentheses. The estimates of the parameters and standard errors corresponding to the exogenous variables are given as β _i × 100, and should be interpreted as percentage change in median house price, conditional on X.

Figure 3.

Estimated transition function G as a function of $\widehat{\gamma }=1.0726$ and $\widehat{c}=0.5118$ . Weights based on queen neighbors.

Figure 4.

Estimated spatial lag parameters in different regions as determined by the transition function G. Natural breaks (Jenks) are used for classification. (a) KNN weights. (b) Queen weights.

Evaluation

To assess the effectiveness of the smooth spatial lag model, we conduct a residual analysis to identify clusters of model over- and under-predictions. Initially, we perform a global cluster analysis by looking at the global Moran’s I. The results are presented in Table 7, with the corresponding numbers for the linear spatial lag reported within parentheses. Figure 5 compares Moran scatter plots between spatial lag and smooth spatial lag residuals when using KNN weights. The global Moran’s I for the residuals of the two models is calculated as 0.1212 and 0.0632, respectively, indicating that the smooth spatial lag model better captures the spatial dependence in the data. However, although the strength of global clustering in the residuals decreases when incorporating smooth transitions, it remains significantly different from zero. In Figure 6, we present the same residual analysis for models using queen weights. The global Moran’s I for spatial lag residuals is 0.0906 with a p-value of 0.0006, while the smooth spatial lag residuals yield a Moran’s I of 0.0268 with a p-value of 0.2841. This essentially implies that the model effectively captures spatial dependence, eliminating clusters of over- and under-predictions.

Table 7.

Global Moran’s Test of Residuals.

Weights	Moran index	Z-score	p-Value
KNN	0.0632 (0.1212)	5.3384 (10.085)	<0.0001 (<0.0001)
queen	0.0268 (0.0906)	1.0711 (3.4442)	.2841 (.0006)

Notes: The table reports results for the smooth spatial lag model. The corresponding numbers from the spatial lag model are in parentheses.

Figure 5.

Moran scatterplot of residuals. Weights based on KNN. (a) Spatial lag. (b) Smooth spatial lag.

Figure 6.

Moran scatterplot of residuals. Weights based on queen’s criterion. (a) Spatial lag. (b) Smooth spatial lag.

Additionally, we conduct a local cluster analysis using Local Indicators of Spatial Association (LISA) on the residuals, employing local Moran’s I as the LISA statistic. The results are visualized in the LISA cluster maps presented in Figure 7 (KNN weights) and Figure 8 (queen weights). These maps illustrate local clustering of model under- and over-predictions. “High-high” census tracts indicate spatial clustering of model under-predictions, while “low-low” indicates spatial clustering of over-predictions. “High-low” or “low-high” observations are considered spatial outliers. Comparing the residuals from the smooth spatial lag model with those from the spatial lag model, we observe an increase in areas with non-significant ³ clustering for both KNN weights (493 compared to 479) and queen weights (489 compared to 483). This signifies an enhancement in accounting for local spatial dependence for both types of weight matrices.

Figure 7.

Spatial clustering of residuals with weights based on KNN. “high-high” indicates clusters of model under-prediction, “low-low” denotes clusters of over-prediction, and “high-low” and “low-high” are spatial outliers. False discovery rates were used to adjust for multiple testing. (a) Spatial lag. (b) Smooth spatial lag.

Figure 8.

Spatial clustering of residuals with weights based on queen’s criterion. “high-high” indicates clusters of model under-prediction, “low-low” denotes clusters of over-prediction, and “high-low” and “low-high” are spatial outliers. False discovery rates were used to adjust for multiple testing.

Conclusion

This paper introduces a novel approach to model spatial regimes for cross-sectional data. We propose using a spatial lag model wherein the spatial lag parameter smoothly varies between regimes. In the resultant smooth spatial lag model, we employ a logistic function as a transition function.

We discover favorable small sample properties for the developed test of nonlinearity across all three weight matrix specifications. The size-adjusted power of the test increases rapidly with the magnitude of the smoothing parameter, reflecting the degree of nonlinearity. While the test exhibits robust power for all weight matrices, it excels when the queen’s criterion is employed. When there is a distinguishable difference between the spatial effects of the two regimes, the power remains robust, even for moderate sample sizes. However, the good power properties do not extend to the cases where the difference between regimes is minimal and/or the sample size is small.

To exemplify the use of the model, an empirical application is carried out, estimating median house values in the Boston area. By subjecting all regressors in the model to the linearity test, the property tax rate is found to be an appropriate transition variable. The smooth spatial lag model is then compared to a linear spatial lag model. The linear model implies that the spill-over effect from neighbors is moderate and constant across all census tracts. On the other hand, the smooth spatial lag model suggests a large positive spill-over effect among census tracts that are above a certain threshold value in property tax. Conversely, a subtler dynamic for census tracts below the threshold can be observed. Here, the model implies that the spill-over effect is much smaller and can even oscillate between positive and negative values, depending on the weight matrix choice. Furthermore, the estimated shift between the two regimes is very smooth, meaning all observations fall somewhere between the extremes. This also implies that this model specification will be preferred over a threshold model with an abrupt shift. The smooth spatial lag further outperforms the linear spatial lag model regarding model fit and ability to account for spatial dependence. The analysis also implies that these results are robust for different choices of weight matrices.