Reasons for Instability in Spatial Dependence Models

Transcription

1 1 Reasons for nstability in Spatial Dependence Models Jesús Mur () Fernando López () Ana Angulo () () Department of Economic Analysis University of Zaragoza Gran Vía, -4. (55). Zaragoza. Spain. () Department of Quantitative Methods and Computing University of Cartagena. Paseo Alfonso X, 5-33 Cartagena. Spain. Abstract There are, at least, two potential sources of instability in spatial dependence models: in the regression coefficients and in the mechanisms of spatial dependence. The problem is that their symptoms are very similar and it is difficult to discriminate between them. This question is important because the consequences, as well as the remedies, of each form of instability are quite different. n this context, the main objective of this paper is to develop a strategy to analyse the hypothesis of stability in the structure of a spatial econometric model, capable of giving us information about its origin. Keywords: Spatial Dependence; Spatial nstability; Monte Carlo. JEL Classification: C1; C5; R15

2 1. ntroduction n spatial data analysis, model specification issues have become of great interest (see, for example Anselin 1988a,b, 199; Blommestein, 1983; Florax and Folmer, 199; Kelejian and Robinson, 199). Among main topics analysed are heterogeneity and spatial dependence. Anselin (1988c) is one of the first papers which proposes to combine the two topics, giving rise to the well-known battery of tests of autocorrelation and heteroskedasticity; Anselin (199) continues along the same lines, adapting the Chow test to models with spatial dependence. Later, Brunsdon et al. (1998a) and Pace and Lesage (4) present specifications where the mechanisms of spatial association are specific for each point in space. The last two papers may be seen as a natural extension of the LWR (Locally Weighted Regressions) algorithm, introduced by the seminal papers of Cleveland (1979) and Cleveland and Devlin (1988) and subsequently developed by the works of, among others, McMillen (1996), McMillen and McDonald (1997), Brunsdon et al. (1998b), Leung et al. ( and 3) and Paez et al. (a and b). Recently, Mur et al. (8) try to bring the two topics even closer by developing several tests of instability in the coefficient of spatial autocorrelation. Although previous literature works separately on different causes of instability in cross-sectional models, as far as we know, there is no literature on the simultaneous analysis of all of them. n this paper, we will try to fill this gap by analysing mainly two issues. On one hand, we focus on deriving several statistics for testing different causes of instabilities in a spatial econometric model, in the presence of other types of instabilities. That is, the resulting tests are robust to the presence of other nuisance parameters that can provoke other instabilities not explicitly considered. On the other, we combine all the battery of diagnostics into a model strategy process so as to define the strategy the researcher can follow to decide the source of instability that a spatial econometric model is suffering from. As spatial models, we will concentrate on results for the Spatial Lag Model (S) and the Spatial Error Model (SEM). Furthermore, and for purposes of comparison, we develop the analogous strategy with the respective non-robust statistics. The structure of the paper is as follows. n Section, the different robust statistics are derived. Section 3 shows the model strategy proposed in the paper in order to decide which model has generated our data. n Section 4, we solve a Monte Carlo experiment for analysing how the different statistics react to a combination of three issues: a) instability in the

3 3 coefficient of spatial autocorrelation; b) instability in the coefficients of the regression; and c) instability in the dispersion parameter (heteroskedasticity). n this section, we also show results on the probability of selecting the true Data Generating Process (DGP) using the strategy proposed in the paper. Finally, we close with some concluding remarks in Section 5.. -tests robust to several sources of instability n this section, we apply to our cases of interest the general approach developed in Bera and Yoon (1993) to test any hypothesis in the presence of nuisance parameters. As mentioned, we will derive several interesting statistics related to a possible instability in a spatial econometric model, whether of type S or SEM. nstability will be incorporated through the coefficient of spatial autocorrelation, the coefficients of the regression and/or the dispersion parameter (heteroskedasticity). Within this framework, the most ample model with instability in all the considered parameters and for the S case is the following: y =ρ HWy+ Xm +ε Ay = Xm +ε ~N(, ); (1) ε Ω Ω = D where { ( zi ) } H = diag h α ;i = 1,,..., R h() =κ (1a) y1 x' 1 xi1 m1 mi1 y y = ;X = x' ; x xi m mi i ;m ; = = mi= y x' x m m R R ik R ik Rx1 RxRk kx1 Rkx1 kx1 g' i= gi1; gi; ; g ip mij= β jpj g' iμ μ' i= μ 1; μ; ; μ p (1b) ε ~N(, Ω); Ω = D Ω = d ' n ii i δ Ω = ;i j (1c) ij d() =κ Underlying (1a) is the hypothesis that there is a basic level of dependence for all space, associated with parameter ρ. f this parameter is zero, the conclusion is that there is no

4 4 cross-sectional dependence and the discussion ends here. Only if coefficient ρ is different from zero is there any sense in asking whether these relations of dependence may not be uniform across space. n the specification of (1a), we propose that the intensity of the dependencies evolves following the distribution of a certain variable z (we assume that only one variable intervenes in vector z; the discussion can be generalised to the case of p variables). Here we are specifically concerned with the question of how to test the break. To advance in this direction, we adopt an approach similar to that of the Breusch-Pagan (Breusch and Pagan, 1979) test, which avoids having to go into details about the form of the h(-) function that intervenes in matrix H. That is, it does not necessarily have to be known, although it must remain stable across space. Also we need to assume that this function, at the origin, verifies: h() = κ being finite constant. n any case, the fundamental piece of information is the indicator of heterogeneity, the variable z, that intervenes in the h[-] function. Our view is that the variability of the variable z generates instability in the measures of spatial dependence. Underlying (1b) is a situation of a structural break in the regression coefficients that is produced according to a well-identified pattern. This means that we explain the break through an indicator, or set of indicators, in a similar way to the method of the expansion of parameters. However, in our proposal, we do not specify the functional form of the expansion equations; on the contrary, we maintain a general specification. The idea is that the vector of parameters changes at each point in space, according to the p j [-] functions and g variables in (1b), one function for each coefficient of regression of the model. As we said before, the argument of this function is composed of a set of variables, the g s, and of parameters associated with the change, the μ s. To abridge the discussion, we assume that, in the functions associated with the different β parameters, the same variables and the same parameters intervene (the restriction can be relaxed). n other words, the break spreads uniformly between the regression coefficients. As before, the form of the p(-) function does not necessarily have to be known. We only need to assume that this function, at the origin, verifies: p j () = κ, 1 p j () = κ 1 and p j () = κ, with 1 p j and p j being the first and second derivatives and κ, κ 1 and κ are finite constants.

5 5 Finally, underlying (1c) heteroskedasticity is introduced into the model assuming that it is generated by the variable n through another function d(-), which at the origin takes a finite value,d() =κ. The log-likelihood function of model (1) is the usual: l(y; θ ) = R ln π R ln + ln 1 ln 1 y Xm ' y Xm ( ) A D ( A ) D ( A ) where θ=[ρ,, β, μ,α,δ] = [γ,μ, α,δ] being the vector of parameters of the model. () For this model, it is interesting to test the null hypothesis of homogeneity in all the parameters: H H : μ=α=δ= (3) :No H A n order to do this, we first need the ML estimation of the S of the null hypothesis H, with which we can obtain the corresponding Lagrange Multiplier (see Appendix A for the details): S Break+Chow+HET 11 = q' q χ (3) (4) where q is a vector of order (3x1) and 11 a square (3x3) matrix: G 'Y ε 1 ' y q tr ε ZW = 11 ρ A ZW V( ) = γ 1 ' ε Nε trn as ε being the vector of error terms of the S obtained under the assumption of homogeneity in all the parameters (S under the null H ), and V( γ ) being the covariance matrix of the parameters of this model, γ= ; ; ρ β '. The SEM case is analogous. The specification now becomes:

6 6 y = Xm + u u =ρ HWu+ε Bu =ε ε ~N(, Ω); Ω = D (5) and the log-likelihood: l(y; θ ) = R ln π R ln + ln 1 ln 1 y Xm ' ' y Xm D (46) ( ) B D ( ) B B( ) The null hypothesis coincides with that of (3) and the Lagrange Multiplier is: SEM 11 Break+Chow+HET = q' χ as q (3) (6) where: G'Y Bu 1 ' u q tr ε ZW = 11 ρ B ZW V( ) = γ 1 ' ε Nε trn ε is the vector of ML residuals filtered by the B matrix: B( ) elements appear in Appendix A. (7) ε= y xβ = B u. The other n the next subsections, we derive, for both models, several statistics to test interesting hypotheses in the presence of nuisance parameters related to the other instability sources that are not tested..1. nstability in the parameter of dispersion robust to instability in the mechanisms of cross-sectional dependence and in the parameters of the model n order to derive the robust tests, it is necessary to make use of the results presented in the previous section referring to the score and information matrix of the most ample model evaluated under the null hypothesis (3). n other words, in this section we will make use of the results presented for the ample models (1) and (5) evaluated in the parameter vector θ = ( ρ,, β,,,)' = ( γ,,,)'.

7 7 To obtain the Lagrange Multiplier statistics for testing the null hypothesis δ= in the presence of the nuisance parameters α and μ, it is useful to use the following partition of the information matrix: = [ δδ] ; 1 δα δμ; 13 δρ δ δβ = = = ρρ ρ ρβ = ; = ; = ρ β βρ β ββ αα αμ αρ α αβ 3 33 μρ μα μμ μ μβ (8) HET The corresponding robust statistics can be derived, in general, as follows: HET g( α) θ g( δ). γ θ. γ θ θ g( μ) θ = χ as 11. γ 1. γ. γ θ 1. γ θ θ θ (1) (9) ( ) where: = V γ 1. γ θ ( ) = V γ. γ θ 3 3 V ( ) = γ 11. γ θ From (9), the derivation for the S model, S HET, requires the use of the score and information matrix elements under the null derived for that model in expressions (A) and (A3). That is, for instance, for the score values, the following expressions must be used: 1 ' y g( ) tr ε ZW α = ρ θ A ZW G 'Y ε g( μ ) θ = 1 ' g( ) ε N ε δ = trn θ (1) Analogously, from (9), the derivation for the SEM model, SEM HET, requires the use of the corresponding results for that model shown in expressions (A6) and (A7). That is, for instance, for the score values:

8 8 1 ' u g( ) tr ε ZW α = ρ θ B ZW G 'Y Bu g( μ ) θ = 1 ' g( ) ε N ε δ = trn θ (11) n both cases, if the respective values [ ] statistics are obtained, which we will call 1. γ = S HET and, the non-robust version of the SEM HET, respectively... nstability in the parameter of the model robust to instability in the mechanisms of cross-sectional dependence and in the parameters of dispersion of the model n this case, we derive the statistic to test the null hypothesis μ = in the presence of the nuisance parameters α and δ. We will use the following partition of the information matrix: = μμ 1 μα μδ = = 13 = μρ μ μβ ; ; ρρ ρ ρβ = ; = ; = ρ β βρ β ββ αα αδ αρ α αβ 3 33 δα δδ δρ δ δβ (1) A general Chow can be derived as follows: Chow ( ) ( ) V ( ) where: = V γ 1. γ θ = V γ. γ θ 3 3 = γ 11. γ θ g( α) θ g( μ). θ γ. θ γ θ g( δ) θ = as χ (1) 11. γ 1. γ. γ θ 1. γ θ θ θ (13)

9 9 As before, the specific statistics for each model, S Chow or SEM Chow, are calculated by introducing the respective information. Also, in both cases, if 1. [ ] the non-robust versions are obtained. That is, if [ S ] S = SEM Chow = SEM Chow. 1. γ =, Chow γ =, and Chow.3. nstability in the mechanisms of cross-sectional dependence robust to heteroskedasticity and instability of parameters of the model The robust statistic to test the null hypothesis α = in the presence of the nuisance parameters μ and δ will be easily represented using the following partition of the information matrix: = [ αα] ; 1 αμ αδ; 13 αρ α αβ = = = ρρ ρ ρβ = ; = ; = ρ β βρ β ββ μμ μδ μρ μ μβ 3 33 δμ δδ δρ δ δβ (14) The corresponding general robust statistic Break is as follows: Break ( ) ( ) V ( ) where: = V γ 1. γ θ = V γ. γ θ 3 3 = γ 11. γ θ g( μ) θ g( α). θ γ. θ γ θ g( δ) θ = as χ 11. γ 1. γ. γ θ 1. γ θ θ θ (1) (15)

10 1 Again, substituting the respective values for the S or the SEM model derived in Appendix A, we will obtain the corresponding robust statistics, in both cases, if [ ] 1. γ = S Break or SEM Break. Also,, the non-robust versions are obtained, that is, S Break = S Break and SEM Break = SEM Break..4- nstability in the mechanisms of interaction and in the parameter of dispersion robust to instability in the parameters of the model An easy representation of the testing of the null hypothesis α =δ= in the presence of the nuisance parameterμ requires the following partition of the information matrix: αα αδ αμ αρ α αβ = = ; 1 = ; 13 = δα δδ δμ δρ δ δβ ρρ ρβ ρ = ; = ; = μμ μρ μ μβ ρ β 3 33 βρ β ββ (16) The general Break+HET can be derived as follows: Break+HET 11 as = q' q χ () (17) Where q is a vector of order (1x) and 11 a square (x) matrix: ( ) θ 1. γ θ ( ). γ θ 3 3 V ( ) 11. γ θ g( α) with: = V γ = V γ = γ q = g( ) g( ) μ δ θ 1. γθ. γθ θ 11 = 11. γ θ 1. γθ. γθ 1. γθ (18)

11 11 As in previous cases, using results for the S model in Appendix A, the robust version for this model, S Break+HET model the robust statistic for this model is derived, non-robust versions, S Break+HET and, is obtained. n the cases of using results for the SEM SEM Break+HET SEM Break+HET, are obtained if 1.. Also, in both cases, the γ =..5- nstability in the mechanisms of spatial interaction and in the coefficients of regression robust to heteroskedasticity Similarly, the testing of the null hypothesis α =μ= in the presence of the nuisance parameter δ is represented analogously by using the following partition of the information matrix: αα αμ αδ αρ α αβ = = ; 1 = ; 13 = μα μμ μδ μρ μ μβ ρρ ρβ ρ = [ ]; = ; = βρ β ββ δδ 3 δρ δβ 33 δ ρ β (19) Break+Chow The general robust statistic is derived as follows: Break+Chow 11 as = q' q χ () () where q is a vector of order (1x) and 11 a square (x) matrix: θ q = g( ) g( ) δ μ θ 1. γ θ γ θ. γ θ θ 11. γ θ 1. γ θ. γ θ 1. γ θ ( ) ( ). γ θ 3 3 V ( ) 11. γ θ g( α) with: = V γ = V γ = γ 11 = (1)

12 1 Again, substituting the respective values for the score and information matrix for the S SEM S or the SEM model, the respective robust statistics, Break+Chow or Break+Chow, S SEM are derived. The respective non-robust versions, Break+Chow and Break+Chow, are obtained if 1. γ =..6- nstability in the parameters of the model and in the parameter of dispersion robust to instability in the mechanisms of interaction Finally, another interesting result comes from testing the null hypothesis μ=δ= in the presence of the nuisance parameter α. n this case, the following partition of the information matrix will be used: μμ μδ μα μρ μ μβ = = ; 1 = ; 13 = δμ δδ δα δρ δ δβ ρρ ρ = [ ]; = ; = βρ β αα 3 αρ αβ 33 ρβ α ρ β ββ () Chow+HET Similarly, the general robust statistic is derived as follows: Chow+HET 11 as = q' q χ () (3) where q is a vector of order (1x) and 11 a square (x) matrix: g( μ) θ q = 1.. g( ) g( ) γ γ α δ θ 11 = 11. γ 1. γ. γ θ 1. γ ( ) with: = V γ 1. γ θ ( ) = V γ. γ θ 3 3 V ( ) = γ 11. γ θ θ θ θ θ θ θ (4) Replacing with the respective values for the S or the SEM model shown in S SEM Appendix A, the robust statistics Chow+HET or Chow+HET are obtained. Their

13 13 respective non-robust versions, the restriction 1. γ =. S Chow+HET or SEM Chow+HET, are derived when imposing 3. Model selection strategies in a spatial model n applied econometrics, in general, and in applied spatial econometrics, in particular, we can say that the selection of the correct DGP that underlie any type of data is the most important task for researchers. However, the list of papers dedicated specifically to comparing specification strategies in a spatial context seems very small, even though Anselin (1988a) had already established the basis for its development (see Part of this textbook). We can only cite the seminal works of Blommenstein (1983) and Bivand (1984), the simulation experiment of Anselin and Florax (1995) and the more recent paper of Florax, Folmer and Rey (3). t is, therefore, no surprise that Anselin, Florax and Rey (4, p. 5) state that much more is needed in terms of comparative studies of competing paradigms and modeling philosophies in this field. n this context, and within the framework of the instabilities we are analysing in this paper, we try to contribute to the literature on this topic. To be precise, our main objective is to know how often we will select the correct DGP when using the strategy and the robust statistic proposed in this paper. As we said before, we will concentrate only on the two spatial models, S and SEM. For purposes of comparison, the results obtained when using the non-robust statistics are also shown. The model selection strategy followed when using the robust statistic is shown in Figure 1 while Figure presents the analogous strategy when using the non-robust versions. (nsert Figures 1 and ) Apart from the use of the respective robust and non-robust statistics, the main difference between Figure 1 and comes from the use in Figure of the statistics RS λ/ρ or RS ρ/λ, proposed in (Anselin and Bera, 1998) (see Appendix B for details). While these statistics make sense for the non-robust case in order to confirm our selection of the S or SEM model, the analogous statistic is very tedious to obtain for the robust case since its calculation requires the estimation of the S or the SEM model under the alternative hypothesis.

14 14 Apart from this issue, in both cases, the model selection strategy starts by testing the null of homogeneity in all the parameters. f this hypothesis is not rejected, the process ends with the selection of the homogeneous model. However, if the restriction is rejected, the procedure continues with the testing of the three individual hypotheses of: i) homogeneity in the regression parameters; ii) homogeneity in the spatial parameter; and iii) homogeneity in the dispersion parameter. When only one of these three statistics is rejected, the conclusion is clear. However, when two out of the three are rejected, the next step will consist of testing the respective joint hypothesis. Only if the corresponding statistic is rejected, can we conclude that there are two sources of heterogeneity. Finally, when the three individual tests reject the null hypothesis, we can conclude that the spatial model presents instability in the three types of parameters. 4. Monte Carlo experiment: design and empirical results n this section, we evaluate the performance of the robust statistics derived in this paper as well as their combination in the proposed model selection strategies. The next subsection describes the characteristics of the experiments and the second focuses on the results Design of the Monte Carlo experiments n this exercise, we concentrate on data obtained from the S model. The problem is to determine which process has intervened in the generation of the data, using only the data. We have simulated all the combinations of the three sources of heterogeneity: spatial parameter, parameters of the model and dispersion parameter. As regards instability in the mechanism of the spatial parameter, H ( z ) { i } = diag h α ;i = 1,,...,R, we have simulated the following situation. We obtained the values of zi from a uniform distribution U(-1;); h(-) was specified as an exponential function and several values for α (,.5 and 1) were used 1. 1 This assumption is equivalent to obtaining the values of z i from a uniform distribution U(1;) and the corresponding values of α (, -.5 and -1). This combination is selected in order to always obtain a spatial dependence parameter lower than 1. The selection of the exponential function responds to the fact that this function satisfied the required condition.

15 15 Regarding instability in the parameter of dispersion, ( n ) { i } D = diag d δ ;i = 1,,..., R, we used an exponential function in d(-), a limited number of values in parameter δ (,.5 and 1) and sequences of random numbers, obtained from an N(,1) distribution, in variable n i. Finally, for instability in the coefficients of regression, we have introduced instability through the following functions: m ij gi = β j 1+μ gmax where the values for g i have been obtained from a uniform distribution, U(1;), g max refers to the maximum of that variable in the sample at hand, and we introduced several values in parameter μ (,.5 and 1). (5) The remaining characteristics of the exercise are the following: a. Only one regressor has been used in the model. The coefficient associated with it takes a value of 3 whereas the value of the intercept is. Both magnitudes guarantee that, in the absence of spatial effects, the R of the model will be close to.8. b. The observations of the regressor and of the random terms ε have been obtained using univariate normal distributions with zero mean and unit variance. That is, cases. = 1 in all c. We have used regular lattices of orders (7x7) and (x) which mean that the sample size is 49 and 4 observations respectively. The weighting matrix has been specified, in the first place, as a binary type using a contiguity criterion and rook-type movements. Afterwards, the resulting matrix was row-standardised in the usual way. d. n each case, two values of parameter ρ have been simulated:.5, and.9. e. Each combination has been repeated 1 times. ε 4. Results of the experiment Firstly, and with the main purpose of analysing the performance of the robust statistics, Tables 1 and show, respectively, the mean of all the tests for the two sample sizes considered. Tables 3 and 4 present the percentage of rejection of the respective null hypothesis, for each sample size. The horizontal divisions of the tables correspond to the

16 16 different simulated processes. Also, we highlight in grey the most important robust statistics for each generated process. (nsert Tables 1 to 4) Secondly, Tables 5 and 6 show the results obtained when the proposed model selection strategy is applied with the robust statistics (Figure 1) in terms of the percentage of times that each model is selected. (nsert Tables 5 and 6) The first block of results in all the tables corresponds to the S model with homogeneity in all its parameters. As expected, for both sample sizes, the statistic for testing the null of homogeneity, S Break+Chow+HET, presents a low mean value and, consequently, the percentage of rejection of the null (empirical size of the test) is quite low. The proposed model selection strategy seems to work quite well, since the percentage of times that the correct model is selected is around 97%, for both the sample sizes considered. The second block of results in the tables presents the results obtained when the data are generated with a different degree of heterogeneity in the spatial parameter. n this case, both the joint statistic, S Break+Chow+HET, and the single one S Break are reacting, especially for the biggest sample size of 4. As a consequence, the probability of making a right decision reaches, in the latter case, 9.8%. The third block of results corresponds to a case when the only cause of heterogeneity comes from the parameters of the model. n this case, the right statistics are also reacting, S Break+Chow+HET and S Chow, but the results are not as good as before, since the statistic for testing homoskedasticity is also reacting. As a consequence, there is no high guarantee of selecting the right model in this case. The following block of results refers to the case of heteroskedasticity. n this case, the results are very promising since only the relevant statistics, S Break+Chow+HET and S Chow, are

17 17 reacting. Then, as shown in Table 6, with a sample of 4, the percentage of correct selections reaches The rest of the results of the tables show the performance when the model is generated by combining at least two out of the three sources of heterogeneity. The combination of instability in the spatial parameter and coefficient regression parameters can sometimes be confused with instability in all the parameters. However, the other combinations (the last three blocks of results in the tables) are properly managed with our proposal. As previously, the probability of selecting the right model is much higher for the largest size sample, and reaches values of 98.%, 97.% and 1%, respectively, for the combinations of instability in the spatial and dispersion parameters, the regression and dispersion parameters and instability in all the parameters. To conclude, we can say that results are very promising because the only problem is the relatively high probability of concluding the existence of a heteroskedasticity problem when it is not the case on two occasions: 1) when the GDP presents instability only in the coefficient of the model; and ) when the GDP is generated as unstable in the coefficient of the model and in the spatial parameter. Also, it is clear that the results substantially improve with the sample size. Finally, we compare our previous results referring to the percentage of times that the right model is selected with the strategy designed for the robust statistics (Tables 5 and 6), with the analogous results calculated when using the strategy for the non-robust statistics (Tables 7 and 8). (nsert Tables 7 and 8) Comparing the figures allows us to draw a clear conclusion. Robust statistics clearly outperform their non-robust versions. With the non-robust version, only the homogeneous model and the heteroskedastic one can be properly identified. For the remaining cases, there is a high probability of concluding that the sources of instabilities are simultaneously the three considered in the paper.

18 18 5. Main conclusions n this paper, we try to generalise the homogeneous spatial models, S and SEM, in order to be able to better capture the heterogeneity of a cross-sectional sample. We propose a battery of robust statistics and a combination of them in an easy strategy to tackle the objective of selecting the right model that underlies the data. The Monte Carlo experiment carried out for the S model has shown that the results are very promising since, in most cases, we select the right specification of the model. These results seem to confirm the good performance of both the proposed strategy and the robust statistics. Further research will be aimed at evaluating the effect of the different combinations of heterogeneity in the correct selection of the S or the SEM model from the static one. That is, we will try to measure whether or not the behaviour of the traditional statistics (Moran, _err, _lag, and the robust _err, _lag) are affected by the different sources of instability. f the answer is positive, it would also be useful to derive the corresponding robust versions. APPENDX A: nstability in all the coefficients of a spatial econometric model For the S specified in (1), the associated log-likelihood function appears in () and the score vector is the following: ( A ) ( A ) ( A ) 1 1' p β D l 1 1' p μ D X' y Xm β X' y Xm l y Xm ' D HWy μ tr A HW + l 1 ( y Xm )' g( ) ρ 1 1 y θ = = A D H ZW l tr ρ A HZW 1 α l 1 ( y Xm )' ( y Xm R A D A ) + 4 l 1 1 δ 1 Ay Xm ' D D tr 1ND Ay Xm D D 1 N ( ) ( ) (A.1)

19 19 Under the null hypothesis of (3), it becomes: G 'Y ε l μ 1 ' y l tr ε ZW α ρ A ZW l δ g( θ ) = H l β = 1 ' l tr ε Nε ρ N l (A.) The obtaining of the information matrix is rather tedious and leads us to the following expression, obviously under the corresponding null hypothesis: x'x 1 x' W ' xβ ββ = βμ = x'yg βρ = A x' ZW ' 1xβ βα =ρ A β = βδ = G 'Y W ' x β G 'Y ZW ' xβ μμ = G ' Y Z μρ = A μα = ρ A R μ = μδ = = 4 trw tr tr ρ = A ZW α = ρ A N δ = 1 trnzw A trn tr 1 δδ = ρδ = NW A αδ =ρ ' tr 1 ( ' β ρρ = + ' ) + A W W A A W A W W A ρα αα 'x ' ' xβ β 'x' ' ' xβ =ρ A W ZW A tr + A ' ZW A ' 'x ' ' X tr ( ' β A W β = ρ + ' ) + Z W A A ZW A ZW ZW A 1 (A.3) We use the following partition of the information matrix: μμ μα μβ μρ ρρ ρ ρβ 11 1 = 11 = αα αδ ; 1 αβ αρ α ; ρ 1 = = δδ δρ δ ρβ ββ (A.4)

20 The results for the SEM model of (5) are obtained similarly. The score vector is: 1 1' 1 p β BD ' l 1 1' p 1 μ BD ' B( ) B( ) ( y Xm 1 )' BD ' HW( y Xm ) X' y Xm β X' y Xm l μ tr B HW + l 1 ( y Xm )' ' 1 ( y Xm g( ) ρ ) θ = = BD HZW tr l ρ B HZW 1 α l 1 ( y Xm )' ' y X R BD B( m) + 4 l 1 1 δ ( y Xm )' ' 1 ( y Xm 1 B D D ND B ) tr D DN 1 (A.5) An under the null of (3), and after rearranging, the score vector is: G'Y Bu l μ 1 ' u l tr ε ZW ρ α B ZW l δ g( θ ) = H = l ρ 1 ε' Nε trn l l β (A.6) Finally, the elements of the information matrix, under the null of (3), are the following: ββ = x' B ' Bx 1 βμ = x' B ' BxYG βρ = βα = β = βδ = μμ G 'Yx ' B ' BxY G = μρ = μα = μ = μδ = = R 4 ρ = trw tr B ZW tr α = ρ B N δ = = tr + ' = trn tr tr δδ = ρδ = NWB αδ = ρ NZWB ρρ αα 1 ( ' B W B W W B ) ρα ρ tr 1 ( ' B ZW B W + W B ' ) tr 1 1 ( ' B ZW ZW B B ZW ) = ρ + (A.7)

21 1 APPENDX B The RS λ ρ test refers to an S model and tests whether there is residual spatial dependence in the errors of the equation: u'wu RS λρ = as χ (1) T T V 1 A [ ρ ] (B.1) T A = tr WW + W 'W ρw and V( ρ ) is the ML estimation of the variance of the where ( )( ) 1 ML estimation of the parameter ρ in the S model under the null hypothesis of no residual spatial dependence (Anselin and Bera, 1998). The RS ρ λ test refers to an SEM model and tests whether we have omitted a spatial lag of the endogenous variable in the main equation of the model: u'( W)'( W)Wy RS θ θ ρλ = as χ (1) H λλ (B.) H λλ being the ML estimation of the variance of the restriction (ρ =) obtained in the SEM model under the null hypothesis of no spatial lag in the main equation. 6. References Anselin L (1988a) Spatial Econometrics. Methods and Models. Kluwer, Dordrecht Anselin L (1988b) Model validation in a spatial econometrics: A review and evaluation of alternative approaches. nternational Regional Science Review 11: Anselin L (1988c) Lagrange Multiplier Test Diagnostics for Spatial Dependence and Spatial Heterogeneity. Geographical Analysis : 1-17 Anselin L (199) Spatial Dependence and Spatial Structural nstability in Applied Regression Analysis. Journal of Regional Science 3: Anselin L (199) Spatial dependence and spatial heterogeneity: Model specification issues in the spatial expansion paradigm. n: J.P. Jones and E. Casetti, eds. Applications of the expansion method (Routledge, London), Anselin, L. and R. Florax, Small Sample Properties of Tests for Spatial Dependence in Regression Models: Some Further Results. n Anselin, L. and R. Florax (eds.) New Directions in Spatial Econometrics (pp. 1-74). Berlin: Springer. Anselin, L. and A. Bera, Spatial Dependence in Linear Regression Models with an ntroduction to Spatial Econometrics. n Ullah, A. and D. Giles (eds.) Handbook of Applied Economic Statistics (pp ). New York: Marcel Dekker.

22 Anselin, L., R. Florax and S. Rey, 4. Econometrics for Spatial Models: Recent Advances. n Anselin, L., R. Florax and S. Rey (eds.) Advances in Spatial Econometrics (pp. 1-4). New York: Springer Bera A and Yoon M (1993) Specification testing with locally misspecified alternatives. Econometric Theory 9: Bivand R (1984) Regression Modelling with Spatial Dependence: An Application of Some Class Selection and Estimation Methods. Geographical Analysis 16: Blommestein H (1983) Specification and Estimation of Spatial Econometric Models. A Discussion of Alternative Strategies for Spatial Economic Modelling. Regional Science and Urban Economics, 13, Breusch T, Pagan A (1979) A Simple Test for Heterocedasticity and Random Coefficient Variation. Econometrica 47: Brunsdon C, Fotheringham S, Charlton, M (1998a) Spatial Nonstationarity and Autoregresive Models. Environment and Planning A 3: Brunsdon C, Fotheringham A, Charlton M (1998b) Geographically Weighted Regression- Modelling Spatial Non-Stationarity. The Statistician 47: Cleveland W (1979) Robust Locally Weighted Regression and Smoothing Scatterplots. Journal of the American Statistical Association 74: Cleveland W, Devlin S (1988) Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting. Journal of the American Statistical Association 83: Florax R, Folmer H (199) Specification and estimation of spatial linear regression models: Monte Carlo evaluation of pre-test estimators. Regional Science and Urban Economics : Florax, R., H. Folmer and S. Rey, 3. Specification Searches in Spatial Econometrics: the Relevance of Hendry s Methodology. Regional Science and Urban Economics, 33, Kelejian HH, Robinson DP(199) Spatial autocorrelation: a new computationally simple test with and application to per capita county policy expenditures. Regional Science and Urban Economics, Leung Y, Mei C, Zhang W () Testing for Spatial Autocorrelation among the Residuals of the Geographically Weighted Regression. Environment and Planning A 3: Leung Y, Mei C, Zhang W (3) Statistical Tests for Local Patterns of Spatial Association. Environment and Planning A 35: McMillen D (1996) One Hundred Fifty Years of Land Values in Chicago: A Nonparametric Approach. Journal of Urban Economics 4: 1-14 McMillen D, McDonald J (1997) A Nonparametric Analysis of Employment Density in a Policentric City. Journal of Regional Science 37: Mur J, Lopez F, Angulo A (8) Symptoms of instability in models of spatial dependence. Geographical Analysis 4: Paez A, Uchida T, Miyamoto K (a) A General Framework for Estimation and nference of Geographically Weighted Regression Models: 1: Location-specific Kernel Bandwidth and a Test for Locational Heterogeneity. Environment and Planning A 34: Paez A, Uchida T, Miyamoto K (b) A General Framework for Estimation and nference of Geographically Weighted Regression Models: : Spatial Association and Model Specification Tests. Environment and Planning A 34: Pace K, Lesage J (4) Spatial Autoregressive Local Estimation. n: Getis A, Mur J, Zoller H (eds.) Spatial Econometrics and Spatial Statistics (pp ). Palgrave, London

23 3 Figure 1. Model selection strategy for the S/SEM model making use of the robust version of the statistics S (SEM) Not reject H S(SEM) Break Chow Het Not reject H STOP: S (SEM) Reject H S(SEM) Chow S(SEM) Break S(SEM) Het Reject H Reject H Reject H S(SEM) Break Chow S(SEM) Break Het S(SEM) Chow Het All possible combinations: S(SEM) Break S(SEM) Chow S(SEM) Het S(SEM) Break Chow S(SEM) S(SEM) Break Het Chow Het ESTRATEGY Reject Not reject Not reject STOP: S (SEM) with instability in the spatial parameter Not reject Reject Not reject STOP: S (SEM) with instability in the coefficients of regression Not reject Not reject Reject STOP: S (SEM) with instability in the parameter of dispersion Reject Reject Not reject Reject - - STOP: S (SEM) with instability in the spatial parameter and in the the coefficients of regression Reject Not reject Reject - Reject - S (SEM) with instability in the spatial parameter and the parameter of dispersion Not reject Reject Reject - - Reject S (SEM) with instability in the coefficients of regression and of dispersion Reject Reject Reject S (SEM) with instability in all parameters

24 4 Figure. Model selection strategy for the S/SEM model making use of the non-robust version of the statistics S (SEM) ( RS ρλ ) RS λρ Reject H STOP: SARMA Not reject H S(SEM) Break Chow Het Not reject H STOP: S (SEM) Reject H S(SEM) Chow S(SEM) Break S(SEM) Het Reject H Reject H Reject H S(SEM) Break Chow S(SEM) Break Het S(SEM) Chow Het All possible combinations: S(SEM) S(SEM) Break Chow S(SEM) Het S(SEM) Break Chow S(SEM) S(SEM) Break Het Chow Het ESTRATEGY Reject Not reject Not reject STOP: S (SEM) with instability in the spatial parameter Not reject Reject Not reject STOP: S (SEM) with instability in the coefficients of regression Not reject Not reject Reject STOP: S (SEM) with instability in the parameter of dispersion Reject Reject Not reject Reject - - STOP: S (SEM) with instability in the spatial parameter and in the the coefficients of regression Reject Not reject Reject - Reject - S (SEM) with instability in the spatial parameter and the parameter of dispersion Not reject Reject Reject - - Reject S (SEM) with instability in the coefficients of regression and of dispersion Reject Reject Reject S (SEM) with instability in all parameters

25 Table 1. Mean values for the robust statistics (R=49) S S S ρ α μ δ Break+Chow +Het Break Chow HET Break+Chow Break +HET Chow +HET S S S S 5

26 Table. Mean values for the robust statistics (R=4) S S S ρ α μ δ Break+Chow +Het Break Chow HET Break+Chow Break +HET Chow +HET S S S S 6

27 7 Table 3. Percentage of rejection of the respective null hypothesis with the robust statistics (R=49) S S S ρ α μ δ Break+Chow +Het Break Chow HET Break+Chow Break +HET Chow +HET S S S S

28 8 Table 4. Percentage of rejection of the respective null hypothesis with the robust statistics (R=4) S S S ρ α μ δ Break+Chow +Het Break Chow HET Break+Chow Break +HET Chow +HET S S S S

29 Table 5. Percentage of times that each model is selected with the robust statistics () (R=49) ρ α μ δ (1) () (3) (4) (5) (6) (7) (8) () (1) S ; () S with instability in ρ; (3) S with instability in μ; (4) S with instability in δ; (5) S with instability in ρ and μ; (6) S with instability in ρ and δ; (7) S with instability in μ and δ; (8) S with instability in all the parameters (ρ, μ and δ). 9

30 Table 6. Percentage of times that each model is selected with the robust statistics () (R=4) ρ α μ δ (1) () (3) (4) (5) (6) (7) (8) () (1) S ; () S with instability in ρ; (3) S with instability in μ; (4) S with instability in δ; (5) S with instability in ρ and μ; (6) S with instability in ρ and δ; (7) S with instability in μ and δ; (8) S with instability in all the parameters (ρ, μ and δ). 3

31 Table 7. Percentage of times that each model is selected with the non-robust statistics () (R=49) ρ α μ δ () (1) () (3) (4) (5) (6) (7) (8) () () SARMA (1) S ; () S with instability in ρ; (3) S with instability in μ; (4) S with instability in δ; (5) S with instability in ρ and μ; (6) S with instability in ρ and δ; (7) S with instability in μ and δ; (8) S with instability in all the parameters (ρ, μ and δ). 31