A comparison of spatial error models through Monte Carlo experiments

Economic Modelling 30 (2013) 743–753

Contents lists available at SciVerse ScienceDirect

Economic Modelling journal homepage: www.elsevier.com/locate/ecmod

A comparison of spatial error models through Monte Carlo experiments Takafumi Kato ⁎ Graduate School of Environmental Studies, Nagoya University, Nagoya, Aichi 464-8601, Japan

a r t i c l e

i n f o

Article history: Accepted 14 October 2012 JEL classification: C21 C51 C52 C53

a b s t r a c t A spatial error model is classified as a geostatistical model or a weight matrix model on the basis of the method of specification of spatial autocorrelation in the disturbance. Specification errors cannot be assumed to be absent, and the robustness of alternative specifications is useful for dealing with potential errors. Previous studies compared several models to arrive at two basic conclusions: (i) all of the models maintain reasonable estimation accuracy, and (ii) the two types of models have well-matched predictive abilities. The present study makes a supplementary comparison to investigate whether these conclusions are true for a broader range of models. Also, implications of our results for the model choice are explored. © 2012 Elsevier B.V. All rights reserved.

Keywords: Spatial autocorrelation Specification error Robustness Model choice

1. Introduction A linear regression model is called a spatial error model when spatial autocorrelation is present in the disturbance. A spatial error model is classified as a geostatistical model or a weight matrix model on the basis of the method of specification of spatial autocorrelation. The geostatistical model is defined with a correlation function, which is a device to specify the autocorrelation in a direct manner, whereas the weight matrix model is defined with a weight matrix, which is a device to specify it in an indirect manner. Unfortunately, errors cannot be assumed to be absent in any specification. As stated in Anselin (2002), even when the type of model is known, the choice of the correct correlation function or weight matrix is not theoretically guaranteed under any circumstances, and only a few incorrect choices can be practically eliminated using validation techniques. Thus, it is useful to explore the robustness of spatial autocorrelation specifications. Choosing a relatively well-performing model is a good policy to allow for specification errors. In a recent study, Dubin (2003) took important steps toward exploring the robustness of alternative specifications. Under a reciprocal procedure, she performed a series of Monte Carlo experiments to compare three geostatistical and five weight matrix models, and found two basic conclusions: (i) all of the models maintain reasonable estimation accuracy, and (ii) the geostatistical models have better predictive abilities than the weight matrix models. Specifically, in each experiment, one of the models was used as the data generator, and the generated

⁎ Tel.: +81 52 789 3502. E-mail address: [email protected]. 0264-9993/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2012.10.010

data were used to examine all of the models with conventional estimators and predictors.1 In a following study, Kato (2008a) took further steps by showing that the conventional predictor applied to the geostatistical models is more efficient than that applied to the weight matrix models. He proposed two alternative predictors for the weight matrix models to make an optimal comparison of the two types of models, and tested the dominance of the geostatistical models over the weight matrix models.2 As expected, in his experiments based on Dubin (2003), conclusion (i) was confirmed to be sound, but conclusion (ii) was shown to be unsound. In a comment on Kato (2008a), although appreciating his results, Dubin (2008) argued that the geostatistical models are still preferable to the weight matrix models. Two reasons were adduced in favor of this argument: (i) the difference in the predictive ability between the best and worst models is smaller for the geostatistical models than for the weight matrix models, and (ii) the predictive ability of the worst geostatistical model is greater than that of the worst weight matrix model. In a response to Dubin (2008), although acknowledging her observations, Kato (2008b) suggested a combined use of the geostatistical and weight matrix models that have the advantage in predictive ability among models of their respective type. Two reasons were produced in support of this suggestion: (i) all of the models offer the best prediction performance only in a limited number of

1 Actually, Dubin (2003) did not use one weight matrix model as the data generator, but used it in estimation and prediction as a substitute for another weight matrix model that did not offer the best prediction performance in any experiment. 2 Bourassa et al. (2007) compared the two types of models through an observed data experiment and concluded that a weight matrix model is not suitable for the purpose of prediction. Kato (2008a) showed that they applied to the weight matrix model a less efficient predictor than Dubin (2003).

744

T. Kato / Economic Modelling 30 (2013) 743–753

experiments, and (ii) neither of the two types of models is dominant. Interestingly, the most advantageous weight matrix model is a model that mimics one of the geostatistical models. The purpose of the present study is to make a supplementary comparison of spatial error models through a series of Monte Carlo experiments based on Dubin (2003). This comparison is meant to explore the robustness of spatial autocorrelation specifications in the following ways. First, we assess a weight matrix model that Kato (2008a) considered to be worth assessing in a future study. This model was proposed by Pace and Gilley (1997) to demonstrate the benefits from using a spatial error model. Second, we propose and assess two weight matrix models to consider the above argument of Dubin (2008). These models mimic two of the three geostatistical models examined in Dubin (2003) and Kato (2008a), respectively, in the same manner as one of the five weight matrix models examined there mimics the third geostatistical model. Observations similar to those made by Dubin (2008) for the geostatistical models could be made for the weight matrix models that mimic those models: the difference in the predictive ability between the best and worst models could be smaller for the three mimics than for a broader range of weight matrix models, and the predictive ability of the worst mimic could be greater than that of the worst weight matrix model. Third, we investigate whether we can choose the same combination of geostatistical and weight matrix models as Kato (2008b) when we consider the three additional models. Different models could be shown to be the most advantageous geostatistical and weight matrix models, and there is a possibility of finding an excellent model. In forming our purpose, we have the hedonic price function in mind, because the spatial error model is finding wide application to that function. This is to follow Dubin (2003) and Kato (2008a), and is not to deny that the model is also applicable to other functions, as intimated in Arbia et al. (2012). If the term spatial error model is always used without relating it to any particular function, it is possible to make our comparison useful for these other functions. This usage is therefore adopted here. We also have reasons for taking up only a few weight matrix models as the additional targets of comparison. One reason is that the developments of the two types of models are different in the hedonic literature. As stated in Palmquist (2005), geostatistical models used are chiefly limited to the three above, whereas weight matrix models proposed are widely diverse. 3 Another reason is that it is not easy to put many models through the above reciprocal procedure. As the number of models compared increases, the amount of resources required jumps. Naturally, the developments of the two types of models in the hedonic literature are reflected in the experiments of Dubin (2003) and Kato (2008a), and are influential in the discussions of Dubin (2008) and Kato (2008b). This indicates that the overall difference in the predictive ability found between the two types of models may be attributable to the difference between the diversities of models considered for their respective type. Unfortunately, the reason for this difference in the predictive ability was not explained in any of those previous studies. Assessment of the above three mimics allows us to discuss that hypothesis. If the overall performance of these weight matrix models is found to be similar to that of the three geostatistical models, the hypothesis may be accepted; otherwise, the difference in the predictive ability should be basically attributable to the difference in the method of specification of spatial autocorrelation. The remainder of the present paper is organized as follows. In Section 2, we provide the aspects of comparison. Models are defined, and statistics are described. In Section 3, we perform Monte Carlo experiments. The design is produced, and the results are discussed. In Section 4, we summarize the points made in the preceding sections and mention topics for future study. 3 Interestingly, Palmquist (2005) saw practical merit in the weight matrix model of Pace and Gilley (1997), which supports our assessment of that model.

2. Aspects of comparison The mathematical notation of Kato (2008a) is adopted, with minor modifications, to provide the aspects of comparison. We first define the geostatistical and weight matrix models before proceeding to describe the estimators and predictors for the respective types of models. 2.1. Models The spatial error model can be expressed as y = Xb + u, where y is the vector of values of the dependent variable; X is the matrix of values of the independent variables, with the first column comprising ones; b is the vector of regression parameters b1, b2, ⋯, and bm; and u is the vector of values of the disturbance. 4 In this model, u is assumed to have a normal distribution with expectation 0 and covariance σ 2K(D;c), where σ 2 is a nuisance parameter, c is the vector of spatial autocorrelation parameters c1 and c2, D is a symmetric matrix of distances separating the locations included in the population, and K( ∘ ; c) is a function in which the input and output are matrices of the same order. The model is classified as a geostatistical model or a weight matrix model on the basis of the method of specification of K( ∘ ; c). In the geostatistical model, K(D;c) is set to [ F(Dij;c) ], where Dij expresses the ijth element of D, which is the distance between the ith and jth locations, and F( ∘ ; c) is a function in which the input and output are scalars. 5 This function is known as a correlation function, and its use indicates that the covariance between the disturbances associated with a pair of locations depends only on the separation distance of that pair. In the weight matrix n o−1 model, K(D;c) is set to ðI−c1 W ðD; c2 ÞÞ′ ðI−c1 W ðD; c2 ÞÞ , where W( ∘ ; c2) is a function in which the input and output are matrices of the same order. The output is known as a weight matrix, and its use indicates that the covariance between the disturbances associated with a pair of locations depends on the separation distances of all pairs. 6 As an expedient for model definition, we assign the same names as Dubin (2003) and Kato (2008a) to their originally compared three geostatistical and five weight matrix models: NEGE, GSS, and SPH to the former models and NN, K, P, LIM, and NEW to the latter. Geostatistical models vary in accordance with the specification of the form of the correlation function, which determines the elements of K(D;c). In the NEGE model, the ijth element is equal to 1 if Dij = 0, and c1exp(− Dij/c2) if Dij > 0, whereas in the GSS model, it is equal to   2  1 if Dij = 0, and c1 exp − Dij =c2 if Dij > 0. The corresponding element of the SPH model is equal to 1 if Dij = 0, c1(1 − 3Dij/ 2c2 + Dij3/2c23) if 0 b Dij b c2, and 0 if Dij ≧ c2. The abbreviations NEGE, GSS, and SPH reflect the choices of the negative exponential, Gaussian, and spherical forms for the correlation function, respectively. Weight matrix models vary in accordance with the specification of the elements of the matrix from which the weight matrix is derived by row standardization. In the NN model, the ijth element is equal to 1 if Dij > 0 and the jth location is one of the c2 locations nearest to the ith location, and 0 otherwise. The abbreviation NN reflects the property that the element is specified with nearest neighbors. In the 4 When the context of the discussion leaves no ambiguity, the dimensions of vectors and matrices are not elucidated. 5 In this setting, a shorthand device is adopted to represent the elements of K(D;c). 6 The dimensions of 0 and I vary according to the context of the discussion. It is noteworthy that if the prevailing definition is applied to the weight matrix, W(D;c2) can be replaced with W(D). Dubin (2003) and Kato (2008a) demonstrated that the parametric definition is superior to the prevailing definition. For an empirical application of the parametric definition, see Pace and Gilley (1997). It is also noteworthy that the above definition of the weight matrix model is based on the simultaneous approach. As stated in Militino et al. (2004), the conditional approach is rarely adopted in the hedonic literature. For reasons for such rarity, see Anselin (1988).

T. Kato / Economic Modelling 30 (2013) 743–753

K model, the ijth element is specified as a weighted sum of the corresponding element of the kth matrix as k ranges from 1 to 15. The name K originates from the notation of the running variable, k. The weight is c2k, where c2 is restricted to lie between 0 and 1. The ijth element of the kth matrix is equal to 1 if Dij > 0 and the jth location is the kth nearest location to the ith location, and 0 otherwise. In the P model, the ijth element is equal to 0 if Dij = 0, and 1/Dijc2 if Dij > 0. The abbreviation P reflects the property that the element is specified with a power. In the LIM model, the ijth element is equal to 1 if Dij > 0 and the jth location lies within the distance c2 of the ith location, and 0 otherwise, with the abbreviation LIM reflecting the property that the element is specified with a distance limit. 7 The corresponding element of the NEW model is equal to 0 if Dij = 0, and exp(− Dij/c2) if Dij > 0. The abbreviation NEW reflects that the model is defined with a negative exponential weight matrix. We also assign the names MAX, GW, and SW to our additionally compared three weight matrix models. The MAX model is what was proposed by Pace and Gilley (1997) and was considered to be worth assessing by Kato (2008a). In this model, the ijth element of the matrix from which the weight matrix is derived by row standardization is equal to 0 if Dij = 0, and max(1 − Dij/c2, 0) if Dij > 0. The abbreviation MAX reflects the property that the element is specified with a maximization. The GW and SW models are what mimic GSS and SPH, respectively, in the same manner as NEW mimics NEGE. In the former model, the ijth element is equal to 0 if Dij = 0, and   2  exp − Dij =c2 if Dij > 0, whereas in the latter, it is equal to 0 if Dij = 0, 1 − 3Dij/2c2 + Dij3/2c23 if 0 b Dij b c2, and 0 if Dij ≧ c2. The abbreviations GW and SW reflect that the models are defined with what might be called Gaussian and spherical weight matrices, respectively.8 2.2. Statistics In estimation, we choose feasible generalized least squares, which is defined with a consistent estimator of c. Let us use the maximum likelihood estimator of c by assuming its existence. This estimator is derived as a vector c˜ ¼ ðc˜1 ; c˜2 Þ′ such that the concentrated log-likelihood function has the maximum value n1 1 ðln 2π−ln n1 þ 1Þ− lnjK ðD11 ; c˜ Þ j 2 2 n ′ ′ −1 − 1 ln y1 HðX1 ; K ðD11 ; c˜ ÞÞ K ðD11 ; c˜ Þ H ðX1 ; K ðD11 ; c˜ ÞÞy1 ; 2

Lðc˜ Þ ¼ −

745

given as     ^ ¼ σ^ 2 X′ K ðD ; c˜ Þ−1 X −1 : V b 11 1 1 The point to observe is that the conventionally used K(D11; ∘ ) is a substitute for the intrinsically desired K(D; ∘ )11, which is the top left submatrix of K(D; ∘ ) that relates to the locations included in the estimation sample. For the geostatistical model, K(D11; ∘) = K(D; ∘ )11, because the covariance between the disturbances associated with a pair of locations depends only on the separation distance of that pair, whereas for the weight matrix model, K(D11; ∘) ≠ K(D; ∘ )11, because that covariance depends on the separation distances of all pairs. Then all of the estimators above are genuine and quasi for the respective types of models. Dubin (2003) used maximum likelihood, whereas Kato (2008a) chose feasible generalized least squares. These methods are not equivalent unless the estimation sample is infinite. Provided that the primary interest lies in the statistical inference about the regression parameters, the former method is inferior to the latter. In prediction, we choose two alternative predictors, which are considered to be optimal for comparison of the two types of models. Let us denote the number of locations included in the prediction sample by n2 and the submatrices of D, y, and X that relate to those locations by D22, y2, and X2, respectively, with D22 situated diagonally to the lower right of D11, and y2 and X2 immediately under y1 and X1, respectively. The first alternative is defined as      ð1Þ ^ þ K D; ^ ;  c˜ K D;  c˜ −1 y −X b y2 ¼ X2 b 1 1 21 11  is a partitioned matrix whose top left and top right are D11 and where D D12, and bottom left and bottom right are D21 and D22, respectively, with D12 as the matrix of distances separating the locations included in the estimation sample and those included  in the prediction   sample,  c˜  c˜ and D21 as the transpose of D12; and K D; and K D; are the 21   11  c˜ , respectively. This predictop left and bottom left submatrices of K D; tor is a simultaneous predictor in the sense that all of the elements of y2 are evaluated collectively. The second alternative is defined as 2

ð2Þ

y2

3 K ðD1 ; c˜ Þ21 K ðD1 ; c˜ Þ−1 11 6 K ðD ; c˜ Þ K ðD ; c˜ Þ−1 7  7 2 2 11 21 ^ þ6 ^ ; ¼ X2 b y1 −X1 b 6 7 4   ⋮  −1 5 K Dn2 ; c˜ K Dn2 ; c˜ 21

where n1 is the number of locations included in the estimation sample; D11, y1, and X1 are the top left submatrix of D and top submatrices of y and X, respectively, with each submatrix relating to those locations; and  −1 X′1 K ðD11 ; c˜ Þ−1 . The feasiH ðX1 ; K ðD11 ; c˜ ÞÞ ¼ I−X1 X′1 K ðD11 ; c˜ Þ−1 X1 ble generalized least squares estimators of b and σ2 are given as     ^ ¼ b^ ; b^ ; ⋯; b^ ′ ¼ X′ K ðD ; c˜ Þ−1 X −1 X′ K ðD ; c˜ Þ−1 y b 1 2 m 11 1 11 1 1 1 and 2 σ^ ¼

   1  ^ ′ K ðD ; c˜ Þ−1 y −X b ^ ; y −X1 b 11 1 1 n1 −m 1

^ is also respectively. The corresponding estimator of the covariance for b 7

Actually, the ith row is not standardized when it comprises zeros. The understanding of each model may be facilitated by visualization of the spatial autocorrelation contained therein. For some images in the form of correlograms, see Dubin (1998). 8

11

where Di for i = 1, 2, ⋯, n2 is a partitioned matrix whose top left and top right are D11 and D12,i, and bottom left and bottom right are D21,i and 0, respectively, with D12,i and D21,i as the vectors comprising the ith column and row of D12 and D21, respectively; and K ðDi ; c˜ Þ11 and K ðDi ; c˜ Þ21 are the top left and bottom left submatrices of K ðDi ; c˜ Þ, respectively. This predictor is a separate predictor in the sense that each of the elements of y2 is evaluated individually.      ∘  ∘ , K D; , K ðDi ; ∘ Þ11 , and The point to observe is that K D; 11 21 K ðDi ; ∘ Þ21 are substitutes for K(D; ∘ )11, K(D; ∘ )21, K(D; ∘ )11, and K(D; ∘ )21,i, respectively, where K(D; ∘ )21 is the submatrix of K(D; ∘ ) that is situated immediately under K(D; ∘ )11, and K(D; ∘ )21,i is the vector comprising the ith row of K(D; ∘ )21. For the geostatistical      ∘  ∘ ¼ K ðD; ∘ Þ11 , K D; ¼ K ðD; ∘ Þ21 , K ðDi ; ∘ Þ11 ¼ model, K D; 11 21 K ðD; ∘ Þ11 , and K ðDi ; ∘ Þ21 ¼ K ðD; ∘ Þ21;i . Then y2(1) and y2(2) are genuine partly because the genuine parameter estimators are used for them, but primarily because their corresponding substitutes are perfect. Technically, these predictors are also equivalent, although the former predictor is preferable in usability to the latter. For the weight matrix model, a limited number of equalities are true, and only for one case. In the no sampling case in which the estimation and prediction samples constitute the      ∘  ∘ entire population, K D; ¼ K ðD; ∘ Þ11 and K D; ¼ K ðD; ∘ Þ21 , 11 21

746

T. Kato / Economic Modelling 30 (2013) 743–753

whereas K ðDi ; ∘ Þ11 ≠K ðD; ∘ Þ11 and K ðDi ; ∘ Þ21 ≠K ðD; ∘ Þ21;i . Then y2(1) is quasi simply because the quasi parameter estimators are used for it, whereas y2(2) is quasi partly because those estimators are used for it, but primarily because its corresponding substitutes are imperfect. In the sampling case in which the two samples constitute only a part of the population, none of the four equalities is true. Then y2(1) and y2(2) are quasi by the same token. Clearly, the two alternative predictors are not equivalent, regardless of whether the case is no sampling or sampling. Dubin (2003) applied y2(2) to the geostatistical model, but a conventional predictor defined as 2

ð3Þ

y2

 3 W D ; c˜ 6  21;1 2  7 6 7  W D21;2 ; c˜2 7 ^ þ c˜ 6 ^ ¼ X2 b 7 y1 −X1 b 16 6 7 5 4  ⋮ W D21;n2 ; c˜2

to the weight matrix model. Unfortunately, y2(3) is suboptimal for comparison of the two types of models, because it is not based on the best linear unbiased predictor. Kato (2008a) considered y2(1) in relation to y2(2) for the geostatistical model while proposing y2(1) in addition to y2(2) for the weight matrix model. Either of the two predictors is optimal for both types of models, because they are based on the best linear unbiased predictor. Naturally, the comparison depends on which predictor is chosen for the weight matrix model. 3. Monte Carlo experiments The experimental design of Dubin (2003) is adopted, with appropriate modifications, to perform Monte Carlo experiments. We first produce a resource-saving design before proceeding to discuss the implication-bearing results. 3.1. Design Ten out of the eleven models defined in the preceding section are adopted to generate data as follows at 280 and 1, 000 locations in Euclidean space for the no sampling and sampling cases, respectively. First, we set m to 3 and choose the values of the parameters. The parameters b1, b2, b3, and σ 2 are set to 25, 10, 1, and 25, respectively. Two alternative values are used for c1: 0.8 in the high autocorrelation case and 0.6 in the moderate autocorrelation case. 9 The parameter c2 is set to 20 in NEGE, GSS, and SPH; 5 in NN; 2.5 in P; 7 in LIM and MAX; and 2 in NEW, GW, and SW. 10 Second, we assign the coordinates and values of the independent variables to each location. The first and second coordinates are randomly drawn from U(0,100). 11 The elements of the second and third columns of X are randomly drawn from U(0,5) and U(0,30), respectively. Third, we calculate the disturbances. For the geostatistical models, u is determined by u = C(D;c)e, where C(D;c) is a lower triangular matrix computed with the Cholesky decomposition of K(D;c), and e is a vector randomly drawn from N(0, 25I). For the weight matrix models, u is determined by u ¼ ðI−c1 W ðD; c2 ÞÞ−1 e. 12 Fourth, we calculate the value of the dependent variable to assign to each location from y = Xb + u. 9 As concerns the hedonic price function, interest focuses on the positive autocorrelation case. Admittedly, consideration of the negative autocorrelation case may also be useful with some other function in mind. 10 The values of c2 for MAX and GW or SW are based on those chosen by Dubin (2003) for LIM and NEW, respectively. 11 Examining the hedonic literature, we find that observations are sometimes made for aggregate units, including census tracts. This implies that the coordinates may be used for identification of the members of an irregular lattice as well as the properties in a market area. 12 Depending on the situation, u can be directly drawn from N(0,25K(D;c)) in either type of model.

A list of models to be fitted to the generated data is selected to save computational resources. When the seven original models NEGE, GSS, SPH, NN, P, LIM, and NEW are used as the data generator, the three models MAX, GW, and SW are examined along with NEW. Dubin (2003) and Kato (2008a) adopted these generators to examine NEGE, GSS, SPH, NN, P, LIM, and NEW along with K as a substitute for NN, 13 and Dubin (2008) and Kato (2008b) made the following observations: (i) the difference between the best and worst models is small for the three geostatistical models, although SPH has a slight advantage, (ii) NEW is prominent among the five weight matrix models, and (iii) the most advantageous geostatistical and weight matrix models are on a par with each other. Taking these observations into consideration, we focus on a comparison of MAX, GW, and SW, with NEW used as a benchmark. When the three additional models MAX, GW, and SW are used as the data generator, the four models NEGE, GSS, SPH, and NEW are examined along with MAX, GW, and SW. Observation (i) must be tested in the experiments in which data are generated with models other than those previously adopted. However, observation (ii) will not be affected by the possibility that an advantage of NN, K, P, or LIM over NEW could be found in these experiments, because the number of experiments with the seven original generators is much greater than that of experiments with the three additional generators, and observation (iii) can be reviewed even when models other than SPH and NEW are shown to be the most advantageous geostatistical and weight matrix models, respectively. Models are examined through the following steps. First, we set n1 and n2 to 200 and 80, and obtain two disjoint samples for estimation and prediction by randomly selecting those numbers of locations, respectively, from the 280 locations in the no sampling case and 1, 000 locations in the sampling case. Arrangement of y, X, and D, and construction of their submatrices are conducted in conformity with these sample selections. Second, we perform a grid search for c˜     ^ σ^ 2 , and V b ^ , with special reference to V b^ , and evaluate b, 1       ^ ^ ^ V b 2 , and V b 3 as the diagonal elements of V b . For NEGE, GSS, and SPH, the value of c˜1 is varied between 0.1 and 1 in increments of 0.1, and that of c˜2 is varied between 10 and 50 in increments of 5. For NEW, MAX, GW, and SW, the value of c˜1 is varied between 0.1 and 0.9 in increments of 0.1, and that of c˜2 is varied between 1 and 4, 1 and 14, 1 and 4, and 1 and 4, respectively, in increments of 0.5.14 The search is refined three times, each time reducing the increments by half. When necessary, the range of search is increased. In order to avoid computational errors, care is taken to maintain the value of c˜1 less than or equal to 1 and not equal to 0 in the geostatistical models and less than 1 and not equal to 0 in the weight matrix models. Third, we apply y2(1) to the geostatistical models and y2(1) and y2(2) to the weight matrix models. The value of the mean squared error is calculated from    ði Þ ′ ði Þ y2 −y2 for i = 1, 2. MSE i ¼ ð1=80Þ y2 −y2 Ten models are adopted to generate highly and moderately autocorrelated data both with and without sampling; consequently, a total of 40 experiments are performed. A set of 200 repetitions constitutes an experiment, and average values over those repetitions are used to assess estimation accuracy and predictive ability. The coordinates and values of the independent variables assigned to each location remain unchanged throughout, but the value of the dependent variable assigned to that location and the configuration of the estimation and prediction samples vary from repetition to repetition.

13

See Footnote 1. The ranges of search for MAX and GW or SW are based on those chosen by Kato (2008a) for LIM and NEW, respectively. 14

T. Kato / Economic Modelling 30 (2013) 743–753

747

Table 1 Experiments using NEGE as the data generator. Experiment/model No sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW Sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW

c˜1

c˜2

b^ 1

b^ 2

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

0.765 0.717 0.732 0.450

4.13 14.48 8.39 6.04

25.00 25.01 25.00 25.00

9.99 9.99 9.99 10.00

1.00 1.00 1.00 1.00

11.39 11.52 11.47 14.42

1.786 1.328 1.446 0.694

0.027 0.026 0.027 0.034

0.0007 0.0007 0.0007 0.0008

11.13 11.44 11.30 14.85

11.03 11.42 11.23 14.88

0.694 0.634 0.656 0.327

5.14 16.57 9.93 5.16

24.98 24.99 24.99 24.99

10.00 9.99 9.99 10.00

1.00 1.00 1.00 1.00

15.66 15.68 15.70 18.41

1.666 1.327 1.419 0.745

0.038 0.037 0.038 0.045

0.0009 0.0009 0.0009 0.0011

15.72 15.93 15.91 19.45

15.64 15.92 15.84 19.46

0.771 0.720 0.741 0.470

4.15 14.27 8.46 6.38

25.01 25.00 25.01 24.99

10.01 10.00 10.01 10.01

1.00 1.00 1.00 1.00

11.31 11.43 11.40 14.18

1.902 1.324 1.497 0.712

0.026 0.026 0.026 0.032

0.0007 0.0007 0.0007 0.0009

10.78 11.05 10.91 13.97

10.72 11.08 10.87 14.01

0.701 0.640 0.664 0.342

5.16 16.40 9.96 5.37

25.04 25.04 25.05 25.05

10.01 10.01 10.01 10.01

1.00 1.00 1.00 1.00

15.55 15.57 15.58 18.16

1.724 1.337 1.461 0.720

0.036 0.036 0.036 0.042

0.0010 0.0010 0.0010 0.0011

15.31 15.52 15.44 18.60

15.26 15.55 15.40 18.63

3.2. Results Tables 1 through 7 present the results obtained for NEW, MAX, GW, and SW in each of the four experiments using their corresponding data generated with NEGE, GSS, SPH, NN, P, LIM, and NEW, respectively. The first nine columns of each table are related to estimation accuracy. The same findings as those made by Dubin (2003) and Kato (2008a) for the five weight matrix models can be made not only for NEW, but also for MAX, GW, and SW.15 The values of b^ , b^ , and b^ are remarkably 1

2

3

close to those of b1, b2, and b3, respectively. This is consistent with the theory that the specification of autocorrelation does not bias the least 2 squares estimators. The value of σ^ is occasionally relatively different   from that of σ2. This has a detrimental effect on the value of V b^ 1 ,     although those of V b^ 2 and V b^ 3 are not adversely affected. Considering that the constant term, b1, is of little interest, we can conclude that the estimates of the regression parameters are accurate. The last two columns of each table are related to predictive ability. In accordance with the explanation given by Kato (2008a), the relative performances of the simultaneous and separate predictors depend on the situation, and practical convenience is likely to be an appropriate criterion for the choice of a predictor. When the correctly specified model is used in estimation and prediction, the simultaneous predictor should be better than the separate predictor, because it is desirable to consider as many locations as possible. This deduction is confirmed by a comparison of the values of MSE1 and MSE2 in the first row of each experiment of Table 7. When an incorrectly specified model is used, the relative performances of the two predictors must be determined through experiments. The findings of the present experiments are slightly mixed in regard to their relative performances. In the remaining three rows of each experiment of Table 7, the value of MSE1 is smaller than that of MSE2, which indicates the superiority of the simultaneous predictor. This superiority is also found, for example, regarding all of the models examined in each experiment of Table 5 and the fourth experiment of Table 6. However, in the first and third rows of each experiment of Tables 1 15 Unfortunately, when an incorrectly specified model other than the data generator is used in estimation, it is impossible to compare the values of c˜1 and c˜2 with those of c1 and c2 used in data generation, respectively. We can only assume the existence of the probability limits of c˜1 and c˜2 to compare with the values of c˜1 and c˜2 , respectively.

through 3, the value of MSE1 is greater than that of MSE2, which indicates the inferiority of the simultaneous predictor. This inferiority is also found, for example, in the same two rows of the first three experiments of Table 4 and the second and third experiments of Table 6. Interestingly, the difference between the values of the two MSEs does not appear to be significant in any situation. The simultaneous predictor is practically appealing, because the separate predictor is computationally demanding. The relative advantages of the four models can be considered on the basis of predictive ability, because each model is shown to maintain reasonable estimation accuracy. Focusing on the value of MSE1, we find the following: (i) the average ranks of NEW, MAX, GW, and SW over the 28 experiments are 1.64, 2.75, 1.79, and 3.82, respectively; (ii) the counts of best and worst models are 18 and 0 for NEW, 2 and 3 for MAX, 7 and 0 for GW, and 1 and 25 for SW; and (iii) the difference in the predictive ability between NEW or GW and SW appears to be significant in most of the experiments. These findings imply the following: (i) MAX is not promising, (ii) observations similar to those made by Dubin (2008) for the geostatistical models cannot be made for the weight matrix models that mimic those models, and (iii) NEW is still the most advantageous weight matrix model. Naturally, focusing on the value of MSE2, we make the same findings. The values of MSE1 and MSE2 for NEW in each experiment of the seven tables are consistent with those of MSE3 and MSE4 reported by Kato (2008a) for the best and worst weight matrix models in the corresponding experiment of his Tables 1 through 7, respectively. 16 Pairwise combinations of the two sets of tables indicate that the difference in the predictive ability between the best and worst models may not be small for the three mimics, 17 and the worst mimic may 16 Kato (2008a) calculated four mean squared errors: MSE1 in the application of the simultaneous predictor to the geostatistical models, and MSE2, MSE3, and MSE4 in the application of the conventional, simultaneous, and separate predictors to the weight matrix models, respectively. 17 Let us express the value of the mean squared error found for the best model as a percentage of that found for the worst model. The higher this percentage, the smaller that difference. Calculating the percentage with the values of MSE1 for the best and worst mimics in each experiment of our Tables 1 through 7, we see that it ranges from 65.62% to 99.57%, with 84.23% as the average over the 28 experiments. Calculating the percentage with the values of MSE3 for the best and worst weight matrix models in each experiment of his Tables 1 through 7, we see that it ranges from 72.25% to 99.41%, with 90.20% as the average over the 28 experiments. Naturally, using the values of our MSE2 and his MSE4, we find the same ranges and averages.

748

T. Kato / Economic Modelling 30 (2013) 743–753

Table 2 Experiments using GSS as the data generator. Experiment/model No sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW Sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW

c˜1

c˜2

b^ 1

b^ 2

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

0.871 0.849 0.860 0.612

4.09 15.32 8.81 9.40

24.97 24.98 24.97 24.98

10.00 10.00 10.00 10.00

1.00 1.00 1.00 1.00

7.73 7.57 7.58 11.00

3.420 2.441 2.796 1.118

0.018 0.017 0.017 0.025

0.0004 0.0004 0.0004 0.0006

7.24 7.10 7.08 10.79

7.22 7.07 7.03 10.87

0.788 0.757 0.773 0.402

5.01 17.43 10.30 6.68

24.97 24.98 24.97 24.96

10.00 10.00 9.99 10.00

1.00 1.00 1.00 1.00

13.14 12.92 12.97 16.99

2.358 1.887 2.030 0.840

0.031 0.030 0.031 0.041

0.0008 0.0008 0.0008 0.0010

12.89 12.75 12.71 17.70

12.85 12.70 12.63 17.74

0.872 0.851 0.861 0.633

4.10 15.31 8.80 9.79

24.98 24.96 24.98 24.95

10.00 10.00 10.00 10.01

1.00 1.00 1.00 1.00

7.73 7.55 7.58 10.52

3.419 2.436 2.802 1.312

0.017 0.017 0.017 0.023

0.0005 0.0005 0.0005 0.0006

7.08 6.94 6.90 10.12

7.08 6.93 6.87 10.17

0.786 0.754 0.772 0.430

5.03 17.20 10.21 7.55

24.96 24.94 24.95 25.00

10.00 10.00 10.00 10.00

1.00 1.00 1.00 0.99

13.26 13.01 13.05 16.68

2.224 1.737 1.910 0.875

0.031 0.030 0.030 0.039

0.0008 0.0008 0.0008 0.0010

12.83 12.67 12.65 16.75

12.80 12.63 12.60 16.72

c˜1

c˜2

b^ 1

b^ 2

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

0.617 0.588 0.599 0.416

2.67 10.44 5.90 5.07

24.98 24.99 24.98 24.97

9.99 9.99 9.99 10.00

1.00 1.00 1.00 1.00

14.93 14.93 14.97 17.23

1.018 0.919 0.967 0.682

0.034 0.034 0.034 0.041

0.0008 0.0009 0.0009 0.0009

14.52 14.82 14.61 17.68

14.34 14.82 14.46 17.74

0.524 0.492 0.501 0.316

3.11 11.12 6.38 4.41

24.98 24.98 24.97 24.97

9.99 9.99 9.99 9.99

1.00 1.00 1.00 1.00

18.50 18.39 18.48 20.29

1.021 0.946 0.976 0.771

0.044 0.043 0.044 0.050

0.0011 0.0011 0.0011 0.0012

18.62 18.84 18.72 21.43

18.49 18.89 18.60 21.43

0.633 0.601 0.617 0.439

2.75 10.53 6.10 5.20

24.96 24.97 24.96 24.98

10.01 10.01 10.01 10.01

1.00 1.00 1.00 1.00

14.85 14.86 14.90 17.07

1.044 0.931 0.986 0.665

0.033 0.033 0.033 0.038

0.0009 0.0009 0.0009 0.0010

14.04 14.32 14.09 17.04

13.87 14.33 13.96 17.09

0.542 0.507 0.516 0.336

3.19 11.25 6.48 4.78

24.99 24.98 24.99 25.01

10.01 10.01 10.01 10.01

1.00 1.00 1.00 1.00

18.43 18.30 18.40 20.12

1.034 0.941 0.978 0.752

0.043 0.042 0.043 0.047

0.0012 0.0011 0.0012 0.0013

18.11 18.33 18.19 20.65

17.98 18.35 18.09 20.64

Table 3 Experiments using SPH as the data generator. Experiment/model No sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW Sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW

offer the worst prediction performance among the weight matrix models. 18 Tables 8 through 10 present the results obtained for NEGE, GSS, and SPH in each of the four experiments using their corresponding data generated with MAX, GW, and SW, respectively. The values of b^ 1 , b^ 2 , and b^ 3 are remarkably close to those of b1, b2, and b3, respectively, 2 whereas the value of σ^ is occasionally profoundly different from that     of σ2. Although the values of V b^ 2 and V b^ 3 are possibly somewhat

18 Comparing the value of MSE1 for the worst mimic in each experiment of our Tables 1 through 7 with that of MSE3 for the worst weight matrix model in the corresponding experiment of his Tables 1 through 7, we see that the former value is almost always greater than the latter. Naturally, the same observation applies to a comparison of the values of our MSE2 and his MSE4.

affected, it can be concluded that the estimation accuracy is reasonable regardless of the model examined. Expressing the value of MSE1 found for the best model as a percentage of that found for the worst model, we see that it ranges from 79.63% to 99.87%, with 94.65% as the average over the 12 experiments. This implies that the difference in the predictive ability between the best and worst models is still small for the geostatistical models.19 The average ranks of NEGE, GSS, and SPH over the 12 experiments are 2.58, 1.75, and 1.67, respectively, and the counts of best and worst models are 1 and 8 for NEGE, 5 and 2

19 Calculating a similar percentage with the value of MSE1 reported by Kato (2008a) in his Tables 1 through 7, we see that it ranges from 92.52% to 99.89%, with 97.83% as the average over the 28 experiments.

T. Kato / Economic Modelling 30 (2013) 743–753

749

Table 4 Experiments using NN as the data generator. Experiment/model No sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW Sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW

c˜1

c˜2

b^ 1

b^ 2

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

0.763 0.725 0.743 0.543

3.16 11.50 6.69 7.36

24.98 24.93 24.97 24.99

10.02 10.02 10.02 10.02

1.00 1.00 1.00 1.00

30.40 29.61 29.74 38.84

4.094 3.061 3.458 2.076

0.068 0.065 0.066 0.089

0.0017 0.0017 0.0017 0.0021

27.42 26.86 26.76 36.82

27.27 26.92 26.65 36.81

0.531 0.486 0.511 0.311

3.32 10.93 6.74 4.95

24.91 24.90 24.91 24.96

10.01 10.00 10.01 10.00

1.00 1.00 1.00 1.00

27.12 26.68 26.85 29.89

1.521 1.347 1.436 1.157

0.065 0.063 0.064 0.074

0.0016 0.0016 0.0016 0.0017

26.93 26.71 26.62 31.38

26.69 26.61 26.39 31.36

0.495 0.480 0.468 0.428

1.70 6.92 3.49 4.43

25.32 25.30 25.30 25.35

9.99 10.00 9.99 9.98

0.99 0.99 0.99 0.99

46.03 45.70 46.10 48.36

2.210 1.960 2.075 1.777

0.102 0.101 0.102 0.108

0.0028 0.0027 0.0028 0.0029

43.67 45.26 44.73 47.67

43.43 45.49 44.52 47.26

0.287 0.300 0.273 0.274

1.69 6.08 2.80 3.92

25.04 25.04 25.06 25.04

10.00 10.00 10.00 10.00

1.00 0.99 0.99 0.99

31.17 30.59 30.92 31.10

1.232 1.157 1.183 1.124

0.074 0.072 0.073 0.074

0.0020 0.0020 0.0020 0.0020

32.02 32.27 32.22 32.60

31.99 32.46 32.27 32.61

Table 5 Experiments using P as the data generator. Experiment/model No sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW Sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

9.98 9.99 9.99 9.99

1.00 1.00 1.00 1.00

34.50 36.05 35.67 37.40

2.176 1.710 1.828 1.403

0.075 0.080 0.077 0.085

0.0019 0.0019 0.0019 0.0019

30.93 35.63 33.45 35.80

31.59 36.27 34.06 36.22

24.90 24.91 24.89 24.91

10.00 10.00 10.00 10.00

1.00 1.00 1.00 1.00

28.21 27.82 28.20 28.27

1.270 1.092 1.157 1.043

0.066 0.067 0.066 0.069

0.0016 0.0016 0.0016 0.0016

27.76 29.33 28.52 29.20

28.03 29.58 28.84 29.33

1.82 4.67 2.32 3.08

25.16 25.13 25.12 25.14

10.00 9.99 9.99 9.98

0.99 0.99 0.99 0.99

56.21 54.83 55.75 55.41

2.334 2.042 2.143 1.968

0.131 0.128 0.130 0.130

0.0035 0.0034 0.0035 0.0035

55.57 57.66 56.70 57.14

56.14 58.29 57.37 57.49

2.05 4.37 2.59 2.79

25.04 25.03 25.04 25.03

10.02 10.02 10.02 10.02

0.99 0.99 1.00 1.00

33.94 32.97 33.58 33.15

1.308 1.202 1.249 1.178

0.082 0.079 0.080 0.080

0.0022 0.0021 0.0022 0.0022

34.66 34.69 34.68 34.53

34.86 34.93 34.91 34.62

c˜1

c˜2

b^ 1

0.603 0.577 0.549 0.552

1.75 6.24 2.87 3.83

24.80 24.76 24.78 24.81

0.410 0.431 0.381 0.426

1.65 4.99 2.44 3.32

0.361 0.407 0.336 0.401 0.234 0.319 0.227 0.312

b^ 2

for GSS, and 6 and 2 for SPH. These imply that SPH is still the most advantageous geostatistical model. Tables 11 through 13 present the results obtained for NEW, MAX, GW, and SW in each of the four experiments using their corresponding data generated with MAX, GW, and SW, respectively. Focusing on the situation in which the correctly specified model is used in estimation, we find that the values of c˜1 and c˜2 are much closer to those of c1 and c2 used in data generation, respectively, in the no sampling experiments than in the sampling experiments. This is also found in the experiments of Table 7 and is quite likely explained by the fact that the ratio of locations included in the estimation sample to those included in the population is higher in the no sampling experiments than in experiments. Examining the values  the  sampling     2 of b^ 1 , b^ 2 , b^ 3 , σ^ , V b^ 1 , V b^ 2 , and V b^ 3 , we can conclude that the estimation accuracy is reasonable regardless of the model examined.

A comparison of the values of MSE1 and MSE2 in the second row of each experiment of Table 11, the third row of each experiment of Table 12, and the fourth row of each experiment of Table 13 confirms that the simultaneous predictor is superior to the separate predictor when they are applied to the correctly specified model. This superiority is also found, for example, in the remaining three rows of each experiment of Tables 12 and 13. However, in several situations of Table 11, the former predictor is inferior to the latter. Considering that the difference between the values of the two MSEs does not appear to be significant in most of the situations, we may choose a predictor for reasons of convenience. The following are found with the value of MSE1: (i) the average ranks of NEW, MAX, GW, and SW over the 12 experiments are 2.42, 2.50, 2.50, and 2.58, respectively; (ii) the counts of best and worst models are 5 and 4 for NEW, 2 and 2 for MAX, 0 and 0 for GW, and 5

750

T. Kato / Economic Modelling 30 (2013) 743–753

Table 6 Experiments using LIM as the data generator. Experiment/model No sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW Sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW

c˜1

c˜2

b^ 1

b^ 2

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

0.700 0.696 0.698 0.591

1.94 8.14 4.72 6.51

25.01 24.99 25.00 25.02

10.01 10.00 10.01 10.00

1.00 1.00 1.00 1.00

39.36 35.10 38.06 45.83

3.485 2.528 3.305 2.278

0.084 0.072 0.080 0.099

0.0021 0.0019 0.0020 0.0024

34.26 30.97 32.33 41.95

34.01 32.61 32.93 43.04

0.515 0.496 0.508 0.371

2.25 8.33 5.05 5.30

24.93 24.93 24.93 24.93

10.02 10.02 10.02 10.02

1.00 1.00 1.00 1.00

30.16 28.71 29.63 33.35

1.591 1.383 1.527 1.315

0.070 0.065 0.068 0.080

0.0017 0.0017 0.0017 0.0019

28.74 27.41 28.02 33.11

28.48 27.63 27.96 33.18

0.543 0.492 0.514 0.313

3.60 11.58 7.05 5.00

25.04 25.04 25.04 25.05

10.00 10.00 10.00 10.01

0.99 0.99 0.99 0.99

31.20 30.81 31.02 34.03

1.795 1.557 1.649 1.279

0.073 0.072 0.072 0.080

0.0020 0.0019 0.0020 0.0022

30.90 31.02 30.80 35.28

30.76 31.14 30.71 35.29

0.308 0.277 0.281 0.189

3.57 9.47 6.03 3.50

25.06 25.07 25.06 25.09

10.01 10.01 10.01 10.00

1.00 1.00 1.00 1.00

27.40 27.08 27.31 27.96

1.123 1.053 1.086 0.996

0.066 0.065 0.066 0.068

0.0018 0.0018 0.0018 0.0018

28.57 28.70 28.63 29.97

28.59 28.90 28.71 30.03

Table 7 Experiments using NEW as the data generator. Experiment/model No sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW Sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW

c˜1

c˜2

b^ 1

b^ 2

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

0.719 0.684 0.693 0.573

2.10 8.80 4.78 5.59

24.87 24.83 24.86 24.86

10.00 9.99 10.00 10.01

1.00 1.00 1.00 1.00

30.66 31.80 31.26 38.74

3.042 2.462 2.717 1.721

0.066 0.067 0.066 0.085

0.0016 0.0017 0.0017 0.0020

27.21 28.89 27.75 35.61

27.67 29.99 28.39 36.51

0.501 0.474 0.469 0.399

2.00 8.01 4.21 4.54

24.89 24.87 24.89 24.90

10.00 10.00 10.00 10.01

1.00 1.00 1.00 1.00

27.16 27.23 27.37 29.18

1.393 1.272 1.323 1.122

0.063 0.062 0.063 0.070

0.0015 0.0016 0.0016 0.0016

26.48 27.22 26.80 29.74

26.49 27.58 26.92 29.90

0.495 0.446 0.452 0.333

2.85 9.54 5.43 4.15

25.11 25.08 25.10 25.17

10.01 10.02 10.01 10.00

0.99 0.99 0.99 0.99

33.17 33.13 33.34 35.53

1.672 1.502 1.558 1.292

0.077 0.076 0.077 0.082

0.0021 0.0021 0.0021 0.0022

32.74 33.52 33.26 36.37

32.74 33.87 33.33 36.44

0.278 0.274 0.250 0.229

2.78 7.46 4.08 3.56

25.08 25.09 25.10 25.09

10.00 10.00 10.00 10.00

0.99 0.99 0.99 0.99

27.77 27.41 27.67 27.92

1.116 1.051 1.072 1.004

0.067 0.066 0.066 0.067

0.0018 0.0018 0.0018 0.0018

28.62 28.92 28.86 29.39

28.64 29.14 28.96 29.44

and 6 for SW 20; and (iii) the differences in the predictive ability among NEW, GW, and SW appear to be significant in many of the experiments. 21 These findings imply the following: (i) MAX is not promising, (ii) observations similar to those made by Dubin (2008) for the geostatistical models cannot be made for the weight matrix models that mimic those models, and (iii) NEW is still the most advantageous weight matrix model. Naturally, the same findings are made with the value of MSE2. Pairwise combinations of Tables 8 and 11, 9 and 12, and 10 and 13 indicate that all of the models, with the single exception of GW, offer the best prediction performance only in a limited number of 20 Interestingly, there is an enormous contrast between GW and SW when they are used in prediction as the correctly specified model. The former model is not the best in any experiment, but the latter is. The SW model is similar to NEW in this regard. See Table 7. 21 Expressing the value of MSE1 found for the best mimic as a percentage of that found for the worst mimic, we see that it ranges from 71.31% to 99.59%, with 87.69% as the average over the 12 experiments. See Footnote 17.

experiments, and neither of the two types of models is dominant. The combined use of the geostatistical and weight matrix models that have the advantage in predictive ability among models of their respective type may be reasonably helpful. Our experimental results indicate that we can choose the same models as Kato (2008b): SPH and NEW. 4. Conclusion A spatial error model is classified as a geostatistical model or a weight matrix model on the basis of the method of specification of spatial autocorrelation in the disturbance. Unfortunately, specification errors cannot be assumed to be absent, and the robustness of alternative specifications is useful for dealing with potential errors. Previous studies compared several models through Monte Carlo experiments and arrived at two basic conclusions: (i) all of the models maintain reasonable estimation accuracy, and (ii) the two types of models have well-matched predictive abilities. The present

T. Kato / Economic Modelling 30 (2013) 743–753

751

Table 8 Experiments using MAX as the data generator. Experiment/model No sampling High autocorrelation NEGE GSS SPH Moderate autocorrelation NEGE GSS SPH Sampling High autocorrelation NEGE GSS SPH Moderate autocorrelation NEGE GSS SPH

c˜1

c˜2

b^ 1

b^ 2

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

0.981 0.895 0.968

10.58 7.93 17.55

24.80 24.89 24.86

10.04 10.04 10.04

1.00 1.00 1.00

120.19 110.12 104.95

7.345 3.020 3.270

0.106 0.083 0.097

0.0024 0.0019 0.0022

40.68 38.57 40.13

– – –

0.871 0.712 0.813

6.06 6.84 14.13

24.88 24.91 24.89

10.03 10.03 10.03

1.00 1.00 1.00

46.35 45.96 45.58

1.870 1.481 1.490

0.075 0.070 0.072

0.0018 0.0017 0.0017

29.02 28.25 28.68

– – –

0.788 0.604 0.708

6.36 7.63 16.21

25.21 25.20 25.21

9.99 9.99 9.99

0.99 0.99 0.99

49.60 49.25 49.59

1.996 1.636 1.671

0.081 0.080 0.079

0.0022 0.0021 0.0021

35.25 35.34 35.48

– – –

0.444 0.338 0.445

4.96 7.47 14.00

25.20 25.19 25.19

9.99 9.99 10.00

0.99 0.99 0.99

31.63 31.68 31.85

1.179 1.140 1.121

0.070 0.070 0.069

0.0019 0.0019 0.0019

30.12 30.11 30.15

– – –

c˜1

c˜2

b^ 1

b^ 2

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

0.998 0.955 0.997

8.86 6.30 14.64

24.84 24.89 24.90

10.02 10.03 10.02

1.00 1.00 1.00

200.13 181.44 172.98

10.140 3.939 4.721

0.194 0.126 0.171

0.0042 0.0026 0.0036

70.37 64.62 68.20

– – –

0.960 0.793 0.929

5.42 5.66 11.81

24.90 24.92 24.91

10.03 10.02 10.03

1.00 1.00 1.00

59.82 57.75 58.34

2.216 1.686 1.685

0.091 0.086 0.085

0.0021 0.0020 0.0019

34.27 33.34 33.69

– – –

0.980 0.680 0.981

4.41 5.03 9.76

25.27 25.28 25.24

9.97 9.98 9.98

0.99 0.99 0.99

107.20 101.81 106.49

3.532 3.043 2.772

0.160 0.173 0.143

0.0043 0.0047 0.0038

75.84 76.48 75.01

– – –

0.541 0.396 0.777

3.50 4.80 7.78

25.19 25.20 25.19

9.99 9.99 9.99

0.99 0.99 0.99

44.66 44.23 45.42

1.578 1.517 1.429

0.098 0.099 0.090

0.0026 0.0027 0.0024

41.65 41.37 40.55

– – –

Table 9 Experiments using GW as the data generator. Experiment/model No sampling High autocorrelation NEGE GSS SPH Moderate autocorrelation NEGE GSS SPH Sampling High autocorrelation NEGE GSS SPH Moderate autocorrelation NEGE GSS SPH

Table 10 Experiments using SW as the data generator. Experiment/model No sampling High autocorrelation NEGE GSS SPH Moderate autocorrelation NEGE GSS SPH Sampling High autocorrelation NEGE GSS SPH Moderate autocorrelation NEGE GSS SPH

  V b^ 1

  V b^ 2

  V b^ 3

88.48 87.26 84.49

3.054 3.078 2.558

0.193 0.211 0.176

0.0044 0.0049 0.0038

64.03 68.32 54.40

– – –

1.00 1.00 1.00

38.16 37.79 37.40

1.394 1.361 1.267

0.094 0.094 0.088

0.0022 0.0022 0.0020

35.14 34.68 31.95

– – –

9.98 9.99 9.99

0.99 0.99 0.99

182.72 180.75 180.29

6.419 6.350 5.621

0.418 0.426 0.360

0.0112 0.0114 0.0097

170.63 169.52 158.59

– – –

9.98 9.98 9.98

0.99 0.99 0.99

56.50 56.22 56.12

2.015 1.999 1.904

0.135 0.134 0.127

0.0036 0.0036 0.0034

55.63 55.22 53.58

– – –

c˜1

c˜2

b^ 1

b^ 2

b^ 3

0.811 0.456 0.962

2.86 3.33 6.04

24.89 24.89 24.88

10.04 10.04 10.05

1.00 1.00 1.00

0.448 0.339 0.730

2.75 3.48 5.65

24.93 24.93 24.93

10.02 10.02 10.02

0.467 0.330 0.820

2.98 4.69 6.32

25.32 25.32 25.29

0.329 0.275 0.589

3.10 5.45 6.75

25.23 25.23 25.22

2 σ^

MSE1

MSE2

752

T. Kato / Economic Modelling 30 (2013) 743–753

Table 11 Experiments using MAX as the data generator. Experiment/model No sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW Sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW

c˜1

c˜2

b^ 1

b^ 2

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

0.697 0.711 0.693 0.642

1.43 7.36 3.74 6.21

25.00 25.03 24.97 25.00

10.02 10.01 10.02 10.02

1.00 1.00 1.00 1.00

41.68 38.26 40.81 46.99

3.549 2.570 3.377 2.368

0.085 0.076 0.082 0.098

0.0021 0.0020 0.0021 0.0023

33.76 31.17 32.56 40.31

35.01 34.52 34.77 42.49

0.508 0.512 0.498 0.437

1.53 7.25 3.64 5.09

24.96 24.98 24.95 24.95

10.02 10.02 10.02 10.01

1.00 1.00 1.00 1.00

30.18 29.11 29.87 32.78

1.533 1.362 1.472 1.295

0.068 0.064 0.066 0.076

0.0017 0.0016 0.0017 0.0018

27.73 26.88 27.45 31.41

27.69 27.46 27.73 31.60

0.511 0.474 0.482 0.366

2.49 8.89 5.12 4.68

25.26 25.23 25.25 25.31

9.98 9.99 9.99 9.98

0.99 0.99 0.99 0.99

36.18 35.80 36.19 38.87

1.836 1.635 1.731 1.435

0.083 0.081 0.082 0.089

0.0022 0.0022 0.0022 0.0024

35.44 36.20 35.79 39.62

35.23 36.41 35.74 39.43

0.285 0.280 0.262 0.235

2.47 7.45 4.20 3.49

25.22 25.22 25.20 25.25

9.99 9.99 9.99 9.98

0.99 0.99 0.99 0.99

29.10 28.66 29.01 29.46

1.157 1.091 1.119 1.051

0.070 0.069 0.070 0.071

0.0019 0.0019 0.0019 0.0019

30.21 30.47 30.36 31.36

30.20 30.66 30.46 31.40

excellent model. The second is to explore the robustness of spatial autocorrelation specifications for the spatial lag model. A linear regression model is called a spatial lag model when spatial autocorrelation is present in the dependent variable. This model has seen an increase in attention similar to that in the spatial error model. Technically, each specification is based on a weight matrix and not a correlation function, which is the point where the former model differs from the latter.

study made a supplementary comparison to investigate whether these conclusions are true for a broader range of models. Three additional models were examined, and results were produced that are consistent with the conclusions. Interestingly, observations similar to those made for the geostatistical models could not be made for the weight matrix models that mimic those models, which indicates that the difference in the predictive ability between the two types of models should be basically attributable to the difference in the method of specification of spatial autocorrelation. We also explored implications of our experimental results for the model choice and support the same combined use of geostatistical and weight matrix models as that previously suggested. Provided that the present comparison of models is accepted, two topics are suggested for future study. The first is to extend the examination to models other than those examined here. This is desirable for a more adequate discussion of the robustness of spatial autocorrelation specifications. Sometime in the future, we may possibly find an

Acknowledgements The research upon which this study is based was supported by a Grant-in-Aid for Scientific Research (C), whose number is 21530199, from the Japan Society for the Promotion of Science. Comments from an anonymous reviewer contributed to refining this paper. The usual caveats apply.

Table 12 Experiments using GW as the data generator. Experiment/model No sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW Sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW

c˜1

c˜2

b^ 1

b^ 2

b^ 3

2 σ^

  V b^ 1

  V b^ 2

  V b^ 3

MSE1

MSE2

0.661 0.681 0.656 0.649

0.68 6.10 2.04 5.28

24.97 24.99 24.94 24.86

10.00 9.99 10.01 10.00

1.00 1.00 1.00 1.00

66.42 71.34 65.80 78.87

4.545 3.692 4.181 3.442

0.128 0.143 0.125 0.163

0.0031 0.0035 0.0031 0.0038

43.96 56.32 44.20 61.65

54.67 63.90 56.23 69.41

0.502 0.529 0.499 0.505

0.69 5.73 1.90 4.53

24.96 24.99 24.94 24.94

10.01 10.01 10.02 10.02

1.00 1.00 1.00 1.00

34.90 35.40 34.63 37.40

1.659 1.513 1.571 1.443

0.075 0.078 0.074 0.085

0.0018 0.0019 0.0018 0.0020

29.11 32.21 29.51 34.04

30.81 33.37 31.34 34.97

0.408 0.443 0.409 0.441

1.18 4.95 1.84 3.86

25.35 25.34 25.32 25.37

9.97 9.96 9.97 9.96

0.99 0.99 0.99 0.99

80.97 78.00 78.93 79.20

3.399 2.939 3.075 2.850

0.182 0.174 0.175 0.178

0.0049 0.0047 0.0047 0.0048

77.87 79.19 79.01 78.67

78.78 79.90 79.74 78.92

0.263 0.334 0.279 0.330

1.24 4.15 1.59 3.38

25.22 25.23 25.21 25.24

9.99 9.99 9.99 9.99

0.99 0.99 0.99 0.99

40.67 39.20 39.86 39.44

1.552 1.422 1.461 1.407

0.097 0.093 0.094 0.093

0.0026 0.0025 0.0025 0.0025

41.47 41.74 41.49 41.32

41.86 42.01 41.89 41.47

T. Kato / Economic Modelling 30 (2013) 743–753

753

Table 13 Experiments using SW as the data generator. Experiment/model No sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW Sampling High autocorrelation NEW MAX GW SW Moderate autocorrelation NEW MAX GW SW

  V b^ 1

  V b^ 2

  V b^ 3

70.33 46.84 54.79 46.83

2.898 1.588 1.927 1.583

0.163 0.113 0.134 0.113

0.0039 0.0025 0.0031 0.0025

59.79 43.96 46.43 43.68

63.26 44.42 46.72 44.23

1.00 1.00 1.00 1.00

35.10 29.22 32.03 29.22

1.374 1.015 1.155 1.014

0.086 0.072 0.080 0.072

0.0021 0.0017 0.0019 0.0017

33.88 28.83 30.42 28.81

34.61 28.87 30.65 28.87

9.98 9.98 9.97 9.98

0.99 0.99 0.99 0.99

171.89 152.82 162.38 152.77

6.478 5.350 5.839 5.341

0.412 0.362 0.387 0.362

0.0111 0.0097 0.0104 0.0097

169.37 154.32 161.89 153.87

172.20 154.76 163.43 154.26

9.98 9.98 9.98 9.98

0.99 0.99 0.99 0.99

54.54 50.37 52.52 50.39

2.019 1.777 1.884 1.772

0.132 0.121 0.127 0.121

0.0036 0.0033 0.0034 0.0033

55.53 52.58 53.97 52.39

56.14 52.72 54.37 52.50

c˜1

c˜2

b^ 1

b^ 2

b^ 3

0.391 0.728 0.540 0.728

0.60 2.03 0.48 2.03

24.93 24.91 24.88 24.91

10.03 10.03 10.04 10.03

1.00 1.00 1.00 1.00

0.261 0.567 0.371 0.565

0.87 2.01 0.76 2.02

24.95 24.92 24.95 24.92

10.02 10.02 10.01 10.02

0.235 0.489 0.308 0.489

1.29 2.26 1.10 2.23

25.37 25.35 25.38 25.36

0.182 0.404 0.247 0.402

1.62 2.44 1.43 2.27

25.24 25.25 25.24 25.26

References Anselin, L., 1988. Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, Dordrecht. Anselin, L., 2002. Under the hood: issues in the specification and interpretation of spatial regression models. Agricultural Economics 27, 247–267. Arbia, G., Bazo, E.L., Moscone, F., 2012. Frontiers in spatial econometrics modelling. Economic Modelling 29, 1–2. Bourassa, S.C., Cantoni, E., Hoesli, M., 2007. Spatial dependence, housing submarkets, and house price prediction. Journal of Real Estate Finance and Economics 35, 143–160. Dubin, R.A., 1998. Spatial autocorrelation: a primer. Journal of Housing Economics 7, 304–327. Dubin, R., 2003. Robustness of spatial autocorrelation specifications: some Monte Carlo evidence. Journal of Regional Science 43, 221–248.

2 σ^

MSE1

MSE2

Dubin, R., 2008. Comment on “A further exploration into the robustness of spatial autocorrelation specifications”. Journal of Regional Science 48, 641–649. Kato, T., 2008a. A further exploration into the robustness of spatial autocorrelation specifications. Journal of Regional Science 48, 615–639. Kato, T., 2008b. Response to comment on “A further exploration into the robustness of spatial autocorrelation specifications”. Journal of Regional Science 48, 651–653. Militino, A.F., Ugarte, M.D., García-Reinaldos, L., 2004. Alternative models for describing spatial dependence among dwelling selling prices. Journal of Real Estate Finance and Economics 29, 193–209. Pace, R.K., Gilley, O.W., 1997. Using the spatial configuration of the data to improve estimation. Journal of Real Estate Finance and Economics 14, 333–340. Palmquist, R.B., 2005. Property value models. In: Mäler, K.-G., Vincent, J.R. (Eds.), Handbook of Environmental Economics, vol. 2. Elsevier, Amsterdam, pp. 763–819.