Spatial Econometric STAR Models: Lagrange Multiplier Tests and Monte Carlo Simulations

Transcription

1 Spatial Econometric STAR Models: Lagrange Multiplier Tests and Monte Carlo Simulations Valerien O. Pede Department of Agricultural Economics, Purdue University West Lafayette, IN 47907, USA, Raymond J.G.M. Florax Department of Agricultural Economics, Purdue University West Lafayette, IN 47907, USA, Department of Spatial Economics, VU University Amsterdam, The Netherlands Dayton M. Lambert Department of Agricultural Economics, University of Tennessee Knoxville, TN 37996, USA, Matthew T. Holt Department of Agricultural Economics, Purdue University West Lafayette, IN 47907, USA, Abstract This paper investigates non-linearity in spatial process models allowing for gradual regime-switching structures in the form of a smooth transition autoregressive process. Until now, applications of the smooth transition autoregressive (STAR) model have been largely confined to time series analysis. The paper focuses on extending the non-linear smooth transition perspective to spatial process models, in which spatial correlation is taken into account through the use of a weight matrix identifying the topology of the spatial system. We start by deriving a non-linearity test for a simple spatial STAR model, in which spatial dependence is only included in the transition function. Next, we propose a non-linearity test for a spatial STAR model that includes a spatially lagged dependent variable or spatially autocorrelated innovations. Monte Carlo simulations of the various test statistics are performed to examine test power and size. Results indicate that the individual and joint nonlinearity tests show power against traditional spatial lag and error autoregressive processes, and the individual and joint spatial dependence tests have power against STAR nonlinearity. These results complicate the design of an effective specification strategy. Keywords: spatial econometrics, non-linearity, autoregressive smooth transition JEL Classification: C1, C1, C51, O18, R11

2 1. Introduction Over the last decade, nonlinear time series modeling has gained considerable attention in the applied economics literature. Nonlinear models are sometimes more appropriate than linear models in describing economic processes. For instance, the presence of asymmetry and nonlinearities in business cycles can be incorporated in smooth transition autoregressive (STAR) models, which generally leads to a better representation of data generating processes (dgp s) compared to linear regression models. The STAR model is a member of the family of nonlinear models that exhibit regime-dependent or regimeswitching behavior. Smooth Transition Autoregressive (STAR) regressions model the nonlinear dynamics in time series by allowing the dgp to depend on one or multiple regimes that typically follow a first order Markov process. In general, STAR models relax the assumption that parameters associated with the data generating processes are fixed throughout the series, allowing for the endogenous determination of structural discontinuities across time. Regime-dependent or regime-switching STAR models are due to Goldfeld and Quant (1973). Recent innovations of the STAR model have modeled nonlinearities and asymmetric responses in industrial production (Terasvirta and Anderson, 199), the hog-corn cycle (Holt and Craig, 006), exchange rates (Baharumshah and Khim-Sen Liew, 006), interest rates (van Dijk and Franses, 000), and unemployment rates (Skalin and Terasvirta, 00). As implied by its name, the STAR framework allows model parameters to take on different values across regimes, following a potentially smooth transition process. Smooth transition regime switching models are familiar to the time series literature,

3 3 although the general approach has seen frequent use in the biometric sciences (Schabenberger and Pierce, 00). Surprisingly, STAR-type models have received little attention in the spatial econometrics literature. In the spatial econometrics literature, structural breaks across space are typically modeled using dummy variables allowing slopes and intercepts to vary over space (Anselin, 1988; Lambert and McNamara, 009), or semiparametric approaches including Geographically Weighted Regression (GWR) (Fortheringham et al., 00), spatial expansion regression (Cassetti, 197), or random coefficients (Anselin, 1988). Structural instability across space may be caused by different response functions or systematically varying parameters. The central assumption behind these approaches is that regression parameters vary over space due to localized or regional differences in consumer behavior, producer supply, or industry technology. Firms may employ different production technologies, skilled labor may be concentrated in certain locations, or benefits from industry information, demand-side, or supply-side spillovers may occur more frequently in or near agglomeration economies. Ad hoc classifications (e.g., metro versus non-metro, north versus south, or treatment versus control ) may also be sources of spatial heterogeneity. Measurement error due to differences in localized, unobserved factors such as cultural preferences, local knowledge or policy, customs, or social networks may cause structural breaks across space as well. The usual consequences of the statistical validity of models result when these issues are overlooked: inferior forecasts, biased coefficients, and compromised inference. Depending on the problem at hand, the above methods may be integrated into more complicated spatial process models simultaneously allowing for autoregressive lags and correlated error structures.

4 4 Whittle (1954) described a broad class of spatial process models where an endogenous variable is specified to depend on spatial interactions between cross sectional units plus a disturbance term. The interactions are modeled as a weighted average of nearby cross sectional units, and the endogenous variable comprising the interactions is usually referred to as a spatially lagged variable. The weights, grouped in a matrix identifying neighborhood connections, form the distinctive core of spatial process models. The model is termed a first order spatial autoregressive lag model (SAR[1]) (Anselin and Florax, 1995). Whittle s SAR(1) model was popularized and extended by Cliff and Ord (1973, 1981), who further developed models in which the disturbances followed a spatial autoregressive process. The general spatial process model containing a spatially lagged endogenous variable, spatially autoregressive disturbances, and exogenous variables is called a spatial autoregressive model with autoregressive (AR) disturbance of order (1,1) (ARAR) (Anselin, 1988; Kelejian and Prucha, 004, 008; Lee, 00, 007); y = ρw 1 y + Xβ + e, e = λw e + u, u ~ iid(0, Ω), where W 1 and W are (possibly identical) nonstochastic, positive definite, exogenous matrices defining relationships between spatial units, and E[uu ] = Ω. The reduced form version is y = A 1 Xβ + A 1 B 1 u; A = (I ρw 1 ), B = (I λw ) where ρ is the spatial lag regressive term and λ is the spatial error autoregressive term. Spatial processes have often been assumed to be linear, although it is much more likely that spatial dynamics exhibit nonlinear features in a way that is similar to time series models (De Graaff et al., 001). 1 1 The implied spatial spillovers actually follow a process of nonlinear decline over space, given the presence of the spatial multiplier in the spatial lag and the ARAR model (Anselin, 003; Pede et al., 009).

5 5 Spatial analogues of the time series STAR approach have been developed in a handful of studies. For instance, Lebreton (005) developed a spatial version of time series STAR model by incorporating spatial autocorrelation in the transition function and allowing for a smooth transition process. The approach proposed in this paper follows the same principle, but in addition incorporates traditional spatial process models in the STAR framework. A spatial analogue to the STAR model was also presented in Gress (004), Basile and Gress (005) and Basile (008), but these studies considered a nonparametric estimator whereas our approach is parametric. Recently, Dorfman et al. (009) developed a model that is quite similar to the STAR approach using a Bayesian perspective, but their approach models hierarchical rather than contagious autoregressive processes. This paper formally develops a series of tests for identifying nonlinear structural heterogeneity across space, allowing for gradual, endogenous regime switching behavior as a smooth transition autoregressive process in the context of traditional spatial process models. We start by deriving a nonlinearity test for a spatial ARAR(1,1)-STAR model. Next, we derive nonlinearity tests for three nested models: the spatial lag STAR (SAR-STAR), the spatial error STAR (SEM-STAR) and the ARAR model (ARAR- STAR). The tests are developed in the maximum likelihood framework focusing specifically on the Lagrange Multiplier variants. Monte Carlo simulations are used to investigate the finite sample performance of the various tests in terms of size and power.. Family of Spatial STAR Models We modify the spatial STAR concept from the time series perspective to spatial crosssectional processes. We start with a basic specification of the STAR model for spatial

6 6 variables in analogy with the model presented for time series (Terasvirta and Anderson, 199; Holt and Craig, 006; Van Dijk and Franses, 000): y = Xα + Xδ o G(, s γ, c) + ε, (1) where y is an N 1 spatial data series, X is an N k vector of explanatory variables, Gs (, γ, c) is a continuous, potentially smooth, real-valued transition function bounded between zero and one, s is the transition variable, γ and c are (respectively) the slope and location parameters, o is the Hadamard product indicating element-by-element multiplication, and ε an N 1 vector of innovations assumed to be independently and identically distributed with constant variance, or alternatively to exhibit a spatially autoregressive pattern. Note that the interaction between the transition function and the exogenous variables allows for parameter variation across space and between regimes identified through G. In analogy with time series STAR models, a possible candidate for the transition variable could be the spatial lag of the dependent variable y or an explanatory variable x. In this paper we only consider the latter. We therefore define the spatial lag of an independent variable x as Wx, where W represents an exogenously defined weight matrix. 3 The spatial weights matrix is usually a Boolean matrix, with each element taking the value 1 when regions are neighbors and 0 when they are not. The weight matrix is typically row-standardized, such that the cross-product Wx represents the average value We did not consider the spatially lagged dependent variable as the transition variable because of additional complications involved in obtaining a reduced form for the STAR model. 3 Wx is an N 1 vector of observations, and simply represents the cross-product of the weight matrix W and the independent variable x.

7 7 of x of neighboring locations. 4 We identify spatial regimes using the logistic transition function 5 given as: 1 GWx (, γ, c) =. 1 + exp ( γ( Wx c) σwx ) () Figure 1 depicts an example of how the transition function G operates. Note that two distinct regimes emerge when γ = 100, whereas there are no regimes identified when γ = 0. Observations located along the curve of the function are transitional, not belonging to either regime as denoted by the observation cluster around the (0, 1) values of the transition function along the y-axis. The parameter c functions as a location parameter. That is, the inflection of the transition function is centered on c. The general spatial ARAR STAR model with spatial autocorrelation in the transition function, a spatially lagged dependent variable, and spatially autoregressive errors is: y = ρwy+ Xα + G( Wx, γ, c) o Xδ + ε, ε = λwε + μ, (3) where ρ and λ are scalar spatial autoregressive parameters, μ are independent and identically distributed errors, and the other symbols are as defined before. Three different spatial STAR models are nested within this general ARAR model, depending on the value of the spatial autoregressive parameters. Restricting λ = 0 leads to the spatial lag STAR model; ρ = 0 the spatial error STAR model; and λ = ρ = 0 the simple spatial STAR model in equation (1). Each of these models can be estimated with maximum likelihood 4 Row-standardization implies dividing each element in the row of a matrix by the sum of the elements in the row. Alternative standardization procedures are also possible, but are not treated in detail here (Bivand et al., 008). 5 Other functions are admissible (Schabenberger and Pierce, 00).

8 8 utilizing nonlinear optimization routines. The null hypotheses of parameter stability across regions and/or specific spatial autoregressive processes can be tested with the spatial ARAR STAR, the spatial lag, and the spatial error STAR models as unrestricted models. 3. ML Estimation of Spatial STAR Models and Lagrange Multiplier Tests In this section, we describe maximum likelihood estimation procedures for the family of spatial STAR models. Individual and joint Lagrange Multiplier tests for nonlinear regime-splitting and/or specific spatial autoregressive processes are developed. 3.1 Spatial ARAR STAR Model Consider first the ARAR specification: 1 y = ρwy+ Xβ + ( I λw) μ, (4) where y is an N 1 spatial data series, X an N k matrix of explanatory variables, μ a vector of innovations, λ and ρ are spatial parameters and W an N N spatial weight matrix. A spatial ARAR STAR model can be constructed from (4) by adding a set of coefficients δ for a second regime, and the explanatory variables interacted with a transition function: 1 y= ρwy+ Xβ+ Go Xδ + ( I λw) μ, (5) where G is the logistic transition function defined in (), and all other variables are as before. As in time series analysis, testing directly for nonlinear regime-switching in each model results in unidentified nuisance parameters under the null hypothesis of no spatial

9 9 regimes. This problem can be solved by considering an appropriate Taylor series approximation of the transition function G( Wx, γ, c) around γ = 0, as suggested by Luukkonen et al. (1988). Considering the first-order Taylor series expansion of G(Wx,γ,c) around γ = 0, the spatial ARAR STAR model is approximated as: y ρwy+ Xα + Xδ o η + ηwx + I λw μ (6) 1 ( 0 1 ) ( ), which, after collecting terms, can be rewritten in reduced form as: ( o ) y ( I W) X + X Wx + ( I W) 1 1 ρ ξ ϕ λ μ ρ β + λ μ (7) 1 1 ( I W) Z ( I W), where Z is the N k matrix defined as Z = ( X, Wx o X ), and β is a k 1 vector defined by the original parameters as = ( + ) 0 ξ ηδ α and ϕ = ( ηδ 1 ). Assuming normally distributed errors that are independently and identically distributed with mean 0 and variance σ, the log-likelihood function for the spatial ARAR STAR model is: N N 1 Ly ( ; β, ρλσ,, ) = ln( π) ln( σ ) + lnw + ln A WR εww, R Rε σ (8) where W = I ρw, A WR = I λw and ε = WA y Zβ. are derived as: Lagrange Multiplier tests for spatial dependence and nonlinear regime switching

10 10 LM LM φ ewy ewe = σ σ + LM ( WZβ) M ( WZβ) σ λ= ρ= 0 λ= 0 = epe σ =0,, [ARAR, spatial dependence] (9) [STAR, nonlinear regime switching] (10) epwx ζ ewy ewe LM = σ σ σ + LM + LM ( WXζ ) M ( WXζ ) σ λ= ρ= φ= 0 λ= 0 φ= 0. [STAR ARAR] (11) where e represents the residuals of the model under the null hypothesis estimated using an appropriate generalized least squares (GLS) estimator, σ = ee N, LM λ = 0 is the usual Lagrange Multiplier test for spatial error dependence (Anselin, 1988), P is a projection matrix 1 Z ( Z Z) Z, and M = I P. The tests are asymptotically distributed with, k and k+ degrees of freedom, respectively Spatial Lag STAR Model To derive the spatial lag STAR model we consider first the basic spatial autoregressive lag specification: y = ρwy+ Xβ + μ, (1) with all symbols as previously defined. A spatial lag STAR model is constructed from (1) by adding a set of coefficients δ for a second regime, interacted with a transition function: 6 The mathematical derivation of the tests is provided in Pede et al. (009).

11 11 y= ρwy+ Xβ + Go Xδ + μ, (13) where G is the logistic transition function defined in (), and all other variables are as defined before. Assuming the errors are independent and identically distributed following a normal distribution, the log-likelihood function for the spatial lag STAR model is: N N 1 Ly ( ; β, ρσ, ) = ln( π) ln( σ ) + ln I ρw μμ, σ (14) where μ = ( I ρw) y f( X; θ ), f is the function defined in equation (1) with θ β, δ, γ,, and σ = μμ / N. parameters = ( c) Considering the first-order expansion of G around γ = 0, the Taylor series approximation for the spatial lag STAR model is: y ρwy+ Xα+ Xδo( η + ηwx) + μ, (15) 0 1 which, after collecting terms, can be rewritten in reduced form as: ( o ) [ ] 1 y ( I ρw) Xξ + X Wx ϕ+ μ + 1 ( I ρw) Zβ μ. (16) Noting that the LM test in (10) would also be used to test for detecting nonlinear regime switching in the ARAR model, individual and joint LM tests for the absence of a spatial lag and spatial lag with nonlinear regime splitting are derived as: LM ρ= 0 1 ewy = σ NJ, (17)

12 1 LM epwz β ewy σ σ epwze NJ σ ρ= ϕ=0 = +, (18) where e are the residuals of the model under the null hypothesis estimated using an adequate GLS estimator, and the term NJ is defined as: ( WZβ) M ( WZβ) NJ = tr ( W + W W ) +. σ (19) respectively. 7 The tests are asymptotically distributed with 1 and k+1 degrees of freedom, 3.3 Spatial Error STAR Model We motivate the STAR model with autocorrelated innovations by considering the typical spatial error autoregressive specification: 1 y = Xβ + ( I λw) μ. (0) Unlike the spatial lag model, the marginal effects associated with a (non-uniform) change in (one of) the exogenous variables are stationary across space in the case of the spatial autoregressive error model. A spatial error STAR model is constructed from (0) by adding a set of coefficients δ for a second regime, interacted with a transition function: 1 y = Xβ + Go Xδ + ( I λw) μ. (1) For estimation we once more apply standard maximum likelihood principles for spatial 7 Details about the derivations of the tests are given in Pede et al. (009).

13 13 error process models. Specifically, if the errors are independently and identically distributed following a normal distribution, the log-likelihood function for the spatial error STAR model is: N N 1 L( θ, ρλσ,, ) = ln( π) ln( σ ) + ln I λw μμ, () σ ( ) where μ ( I λw ) y f ( X; θ) =, f is the function in equation (1) with parameters (,,,c) θ = β δ γ, and σ = μμ / N. Since there is no analytical expression forθ, optimization cannot be based on a one-shot maximization of the concentrated likelihood. An iterative feasible generalized least squares approach is, however, appropriate as long as a consistent estimate of λ is attained through optimization of () given μ. In order to construct the appropriate LM tests, we again apply the first-order Taylor series approximation of the transition function, which for the spatial error model is: 1 ( o ) ( ) y Xξ + X Wx ϕ+ I λw μ 1 Zβ + ( I λw) μ. (3) Lagrange Multiplier tests for spatially autoregressive errors and nonlinear regime switching are derived following the general principles outlined in Anselin (1988): LM = LM + LM (4) λ= ϕ= 0 λ= 0 ϕ= 0.

14 14 The tests are asymptotically distributed with 1, k and k+1 degrees of freedom, respectively. 8 The family of LM tests is summarized in Table Monte Carlo Experimental Design Although the asymptotic properties of the derived LM tests are well-known, it is of crucial interest to investigate their performance in small samples in terms of size and power. The performance of the LM tests are therefore examined under various data generating assumptions. The experimental design follows standard practice for Monte Carlo simulations typically used in a spatial econometric context (Anselin and Rey, 1991; Florax and Folmer, 199, 1994, Anselin and Florax, 1995; Anselin et al., 1996; Anselin and Griffith, 1998). 9 We first generate a random spatial pattern where (x, y) coordinate pairs are drawn from a uniform (0, 1) distribution. The sample sizes we investigate are N = 5, 49, 100 and 400. The data generating process is: ( o ) y= I W X + G X + I W 1 1 ( ρ ) β δ ( λ ) μ, (5) where X is an N 3 matrix of explanatory variables consisting of a constant term and two variables drawn from a (0, 1) uniform distribution, μ is a vector of innovations assumed to have a standard normal distribution, and the parameter vectors β and δ are fixed to unity. Assuming that one of these explanatory variables (say x 1 ) governs the regime transition, the transition variable is generated as Wx 1, the spatial lag of x 1. We use 500 Monte Carlo replications, looping over different values for the spatial autoregressive parameters ρ and λ, and the spatial nonlinearity parameters γ and c. Six values of the 8 Details about the derivation of the tests are given in Pede et al. (009). 9 An extensive overview of the available simulation studies is provided in Florax and De Graaff (004).

15 15 smoothing parameter γ are considered; 0, 1,, 3, 5 and 10. The values for the location parameter c are: 0, the mean of Wx, and one standard deviation above and below the mean of Wx. Seven parameter values are used for the spatiall autoregressive parameters ρ and λ; 0, 0.1, 0., 0.4, 0.6, 0.8 and Three weight matrices were constructed to identify different neighborhood structures, given the random set of spatial coordinates of size N. The first weight matrix is based on the queen contiguity criterion, where all cells sharing a common edge or vertex are considered neighbors. The second neighborhood structure is based on a k- nearest neighbor (KNN) definition. We use the square root of the number of observations in the sample to define the number of nearest neighbors. The weight attributed to each observation is the inverse of the number of nearest neighbors. The third neighborhood definition also follows the KNN criterion, but the weight attributed to each observation is the inverse of the Euclidian distance between neighbors. All weight matrices were rowstandardized such that the sum of each row equaled one. Following the ARAR STAR specification described in equation (5), N observations on the dependent variable were generated from the explanatory variables, the spatial parameters, the transition variable, the smoothing and location parameters, and the standard normal random error vector μ. The matrix of variables Z = ( X, Wx o X ) corresponding with the first-order Taylor expansion of the transition function G was maintained fixed in the 500 replications. The model under the null hypothesis is a classical linear regression model with no spatial dependence: y = Xξ + μ. (6)

16 16 For each replication, the LM tests were computed and compared to their asymptotic critical value at α = We report the proportion that the null hypothesis was rejected, which is defined as the number of times the calculated value of a test statistic exceeds its asymptotic critical value, scaled by the total number of replications. Table 1 summarizes the LM tests evaluated during the Monte Carlo simulation. The Monte Carlo standard errors (MCSE) for a nominal significance level p with M replications are estimated as: MCSE = p(1 p). M (7) Using the 0.05 critical value for p plus or minus two times MCSE and considering 500 replications implies that the 95% confidence interval centered on p = 0.05 includes rejection frequencies between and The power and size of the respective tests are examined, and the simulation results for the Lagrange Multiplier LAG, ERR and ERR+LAG tests are also compared to those obtained in previous simulation studies (Anselin and Rey, 1991; Anselin and Florax, 1995; Anselin et al., 1996) Monte Carlo Simulation Results 5.1 Empirical Size of the Tests To determine the size of the respective tests, we use the simple linear model with no spatial dependence as the restricted model. The rejection frequencies for the null hypotheses of the different tests are obtained from the simulation output for the case 10 R code for the Monte Carlo simulations is presented in Pede et al. (009).

17 17 where the parameters ρ, λ, γ and c are set equal to zero. The rejection frequency of this general null hypothesis, when it is true, is reported in Table for the three weight matrices used in the simulation. It is interesting to observe that all tests except the ERR+LAG test yield rejection frequencies within the 95% confidence interval for the four sample sizes used in the simulation. These results, to the extent that they overlap, correspond to earlier findings reported in Anselin and Florax (1995) and Anselin et al. (1996). 11 The NLIN test overrejects the null hypothesis for N = 100 with the queen weight matrix, but behaves well in all other cases. It should be noted that the empirical size of the tests varies across different weight matrices. Anselin and Rey (1991) and Anselin and Florax (1995) also noted the influence of the choice of weight matrix on the empirical size of the tests. A priori, the influence of the specification of the weight matrix on empirical size may appear counterintuitive because the null hypothesis assumes no spatial dependence, but the weights matrix is obviously used in the definition of the various tests. The ERR+LAG test under-rejects the null hypothesis for all three weights matrices and samples sizes considered. Anselin and Florax (1995) observed similar results for the ERR+LAG in the case of a misspecification in the form of lognormal errors. They stressed that an underrejection of the null hypothesis when no spatial dependence is present does not have any consequences, since the standard estimation results are interpreted as they should be. 11 Anselin and Florax (1995) and Anselin et al. (1996) used 5,000 replications and p = 0.05, which corresponds to rejection frequencies between and for the 95% confidence interval and they observe correct sizes for these tests.

18 18 5. Power of the Tests In this section we discuss the statistical power of the tests. The power of a test is defined as the probability of rejecting the null hypothesis when the null hypothesis is false. The small sample power of the various tests is summarized in Figures through Power against Spatial Autoregressive Processes We first describe the power of the various LM tests against spatial autoregressive processes, and then discuss the small sample power of the tests against nonlinear spatial regimes. With respect to the spatial autoregressive processes, we discuss the spatial error, the spatial lag, and the spatial ARAR data generating processes, successively. In the case of spatial autoregressive error dependence, the small sample characteristics are obtained from the data generating process in which the parameters ρ, γ and c are set to zero. The power functions of the seven tests are shown in Figure for N = 5 and 400 using the queen weight matrix, and in Figure 3 for N = 400 using the KNN(1/N) and KNN(1/d ij ) weight matrices. For all tests the rejection frequencies increase as the magnitude of the spatial autoregressive parameter and sample size increase. In general, the ERR test has the highest power, followed by the LAG test. It should be noted that all tests perform rather poorly in the smallest sample, achieving the 95% rejection mark only for λ > 0.9. For a reasonably large sample size (N = 400) the power of the tests is comparable, but slightly higher for the weights matrices based on the queen and the KNN(1/d ij ) specification as compared to the KNN(1/N) version. To the extent that the setup of the simulations is comparable, these findings are consistent with the simulation results of Anselin and Rey (1991) and Anselin and Florax (1995). 1 For completeness, the rejection frequencies for various data generating processes are listed in Tables 3 through 6 of Pede et al. (009).

19 19 The most noteworthy conclusion from the rejection patterns presented in the respective figures is that all tests have remarkably high power against spatially autoregressive errors, even although most of the tests have a different alternative hypothesis. This is partly due to the similarity between the spatial error and the spatial lag process (Anselin et al., 1996). The NLIN test is least affected, although its power is substantially larger than the nominal significance level associated with the size of the test. As with the LAG and ERR, test this is most likely due to the similarity in the underlying spatial processes, especially since Wx is used as the transition variable. The small sample characteristics in the case where the spatial lag model is the true underlying data generating process are very similar to those described above. Figures 4 and 5 document the simulation outcomes. In accordance with previous simulation work (Anselin and Rey, 1991; Anselin and Florax, 1995) the LAG test appears most powerful. The NLIN test has substantial power against an erroneously omitted spatially lagged dependent variable, and in fact this is more pronounced than in the case of the spatial error model being the true model. The latter result is likely due to the fact that the spatial lag term Wy implicitly includes the transition variable Wx. Figures 6 through 1 provide three-dimensional power functions against the spatial ARAR model for various values of ρ and λ. 13 We again encounter the general pattern identified above. All tests have power against the ARAR process, with the NLIN test s power being the lowest. One should note, however, that the extent to which the NLIN test wrongfully points to the STAR model as the correct alternative is substantial. 13 The surfaces are generated using 36 data points corresponding to the nonzero values for the spatial autoregressive parameters.

20 0 5.. Power against Spatial STAR Nonlinearity The power against a spatial STAR process is obtained by setting the parameters ρ and λ equal to zero and extracting the corresponding rejection frequencies from the simulation output. 14 The power functions of the seven tests against spatial STAR nonlinearity are shown in Figure 13 for N = 5 and 400 using the queen weight matrix, and in Figure 14 for N = 400 using the KNN weight matrices. In each figure, the location parameter is set equal to the mean of Wx. As expected, the individual NLIN test shows the highest power against the null, followed by the joint ERR+NLIN, LAG+NLIN and ERR+LAG+NLIN tests, respectively, in the smallest sample. It is interesting to see that the ERR, LAG and ERR+LAG tests all have very little power against the null in the smallest sample. This is as expected, because no spatial dependence is present. However, in the largest sample, the ERR, LAG and ERR+LAG tests do show considerable power, even although in terms of true parameters ρ = λ = 0. For instance, when N = 400 with a KNN(1/N) weight matrix the power function of all seven tests are almost indistinguishable (see Figure 14). In general, the individual and joint tests for spatial nonlinearity perform well in the largest sample as well as in moderately sized samples, achieving the 95% rejection mark for values of γ greater than 1. In the smallest sample the 95% rejection mark was reached for values of γ greater than 5. We also investigated the power of the tests against spatial STAR nonlinearity in the case where the location parameter is set equal to the values of 0, one standard deviation above the mean of Wx, and one standard deviation below the mean of Wx. The power functions of the seven tests against spatial STAR nonlinearity for various 14 For completeness a numerical overview of the rejection frequencies against STAR nonlinearity are reported in Table 4 of Pede et al. (009).

21 1 configurations of the true parameters, sample sizes and weights matrices are shown in Figures 15 and 16. Figure 17 shows that for a very small sample (N = 5) we again observe the ideal situation that the individual and joint nonlinearity tests have power, whereas the individual and joint spatial dependence tests do not. The power of the NLIN tests is however rather low, and even deteriorating for higher values of γ in the case where the location parameter is set to one standard deviation above the mean of the transition variable. In Figure 16 we observe that the power increases with a larger sample size (N = 400). However, this is the case for the individual and joint tests for spatial dependence processes as well, although in reality ρ = λ = 0. The behaviors described above is also observed when the spatial error or the spatial lag model is taken as the restrictive model. 15 In sum, we are left with a situation that warrants further investigation. All tests have a reasonable size in the case where the non-spatial model is taken as the restricted model. The power of the tests is also quite satisfactory, especially in relatively larger samples (for instance, N = 400). However, the individual and joint nonlinear regime switching tests show power against the traditional spatial autoregressive processes (error and lag), and the individual and joint spatial dependence tests have power against STAR nonlinearity, operationalized with a spatially lagged exogenous variable as the transition variable. Given this behavior, it remains to be seen whether a well-performing specification strategy can be designed. Lundbergh et al. (003) found similar results with the time series STAR. They studied time-varying smooth transition (TV-STAR) models, which describe simultaneously nonlinearity and structural change by using two modeling strategies: the specific-to-general-to-specific procedure which consists of testing for 15 For completeness, Tables 5 and 6 in Pede et al. (009) report the associated rejection frequencies.

22 nonlinearity and/or structural change, and the specific-to-general procedure which consists of specifying a model with nonlinearity and/or time-varying properties. Their Monte Carlo results also suggest that neither of the two procedures dominates the other. Moreover, their empirical application on U.S. macroeconomic time series reveals the merit of both modeling strategies. Given the complication in designing an effective specification strategy, further investigation may be needed. In empirical applications, the consistency in model specification could be verified using both approaches. That is, investigate nonlinear structural breaks across space first and then decide on the spatial process, or proceed the other way around. 5. Conclusion In this paper we developed a new method to evaluate and incorporate nonlinear spatial dynamics and regime switching behavior into conventional spatial lag and error process models, based on smooth transition autoregressive models developed in time series research. The spatial STAR model belongs to the family of spatial regression models allowing for parameter variation across space. In particular, the STAR model allows for parameter variation across spatial regimes, where the membership of the spatial regimes is determined endogenously. The spatial STAR model is relatively easy to estimate and substantially more parsimonious than previous spatial regression methods allowing for spatial parameter variation. Moreover, the STAR model proposed here provides the opportunity to incorporate traditional spatial process models such as spatially

23 3 autoregressive errors, a model with a spatially lagged dependent variable, or a combination of lag and error processes in the ARAR model. Four types of spatial STAR models were developed in this study: the basic spatial STAR model, and the spatial error (SEM-STAR), the spatial lag (SAR-STAR) and the spatial ARAR STAR models. Maximum likelihood estimation procedures for each model were presented, and a series of new Lagrange Multiplier tests was developed to investigate nonlinear spatial regime switching and/or spatial dependence. Monte Carlo simulations of these tests reveal that all tests perform well with respect to size. Each tests produce rejection frequencies within a 95% confidence interval under the null hypothesis of an aspatial linear model. The tests also show relatively good power against spatial dependence (spatial error and spatial lag processes) and STAR-type nonlinearity. Almost invariably, the test that was designed for a specific alternative has the highest power. Unfortunately however, the spatial processes associated with the different tests appear to be rather similar in nature. As a result, the individual and joint nonlinear regime tests have power against the traditional spatial error and/or lag processes, and the spatial dependence tests have power against spatial STAR-type nonlinearity. This obviously complicates the design of an effective specification strategy for cross-sectional models with spatial autocorrelation and endogenous regime-switching behavior. The design and investigation of the performance of feasible alternative specification strategies is left for future research, but it is foreseeable that a specific-to-general strategy based on the estimation of the non-spatial model and a subsequent choice for an unrestricted model on the basis of p-values of the different LM tests may very well be effective.

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28 8 Table 1: Lagrange Multiplier Tests for Spatial Dependence and/or Nonlinearity. Test ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Test statistic and asymptotic distribution LM 1 ewe = T σ λ = 0 1 ewy σ LM ρ= 0 = NJ LM =0 = epe ϕ ~ χ (3) σ ~ χ (1), ~ (1) χ λ= ϕ= = λ= + ϕ= χ LM 0 LM 0 LM 0 ~ (4) LM LM epwz β ewy ( WZβ) M ( WZβ) ( + ) T= tr W WW σ σ ρ= ϕ= 0 = + T + σ ewy ewe = σ σ + LM ( WZβ) M ( WZβ) σ epwze σ λ= ρ= 0 λ= 0 ~ (4) χ ~ χ () epwx ζ ewy ewe LM = σ σ σ + LM + LM ( WXζ ) M ( WXζ ) σ λ= ρ= φ= 0 λ= 0 ϕ= 0 ~ χ (5)

29 9 Table : Empirical Size of the LM Tests. Queen contiguity KNN (1/N) KNN (1/d ij ) Tests N = 5 N = 49 N = 100 N = 400 N = 5 N = 49 N = 100 N = 400 N = 5 N = 49 N = 100 N = 400 ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Note: The 95% confidence interval for the rejection proportion (prop) with 500 replications is < prop <

30 Figure 1: Example of the transition function G(Wx; γ, c), and different levels of the smoothing parameter, γ. Note that two distinct regimes emerge when γ = 100, whereas there are no regimes identified when γ = 0. The parameter c functions as a location parameter. That is, the inflection of the transition function is centered on c. 30

31 ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Lambda ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Lambda Figure : Power against Spatial Error Model, N = 5 (top) and 400 (bottom), Queen.

32 ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Lambda ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Lambda Figure 3: Power against Spatial Error Model, N = 400, KNN(1/N) (top) and KNN(1/d ij ) (bottom).

33 ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Rho ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Rho Figure 4: Power against Spatial Lag Model, N = 5 (top) and 400 (bottom), Queen.

34 ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Rho ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Rho Figure 5: Power against Spatial Lag Model, N = 400, KNN(1/N) (top) and KNN(1/d ij ) (bottom).

35 Rho Lambda Figure 6: Power of ERR against ARAR Process, N = 100, Queen Rho Lambda Figure 7: Power of LAG against ARAR Process, N = 100, Queen.

36 Rho Lambda Figure 8: Power of NLIN against ARAR Process, N = 100, Queen Rho Lambda Figure 9: Power of ERR+NLIN against ARAR Process, N = 100, Queen.

37 Rho Lambda Figure 10: Power of LAG+NLIN against ARAR Process, N = 100, Queen Rho Lambda Figure 11: Power of ERR+LAG against ARAR Process, N = 100, Queen.

38 Rho Lambda Figure 1: Power of ERR+LAG+NLIN against ARAR Process, N = 100, Queen.

39 ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Gamma ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Gamma Figure 3: Power against Spatial STAR Model, N = 5 (top) and 400 (bottom), with c equal to the Mean of Wx, and the Queen Matrix.

40 ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Gamma ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Gamma Figure 34: Power against Spatial STAR Model, N = 400, c equal to the mean of Wx, KNN(1/N) (top) and KNN(1/d ij ) (bottom).

41 ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Gamma ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Gamma Figure 15: Power against Spatial STAR Model, N = 5, c equal to Mean Wx+SD (top), Mean Wx SD (bottom), Queen Matrix.

42 ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Gamma ERR LAG NLIN ERR+NLIN LAG+NLIN ERR+LAG ERR+LAG+NLIN Gamma Figure 46: Power against Spatial STAR Model, N = 400, c equal to Mean Wx+SD (top), Mean Wx SD (bottom), KNN(1/N).