Cliometrica (2008) 2:49–83 DOI 10.1007/s11698-007-0012-6 ORIGINAL PAPER

Minimum distance estimation of the spatial panel autoregressive model The´ophile Azomahou

Received: 20 September 2006 / Accepted: 18 January 2007 / Published online: 23 March 2007 Ó Springer-Verlag 2007

Abstract This paper contributes to the interface literature of new methodological foundation of analyzing historical data with space and spatio-temporal phenomena. In particular, I consider estimating the spatial panel autoregressive model using the minimum distance estimator. Spatial autoregression has important implications for economic system that typifies correlatedness across many spatial locations and which could evolve over long span of time. To overcome computational difficulties, I suggest a two-stage estimation procedure based on minimum distance estimators. A striking feature of the proposed model is that minimum distance estimates are derived under common slopes and complete equality of parameters across spatial units. Assumption of common slopes across spatial units is an empirical and theoretical plausibility as many spatial units are observed to share common trend and typology of changes occurring to the individual system under which equality of parameters are possibilities. The estimation strategy allows various restrictions on time-varying vector parameters. Moreover, those restrictions can easily be tested. I apply this procedure to the residential demand for water of 115 French municipalities over the biannual period 1988–1993. The primary contribution of the paper is to the methodological side of cliometrics while the empirical application (with shorter time period) has been presented for illustrative purpose although, it can nonetheless be readily applied to historical data with long-time horizon allowing for restrictions such as spatio-temporal common vector and structural break in parameter estimates.

Keywords Spatial dependence
Panel data
Minimum distance estimator
Residential demand for water T. Azomahou (&) Bureau d’E´conomie The´orique et Applique´e (BETA-Theme), Universite´ Louis Pasteur, 61, avenue de la Foreˆt Noire, 67085 Strasbourg Cedex, France e-mail: [email protected]

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50

JEL Classification

T. Azomahou

C13
C23
D12
Q25

1 Introduction Study of dependence structure of economic variables is a challenging task. The task gets more arduous when the dependence is characterized in the spatial domain while still accounting for temporal variations. Analysis of spatial dependence is now getting increasing attention in the face of discovery and development of new methodological tools. While spatial structure has an outright statistical and/or econometric dimension, its ready application and comprehension for analyzing historical data and eking out the underlying dynamics, cannot be undermined. Rather, with the help of rich spatial econometric models that describe spatial dependence and its evolution over time, it is more the way useful to describe how at a particular point of time, some historical characteristics at one point have had considerable impact on other locations, and which have evolved over time. This is precisely the subject of scrutiny in this study. I propose to construct a two stage estimation procedure for spatial autoregressive models within the framework of minimum distance estimators, and apply this methodology to examine some determinants of the residential demand for drinking water. Specification of spatial autoregressive model in fact justifies how some economic characteristics observed at one point in space can also be observed to have persisted in another location. Precisely, this implies the continuation of historical attributes in the spatial domain. For the purpose, I specify a spatial autoregressive model using a row-standardized spatial weighting matrix, i.e. a spatial weighting matrix (based on the geographical distance between points) which is normalized so that the rows sum to unity. This standardization produces a spatially lagged dependent variable which represents a vector of average values from neighboring observations. In a regression framework, spatial dependence (vs. spatial correlation) occurs when the response variable (vs. error terms) is spatially correlated at each location with the values of the response variable (vs. residuals of estimation) at other locations. As pointed out by Anselin (1988b) and by Anselin and Bera (1988), ignoring this structure when it actually exists results in mis-specification and estimation bias. Examples of empirical work that explicitly incorporates spatial considerations include, among others, the estimation of spatial effects in household demand for rice in Indonesian districts (Case 1991), the analysis of the influences of spatial contiguity on state price level formation (LeSage and Dowd 1997), the variation of real wages to local and aggregate unemployment rates over time (Ziliak et al. 1999), the estimation of a hedonic model for residential sales transactions (Bell and Bockstael 2000), the forecasting of cigarette demand in the U.S. (Baltagi and Li 2002), the spatial price competition (Pinkse et al. 2002). The goal of this paper is to propose a new methodological tool to examine some of the intriguing cliometric problems, such as impact of residential demand, etc. For the purpose, I investigate spatial dynamics using panel data and suggest a two-stage minimum distance estimation. While most studies focus on cross-sectional specifications, spatial models for panel data have not received much attention. As

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Minimum distance estimation of the spatial panel autoregressive model

51

pointed out by Case (1991), fixed effect specifications can be used to check spatial components. But, in some cases, when there is no intra-regional variation in variables of interest, a spatial approach may however be more appropriate. This would be the case when variations in variables of interest depends upon the distance between points. Then, there is a perfect correlation between these variables and the fixed effects. Case (1991) discussed also the gains in information and efficiency which are achieved by modelling spatial random effects, and showed that when specific effects are uncorrelated with the right hand side variables, there are clear benefits to a spatial specification. More generally, it can be argued that the equicorrelated structure of individual dependence that is typically specified in errorcomponent models for panel data does not allow for distance decay effects (Anselin 1988b). Moreover, this equicorrelation is associated with the time dimension and not with the individual one. As a result, such a structure is not adequate for estimating spatial patterns in panel data. The main contribution of this study, as mentioned earlier is to propose a two stage estimation procedure for spatial autoregressive models within the framework of minimum distance estimators. In order to overcome the computational difficulties that beset spatial processes, in a first stage, the data are treated as T cross-sections. Under the assumption of normally distributed errors, the parameters are estimated by a maximum likelihood procedure.1 Under suitable regularity conditions this stage provides both unrestricted consistent parameter estimates, including the spatial coefficient, and elements of scores which are used to compute a consistent asymptotic covariance matrix for the second stage. The minimum distance method is then applied considering two sets of restrictions: the common slopes (or fixed slopes) and the complete equality of parameters (or all identical parameters). The minimum distance estimates are computed for each case and are consistent and asymptotically efficient. This framework is applied to the empirical analysis of spatial variations in the residential demand for water. We have collected a panel over 115 French municipalities from 1988 to 1993. Apart from the fact that we are handling dependent observations, two important reasons motivate the application of a spatial methodology to this particular data and then to issue of residential demand for water. Firstly, as noticed in INSEE (1998), for the municipalities concerned by the sample, a regional specificity in households’ consumption of drinking water may be observed.2 Such a behavior is mainly due to the availability of water resources. Furthermore, as will be seen later, these municipalities have been grouped into spatial sectors for the purpose of water network management. In this context, our suggested specification may be viewed as a model of endogenously changing water 1

The maximum likelihood method to estimation and hypothesis testing in spatial econometrics is by far the better known methodological framework and the most widely implemented. We also use this technique in the first step of our estimation procedure (Lee 2004). Kelejian and Prucha (1998, 1999) propose alternatives estimators to avoid some of the computational difficulties associated with the ML estimation. Other recent theoretical developments are, among others, Driscoll and Kraay (1998), Conley (1999) and Chen and Conley (2001). 2 See Tableaux de l’E´conomie Lorraine 1997/1998 (Tables of Lorraine Economics).

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T. Azomahou

consumption behavior, which allows to check for social interdependence by testing the extent to which households look to a reference group when making water consumption decisions. This may also be thought of as indicating the magnitude and the direction of interactions between consumers with respect to the availability of water resources. The second reason is explicated in the theoretical framework. As will be described in the data, there is no intra-regional variation in the price of water. As pointed out by Deaton (1990) and by Case (1991), in such a situation, a spatial approach seems more appropriate and should be preferred to the pure fixed effect modelling in order to evaluate own- and cross-price elasticities from spatial variation in prices. In demand analysis, the presence of variables and errors which are spatially dependent is quite intuitive. Akerlof (1997) outlined the role of spatial interdependence stressing how social decisions may include dependence of individuals’ utility on the utility or the action of others. Case (1987) noticed that the formation of preferences through replication of neighbours’ behavior, may lead to spatially dependent variable. So, the availability of complementary goods, the diffusion of information, the natural character of resources, and soil and climate conditions may all be unobservable variables which are spatially correlated and contribute to spatial correlation in demand errors. Most of the literature on the residential demand for water focuses on the estimation issue resulting from water pricing structures, and how pricing modulations may impact the consumption behavior; see e.g. Hanke and de Mare´ (1982), Howe (1982) and Hewitt and Hanemann (1995). The importance of energy effect in estimating determinants of the residential demand for water has been pointed out by Hansen (1996). He noticed that we may expect to detect the indirect effects of energy variables with respect to the consumption of water among different water-using tasks. Indeed, water is consumed by households using appliances that in most cases consume large amounts of energy.3 In Nauges and Thomas (2000), a static demand function was first analyzed. Then, a temporal dynamics demand function was used to measure the adaptation of households to a persistent increase in the price of water. The issue regarding the choice of functional forms to estimate water demand had received a closer attention in the study of Yoo and Yang (2000) who used a more flexible semiparametric approach. Three main empirical findings emerge from this study: (1) the residential demand for water displays spatial patterns, i.e. the estimated spatial coefficient is positive and strongly significant; (2) the average price of water has a negative and significant effect on the demand whereas the average price of electricity (a proxy of energy variable) impacts positively and significantly the demand; (3) income, water ownprice and spatial elasticities at means appear to be weak.

3

In France for example, it is observed that about 50% of the daily distribution of the residential consumption of water is concerned with water heating (mainly by electricity), see Table 12 for a description.

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Minimum distance estimation of the spatial panel autoregressive model

53

The paper is intended to elevate some of the methodological incongruities and complexities involved in the analysis of cliometric theories. And in this context, the paper attempts to make a methodological contribution and for empirical illustrative purpose the residential demand function has been examined. The framework developed in the study, nonetheless can be readily applied to historical data in some other branches of cliometric theories. The outline of the paper is the following. In Sect. 2, we present the econometric analysis. The specification combines elements of spatial modelling and panel data framework using a minimum distance estimator. Section 3 presents parameter vector restrictions. We also suggest a specification test based on the estimated minimum distance method. Section 4 is dedicated to the data. This includes the description of the sample and basic descriptive statistics. We also compute spatial correlograms to check for spatial effects in data, and nonparametric density estimation to identify ‘spatial sector tendencies’ in the distribution of the average price of water. The estimation results are discussed in Sect. 5. Concluding remarks are given in Sect. 6.

2 Econometric specifications In this section, we develop the statistical model and the estimation strategy. Inference details are also provided. 2.1 The Model We consider a spatial autoregressive model for panel data containing a spatial lag of the response variable. For i = 1, ..., N; j = 1, ... , N and t = 1, ... , T, the model has the following structure: yit ¼

N X j¼1

qxij yjt þ

K 1 X

ðkÞ

xit bk þ eit

jqj<1;

ð1Þ

k¼1

i6¼j

where yit is the i-th observation on the dependent variable at period t, x(k) it is the i-th observation for the kth explanatory variable; yjt is the jth observation on the dependent variable; q is a scalar (the spatial coefficient), and the b’s are the (K1) coefficients of the remaining explanatory variables; q and b are the parameters of primary interest to be estimated; eit denotes an i.i.d. disturbance. For the reasons described in Sect. 1 regarding fixed effects specifications in a spatial context, we assume that eit has random effect components; xij is an element of a spatial weighting matrix, say W, whose computation is described in Appendix B.4 4

Here, we assume that W is time invariant. This assumption seems reasonable since in the empirical analysis, we work with regionalized data and the computation of W is based on the geographical distance between municipalities; the spatial delineation of these municipalities had not been modified. As a result, their geographical position remain unchanged.

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Equation (1) can be rewritten in a more convenient form by stacking as vectors and matrices. For each time period, let y = (y1 , ... , yi , ... , yN)0 , X = (X1 , ... , Xi , ... , XN)0 and e = (e1 , ... , ei , ... , eN)0 . Thus, each wave t contains N cross-sections. We organize the data as such due to the time-invariance of the spatial weighting matrix. Moreover, as we will see in the next sub-section, this will contribute to ease the estimation strategy. Hence, for each period, y and e are of dimension (N x 1) and X is N x (K1). The structure of the model implies that each cross-section follows a spatial autoregressive process. Then, in stacked form the model is y ¼ ½Wy; X h þ e;

0

h ¼ ðq; b0 Þ :

ð2Þ

We assume that the errors are normal with Eðet Þ ¼ 0; Eðet e0s Þ ¼ r2t I; for t = s and Eðxit eit Þ ¼ 0: The assumption of normally distributed errors allows us to estimate, in a first stage, the parameters of each cross-section separately by maximum likelihood. Then the conditional distributions of et given Xt in this stage are assumed to be time varying. In a second stage the minimum distance method can be applied. This second stage provides efficient estimates, since the quadratic form to be minimized is optimally specified in the sense of Hansen (1982).5 We now describe the estimators. 2.2 Estimation and inference 0

2 ^0 ; r Let ^ ht ¼ ð^ qt ; b stage unrestricted maximum likelihood t ^ t Þ denote the first 0 estimates for parameters ht = (qt , bt, r2t )0 for each cross-section. That is

^ ht ¼ arg max ht 2H

N X

wðyit ; Xit ; W; ht Þ

t ¼ 1; . . . ; T;

ð3Þ

i¼1

where w (
) denotes the log likelihood function computed as N N 1 wðy; X; W; q; b; r2 Þ ¼  lnð2pÞ  ln r2 þ ln jUj  2 g0 g; 2 2 2r

ð4Þ

with g = U y  X b, U = IN  q W and |
| denotes the determinant. In the second stage, one can use the unrestricted maximum likelihood estimates of the first stage to form the restricted minimum distance estimates by imposing restrictions of the form gðbð^ hÞ; aÞ ¼ 0: These restrictions link the set of parameters of interest A ¼ aðPÞ  RK ; a0 ¼ aðP0 Þ; 8P 2 P; where a0 denotes the true value of a, and the set of auxiliary parameters: B ¼ bðPÞ  RH and b0 ¼ bðP0 Þ; 8P 2 P the data generating family law, and where b0 denotes the true value of b. The estimating equations are such that

5

Note that this specification belongs to the class of exponential models. As a result, applying the minimum distance method allows to achieve the lower bound of Cramer-Rao (Gourie´roux and Monfort 1989, Chap. 9).

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Minimum distance estimation of the spatial panel autoregressive model

gðb; aÞ ¼ 0;

with

gðb; aÞ ¼ 0 ) a ¼ aðPÞ;

55

8P 2 P:

ð5Þ

^ Relation (5) means that there exists a sequence b^n of estimators for b such pffiffiffiffi that: (1) b converges towards b0, for P0 a.s., (2) the asymptotic distribution of N ðb^n  b0 Þ; with a covariance matrix R0, is pffiffiffiffi D N ðb^n  b0 Þ ! Nð0; R0 Þ: N!1

D

where ! means convergence in distribution. Under these conditions, the minimum distance estimator is obtained by choosing a^n to minimize h i0 h i a^ ¼ arg min gðbð^ hÞ; a0 Þ Sn gðbð^hÞ; a0 ;

ð6Þ

a2A

a.s. where Sn ! S0 ; a positive definite symmetric matrix. The optimal choice of S0 is known to be the inverse of the asymptotic variance matrix of gðbð^hÞ; a0 Þ; see Gourie´roux et al. (1985) and Kodde et al. (1990) for a detailed and rigorous exposition of the minimum distance method. Under regularity conditions, the estimator a^ðSn Þ exists and is consistent for a0. Furthermore, the following asymptotic distribution holds pffiffiffiffi D N ða^n  a0 Þ ! Nð0; X0 Þ: N!1

ð7Þ

The matrix X0 is estimated as X0 ¼

1  1 1  J I0 J0 ; N 0

ð8Þ

where J0 = diag {J1 , ... , JT } is a block diagonal matrices with 

 @ 2 wðyt ; Xt ; W; ht Þ Jt ¼ E  ; @h@h0

ð9Þ

and where the out of block elements corresponding to waves t and s for t = s are computed as   @w @w ðyt ; Xt ; W; ht Þ ðys ; Xs ; W; hs Þ : Its ¼ E @h @h

ð10Þ

^ of X is computed as J^1 I^J^1 using individual scores and A consistent estimator X by replacing theoretical expectations sample means. 2.3 Individual scores and empirical variance We now provide the computation of individual scores and empirical variance. Let wi (.) denote the log-likelihood for one observation. We have

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1 1 1 wi ðy; X; W; q; b; r2 Þ ¼  lnð2pÞ  ln r2 þ ln jUj 2 2 N " #2 X X ðkÞ   1  2 1½i¼j  qxij yj  Xi b k ; 2r j2J k

ð11Þ

where j = 1, ..., J denote units contiguous to a unit i and 1[i=j] denotes an indicator function. Taking partial derivatives of (11) with respect to the parameters yields " # X ðhÞ  @ 1 X ðkÞ w ð
Þ ¼ 2 1½i¼j  qxij yj  Xi bh Xi ; ð12Þ @bk i r j2J h " # X X ðhÞ  @ 1 X wi ð
Þ ¼ 2 1½i¼j  qxij yj  Xi bh xij yj þ N 1 n; ð13Þ @q r j2J j2J h " #2 X ðkÞ  @ 1 1 X w ð
Þ ¼  2 þ 4 1½i¼j  qxij yj  Xi bk ; ð14Þ @r2 i 2r 2r j2J k with n¼

  @ ln jUj ¼ tr U1 W : @q

The compute the empirical variance matrix of individual scores, we take the cross products   @w @w I^ts ¼ E ðyt ; Xt ; W; ht Þ ðys ; Xs ; W; hs Þ 8 t 6¼ s: @h @h h¼^h 3 Restrictions on parameters and specification test The minimum distance procedure implemented in the second stage is based on the assumption that H0 : fh=9a 2 A  Rq : gðbðh; aÞÞ ¼ 0g;

ð15Þ

where the function g(.) is valued in Rr : It is assumed that @g=@h0 and @g=@a0 are respectively of rank r and q. For empirical estimation one needs to assume a functional form for the estimating equations g(b(h), a). Here, we assume that g(b(h), a) is linear with respect to the parameters of interest, a. From this specification, minimum distance estimates are obtained by imposing two restrictions. The first restriction is that of common slopes and can be expressed as 0 ^1 1 hx  a 2 B^ C B hx  a C gðbð^ hÞ; aÞ ¼ B . C; @ .. A ^ hT  a x

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ð16Þ

Minimum distance estimation of the spatial panel autoregressive model

57

where h^t ; t ¼ 1; . . . ; T denotes the vector of parameter of varying intercepts and slopes for each period, and a the vector of common slopes. The second restriction is that of complete equality of parameters, i.e. 0 1 ^ h1  a B^ C B h2  a C gðbð^ hÞ; aÞ ¼ B . C; ð17Þ @ .. A ^ hT  a with ^ h ¼ ð^ h1 ; . . . ; ^ hT Þ0 : For each case, the minimum distance estimates are computed using relation (6). The assumption H0 the relation (15) can be empirically tested. Indeed, assuming that g(.) is linear in a, relation (15) turns out to be H0 : f9a : bðhÞ ¼ HðhÞag;

ð18Þ

where b(h) is a vector of dimension r and H(h) is a matrix of one and zeros of dimension r x q. ^ n ½bð^hn  Hð^hn Þ^ The statistic test is Tn ¼ NT½bð^ hn  Hð^ hn Þ^ an 0 X an  with distribu2 tion v1a (rq). One of the interest of this approach is to allow for several restrictions on parameter estimates and to test easily these restrictions. In particular, in the case of historical data with long-time horizon (for example in cliometrics), one can test for structural break in parameter estimates, spatio-temporal common vector parameter, etc. In what follows, we provide a empirical illustration of the above developed methodology.

4 Data and variables The department ‘Moselle’ consists of about 730 municipalities out of which 115 have been selected for the empirical study of households’ demand for drinking water.6 Households living in these municipalities are supplied with drinking water by a private operator. The data considered here represents the first lattice collected from the French network of drinking water distribution. The data is collected biannually from 1988:1 to 1993:2, a balanced panel of 1,380 spatial observations. Some variables do not require important changes before being used. Others have been constituted from information available in the last municipal inventory.7 In this section, we describe the sampling and relevant features of the variables; see Appendix A for further details on data and the definition of variables. 6

The department ‘Moselle’ is located in the north-east of France. The retained municipalities are those for which we succeeded obtaining reliable information on the residential consumption of drinking water.

7

All information related to the municipalities’ characteristics come from the municipal inventory of 1990. The municipal inventory is a document which provides the characteristics of French municipalities. The study is conducted by the ‘Institut National de la Statistique et des E´tudes E´conomiques’ (INSEE), National Institute of Statistics and Economic Studies. The last recorded dates from 1998, were not yet available when this empirical work was conducted.

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4.1 Sampling and descriptive statistics The first step of this study was the collection of data. Since this kind of data had never been collected before, two important issues arose from a closer look of the consumption values. The first was the identification of households’ consumption. The network manager (a private operator) provides water services to the subscribers, i.e. citizens living in individual houses or in collective blocks of flats (for instance council flats), as well as industrial consumers and businesses. The households’ demand gathers together individual user consumption and collective user consumption. Most of the households living in collective lodgings do not yet have meters that indicate accurately the amount of their consumption. Also, for these consumers the charge for water is included in the rent. We can then suppose that the households concerned are not aware of the necessity to control their budget with respect to water expenses. Moreover, there are also blocks of flat sheltering small businesses. In the case when a household living in a collective lodging has also a business linked to his subscriber regime, the former’s consumption of water cannot be distinguished from the latter’s. A similar issue occurs for some households living in individual houses. For farmers for instance the identification of purely domestic volumes is difficult. For all of these reasons, and in order to reduce the evaluation errors as well as to be sure that the target sector corresponds to the residential one, we have selected subscribers connected to the network of drinking water with a main water capacity of 15 mm in diameter, when this information was available. Despite this choice, we cannot be absolutely confident that some marginal values of consumption (coming from small businesses or a consumption different from a domestic one) are still in the collected data. The second problem concerns the reconstruction of some consumption values, either because they disappeared during floods (as in the case of 1990s data), or because they existed only at a high level of aggregation. This concerns only very few unionized municipalities. The non-unionized municipalities display a half year water volume. Unions result from the gathering of municipalities; we use union data to estimate the volume consumed when municipalities’ data is missing. The data used to reconstruct consumption values, as far as municipalities are concerned, come from a document termed ‘water products’. The volumes looked for are semester values. When semester data are missing, we face two possible cases: either only some municipalities composing the union are considered or the details of the volume consumed are not available. In the former case we suppose that the consumption in the other municipalities varied in the same proportion. In the latter case, the average weight of each municipality in the union is computed. As a result, the data possess two characteristics which make their biannual use delicate. On the one hand, the water reading frequency ran from at least a quarterly period to an annual one whilst the pricing remains biannual. The accurate biannual readings are available for 1988:1 for all the municipalities, as well as the readings of 1993:1. From 1990:1 to 1992:2 some municipalities adopted an annual reading. In this case, and to reduce the cost induced by meter readings, the volumes for one semester are estimated from the consumption of a preceding year, where the

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Minimum distance estimation of the spatial panel autoregressive model

59

Fig. 1 Distribution of the residential consumption for water based on kernel density estimates. Epanechnikov kernel and the cross-validation method for bandwidth calculation were used. The figure shows a mainly uni-modal distribution between 60 and 80 m3 for each period

duration between two readings does not always equal 52 weeks. Moreover, the calendar year is no longer taken into account, instead the period stretching from June to June is considered. On the other hand, we face differences in the frequency of data collection. The consumption reading frequency may vary from one year to another because of climatic hazards or other unforeseeable parameters. To correct these biases, the consumption values presented in this study are corrected to lead to a frequency of 52 weeks. These two characteristics, estimated values and differences in the reading frequency, are possible sources of measurement errors. Despite this, it is important to note that the percentage of the initial sample that cannot be used because of identification problems is about 3% and the percentage of the data that had to be reconstructed in order to obtain consumption values is about 5%. Then, we compute a nonparametric density estimation to have a closer look at the distribution of consumption values. Figure 1 shows estimation results.8 We notice mainly a unimodal distribution between 60 and 80 m3 for each period. This result reinforces our recording target, i.e. the consumption of residential subscribers. We have recorded the aggregate water consumption per municipality in cubic meters per house. Since urban municipalities are larger than rural ones, each consumption value has been divided by the total number of households per 8

We use the Epanechnikov kernel and the cross-validation method for bandwidth computation; see e.g., Silveramn (1986) and Wand and Jones (1995) for details. All calculations are implemented using GAUSS program.

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Table 1 Descriptive statistics of water consumption and the average price Period

Consumption in cubic meters

Average price in FF

Mean

SD

Min.

Max.

Mean

SD

Min.

Max.

1988:1

69.68

27.75

1.11

1988:2

70.13

23.56

1.04

153.15

6.28

2.11

3.24

11.29

148.74

6.37

2.14

3.27

1989:1

72.28

28.37

11.38

1.00

186.78

6.69

2.31

3.09

1989:2

74.55

11.52

27.11

0.88

175.28

6.79

2.35

3.09

11.64

1990:1 1990:2

73.47

27.43

0.96

162.52

7.05

2.44

3.40

12.40

72.67

26.33

0.86

163.37

7.22

2.53

3.41

12.54

1991:1

75.56

29.17

0.90

179.48

7.70

2.65

3.47

13.08

1991:2

75.04

28.90

0.86

187.81

7.95

2.91

3.56

16.19

1992:1

71.94

27.07

0.73

155.81

8.66

3.40

3.63

17.59

1992:2

72.75

27.68

0.87

170.37

9.01

3.51

3.67

18.10

1993:1

72.14

26.51

0.81

157.33

9.97

3.97

4.09

19.46

1993:2

71.24

29.26

0.83

176.19

10.58

3.50

4.77

19.50

Data source ‘Compagnie Ge´ne´rale des Eaux’

community in 1990, the year of the last inventory available to us. This is also when the last general population census was conducted by he research centre INSEE. Descriptive statistics on the variables are shown in Tables 1, 2, 3 and 4. National statistics indicate an average water consumption tendency of about 120 m3 per house per year. These figures display somewhat varying pattern: old houses consume less water whereas high standing dwellings with gardens can consume around 180 m3. When we compare these indicators with those computed from the sample, we notice that the averages of recorded consumption are of the same magnitude. Minimum values can be considered as the consumption of rural municipalities. These tendencies are also indicative of the standard of living of the population considered. As a whole, there are no outliers in consumption values. Note, however, that some high values exist for 1989:2, 1991:1 and 1991:2 where we observe 74.55, 75.56 and 75.04 m3, respectively. This may result from extra consumption in addition to purely domestic consumption. Again these statistics support, on average, our recording target sector: the water consumption of residential subscribers. Income data is collected from the regional center of taxes in the department Moselle. Information is drawn up from the statistical report of the ‘Impoˆt sur le Revenu des Communes’ (IRC), income tax of the Municipalities or municipalities’ fiscality (see Appendix A for further details).9 Disposable income statistics are 9

IRC, No 1, re´pertoire permanent des statistiques. The statistics are drawn up on the ‘12th emission’ of the ‘roles’ of taxation, describe the taxation of the municipalities. The ‘roles’ are ‘titles’ in virtue of which the services of the treasury (public accounts) carry out the covering of the direct tax. It is a list of taxpayers liable to the tax comprising in particular for each of them the taxable amount, the nature of the contributions and the taxes, the tax rate and the amount of the contributions and the total of taxes. This list is drawn up by the director of the tax center on the basis of the elements provided by the tax authorities. The data taken into account correspond to the impositions carried out to No 55 included.

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Minimum distance estimation of the spatial panel autoregressive model

61

Table 2 Descriptive statistics of meteorological variables Period

Rainfall in mm

Mean temperature in Celcius

Mean

SD

Min.

Max.

Mean

SD

1988:1

8.76

0.65

7.38

1988:2

7.29

0.70

5.94

1989:1

6.17

0.56

5.12

1989:2

6.55

0.50

5.69

1990:1

6.94

0.66

5.84

1990:2

6.93

0.77

5.50

1991:1

4.65

0.40

1991:2

5.95

1992:1 1992:2

Min.

Max.

11.10

8.91

0.33

8.00

9.58

8.87

11.47

0.29

10.68

12.15

7.69

8.78

0.32

8.21

9.50

8.92

11.86

0.36

10.83

12.55

8.24

9.33

0.26

8.66

10.00

9.41

11.48

0.32

10.56

12.20

3.72

6.80

7.02

0.28

6.25

7.65

0.89

4.76

7.51

11.99

0.27

11.15

12.58

5.46

0.67

4.14

7.51

8.70

0.28

8.10

9.28

7.97

1.20

5.26

10.25

11.92

0.18

11.30

12.46

1993:1

4.50

0.81

2.77

5.77

8.81

0.22

8.31

9.46

1993:2

10.14

0.72

8.78

12.35

10.71

0.26

9.90

11.28

Data source ‘Centre De´partemental de la Me´te´orologie de la Moselle’ Table 3 Descriptive statistics of disposable income Period

Mean

SD

Min.

Max.

1988

57.51

8.28

33.38

75.24

1989

59.07

8.65

37.47

79.23

1990

62.06

9.31

31.82

85.16

1991

3.82

10.31

33.66

92.14

1992

65.49

11.33

34.18

104.16

1993

66.97

11.76

34.33

97.68

Data source ‘Direction Re´gionale des Impoˆts’ Values are expressed in thousands of FF Table 4 Descriptive statistics of municipalities’ characteristics Variable

Mean

SD

Proportion of Persons below 19 years

0.28

0.04

0.13

Density of Population

1.10

2.60

0.0038

29.96

4.39

11.92

9.78

4.03

2.70

23.62

61.87

6.86

30.24

76.84

Proportion of employees Proportion of Unemployed Index of Equipment

Min.

Max. 0.31 14.61 37.82

Data source INSEE Statistics are computed for 1990, the year of reference

characterized by very low values. Consider for example the year 1990 where the minimum values are the lowest, i.e. 31,820 FF per taxed household. This gives a monthly disposable income of 2,651.66 FF. Supposing that this household is made

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up of a single member, the latter roughly earns the so-called ‘minimum insertion income’ in France. This shows the difficulty usually encountered in recording income data. Other reasons explain these low values. Indeed, various studies conducted by INSEE show that in the department of ‘Moselle’, taxable incomes under-estimate actual household incomes by 30% on average.10 This underestimation is extremely high for the self-employed (43%), even higher so for selfemployed farmers (57%). Moreover, even if we know that the consequences of the economic crisis on the evolution of global wages has been compensated by a strong increase in social benefits and only a slight increase in taxes, the department ‘Moselle’ is below national income indicators. The rate structures for water are simple two-part tariffs with a marginal price, that is the price of 1 m3, and a fixed charge that is independent of the volume consumed. The average price of water is then computed to include these two components (see Appendix A for definition). Average price values clearly indicate relevant patterns. The average price continuously increases over the 12 biannual periods. From 1988:1 to 1989:2 the average price is below 7 FF; from 1990:1 to 1991:2 it is between 8 and FF, and above 8 FF from 1992:1 onwards. This last tendency indicates an important modification in the structure of the price of water. As a result, the price variable suggests a clustering pattern. It also presents an increasing dispersion within clusters with stable minimum values (around 3.5 FF). All these figures are examined more carefully in the next section. Finally, note that the meteorological variables (rainfall and temperature) presented here are not limited to dummy variables as is usually the case in the literature on water demand. The values of these variables were recorded by the Regional Center of Meteorological Studies of ‘Moselle’. As we may expect, for the temperature, the first semester values are lower than those of the second semester. Further details on these variables are also provided in Appendix A. 4.2 Distribution of the average price of water For various reasons that are described below, it seems relevant to study the distribution of the average price of water for the period under study. Indeed, the organization and the management of water distribution in France pertains to public service liability. The price of water results from a negotiation between local authorities and the water distributor who may be the local collectivity itself or a private company. Municipalities and households concerned by this study are supplied with drinking water by a private firm. According to the water supplier, the municipalities are split into two sectors, however the exact number of sectors is unknown to us. We denote each sector by a dummy variable (dummy 1 for sector 1 and zero for sector 2). Out of the 115 municipalities, 65.2% belong to sector 1. The sectors correspond to two distinct areas of water management. This spatial arrangement is mainly due to network management issues (water transportation, treatment to make water drinkable, etc.) 10 These figures can be found e.g. in ‘Tableaux de l’E´conomie Lorraine 1997/1998’ (Tables of Lorraine Economics), INSEE (1998).

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Minimum distance estimation of the spatial panel autoregressive model

63

and is closely linked to elements of water prices.11 The marginal price of water is the same within a given sector but varies between sectors. Thus, we know that there is no intra-regional variation in the marginal price. However, the average price of water varies from one community to another when the fixed charges of water are included. Moreover, the laws on water of November 1992, the so-called ‘M-49 directive’, have strongly modified the working orders of water agencies.12 This modification translated into high increases in water prices. The aim is to let customers pay for the effective price of water, instead of for the water service. To check for the persistency of sector design effects in the distribution of the average price (having incorporated the fixed charges of water), we use nonparametric estimation for data analysis and identification purposes. Figure 2 shows the kernel density estimate of the average price of water for each time period. Two main conclusions emerge. Firstly, we notice that up to 1991:2, the distribution displays three modes. From 1992:1 onwards the central mode starts disappearing and by 1993:2 there are only two modes left. This distribution can mainly be explained by the modifications that occurred in water pricing in 1992. These modifications may be due to the ‘M-49 directive’ which resulted in a change in water pricing. Not only did the price increase continuously as indicated by descriptive statistics, but now, two sectors appear clearly from 1992. Secondly, the distribution reveals that there may also be three sectors up to 1992:1. Thus, sector design effects clearly appear in the average price of water. As a result, we may expect a within-sector behavior regarding water consumption as well as a spatial effect. 4.3 Testing spatial autocorrelation in water consumption We introduce various analytic methods which are of value in assessing the spatial scale of a process. The variables of interest is the consumption of water. We use Geary ratio or G-statistics which provide a measure of overall spatial association as well as observation-specific spatial association. The G-statistic is a paired comparisons coefficient (Cliff and Ord 1981; Griffith 1988); see Appendix B for a formal definition. These statistics are computed by defining a set of neighboring municipalities. For each location, neighboring municipalities are considered as those which fall within a distance band as computed in Appendix B. We test for a specific spatial association, i.e. the extent to which a location is surrounded by a cluster of high or low values for the variables of interest for each period. The results of the tests are reported in Table 5. We observe a significant value for consumption (except for 1992:1 and for 1993:1) which is indicative of a spatial clustering.

11 The correlation coefficient between the average price of water and the sector dummy for the 12 time periods ranges between 0.41 and 0.32, there is evidence of correlation. 12

Set up on November 10th, 1992 (its implementation date) the ‘M-49 directive’ imposes the rule of budget balance to water services (supply and cleaning up). Expenditures on water spending (building up and maintenance of the network, equipment, cleaning up...) are no longer allowed to be included in their general budget.

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Fig. 2 Distribution of the average price of water based on kernel density estimates. Epanechnikov kernel and the cross-validation method for bandwidth calculation were used. We observe three modes from 1988:1 to 1991:2. In 1991:1 and 1992:2, the central mode starts disappearing. From 1992.1 on, only two modes remain

Although the interaction between spatial units may be strong between immediate neighbours, the strength of interaction will often vary in a complex way with distance. The G-statistic test has a static aspect and does not provide information on the spatial dynamics of the process. We test for the difference of spatial autocorrelation for consumption values over different weighting matrices using spatial correlograms (see Appendix B for definition). Higher order contiguity is used to compute spatial correlograms. The contiguity matrices are obtained by taking powers of the unstandardized form of the first order contiguity matrix and by correcting for circularity. The spatial lag length is eight. It corresponds to the point

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65

Table 5 G*(d)-test for specific spatial autocorrelation Period

Water consumption G-stat.

Prob.(%)

1988:1

0.329

0.7

1988:2

0.337

3.0

1989:1

0.332

1.6

1989:2

0.335

2.3

1990:1

0.332

1.2

1990:2

0.331

0.6

1991:1

0.329

0.5

1991:2

0.328

0.4

1992:1

0.341

11

1992:2

0.332

1.1

1993:1

0.342

13

1993:2

0.328

0.6

where the higher order contiguity results in unconnected spatial units, i.e. spatial units for which the corresponding row in the contiguity matrix consists only in zeros, i.e. W9 = 0. The results of the estimated spatial correlograms for each time period are reported in Figs. 3 and 4. Spatial lags are reported on the X-axis (up to eight lags are computed), and the t statistics on the Y-axis. To ease presentation, other statistics related to spatial correlograms (expectation, standard deviation etc.) are not reported here. A significant and strong indication of spatial clustering for the first and second orders of contiguity is evident (except for 1993:1). Except lag 4, in general, we notice a decreasing spatial autocorrelation with increasing orders of contiguity, which is typical of many spatial autoregressive processes. The significant and negative spatial autocorrelations at lag 2 contrast with the significant and positive spatial autocorrelations at lag 1. For the average price of water, lag 7 is nearly significant and lags 5–8 are nearly significant for 1991:1. These results clearly indicate potential spatial dependence in the consumption of water. Thus, it seems relevant to include the spatial dimension in the model specification.

5 Estimation results We use the theoretical framework sketched in Sects. 2 and 3 to carry out the empirical estimation using the data described in the previous section. Tables 6 and 7 present the unrestricted maximum likelihood estimates for the 12 time periods.13 13 A crucial assumption behind the maximum likelihood method in our first stage estimation procedure is that the disturbances are normally distributed. Violation of this can arise from outliers or spatial enclave effects where a small cluster of observations display aberrant behavior.

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Fig. 3 Estimation of spatial correlograms for the residential consumption of water from 1988:1 to 1990:2. Up to eight spatial lags on the X-axis and the t-value of the G-statistics on the Y-axis. The first two lags of each correlogram are highly significant (significant level = 2), indicating spatial dependence which decreases with spatial lags. However, lag 4 is significant for 1990 and nearly significant for 1988:1 and 1989

We use a Lagrange multiplier statistic to test for spatial error autocorrelation in the spatial lag model. The LM statistics suggested by Anselin (1988a) takes on the form 2

LMerr ¼

123

ðe0 We=s2 Þ ; 0 trðW W þ W 2 ÞA1 varðqÞ

Minimum distance estimation of the spatial panel autoregressive model

67

Fig. 4 Estimation of spatial correlograms for the residential consumption of water from 1991:1 to 1993:2. Up to eight spatial lags on the X-axis and the t-value of the G-statistics on the Y-axis. The first two lags of each correlogram are highly significant (significant level = 2), indicating spatial dependence which decreases with spatial lags. However, lag 4 is significant for 1992 and nearly significant for 1991 and 1993

where A1 = (I  q W)1, e are the residuals in the ML estimation, s2 the estimated error variance, W the spatial weighting matrix and varðqÞ the estimated asymptotic variance for the spatial autoregressive coefficient. The statistic is asymptotically distributed as v2 with one degree of freedom. For technical

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T. Azomahou

Table 6 Unrestricted maximum likelihood estimates (continued) Variable

Cross-section estimates (and standard errors) 1988:1

1988:2

1989:1

1989:2

1990:1

1990:2

128.33

157.20

190.32

58.00

653.15

160.67

(157.1)

(171.1)

(267.6)

(244.2)

(310.4)

(256.6)

0.53

0.04

0.56

0.89

0.24

0.06

(0.6)

(0.5)

(0.7)

(0.7)

(0.6)

(0.6)

1.35

0.86

1.51

1.71

2.09

1.39

(1.2)

(1.2)

(1.1)

(1.2)

(1.1)

(1.1)

0.19

0.18

0.48

0.13

0.59

0.35

(0.1)

(0.1)

(0.3)

(0.2)

(0.3)

(0.2)

0.95

0.27

1.22

1.38

0.86

0.55

(0.4)

(0.4)

(0.3)

(0.4)

(0.4)

(0.3)

1.14

16.08

3.42

8.37

18.22

2.92

(6.4)

(8.5)

(6.5)

(6.3)

(9.1)

(6.9)

1.39

1.69

1.03

1.24

0.58

0.88

(0.6)

(0.5)

(0.6)

(0.6)

(0.6)

(0.5)

Proportion of employees

2.66

1.48

2.19

0.73

2.72

1.72

(0.6)

(0.5)

(0.7)

(0.7)

(0.6)

(0.6)

Proportion of unemployed

2.97

2.62

3.32

2.74

3.48

3.12

(0.6)

(0.5)

(0.6)

(0.6)

(0.6)

(0.6)

Index of equipment

0.41

0.68

0.28

0.21

0.08

0.09

(0.3)

(0.3)

(0.4)

(0.4)

(0.3)

(0.3)

Intercept Disposable income Price of water Price of electricity Rainfall Temperature Persons below 19 years

Density of population Spatial laga

0.16

0.12

0.22

0.01

0.32

0.08

(0.4)

(0.3)

(0.4)

(0.4)

(0.4)

(0.4)

0.28

0.12

0.28

0.32

0.01

0.31

(0.2)

(0.3)

(0.2)

(0.2)

(0.3)

(0.2)

Diagnostics testb LM error.c Spatial B–P.d

0.70

3.91

0.22

0.83

0.27

0.37

(0.4)

(0.0)

(0.6)

(0.4)

(0.6)

(0.5)

13.75

7.77

19.82

13.54

16.63

25.22

(0.1)

(0.5)

(0.0)

(0.1)

(0.0)

(0.0)

Number of observations a b

115

‘Spatial lag’ means the spatially lagged dependent variable, computed as Wy p-values are in parenthesis

c

Lagrange multiplier test for the spatial model, v2 (1)

d

Spatial Breusch–Pagan test for spatial heteroskedasticity, v2 (2)

details, see Anselin (1988a). The results of the test rejects the alternative spatial error specification for most cases except for 1988:2, 1991:1, 1992:1 and 1993:1. For these cases, spatial dependence remains in the residuals and our specification is clearly rejected. Thus, a mixed autoregressive spatial moving average model, i.e. a model with a spatial lag dependent variable as well as a spatial moving

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Minimum distance estimation of the spatial panel autoregressive model

69

Table 7 Unrestricted maximum likelihood estimates (end) Variable

Cross-section estimates (and standard errors) 1993:1

1993:2

21.60

50.34

320.07

(317.1)

(322.7)

(350.2)

0.22

0.15

0.27

(0.4)

(0.4)

(0.4)

(0.5)

3.59

2.18

0.73

1.95

3.09

(1.1)

(0.9)

(0.9)

(0.7)

(0.7)

0.44

0.50

0.17

0.21

0.15

0.48

(0.3)

(0.2)

(0.3)

(0.2)

(0.3)

(0.2)

1.38

0.13

0.01

0.91

0.01

0.36

(0.6)

(0.3)

(0.4)

(0.3)

(0.2)

(0.3)

9.58

10.83

10.02

3.24

13.72

7.66

(8.2)

(8.5)

(7.6)

(12.1)

(9.3)

(8.2)

0.83

0.46

0.69

0.46

1.14

0.44

(0.6)

(0.6)

(0.5)

(0.5)

(0.5)

(0.6)

Proportion of employees

2.53

2.08

2.20

1.61

1.79

2.52

(0.7)

(0.6)

(0.6)

(0.6)

(0.6)

(0.6)

Proportion of unemployed

3.08

2.93

3.07

2.87

2.48

3.13

(0.7)

(0.6)

(0.6)

(0.5)

(0.6)

(0.6)

Index of equipment

0.02

0.13

0.16

0.01

0.55

0.20

(0.4)

(0.3)

(0.3)

(0.3)

(0.3)

(0.4)

Intercept Disposable income Price of water Price of electricity Rainfall Temperature Persons below 19 years

Density of population Spatial laga

1991:1

1991:2

1992:1

248.72

423.66

72.88

(348.1)

(330.1)

(346.5)

0.07

0.10

0.13

(0.6)

(0.5)

1.21 (1.2)

1992:2

0.09

0.45

0.11

0.19

0.01

0.52

(0.4)

(0.4)

(0.4)

(0.3)

(0.4)

(0.4)

0.38

0.13

0.48

0.26

0.23

0.28

(0.2)

(0.3)

(0.2)

(0.3)

(0.3)

(0.2)

Diagnostics testb LM error.c Spatial B-P.d

6.38

0.07

7.86

1.19

9.79

1.09

(0.0)

(0.7)

(0.0)

(0.2)

(0.0)

(0.2)

13.46

20.64

21.72

15.66

19.71

42.13

(0.1)

(0.0)

(0.0)

(0.0)

(0.0)

(0.0)

Number of observations a b

115

‘Spatial lag’ means the spatial lagged dependent variable, computed as Wy p-values are in parenthesis

c

Lagrange multiplier test for the spatial model, v2 (1)

d

Spatial Breusch-Pagan test for spatial heteroskedasticity, v2 (2)

average process in the error will be more appropriate. In other cross-sections the spatial dependence has been adequately dealt with. A spatial Breusch–Pagan test for spatial heteroskedasticity clearly indicates that heteroskedasticity patterns remain in the specification: the null of homoskedasticity is rejected for all waves, except 1988:2, 1989:2 and 1991:1.

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The conditioning characteristics of municipalities are: proportion of persons below 19 years, proportion of employees, proportion of unemployed, municipalities’ equipment and the density of population. Some of them (proportion of persons below 19 years, proportion of employees and proportion of unemployed) are highly significant in the unrestricted cross-sectional estimates. For instance the proportion of employees and the proportion of unemployed have a negative and significant effect except for the proportion of unemployed in 1989:2. The density of population is not significant and the equipment is significant only for 1988:2 with a negative sign. The intercept varies widely but is not significant. Particular attention must be paid to the spatially lagged dependent variable Wy. We observe that it is never significant, if really necessary, except for 1992:1. The results concerning economic variables are also modified. Income is never significant. This is maybe due to the joint conditioning on the proportion of employees and the proportion of unemployed. The average price of water has a negative effect on the residential demand for water, but it is significant only from 1991:2, 1992:1 and 1993:1. On the contrary, the average price of electricity has a positive impact on demand, but it is only significant for 1990:1. Table 8 reports the results from the minimum distance estimates. To check the validity of the specification g(.) = 0, the minimum distance tests are computed for each restriction. The computed values are 94.165 and 133.972, respectively, for the two restrictions. Given the associated degree of freedom, there is no rejection of the adopted specification both under fixed slopes and all identical parameters restrictions. For the first restriction, the estimated coefficients appear to be significant except for disposable income and the density of population variables. The other coefficients have the expected sign, except perhaps for the coefficient of the electricity price variable which is positive. This seems a priori, surprising. Indeed, although complementarity between the two commodities (water and electricity) may be expected, the positive sign for the parameter of the average price of electricity indicates that, for the sample concerned, water and electricity display substitutability patterns. This means that an increase in the average price of electricity may result in more water consumption by residential consumers. This a priori surprising result is in contradiction with the study of Hansen (1996) where the energy crossprice parameter is found to be negative. But, it should be noticed that the estimation of Hansen (1996) does not incorporate equipment variables in estimations. Also, our result may come close to the fact that the equipment has a negative effect. The cross-effects estimates indicate that changes in the average price of electricity may induce modifications in the distribution of residential water consumption for different uses. That is to say, the share of residential water consumed in connection with electricity may decrease with the price of electricity, whereas the remainder (the share of residential water consumed without energy) increases. We noticed earlier that about 50% of daily residential water consumption in France is used for heating. Hence, the remaining 50% may partly explain our result.

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71

Table 8 Minimum distance estimates Variable

Intercept Disposable income Price of water Price of electricity Rainfall Temperature Persons below 19 years

Restriction 1 (common slopes)

Restriction 2 (equality of parameters)

coef.

coef.

—

std.err

t-stat.

std.err

t-stat.

—

—

4.49

40.88

0.09

0.14

0.6

0.20

0.17

1.2

1.99

0.25

7.8

2.38

0.26

9.1

0.24

0.05

4.3

0.20

0.04

5.2

0.08

0.04

1.9

0.59

0.43

1.4

0.95

0.18

5.2

0.37

0.08

4.5

5.78

1.99

2.9

0.99

0.15

6.4

0.1

Proportion of employees

2.11

0.17

11.9

2.13

0.20

10.6

Proportion of unemployed

2.93

0.17

17.5

2.80

0.19

14.2

Index of equipment

0.22

0.09

2.3

0.27

0.12

2.3

Density of population

0.15

0.11

1.3

0.17

0.13

1.3

Spatial laga

0.27

0.08

3.4

0.28

0.09

3.2

v2(5%) distance

94.16

133.97

Degree of freedom

143

121

p-value

0.99

0.19

Number of obs. (N x T)

1380

All estimates are carried out with a significance level of 5% a

Spatial lag means the spatial lagged dependent variable, computed as Wy

For the second restriction, meteorological variables (rainfall and temperature) are no longer significant but are of the expected sign. The spatial coefficient is also highly significant, which confirms the modelling framework. Here, the spatial behavior may be viewed in two ways. First, we can argue that households actually influence their neighbours. Water consumption behavior of other households affects the consumption of a given household through social proximity. In this sense, the estimated spatial coefficients represent a direct measure of an externality. The significant spatial pattern may also be interpreted as the reaction of households with respect to the availability of water resources. The time varying spatial elasticities at means are computed as  X
kt ^ Ekt ¼ bk
; k ¼ 1; . . . ; K; ; t ¼ 1; . . . ; T: Yt This relation is termed ‘spatial elasticity’ as it makes use of the variation in variables of interest between municipalities. Results of the elasticities for the disposable income, the average price of water and the spatial lagged dependent variable are reported in Table 9. Although the coefficients (from Table 8) used in computing the elasticities are highly significant, these elasticities are very weak. They do not exceed 1% in absolute value. Elasticities related to the restriction of

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Table 9 Spatial elasticities at means Period

Restriction 1 (common slopes)

Restriction 2 (equality of parameters)

Income

Income

Price

Spatial var.*

Price

Spatial var.*

1988:1

0.038

0.180

0.265

0.085

0.215

0.282

1988:2

0.037

0.181

0.264

0.084

0.217

0.281

1989:1

0.036

0.185

0.264

0.083

0.221

0.282

1989:2

0.036

0.181

0.265

0.081

0.217

0.282

1990:1

0.038

0.192

0.265

0.086

0.229

0.283

1990:2

0.039

0.198

0.264

0.087

0.236

0.281

1991:1

0.038

0.204

0.265

0.086

0.243

0.282

1991:2

0.039

0.212

0.266

0.087

0.253

0.284

1992:1

0.042

0.241

0.266

0.093

0.287

0.284

1992:2

0.041

0.247

0.267

0.092

0.295

0.283

1993:1

0.042

0.276

0.266

0.095

0.330

0.284

1993:2

0.043

0.297

0.267

0.096

0.354

0.285

* Spatial means the spatially lagged dependent variable

complete equality of parameters are higher than those associated with the common slopes restriction. We also observe that all elasticities have been increasing in absolute value since 1991:1 and the price elasticity since 1989:2.

6 Conclusion Led by the idea that spatial autocorrelation reveals much about the prevalent dynamics of feedback effect and learning of an economic agent at one point in space, this paper investigated how individual decisions at other locations in the space appear to have common features. Similarity of consumption and/or many economic decision patterns are heuristically shown to have spatial dependence with enormous implications. In that context, this paper developed a new methodological tool to explain the spatial patterns of economic decisions with the help of designing a consistent and asymptotically efficient minimum distance estimator for the spatial autoregressive model. We applied this methodology to investigate spatial dynamics in the residential demand for drinking water. In particular, we attempted to answering the following concerns: (1) whether households look to a reference group when making decisions on water consumption and (2) what we learn about the residential demand for water. By designing a simple but powerful tool, we showed that spatial patterns matter for decisions about water consumption. The same can be said about similar economic problems and decision making. With respect to the previous concern, we showed that households living in the same geographic area have approximately similar water consumption behaviors. This result provides us with a measure of

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externality which is not usually observable. We also find evidence that consumers respond jointly to the average price of water and to the average price of electricity, although the related elasticities are weak. The methodology we developed has the advantage to not be computational demanding and can also be applied to very long historical data for which one may need some identification restriction for parameters of interest. A well known identification restriction following from the reduction of the time dimension is that of common vector parameters. Further empirical applications (in cliometrics) based on historical data with long-time horizon may deserve immediate attention. Acknowledgments I gratefully acknowledge Franc¸ois Laisney, two referees and the editor of this journal for comments leading to improvements in the paper. The usual disclaimer applies.

Appendix

Appendix A: Details on data collection In France, the municipalities in charge of water utilities can choose to delegate the management of the service of water or to manage it themselves. They still have the possibility of combining to ensure water management. There are three scheme of water management: ‘direct management’, ‘delegated management’ and ‘intermediate management’. In the case of a direct management, the service of water is ensured by the municipal agents. For a delegated management, the municipality entrusts water utilities to a private operator (called the farmer) and then perceives a contractual remuneration. There are two principal types of contracts specific to a delegated management: ‘concession’ and ‘leasing’. For the leasing contract, the municipality is in charge of water equipment (network, etc). The municipality remains the owner of this equipment, but temporarily entrusts them to the farmer. The latter is responsible for the maintenance and sometimes the renewal of operating expenses of the network. The intermediate management differs from the concession and the leasing by its legal principle. In such a situation, one distinguishes: the ‘management’ and the ‘interested control’. The manager is an ‘employee’ of the municipality and for this reason, he or she is remunerated according to a price determined at the time of the contract. In the interested control, the municipality keeps the financial responsibility of water utilities, the exploitation of which is entrusted to an external person or entity receiving benefits. The data we have collected comes from different sources. They were provided by: ‘Compagnie Ge´ne´rale des Eaux, Direction Re´gionale Est’, ‘Direction Ge´ne´rale des Impoˆts de la Moselle’ (Regional Tax Center), ‘Centre De´partemental de la Me´te´orologie de la Moselle’ (Regional Center of Meteorological Studies) and ‘Institut National de la Statistique et des E´tudes E´conomiques’. In the following, we provide more details on data and the way some of the variables used in the empirical analysis are computed.

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A1. Consumption of drinking water The consumption of drinking water is measured in cubic meter per municipality. These measurements are thus aggregate observations. Some municipalities are much longer than others, mostly rural. Thus, in order to consider observations which are not too heterogeneous and to reduce the size effect, we have divided each observation of consumption by the total number of households in the municipality in 1990. The basic year 1990 was selected because it is the year of the last communal inventory available. It also constitutes the last period of the general census of the population. The demand for drinking water is defined as the quantity of water taken from natural environment at each moment to meet specific needs, taking into account water losses.14 Thus, the consumption statistics produced in Table 1 include the losses which have occurred on the network and those the supplier distributes on the effective consumption of domestic users. The losses at the subscriber level correspond to the leak in the network as well as the leak at the points of distributions of which 80% come from the water closet, Valiron (1994).15 From Table 1, we observe that the average values of water consumption are stable, about 70 m3 in the first half of the year 1988, and 76 in the first half of the year 1991. Meanwhile, the average consumption in the first half of 1988 is the lowest over all periods, that is to say 69.68 m3 with a standard deviation of 27.75. The highest standard deviation is 29.26 and corresponds to the second half of 1993. In contrast, one can expect that the consumption in the first half of 1988 is one of the highest with respect to the average price of water. The statistics of 1992 suggest opposite indications. This is guided by the fact that the year 1992 is the year of the application of the accounting instrument (directive) ‘M-49’ which involved a substantial increase in prices. Indeed, if the average price of water regularly increased over the whole period of study, it is noticed however that the largest increases occur from 1992 on. A2. Price of water The tariff modulation used by the farmer is decided during the making of the leasing contract between this latter and the local authority. This contract is termed a ‘treaty’. We then face a ‘delegation’ mode of the public utility of drinking water distribution. The contract establishes the clauses governing the tariffs, their distribution as well 14 The demand for water is defined as the quantity of water necessary to a specific use, for example for a shower or a cycle of a washing machine. The variation between the demand and the need is thus a function of the effectiveness of the distribution network to reduce the losses on the chain of supply. This variation also depends on the adjustment of the internal device placed at the disposal of each user. The maintenance of the internal devices holds with a subjective criterion which is the perception of consumers about water. 15

All of the escapes from the adductions and the network varies according to the material (pipes), their antiquity, the nature of the ground and the quality of maintenance. It is considered difficult to reduce the losses to less than 10% , even for a new network. 15% of losses characterize a network in good condition with good maintenance.

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Minimum distance estimation of the spatial panel autoregressive model

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as the principle of their variation over time. A ‘treaty’ can cover one or more municipalities. In this last case, the same tariff applies in all the municipalities concerned. The tariff scheme consists of a two-part tariff: a fixed part (FC) which corresponds to the charges (or service charge, it does not vary with the consumption) and a part which is the marginal price of water (MP) per units measured in cubic meter. The service charge corresponds to investments and the fixed charges of exploitation, i.e., the maintenance of connection to the network and the maintenance of the meter. The variable part consists of the farm price, other taxes, the royalty, the VAT, etc. The variable part of the price of water is divided in the following way: 46% for the distribution of water, 33% for the collection and treatment of used, 14.5% of water agencies, 5.5% of VAT, 1% on behalf of the ‘Fonds National pour le De´veloppement des Adductions d’Eau’, (FNDAE, National Funds for the Development of the Water Conveyance). This last royalty usually serves as development assistance for the rural networks. The part of the fixed charges in the total price proves very important. It can reach 90% of the total. From this tariff information, we compute the average price of water for each time period (AP) for each municipality as AP ¼

ðQ  MPÞ þ FC Q

where Q indicates the domestic volume of water consumed in a given municipality. As a result, we are able to avoid the theoretical and estimation problems of more complex rate structures that qualify many other studies in the field. During the period of study, some legislative texts came into effect and had major effects on the price of water. Most important is the law of January 3th, 1992, which lays down new rules of organization and management of the public services of water distribution. This regulation aims to reveal the truth and transparency of the water prices. Two new principles thus appeared. The first is concerned with the removal of tariffing all-in price. It stipulates that the price of water corresponding to the service of distribution of drinking water must be calculated starting from the consumed volume. It is possible however to impose a binary tariff comprising a fixed part and a variable part. This provision is that which was adopted by the water supplier. It clearly underlines the choice to make the consumer pay rather than the taxpayer. There is nevertheless an exemption from this principle.16 The exemption is adopted by municipalities located in tourists zones where the population shows a strong seasonal variation. It should be noted that some municipalities circumvent the regulation and charge very highly for the fixed part, which in fact means they practice an all-in price tariffing.

16 The exemption appears in the second subparagraph of article 13-II and the decree No 93-1347 of December 28, 1993. It makes it possible to the prefect (chief administrator in a department) to authorize tariffing all-in price under two conditions: firstly if the water resource of the municipality is abundant and the number of inhabitants lower than 1,000; secondly when a municipality is subjected to important demographic changes. It is this last provision which targets the tourist zones.

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The second principle is the accounting instruction ‘M-49’. Founded on November 10th, 1992, this accounting instrument has as main objective to clarify the accounts of water utilities (water distribution and cleansing) by aligning their operation on the chart of accounts of 1982. By doing so, it restores the truth of the prices while making it possible to better identify costs inherent in the service and to define suitable tariffs. With this intention, the accounting instruction ‘M-49’ imposes the balance of budget on the water services (distribution and cleansing), as well as their individualization in an additional budget. It is a question of applying the accounting policy according to which ‘water pays water’. The management of the service of water in an independent budget implies a balance between receipts and expenditure (including the amortizing of the investments). Thus, the municipalities cannot make any more support by their general budget, i.e. by the tax, the expenditure intended for water.17 These various regulations in particular the accounting instruction ‘M-49’ resulted in a strong increase of prices of water and sometimes even a modification of the tariff distribution as can be observed in Fig. 2. A3. Fiscality of municipalities: income variable In the documents we exploited, the data are not provided for municipalities having less than ten taxpayers. Records available are the following. – –

–

–

Total number of taxable households (A): taxable persons or those who are given a partial restitution of taxes. Total number of non taxable households (B): non taxable persons including those who are given a total restitution of taxes (this concerns roughly a tax lower than 380 FF or free tax lower than 80 FF). Taxable or non taxable income (C): taxable free income of taxpayers including the appreciations and the exceptional agricultural benefit taxed according to the mode with the quotient counted for the 5/5, including the appreciations taxed according to an exceptional rate. The amount of the total deficits is not deduced from the total of the incomes. It should be noted that the taxable incomes underestimate the actual household incomes by 30% on average. The underestimation is very important for non-farmer-free-workers (about 43%) and still more for the farmers (57%). This partly explains the low level of the disposable incomes especially in the rural municipalities. Free tax (D): free tax on the income to be paid including the tax on the proportional appreciations. The amount of the partial restitutions is not deduced from the total of the tax. The social contribution (0.4 %) is not taken into account in the statistics.

17 Other legislative texts supplement this device. Particularly, they relate on the information to the consumers and the contractual relationship between municipalities and farmers. The Barnier law of the 02-02-1995 provides in particular that the mayor or the president of the inter-municipality presents at his assembly an annual report on the price and quality of the service of water. This report must include the details of the tariffs and their methods of variation, as well as a standard invoice for a consumption of 120 m3. The laws Sapin of the 29-01-1993 and Mazeaud of the 08-02-95 supplement this regulation.

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Based on this information, we compute the following quantities for each municipality: – – –

the total number of taxable households: NTFF: = (A) + (B), the disposable income RD:=(C)  (D), the average disposable income RMD: = RD/NTFF.

For each municipality and per year, the average disposable income by taxed household is divided by two to obtain biannual observations, assuming that the income is stable over time. It is this variable (a proxy variable of the average of disposable income of municipalities by household) measured in thousands of French Francs which is used in the empirical estimation. A4. Weather variables The data of the Departmental Center of the Meteorology of the ‘Moselle’ provides indications on the environmental factors. Contrary to the preceding data, those were not raised by the 115 municipalities, but starting from ‘meteorological stations’. A station is a point of observation which covers several municipalities. The rough data are monthly observations which are aggregated to form biannual values. At the beginning we have monthly rainfall at 12 stations, monthly averages of minimum and maximum temperatures at 7 stations, monthly sunstroke in Metz (1 station), monthly averages of minimal and maximum moisture in Metz (1 station).18 For the 115 municipalities considered, we then determine the thermometric and pluviometric station of reference. Rainfall is measured at 6.00 AM, universal time (UT). A height of water of 1 mm is equivalent to 1 l per square meter or 10 m3 by hectare.19 The temperatures indicated are taken with the shelter of 1.50 m groundlevel above. Rainfall is measured in 100 mm, of the temperatures maximum minima expressed of 108C. A5. Characteristic variables The data describing the characteristics of the municipalities are provided by INSEE. They come from the communal inventory and the general census of the population. The general census of the population was carried out in Metropolitan France and in the overseas departments in March 1990. The listed population is the whole population resident in France, whatever their nationality. From these documents, we compute some variables gathering four types of information per municipality: the demographic structure of the population by age, the density of population, the proportion of employees and the index of equipment. Certain components were aggregated to obtain a more adequate representation. The demographic structure of the population gathers the proportion of people of age 18

In meteorological terminology, sunstroke is defined as the duration of sun appearance.

19

Is considered as day of rainfall any day when the quantity of water collected in the pluviometer corresponds to a height of at least 0.1 mm. The fogs and condensations involving a water deposit in the pluviometer are not regarded as rainfall.

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ranging between 00 and 19 years. They are calculated as (NTH + NTF)/PT, where NTH indicates the total number of men of age ranging between 00 and 19 years, NTF the total number of women of age ranging between 00 and 19 years, PT the total population. The age indicated is the age in completed years at December, 31st of the year of census. The density of the population of the households (DENSM) per municipality is calculated as DENSM = (NTM)/(SURF), where NTM is the total number of households and SURF denotes the surface of the municipality measured in hectares. A household is defined as occupants of the same house or flat, whatever the bonds between them. A household can be only one person. It also includes the people who have their personal residence in the house but who remain at the time of the census in certain establishments known as ‘population counted separately’ (boarders of educational establishments and national service) which are thus ‘reinstated’ in the population of the households of the municipality where they have their personal residence. Information regarding employment take into account the percentage of employees (TXS) and the percentage of the unemployed (TXCHM). These two variables are calculated as TXS = (NTS/PT) x 100, with NTS being the total number of employees, and TXCHM = (NTC/PTA) x 100, with NTC indicating the total number of unemployed and PTA the active total population. The ‘unemployed’ are defined as declared people ‘unemployed’ or without ‘employment’ except if they explicitly stated they are not seeking a job. The mothers and ‘housewives’ who explicitly stated to seek a job are also classified in this heading. The working population includes people having an employment and the unemployed. They are at least 15 years old. The working population having an employment are people who have a profession and pursue it at the time of the census. They are classified in this heading as people who help a member of their family in their work (for instance on a farm, a trade, liberal profession, etc.) provided that the helped person is not paid for. The apprentices, except if they are pupils of a technical school, the people under contract of adaptation or qualification, temporary worker, the remunerated trainees which work in a company or in a center of formation, young employees in community work, ‘Travaux d’Utilite´ Publique’ (TUC) and people who, while continuing their studies, carry on a professional activity, form also part of the working population having an employment. To compute the index of equipment, we sum the numbers of various health facilities and divide it by the total population to obtain an index of the level of health facilities as TXES = (NTW + NTB + NTD) x 100/PT, where NTW indicates the total number of toilets, NTB the total number of bath-tubs and NTD the total number of showers.

Appendix B: Spatial arrangement of data The computation of the binary spatial weighting matrix W is based on information regarding the distance between municipalities. First, a matrix of distances D with elements dij based upon latitude–longitude coordinates of the centroids from each

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Table 10 Characteristics of the distance matrix Variables

Statistics

Dimension (number of points)

115

Average distance between points

28.665

Minimum distance between points

1

Maximum distance between points

86.135

Quartiles First

13.317

Median

29.273

Third

41.641

Minimum allowable distance cutoff

5.362

Statistics were computed using SpaceStat (Upgrade 1.90) of Luc Anselin

municipality is computed using the Euclidean metric. Characteristics of the distance matrix are summarized in Table 10. In a second step, the information in the distance matrix is used to create a row-standardized spatial weighting matrix W whose elements xij are defined as follows:  xij ¼

1 0

if

dij 2 ½d1 ; d2  otherwise



where [d1 ; d2] is a specified critical distance band. Here we do not have any prior notion of which distance ranges are meaningful. Hence, we choose a statistically meaningful one, i.e. the first and third quartiles: d1 = 13.3 km and d2 = 41.6 km. The reason for using such a construction and not the usual common border criterion is that the municipalities considered here are not all contiguous. The list of the municipalities is given in Table 11 which provides the X–Y centroids from which the distance between municipalities have been computed. The first column gives the name of the municipalities. The second and the third report, respectively, the Xcentroid and the Y-centroid (Table 12). B1. G-statistics For further technical details and discussions on the G-statistics and spatial correlograms, see e.g. Cliff and Ord (1981), Cressie (1991) and Getis and Ord (1992). For a cutoff distance d, the G-statistic, denoted G(d) is defined as P P i j wij ðdÞzi zj GðdÞ ¼ P P ; i ¼ 1; . . . ; N; j 2 JðiÞ; i j zi zj where zi is the value observed at location i, wij(d) stands for an element of the symmetric (unstandardized) spatial weighting matrix for distance d and J(i) is the set of neighbours to i. Inferences on G(d) are typically based on a standardized

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Table 11 List of municipalities and corresponding X–Y centroids Municipality

X-centroid

Y-centroid

Municipality

X-centroid

Y-centroid

Amelecourt

904,000

2,435,100

Hundling

939,200

2466,000

Arry

872,100

2,450,200

Hunting

889,300

2498,000

Audun le Tiche

862,600

2,502,000

Ippling

940,700

2466,300

Behren les Forbach

935,700

2,472,900

Kappelkinger

934,600

2451,200

Berg sur Moselle

888,200

2,500,200

Kemplich

895,100

2488,400

Bertrange

880,900

2,486,700

Kerbach

938,200

2472,400

Betting les St Avold

926,700

2,468,300

Kerling lesSierck

893,000

2496,200

Bitche

973,600

2,461,600

Kirsch lesSierck

895,100

2500,900

Blies les Ebersing

950,800

2,467,900

Kirschnaumen

897,200

2497,500

Blies Guersviller

947,000

2,470,900

Laudrefang

914,600

2462,300

Bousbach

936,200

2,470,300

Laumesfeld

897,900

2493,500

Bousse

881,300

2,482,000

Liederschiedt

976,200

2470,500

Boust

879,500

2,499,800

Lubercourt

906,600

2434,800

Brehain

908,300

2,442,400

Marthille

909,400

2444,900

Briestroff la Grande

882,200

2,501,800

Metzing

937,500

2465,300

Bulding

891,000

2,490,600

Monneren

895,900

2490,900

Burlioncourt

911,000

2,437,400

Morville les Vic

909,000

2432,300

Carling

919,200

2,471,900

Moyeuvre Grande

869,400

2479,700

Cattenom

882,900

2,497,800

Moyeuvre Petite

867,900

2481,900

Chateau Brehain

906,500

2,441,500

Neufrange

945,400

2463,400

Chateau Salins

906,200

2,432,400

Nousseviller St Nabor

938,100

2,468,900

Chicourt

905,400

2,443,600

Oeting

933,300

2,473,300

Cocheren

929,700

2,469,800

Oudrenne

891,200

2,492,800

Cornysur Moselle

872,900

2,454,500

Petite Rosselle

929,500

2,477,500

Crehange

909,700

2,457,900

Puttelange aux Lacs

934,800

2,460,300

Dalhain

910,200

2,440,200

Puttigny

909,100

2,435,600

Diebling

935,800

2,466,100

Remelfing

946,700

2,464,900

Ernestviller

938,600

2,461,300

Remeling

900,200

2,499,200

Etzling

936,800

2,474,200

Remering les Puttelange

936,100

2,457,700

Evrange

879,200

2,507,400

Basse Rentgen

880,300

2,505,600

Fameck

874,100

2,484,700

Rettel

890,000

2,500,600

Farebersviller

930,400

2,467,000

Richeling

938,600

2,457,900

Faulquemont

911,600

2,456,900

Roppeviller

978,300

2,467,600

Fixem

885,500

2,501,000

Rosbruck

929,500

2,471,600

Fletrange

908,600

2,460,900

Rosselange

871,000

2,480,200

Florange

875,400

2,487,300

Rouhling

940,300

2,469,100

Folking

933,500

2,471,000

Roussy le Village

878,400

2,502,600

Folschviller

917,400

2,460,900

Rurange les Thionville

883,900

2,481,700

Forbach

931,600

2,475,200

Russange

861,700

2,504,900

Frauenberg

948,800

2,470,100

St Jean Rohrbach

932,300

2,456,900

Fresnes en Saulnois

901,000

2,434,600

Sarrable

941,700

2,454,400

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Table 11 continued Municipality

X-centroid

Y-centroid

Municipality

X-centroid

Y-centroid

Freyming Merlebach

925,300

2,470,400

Sarreguemines

946,200

2,466,800

Gavisse

886,600

2,500,100

Sarreinsming

949,400

2,465,500

Grindorff

903,100

2,495,000

Seremange Erzange

873,100

2,487,100

Grundviller

940,500

2,460,100

Sierck les Bains

891,700

2,500,700

Guebenhouse

936,800

2,462,500

Spicheren

937,700

2,476,200

Le Val de Gueblange

938,300

2,452,100

Stiring Wendel

935,100

2,477,200

Guenange

881,400

2,484,700

Tenteling

934,900

2,467,700

Hagen

877,600

2,506,900

Teting sur Nied

916,600

2,458,500

Halstroff

900,600

2,495500

Theting

932,200

2,468,600

Hambach

943,400

2,460,400

Valmont

918,900

2,462,600

Hampont

911,600

2,434,400

Vannecourt

908,000

2,439,300

Hazembourg

936,400

2,450,500

Vaxy

907,300

2,437,300

Hilsprich

934,500

2,454,700

Veckring

892,800

2,489,700

Holving

938,600

2,455,400

Waldweistroff

902,100

2,492,200

Hombourg Haut

924,000

2,468,100

Willerwald

943,900

2,457,400

Hopital

920,700

2,471,200

Woustviller

940,900

2,463,400

Wittring

950,400

2,460,800

Source ‘Institut National de Ge´ographie (IGN)’

Table 12 Water-using tasks (share of various uses in residential water consumption in France. About 50% is concerned with heating) Water consuming tasks Drink Cooking (heating) Dish washing (heating)

Proportion (%) 1 6 10

Clothes washing (heating)

12

Toilets

39

Personalhygiene (heating)

20

Outdoor use (including sprinkling)

6

Other uses

6

Source ‘Compagnie Ge´ne´rale des Eaux’

t-value tG = { G(d)  E[G(d)] } /SD[G(d)], with the notation SD denoting the standard deviation. Based on asymptotic considerations, the t-value follows a standard normal distribution. For each observation i, the G*i (d) statistic for a specific spatial association indicates the extent to which that location is surrounded by high values or low values of the variable of interest. For a given distance d

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P 

G ðdÞ ¼

wij ðdÞzj P : j zj

j

Inference about the significance of G*(d) is derived as for G(d). B2. Spatial correlograms Consider a system of n sites with random variables x1 , ... , xn and let the sites i and j be d-order neighbours. Then, the d-order sample spatial autocorrelation is given by CðdÞ ¼

N z0 Wz ; DðdÞ z0 z

where z0 ¼ ðz1 ; . . . ; zn Þ; zi ¼ xi  x
; i ¼ 1; . . . ; N; and DðdÞ ¼ natively, this statistic may be rewritten as N CðdÞ ¼ DðdÞ

P

P P i

j

wij ðdÞ: Alter-

wij ðdÞzi zj P 2 : i zi

ði;jÞ

Observe that the symmetric form of W in the statistic means that each term appears twice in the find sum. Readers are referred to the literature mentioned above for the computation of the means and the variances of these measures. The plot of C(d) against d yields the spatial correlogram.

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