econometric modeling and analysis of residential ... -...

F. Carlevaro, C. Schlesser,

M.-E. Binet, S. Durand, M. Paul

Econometric Modeling and Analysis of ResidentialWater Demand Based on Unbalanced Panel Data

This paper develops an econometric methodology devised to analyze a sample of time unba-

lanced panel data on residential water consumption in the French island La Reunion with the

purpose to bring out the main determinants of household water consumption and estimate the

importance of water consumption by uses. For this purpose, we specify a daily panel econometric

model and derive, by performing a time aggregation, a general linear regression model accoun-

ting for water consumption data recorded on periods of any calendar date and time length. To

estimate efficiently the parameters of this model we develop a feasible two step generalized least

square method. Using the principle of best linear unbiaised prediction, we finally develop an ap-

proach allowing to consistenly break down the volume of water consumption recorded on hou-

sehold water bills by uses, namely by enforcing this estimated decomposition to add up to the

observed total. The application of this methodology to a sample of 437 unbalanced panel obser-

vations shows the scope of this approach for the empirical analysis of actual data.

1. Introduction

Astudy carried on in 1997 in the French island La Re únion, brought out an over-consumpti-on of water, namely a consumption of 246 liters/day/inhabitant against 145 liters/day/inhabitant in continental France.

In order to explain this global figure, the Regional Directorate for the Environment (DIREN)commissioned the «Centre de recherches e ćonomiques et sociales de l’Universite ´ de La Re únion»(CERESUR) to carry out a study aiming at:

� measuring domestic water consumption, stricto sensu, in order to assess the quantitativeimportance of the over-consumption of water;

� explaining the determinants of water consumption for the residential sector, in particular,the role played by the uses, the equipments, and the behaviour of individuals, in the over-con-sumption of water, in order to develop policy measures of rational use of water resource, allowingto reduce waste.

To achieve these goals, a stratified random survey was designed and conducted in 2004 ona sample of 2000 households, in order to collect the necessary information to perform theseanalyses. Carried out by telephone, this first survey was completed with a mailing to 1000 volun-teer households, intended to collect the volume of water consumption displayed in the last threebills. Unfortunately, this mail survey provided only 173 reliable responses.

Our study follows the first analysis of the data of this survey carried out by CERESUR’s team incharge of the DIREN study [Binet et al. (2005)]. It intends to go deeper in the analysis of this

� Российско�швейцарский семинар по эконометрике и статистике81

information by developing a microeconometric model allowing to bring out the main climatic,demographic, economic, and technological determinants of household water consumption andto estimate the share of each type of use in the total volume of water consumed by households.

Our presentation is organized as follows. In Section 2, we specify a causal econometric modeldescribing the daily water consumption of a panel of households according to the distinct uses ofwater they make at home. The impact of unobserved factors of heterogeneity in householdbehaviour is taken into account by means of an error component structure of model disturbances.In Section 3 we derive, from this model, a conventional general linear regression model explai-ning the available time unbalanced panel data for empirical analysis. Based on this conventionalregression model, we develop, in Section 4, an efficient two step feasible generalized least squareestimator for the model parameters. Using this model, we analyse, in Section 5, the DIREN surveydata in order to bring out the determinants of two broad categories of home water uses, namely«essential water uses» and «leisure water uses», respectively1. Finally, we develop, in the last Sec-tion 6, an efficient method to decompose the observed volumes of water consumption accor-ding to these two categories of uses and we analyse the results of its empirical application to oursample of observations. To conclude we comment on the scope of our methodology for the em-pirical analysis of actual data and outline the most promising developments we plan to carry out.

2. Modeling Daily Water Consumption

Our model decomposes the total daily water consumption in a given housing, Y ti ( ), where i isthe index of a housing and t that of the day, according to h H� 1, . . . , distinct uses of water ina home. This decomposition is expressed by the following identity

Y t Y ti ih

h

H

( ) ( ),��

�1

(1)

where Y tih ( ) stands for the quantity of water consumption in housing i, during day t, for the use of

type h.

In turn, we decompose water consumption per use according to the following identity

Y t Q t Nih

ih

ih( ) ( ) ,� (2)

where Q tih ( ) expresses a daily water consumption per user and N i

h the size of water users’population (individuals or equipments) for use h in housing i.

From identities (1) and (2) we infer a causal econometric model explaining variable Y ti ( ),by modeling the unobserved variable Q ti

h ( ) according to the linear error components model,namely

Q t x t e e tih

ih h

ih

ih( ) ( )' ( ),� � �� (3)

wherex ti

h ( ) � vector of K h exogenous explanatory factors,� h � vector of K h impact parameters on Q ti

h ( ), for explanatory factors x tih ( ),

e ih � random disturbance, expressing the households’ heterogeneity with respect to water

consumption per user for use of type h,

82Российско�швейцарский семинар по эконометрике и статистике �

EconometricModelingandAnalysisofResidentialWaterDem

andBased

onUnbalancedPanelData

1 Empirical results are drawn out of a Master thesis of [Schlesser (2006)].

e tih ( ) � random disturbance, expressing the impact of unspecified factors influencing water

consumption per user for use of type h during day t.

Inserting (3) into (2), and the result into (1), we obtain a linear regression model

Y t x t ti i i( ) ( ),� � �( ) � 1 (4)

where x t x t Ni ih

ih

h Hh

h H( ) [ ] [� �� ( ) ,, . . . , , . . . ,]1 1� �

and 1 i ih

ih

h

H

iht e e t N( ) ( ) .( )� �

��

1

(5)

The moments of the random disturbances 1 i t( ), derived from the moments of the jointdistribution of random vectors e ei i

hh H� �[ ] ,.. . ,1 and e t e ti i

hh H( ) [� �( )] ,.. . ,1 , are

� centered,E e E e ti i( ) ( ) ;( )� � 0 (6)

� homoscedastic,V e S V e ti i( and ;() ( ))� � 2 (7)

� non-autocorreled,E e t e ti j( )( ) ( ) ;3 3� � �0, (8)

� non-intercorreleted,

E e e E e t e t i j E e e ti j i j i j( ( ) ( ) and ( )( ) ( )� � � � � � �) , .0 0 (9)

Expressing disturbances 1 i t( )as a linear combination of random vectors e i and e ti ( ), namely as

1 i i i it n e e t( ) ( )� � �( ) , (10)

where n Ni ih

h H� �[ ] ,, . . . ,1

we conclude that these disturbances are

� centered,E ti( )( )1 � 0; (11)

� heteroscedastic,

V t n S n si i i i i( ) ) 2 21 �( ) (� � � � �2 ,

where s n Sni ii2 � � and � i

2 � �n ni i2 ; (12)

� equicorreleted (for a given housing),

Cov , ;( )1 1 3 3i i i it n Sn t( ); ( ) � � � (13)

� non-intercorreleted,E t i ji j( )( ) ( ) .1 1 3 � �0, (14)

Heteroscedasticity (12) follows from the variability of size ni across the population of waterusers but does not change in time. Similarly, the correlation between the terms of time series


F.Carlevaro,C.Schlesser,M.�E.Binet,S.Durand,M.Paul

disturbances 1 i t( ), which is constant (equicorrelation) for a given household, varies across house-holds as a function of the size ni of the population of users. As far as the absence of correlation ac-ross the terms of two distinct disturbances times series is concerned, 1 1i jt t i j( ) ( ),and ,� it is theconsequence of assumptions (8) and (9).

3. Modeling Observed Water Consumption

Residential water consumption from DIREN’s survey is not observed on a daily base, but fordurations and periods of time that change across the surveyed households.

Let 4 � �{ , , . . . , }t t t0 0 11 be a period of observation, where t 0 and t1 are the first and last day ofthe period, respectively. The water consumption of housing i observed during such a period is, bydefinition,

Y Y ti i

t

( ) ( ).44

�� (15)

The econometric model explaining such an observation can be derived from the daily econo-metric model (4)–(5) by performing a time-aggregation of this model over the observation period4, which leads to

Y xi i i( ) ( ),4 4 4� � �( ) � 1 (16)

wherex x t x N x x ti i

t

ih

ih

h H ih

ih( ) ( ) ( ) , ( ) ( )[ ] , . . . ,4 4 4

4

� � ��

�� 1

t��

4

(17)

and1 1i i

t

i i i it n d e e d t t e( ) ( ) ( ) ( ) , ( ) (( )4 4 4 44

� � � � � � �� 1 0 1, 4

4

) ( ).�� e ti

t

(18)

Note that if an exogeneous factor x tihk ( )does not vary during the observation period’s 4, i.e. if

x t xihk

ihk( ) � ,

we obtainx d xi

hkihk( )4 4� ( ) .

To compute the moments of random disturbances 1 i ( )4 , it is useful to first compute those ofrandom vectors e i ( )4 . Using moments (6) to (9), we derive

E e E e ti i

t

( ) ( )( ) ( ) ,44

� �� 0 (19)

V e E e t e V e t di

t

i i i

t

( ) ( ) ( )( ) ( ) ( ) ( )4 444 4

2� � � �� 3

3 ( ) ; (20)

Cov( ) ( ) ( )( ); ( *) ( ) ( ) ( )*

e e E e t e V e ti i

t

i i i4 444

� � ��3

3t

d� 5� � 54 4

4 4 2*

*( ) ; (21)

Cov if( ) ( )( ); ( *) ( ) ( )*

e e E e t e i ji j

t

i j4 444

� � � ��3

3 0 ; (22)

Cov if .( ) ( ); ( ) ( )e e E e e t i ji j i j

t

44

� � � �� 0 (23)

From these moments, we easily derive those of the random disturbances 1 i ( )4 of the model(16), namely



andBased


E n d E e E ei i i i( ) ( )( ) ( ) ( )( )1 4 4 4� � � �( ) 0; (24)

V n V d e e n n d S d ni i i i i i i( ) ( )( ) ( ) ( ) ( )[ ]1 4 4 4 4 4 2� � � � � �( ) 2 ; (25)

Cov Cov ( )( ) (( ); ( *) ( ) ; ( *) (1 1i i i i i i in d e e d e e4 4 4 4 4 4� � � � *) ( ) ( *) ( *)) [ ]n n d d S d ni i i� � � 54 4 4 4 2 ; (26)

Cov Cov ( )( ) (( ); ( *) ( ) ; ( *) (1 1i j i i i j jn d e e d e e4 4 4 4 4 4� � � � *) , .)n i ji � �0 (27)

Compared to random disturbances of the daily water consumption model (4), those of the ob-served water consumption model (16) are subject to a new source of heteroscedasticity and auto-correlation, following from the possible variability of the duration of observation periods d( )4 andd( *)4 .

4. Generalized Least Squares Estimation

Relation (16) leads to a generalized linear regression model allowing estimating the parame-ters � for the DIREN’s sample of observation, denoted by

Y Y x x t T i mit i it it i it i� � � �( ) ( ) ,..., ,..., ,4 4, , ,1 1 (28)

where 4 it it T, ,� 1, . . . , refer to the available disjoint periods of observation ( )4 4it i t5 � 6 �3 3, forhousing i.

The observations of this unbalanced panel are explained by the generalized linear regressionmodel

Y x t T i mit it it i� � � � �� 1 , ,1 1,..., ,..., , (29)

where random disturbances 1 1it i it� (4 ) are

� centered,E it( )1 � 0, (30)

� heteroscedastic,E d s dit i it it( )1 �2 2� �2

i2 , where d dit it� (4 ), (31)

� serially correleted (for a given housing),

E s d d tit i it i( ) ,i21 1 33 3� � , (32)

� uncorreleted (accross housings),

E i jit j( ) 0,1 1 3 � � . (33)

To write this model in the classical matrix form, we first specify the models explaining theavailable individual housing time series of observations. These models are written as

y X i mi i i� � �� 1 , 1,..., , (34)

wherey Yi it t Ti

� �[ ] , . . . ,1 , X xi it t Ti� � �[ ] , . . . ,1 , 1 1i it t Ti

� �[ ] , . . . ,1 .

Moments of random disturbances 1 i may be written in the matrix form:

E E D s d d E i ji i i i i i i i i i j( ) ( ) ( ) ,1 1 1 � 1 1� � � � � � � � �0, ,2 2 07 , (35)

where d di it t T i� �[ ] ,.. . ,1 and D di i� diag( ).



Finally, we derive a compact matrix form of a generalized linear model for the whole observa-tions panel by stacking the m housing time series models

y X� �� 1, (36)

where y y i i m� �[ ] , .. . ,1 , X X i i m� �[ ] , .. . ,1 , 1 1� �[ ] , .. . ,i i m1 , E( )1 � 0and

E m( ) ) .11� � 8 8 �diag(7 7 71 � (37)

The BLUE estimator for � of this model is the GLS estimator

� )� GLS i i i

i

m

iX X X y X X X� � � � �

��

�

��

�

�

�( 7 7 71 1 1 1

1

1

7 i i

i

m

y�

�� 1

1

. (38)

It may be computed by using an analytic expression of 7 i�1 obtained through the application

of the inversion formula:

( M MM M

M� � � �

�

� ��

� �

�9�

9�

9 �) 1 1

1 1

11,

where M D s d di i i i i� � �� 9 �2 2, , .

This formula may be written as

7 i

i

ii

i i i

i iDs

s

� ��

�

��

�

��1

2

12

2 2

1

� � ': : , (39)

where : i indicates the unit vector with Ti components and ' :i i i it

t

T

d di

� � ��

1

denotes the totalduration of observation (in days) of water consumption in housing i.

By transforming the matrices and the vectors of observations as follows:

� �X D X

dx y D y

Yi i i

i

it

t T

i i ii

i

� � �)

*+

,

-/ � �

�

�

�1

2

1

1

21

,. . . ,

, t

i t Td

i

)

*+

,

-/

�1, . . . ,

; (40)

~ ~x X x y y Yi i i it

t

T

i i i it

t

Ti i

� � �)

*+

,

-/ � � �

� �� : :

1 1

, ; (41)

we can compute matrices and vectors in formula (38) directly from these transformed data:

� � � ��

�

��

�

�� X X X X

s

sx x Mi i i

i

i ii

i i i

i i i7 1

2

2

2 2

1

� � '

� � ~ ~ ; (42)

� � � ��

�

��

�

�� X y X y

s

sx y mi i i

i

i ii

i i i

i i i7 1

2

2

2 2

1

� � '

� � ~ ~ . (43)

Finally, we derive the simplifed GLS estimation formula:

�� GLS � �M m1 , (44)

where M Mi

i

m

��

1

and m mi

i

m

��

1

.



andBased


In practice the variance-covariance matrices S and 2 are not known, and, therefore, our GLSestimator is not feasible. To develop a feasible GLS estimator we rely on a two step method, startingwith a consistent estimation of these matrices, �S and �2, using the ordinary least squares residualsof the model (36), followed by a GLS estimation of � based on the previous estimation of S and 2.

According to a suggestion of [Vonesh and Chinchilli (1997)], consistent estimators of S and 2can be obtained by using the following two steps procedure.

� In the first step, we look for consistent estimators of S and 2, assuming � to be known. Suchestimators, denoted by �( )S � and � ( )2 � , may be derived by applying OLS to the following linear re-gression model with respect to the vector ; of the independent components of S and 2

1 � 1 � ' < ;3 3 3 3it i it i i i t it i i it itd d n Sn d n n z( () ) � � � � � � �2 2 � � � �< 33it it T i m, 1 1,..., , ,..., , (45)

where 1 � �it it itY x( ) � � � , ' 3t is the «Kronecker delta» (1 if t � 3, 0 otherwise) and <it a centered ran-dom disturbance. For the true value of �, the OLS estimator �( ); � is consistent for ;.

� In the second step, we estimate the dependent variables of the model (45) using the resi-duals of an OLS estimate of the model (29), namely

� �1 �it it itY x� � �MCO , (46)

where � ) .�MCO (� � ��X X X y1

Then we estimate parameters ; of this model by OLS, which provides consistent estimates,� �( � ); ; �� MCO , for the elements of matrices S et 2.

This FGLS estimator for � is consistent and asymptotically distributed according to the normaldistribution

NT

X X� ; )1 1 1( �

��

�� 7 , (47)

where T Ti

i

m

��

1

.

Therefore, statistical inference about parameters � can be based on this asymptotic distributi-on, using the following consistent estimator for the asymptotic variance-covariance matrix of�� FGLS

� ( � ) ( ' � )VT

X XA �FGLS � � �1 1 17 , (48)

where � ( � ).7 7� ;

5. Empirical Application

Due to data limitations, our present application of the model considers only two broadcategories of water uses, namely «essential uses of water» and «leisure uses of water».

We measure the size of the population of water users for the first category of uses by means ofan equivalence scale, where the adult is chosen as unit of measurement and any child is weightedas a fraction of an adult. Therefore, the size of this population of users is computed as a number ofequivalent adults according to the following formula:

N A aEi i i1 � � , (49)



where Ai refers to the number of adults of household i, E i to the number of children of the house-hold, and 0 1& &a to the weighting coefficient for children. Experiments performed by usingdifferent values of the equivalence coefficient a, ranging between 0.25 and 1, have convinced usthat 0.5 is the optimal choice for this coefficient, both from the point of view of model fitting andwith respect to the economic relevance of parameter estimates.

When measuring the size of the population of users of the second category of uses, we wouldhave liked to use data on the size of the garden and the volume of the swimming pool, but DIREN’ssurvey gives only information on the presence or not of these equipments. This led us to assumethat the size of these equipments is a function of household income per equivalent adult as suchequipments are present only in well-off households. As income information provided by DIREN’ssurvey is recorded in terms of income intervals, we use an imputation model (presented in the Ap-pendix) to estimate the household’s level of income. Therefore, the size of the population of waterusers for the second category of uses is defined as

N

R

Ni

i

i

2 1

0

�for housings with private garden

otherwi

,

se,

=>?

@?(50)

where Ri refers to the imputed income for household i in thousand of euros per month (kˆ).

The daily water consumption per user of the «essential uses of water» is modeled as a linearfunction of two explanatory variables, namely:

� the proportion of employed adults, X i1, defined as the number of employed adults of house-

hold i divided by the household number of equivalent adults. We expect a negative impact of thisvariable on household water consumption, as employed adults spend more time outside homethan the other members of the family;

� the number of rooms per equivalent adult, X i2 , defined as the number of rooms of housing

i divided by the household number of equivalent adults. We expect a positive impact of this va-riable on household water consumption, as water consumption for keeping up the house increa-ses with the size of the housing;

� we experimented with other explanatory variables, such as the use of a dishwasher, the pre-sence of a vegetable garden, the type of housing, etc. but the impact parameters of all these varia-bles turned out to be non-significant or of unsuitable sign.

We modeled the daily water consumption per kˆ/equivalent adult for leisure needs, as a linearfunction of two explanatory variables:

� the rainfall, X ti3 ( ) (when considering the maintenance of the private garden), defined as the

precipitations in centimetres (cm) for the day t. We expect a negative impact of this variable onhousehold water consumption as garden watering decreases when it rains;

� the presence of a private swimming pool, X i4 , defined as a dichotomous variable

X i4

1

0�

if the housing has a private swimming pool

o

,

therwise.

=>@

(51)

Using these assumptions, the specified model of household water consumption is written as



andBased


Y d N X d N X d N d Ni i i i i i i( ) ( ) ( ) ( ) ( )4 4 4 4 4� � � �� 1 1 2 1 1 3 2 1 4 2 5 3 2 6 4 2� � �� 1X d N X d Ni i i i i( ) ( ) ( ) ( ).4 4 4 4 (52)

Here Yi ( )4 stands for the water consumption of household i in liters (lt) during the period 4,d( )4 represents the duration in days of period 4 and X i

3 ( )4 symbolizes the average precipitationsin millimeters for the period 4. This last variable allows expressing the total precipitations X i

3 ( )4during the period 4, as the product: X di

3 ( ) ( )4 4 . To make the interpretation of the modelestimates easier, we present in Table 1 the economic meaning and the unit of measurement ofthe parameters of the model.

Table 1

Economic meaning and unit of measurement of model parameters

Para-meter

Economic meaning Unit of measurement

�1 Constant daily water consumption per equivalent adult lt/[day · equivalent adult]

�2 Marginal impact of employed adults on daily water consumption lt/[day · employed adult]

�3 Marginal impact of number of rooms on daily water consumption lt/[day · room]

�4 Constant daily water consumption per unit of household income lt/[day · kˆ/equivalent adult]

�5 Marginal impact of rainfall on daily water consumption per unit of

household income

lt/[cm rain · day kˆ/equivalent adult]

�6 Impact of owning a private swimming pool on daily water

consumption per unit of household income

lt/[day · kˆ/equivalent adult]

To estimate this model, we used the data from the DIREN survey complemented by dailymetereological observations recorded by the metereological stations (about a hundred) set up inLa Re ´union island by Meteo France Agency. These geographically distributed metereologicalobservations allowed us to compute the rainfall to be used for each household according to theobservations recorded at the closest metereological station.

While the socio-economic characteristics of the household as well as the characteristics of itshousing used to quantify the non-metereological explanatory variables of the model have beencollected from a random sample of 2000 households interviewed in 2004 by telephone, the volu-mes of water consumption for analysis have been provided only by some volunteers of this sam-ple, having accepted to fill out a questionnaire concerning the amounts and volumes of their lastthree water consumption bills. Unfortunately, this approach allowed us to collect only 173 reliablehousehold answers providing information on one to three bills staggered over the years 1998 to2004, namely 437 observations of which 340 from households with a private garden, and only 35(12 households) with a private swimming pool as well. Moreover, while a comparative analysis ofthe caracteristics of the telephone survey sample and that of the volonteers with data provided bythe 1999 national census for La Re ´union island [Binet et al. (2005)], confirms the representative-ness of the first sample with respect to the household size and the distribution by district and wea-ther, nevertheless, we observe in the second sample an overrepresentation of households withtwo persons and of those located in some urban districts. Therefore, our sample may have beengenerated by some self-selection mechanism that can flaw the optimal sampling properties ofour FGLS estimator.



Finally, the household income level used to quantify a proxy for the size of the user populationfor «leisure uses of water», which was not observed directly, but was indirectly estimated with theimputation model described in the Appendix. In order to assess the sensitivity of the estimates ofthe model (52) with respect to the assumptions underlying the imputation model, we conductedthree evaluations of the household income level based on differents specifications of the imputa-tion model. Table 2 presents the FGLS estimates of the impact parameters of model (52), usingthese three imputed values of household income level, namely:

� income 1: household income imputed without income indicators;� income 2: household income imputed using household standard of living;� income 3: household income imputed using household head profession.

Table 2

Parameter estimates of model (52)

ParameterParameter estimatesa according to imputed household income

Income 1 Income 2 Income 3

�1 80.5*

(19.6)

85.8*

(19.8)

87.6*

(19.4)

�2 –78.5*

(23.6)

–72.3*

(23.8)

–65.7*

(23.3)

�3 104.0*

(12.2)

99.1*

(12.5)

94.0*

(12.2)

�4 190*

(45.3)

200*

(49.0)

200*

(48.7)

�5 –94.5*

(2.62)

–76.3*

(2.42)

–43.4*

(1.99)

�6 69.0

(90.7)

39.9

(94.8)

45.0

(98.8)

a Figures in brackets are estimated asymptotic standard errors.

* Indicates statistical significance against a one-sided alternative at the 1% level.

These results call for the following comments:

� Estimates are robust with respect to the model specification used to impute an income levelto households. Any household income imputation leads to very narrow parameter estimates, sho-wing the expected sign and a similar level of statistical significance against the relevant one-sidedalternative. Clearly, robustness of the parameter estimates is stronger for the parameters descri-bying the daily water consumption for «essential uses of water» as the household income is usedto quantify a proxy for the size of the population using water for leisure needs.

� Water consumption per equivalent adult for essential uses consists of a committed consum-ption �1 estimated to 80/88 liters per day. Every employed adult decreases the household com-mitted consumption by a quantity � 2 estimated to 65/79 liters per day, while an extra room in thehousing increases the household committed consumption by a quantity �3 estimated to 94/104liters per day.



andBased


� In turn, water consumption per kˆ/equivalent adult for leisure uses consists of a committedconsumption for garden watering �4 estimated to 190/200 liters per day. This fixed consumptionis reduced by rainfall by a quantity �5 estimated to 43/95 liters per day and per cm of rain. Whenthe housing is equipped with a private swimming pool, one must add to the committed consum-ption �4 an extra consumption for leisure �6 estimated to 40/69 liters per day. Although non-sig-nificant, due to the small number of observations, in our sample, for housings equipped by a pri-vate swimming pool, this parameter estimate is certainly economically relevant.

6. Decomposing Observed Water Consumption by Use

Inspired by former studies, [Chow and Lin (1971)], [Carlevaro (1987, 1994)], [Carlevaro and Ber-tholet (1998, 2000)], in particular, we develop in this section a methodology allowing to decom-pose, by uses, the observed water consumption data. More specifically, this approch provides anestimate � ( ), , . . . ,Y h Hi

h 4 � 1 , of unobserved water consumption by uses Y h Hih( ), , . . . ,4 � 1 , consis-

tent with the observed water consumption Yi ( )4 , as it enforces the identity:

Y Y Yi ih

h

H

ih

h

H

( ) ( ) � ( )4 4 4� ��

� �1 1

. (53)

This result is obtained by formulating the decomposition problem as an optimal linearprediction problem in the frame of the joint distribution of observed variables Yi it( )4 and ofunobserved variables Y h Hi

hit( ), , . . . ,4 � 1 , where t T i mi� �1 1, . . . , , , . . . , .

We already modeled in Section 4 (Equations (29) to (33)) the distribution of observed variablesY t T i mi it i( ), , . . . , , , . . . ,4 � �1 1 . To model the latent variables Y h Hi

hit( ) , . . . ,4 , � 1 , for t Ti� 1, . . . , , i m� 1, . . . ,

we rely on the following relation:

Y N xih

it ih

ih

ith

ih

it( ) ( ),4 4 4� � �( ) � 1 (54)

where1 i

hit i

hit i

hih

itN d e e( ) ( ) ( )( )4 4 4� � and e e tih

it ih

t it

( ) ( )44

�� . (55)

From this relation, we derive the matrix expression of the regression model explaining the timeseries of the vector of the H uses of water by the household i, namely

y X t Ti it i it i it i( ) ( ) ( ) ,..., ,4 4 4� � �� 1 , 1 (56)

wherey Y X i N xi it i

hit h H i it h i

hih

i( ) ( ) , ( ) ([ ] [, . . . ,4 4 4 4� � � A�1 t h H�

�) ] , . . . ,1(57)

and1 1i it i

hit h H i it i i itN d e e( ) ( ) ( ) ( )[ ] , . . . , ( )4 4 4 4� � ��1

, (58)

whereN ni i� diag( ).

The symbol �ih used in (57) stands for row h of the unit matrix of size H.The moments of vector random disturbances 1 i it( )4 can be directly computed from the mo-

ments of random vectors ei and ei it( )4 obtained in Section 3. From formulas (19) to (23), we derive:

� E N d E e E ei it i it i i it( ) ( )( ) ( ) ( ) ;( )1 4 4 4� � �( ) 0 (59)

� V N V d e e N N d S di it i it i i it i i it( ) ( )( ) ( ) [ ( ) (1 4 4 4 4 4� � � �( ) 2it iN) ] ;2 (60)



� Cov Cov ( )( ) (( ); ( ) ( ) ; ( )1 1 3 3i it i i i it i i it iN d e e d4 4 4 4 4� � e e N d d N SN ti i i i it i i i� � �( ) ( ) ;)4 4 43 3 3( ) , (61)

� Cov ,( )( ); ( ) .1 1 3i it j j i j4 4 � �0 (62)

To specify the regression model of unobserved water consumption by uses in a compact form,we first write, for each household, their time series of water consumption (56) in a vector form:

y X i mi i i� � �� 1 , 1,..., , (63)

where

y yi i it t Ti� �[ ]( ) , . . . ,4 1 , X Xi i it t Ti

� �[ ]( ) , . . . ,4 1 , 1 1i i it t Ti� �[ ]( ) , . . . ,4 1 .

These models display vector random disturbances

1i T i i i iiN d e e� A A �( )( )I , (64)

where e ei i it t T i� �[ ]( ) ,.. . ,4 1 .

These random vectors are centered and non-intercorrelated with a variance-covariance matrixthat can be written as

V N V d e e N

N d d

i T i i i i T i

T i i i

i i

i

( ) ( ) ( )( )

( )[

1 � A A � A �

� A �A

I I

I S D N

d d N SN D N N

i T i

i i i i i i i i

i� A A �

� �A � A �

2

2 7

]( )

.

I (65)

Finally, we obtain a compact matrix expression of the regression model by stacking togetherthe time series models (63)

y X� �� 1, (66)

withy y i i m� �[ ] , . . . ,1 , X X i i m� �[ ] , . . . ,1 , 1 1� �[ ] , . . . ,i i m1

andE V m( ) , ( ) )1 1� � 8 8 �0 1diag( 7 7 7� . (67)

Following [Goldberger (1962)] we estimate the vector y of water consumption by uses witha predictor �y satisfying the following statistical properties:

� linearity, with respect to the vector of observed water consumption y, namely of the form:

� ,y Ay� (68)

where A is a nonstochastic matrix;

� unbiasedness, i.e. having prediction errors

� � ( )1 � 1 1� � � � � �y y AX X A (69)

with zero expectation, implying the restrictions:

AX X� and � [ ]11

1� �

)

*+

,

-/A I ; (70)



andBased


� efficiency, i.e. having, for any linear combination of the prediction errors �z �1 (where z standsfor a nonzero vector of arbitrary weights) the following variance:

V z z V z( � �� 1B 1 , (65)

where

V A VA

( � [ ]1B1

1� �

)

*+

,

-/

��

�

��

�

�

)

*+

,

-/II

which does not excede that of any other linear and unbiased predictor y .

To find this best linear unbiased predictor (BLUP) for y , we rely on the prediction model

y

y

X

X

)

*+

,

-/ �

)

*+

,

-/ �

)

*+

,

-/�

1

1, E

1

1)

*+

,

-/ � 0 and V

R

R

1

1)

*+

,

-/

��

�

��

)

*+

,

-/

7

7, (66)

where the covariances matrix R is bloc-diagonal

R E R Ri j i j m m� � � � ��( ) ) ( ,..., )] , , . . . ,11 1 1[E( diag1 1 . (67)

Writing the disturbance vector 1 i in the following compact form:

1 i i it i i it t T T i in d e e n d ei i

� � � � A � A�[ ( ( ) ( ) ( )(] , . . . ,4 4 1 I i ie� ), (68)

it is easy to compute analytically the expression for the matrix Ri :

R n E d e e d e e N

R

i T i i i i i i i T i

i

i i� A � A � � A � � � A �

�

( ) ( ) ) ( )( ( )I I

( )( )( )I IT i i i i T i i i i i ii in d d S D N d d n SN D nA � � A � A A � � A � � A �2 i N2 .

(69)

For this prediction model, predictor �y for y :

� � �,y X MCG� �� 1 (70)

where � � , � � .1 1 1 �� R y X MCG7 1

As a consequence of the bloc-diagonality of matrices R and 7, we derive the independentpredictors for each vector y i :

� � �y Xi i MCG i� �� 1 , (71)

where � �1 1i i i iR� � �7 1 and � �1 �i i i MCGy X� � .

This decomposition formula is the sum of two terms, which can be interpreted, according to[Nasse (1973)], as a calibration term and an adjustment term, respectively. The calibration termX i MCG

�� provides a first estimate of water consumption by uses for the household i, whereas theadjustment term corrects the calibration term, in order to reset the equality (53) between each to-tal of observed water consumption and the sum of estimated water consumption by uses.

This intrinsic adding-up property of �y i can be checked by premultiplying the predictor bythe matrix ( )IT Hi

A �: which transforms the vector y i of unobserved water consumption by usesfor the household i into the vector of observed water consumption for this household, y i . Thisleads to

( ) � � �IT H i i MCG i iiy X yA � � � �: � 1 , (72)



because

( ) ( )[ ] [ ], . . . , , . . . ,IT H i H i it t T it t Ti iX X xA � � � � �� : : 4 1 1 i

X i� (73)

and

( ) ( )( )I IT H i i T H i i i i i i i

i

i iR d d N Sn D N nA � � A � � A � A�: :

�7 21 1

2

12

2 2

12

Ds

s

Rs

ii

i i i

i i

T H i ii

i

�

�

��

�

��

�

��

A � �

� ': :

:�

( )I 7i

i i i ii

i i i

i i

T

d d D Ds

s

i

2

12

2 2� �

��

�

��

��

��

�

��

A

�

� ': :

( I � � � � ��

�:�

:'

� ' �'H i i

i

i

i i Ti i

i i i

i

i

iRs

ds

s

sd

i) 7 1

2

2

2

2 2

2

2I i i i i Td

i� � �

��

�

�� : : I .

(74)

Formula (71) shows that this adjustment is a distribution process of the vector of residuals �1 i

between the components of the calibration vector X i MCG�� using, as distribution coefficients the

elements of matrix � �Ri i7 1. The main weakness of the predictor �y i comes from the fact that itcannot avoid negative estimates of water consumption by use.

Table 3

Distribution of water consumption by use predictions

Distributioncharacteristics

Parameter estimates of model (5)

Income 1 Income 2 Income 3

Mean valuea

Essential usesb 215 214 210

Leisure uses 52 54 59

All uses 256 256 256

Median valuea


Leisure usesc 16 15 18

All uses 180 180 180

Minimum valuea


Leisure usesc –512 –527 –477

Maximum valuea


Leisure usesc 2010 2075 2026

a In liters per person and per day.b Computed for the full sample of 437 observations.c Computed for the subsample of 340 observations of households using water to satisfy both essential and leisure

needs.

Table 3 summarizes the results of the application of this approach to our sample of 437 waterconsumption observations, of which 340 observations were provided by households using water



andBased


to satisfy both essential and leisure needs. Predictions of water consumption by use are carriedout using the three FGLS estimates of the model (50) presented in Section 6. This enables to assessthe sensitivity of these predictions to the assumptions underlying the household incomeimputation model.

Not surprisingly, these model estimates lead to very narrow predictions of average andmedian water consumption by use, namely

� for the essential uses of water, within 210 and 215 liters/person/day in the average andwithin 175 and 177 liters/person/day in the median;

� for the leisure uses of water, within 52 and 59 liters/person/day in the average and within 15and 18 liters/person/day in the median.

We conclude that the use of water for leisure needs (gardening and swimming pools)represents only a quarter of the average consumption for essential uses and a fifth of the averagewater consumption of the sample of 256 liters/person/day. Therefore, according to these figures,water consumption for leisure cannot explain the residential over-consumption of water in LaRe únion with respect to that of continental France, estimated to 145 liters/person/day.

The comparaison of median values to mean ones sheds some light on the shape of personalwater consumption distribution in La Re únion. Indeed, median water consumption is noticeablylower than mean water consumption and much closer to the average figure for continentalFrance, showing by the same token that the personal water consumption in La Re únion isstrongly asymmetric towards high figures, as shown by the maximum predicted consumption of1265 to 1371 liters/person/day for essential uses and of 2010 to 2075 liters/person/day for leisureuses. Therefore, the average over-consumption of water by La Re únion inhabitants seems to beattributable to a quite small fringe of heavy residential consumers.

8. Conclusion

The design of policy measures intended to foster a rational use of water in the residential sectormust rely on quantitative knowledge of the uses, which households are making of this naturalresource. In this paper we develop an econometric methodology aiming at providing suchinformation to environmental decision makers. The application of this methodology on a data setcollected from a sample of households shows that our methodology can be an effective tool toprovide economically and statistically reliable information to this end.

Regarding the terms of the mandate the study was intended to fulfill, our analysis points outthe necessity to extend our database by collecting more detailed and reliable informationallowing a more precise breakdown of household water consumption by uses. This is the maindirection in which we are currently developing this research project.

Appendix

Imputing an Income Level to DIREN’s Survey Households

The DIREN survey, conducted in 2004 on a sample of 2000 households, records householdincome level as an ordered qualitative variable, namely as belonging to one of the following fiveincome intervals (in ˆ/month): I1 0 750� [ ; ], I 2 750 1500� [ ; ], I3 1500 3000� [ ; ], I 4 3000 4500� [ ; ]and I 5 4500� #[ ; ]. But to quantify a proxy for the size of the population using water for leisure, we



need to measure household income levels as a quantity. Using the middle point of these intervalsto quantify this proxy proved to be inappropriate for mainy reasons. From a theoretical point ofview, such an imputation method provides a biased estimate of the average income for any in-come interval in which the household income distribution is not uniform. Moreover, this methodis uneffective for the highest unbounded income interval. From a practical point of view, an appli-cation of this method brought about nonpositive definite estimates of the matrices S and 2, pre-venting the proper course of the model estimation and that of the decomposition by use of waterconsumption observations.

Therefore, we decided to develop an imputation method based on an econometric model de-scribing the observed qualitative information on household income according to an ordered po-lychotomous econometric model, where the unobserved household income level is specified asa latent variable.

This model is specified as follows:

y h yi i h� C �* I , (75)

where y i stands for the numerical encoding of observed income interval of household i, h for theindex of income interval I h h, , . . . ,� 1 5, and y i

* for the unobserved (latent) income level ofhousehold i (in ˆ/month).

We also assume that unobserved incomes y i* are distributed, within the household

population, according to the lognormal random variable defined by the following regressionmodel:

ln ,*y xi i i� � �� 1 (76)

where x i refers to the vector of indicators of income level for the household i, � to the vector ofparameters including the constant term, and 1 i to the N( , )0 2� random variable, identically andindependently distributed within the DIREN sample of households.

To impute an income level to household i, we use the mean square error (MSE) predictor for y i*

given the available information, $ i , namely

� ( | )* *y E yi i i� $ , (77)

whereE y i i( | )* $ stands for the expected value of y i* computed according to its conditional density

f y f yi i( | ) ( )$ � .

Two distinct cases should be considered, according to whether household i has or not decla-red its income interval y i . In the first case, the available information is given by$ �i i ix y, , ,� � andthe conditional density f yi ( ) is that of a truncated lognormal random variable, LN x i h( , | )�� 2 I ,defined on support I h , namely

f yy

y x

y xi

i

h( )

expln

ln�

��

��

�

��

=>?

@?

DE?

F?

� ��

12

1

�

�

! ih

i

hy x

y�

�

�

�

��

�

��

� �

��

�

��

�

!ln

, I , (78)



andBased


where !( )� stands for the standard normal cumulative distribution function and y h , y h�1 for thelower and upper bounds of income interval I h , respectively. Hence, the MSE predictor (84) is gi-ven by

�

expln

ln

*y

y xdy

yi

i

y

y

h

h

�

��

��

�

��

=>?

@?

DE?

F?

�

�

�

21

!h

ih

ix y x� � �

��

�

��

� �

��

�

��

1 �

�

�

�!

ln. (79)

Notice that computing this predictor requires a numerical integration.When the household income interval y i is unknown, the available information is given by

$ �i ix{ , , }� � , and the conditional density f yi ( ) is simply that of an untruncated lognormalrandom variable, LN x i( , )�� 2 , leading to the MSE predictor

� * expy xi i� � �=>@

DEF

�� 2

2. (80)

To practically implement these imputation formulas, we need to estimate the unknown para-meters � and �. We can provide efficient estimates of these parameters, by applying the maxi-mum likelihood principle to the discrete outcome random variable y i . This leads to maximizingthe following logarithmic likelihood function of the random sample y i ni , , . . . ,� 1

ln ( , | ,..., , ,..., ) ln |{ }L y y x x y P y h xn n ih

h

H

i i

i

� � 1 1

0

� ��

��

�1

n

, (81)

where

yy h

y hih

i

i

��

�=>@

1

0

if

if(82)

and

P y h x P y y y x

P y h x P

i ih

ih

i

i i

{ } { }

{ } {

| ln ln ln |

| ln

*� � & � �

� �

�1

y x y x x

P y h xy x

hi i

hi i

i i

hi

� � & � � � �

� ��

�

�

� 1 �

�

ln |

|ln

}

{ }

1

1

!�

�

�

��

�

��

� �

��

�

�� !

ln, ,..., ,

y xh

hi 1 5

(83)

where !ln 'y x i

1

0�

��

�

��

�

�and !

ln 'y x i6

1�

��

�

��

�

�.

Inserted in formulas (86) and (87), the maximum likelihood estimators for parameters � and �provide an asymptotically efficient estimator for the MSE predictor � *y i .

We tested three specifications of this income imputation model.

� The first specification assumes that the form of the personal income distribution is not influ-enced by some household income indicator recorded by the DIREN survey. It is, therefore,



a «na¿ve» specification used as a reference, with which the other two more informative specificati-ons may be compared. This specification is written as

ln *y i i� �� 10 . (84)

� The second specification assumes that the form of the personal income distribution isinfluenced by ordinal measure of household standard of living (HSL), stated by households ona four level scale, namely

HSL=1 if «The household hasn’t enough money to live. It cannot make ends meet».HSL=2 if «The household has just enough to live, but by making great sacrifices».HSL=3 if «The household has enough money to live without making great sacrifices».HSL=4 if «The household makes no sacrifices, in any case nothing important».

This specification is written as

ln *y Xi k ki

k

i� � ��

�� 11

2

4

, (85)

where

Xk

ki �G=

>@

1

0

if HSL

otherwise.

,(86)

� The third specification assumes that the form of the personal income distribution isinfluenced by the household head’s occupation (HHO), recorded according to a seven attributeclassification, namely

HHO=1 if «Senior executive or profession».HHO=2 if «Farmer, trader, or middle manager».HHO=3 if «Others, student».HHO=4 if «Employee».HHO=5 if «Worker».HHO=6 if «Foreman or independent profession».HHO=7 if «Retired».

This specification is written as

ln *y Zi k ki

k

i� � ��

�� 11

2

7

, (87)

where

Zk

ki ��=

>@

1

0

if HHO

otherwise.

,(88)

Table 4 presents the maximum likelihood estimates of these specifications, and Table 5 showsthe imputed household income level derived from these estimates using prediction formulas (86)and (87). The model estimates are based on the data set provided by the volunteers whoparticipated to the mailing survey aiming at collecting water consumption data from water bills.


andBased



We shall not comment on these empirical results, but simply notice that parameter estimates areeconomically plausible and statistically significant. Moreover, from an empirical point of view,specification 2, using stated household standard of living as an income level indicator, performsbetter than specification 3 based on household head occupation.

Table 4

Parameter estimates of income imputation model

ParameterSpecificationa

(91) (92)–(93) (94)–(95)

�1 7.265

(0.072)

6.312

(0.242)

8.251

(0.144)

�2 0.486

(0.258)

–0.665

(0.225)

�3 0.778

(0.116)

–2.014

(0.224)

�4 0.760

(0.191)

–1.025

(0.213)

�5 –1.177

(0.234)

�6 –0.480

(0.245)

�7 –1.065

(0.171)

� 0.834

(0.064)

0.588

(0.044)

0.606

(0.045)

Log likelihood

HAIC2 b

–233.31

0

–187.96

0.181

–189.36

0.162

a Figures in brackets are estimated asymptotic standard errors.b Akaike information criterion (AIC) adjusted pseudo-R2.

Table 5

Imputed household income (ˆ/month)

Income levelindicator

Conditional imputation UnconditionalimputationI1 I2 I3 I4 I5

None Model specification (91)

498 1100 2116 3632 7041 2024

HSLa Model specification (92)–(93)

1 444 1006 1881 3449 5254 655

2 522 1061 1970 3506 5455 1065

3 599 1153 2142 3609 6044 2319

4 640 1232 2321 3717 7477 4957



Income levelindicator

Conditional imputation UnconditionalimputationI1 I2 I3 I4 I5

HHOb Model specification (94)–(95)

1 631 1217 2294 3705 7374 4603

2 593 1150 2146 3616 6149 2368

3 426 1003 1882 3454 5279 614

4 564 1110 2067 3569 5796 1651

5 548 1093 2035 3550 5684 1419

6 606 1170 2187 3640 6395 2848

7 560 1106 2058 3564 5765 1587

a Household standard of living.b Household head’s occupation.

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potable a` La Reúnion., Etude reáliseé pour la DIREN, CERESUR, Universite´ de La Reúnion, Saint-Denis de La

Reúnion. 2005.

Carlevaro F. Re´gionalisation d’agre´gats nationaux au moyen d’indicateurs: une me ´thode e ćonome´trique.,

in: Modelisation spatiale. Theorie et applications, e´d. B. Guenier et J. H. P. Paelinck, Institut de Mathematiques

Economiques et Librairie de l’Universite´ de Dijon, Dijon. 1987.

Carlevaro F. Dećomposition d’agre´gats au moyen d’indicateurs. Une me´thode ećonome´trique // Revue

suisse d’Economie politique et de Statistique. 1994. ¹ 130(3).

Carlevaro F., Bertholet J.-L. Indices de consommation e´lectrique pour les services geńe´raux d’immeu-

ble et e ´valuation de gisements de ne ´gawatts., Revue suisse d’Economie politique et de Statistique. 1998.

¹ 134(3).

Carlevaro F., Bertholet J.-L. Aggregating Time Unbalanced Panel Observations: Methodology and Applica-

tions to Oil Consumption of Boiler Rooms., Chapter 8 in: Panel Data Econometrics: Future Directions, e´d. J. Krish-

nakumar and E. Ronchetti, Elsevier Science B. V. 2000.

Chow G. C, Lin A. Best Linear Unbiased Interpolation, Distribution and Extrapolation of Time Series by

Related Series // The Review of Economics and Statistics. 1971. LIII(4).

Goldberger A. S. Best Linear Unbiased Prediction in the Generalized Linear Regression Model // Journal of

the American Statistical Association. 1962. ¹ 57. P. 369–375.

Nasse P. Le syste´me des comptes nationaux trimestriels // Annales de l’INSEE. 1973. ¹14. P. 119–161.

Schlesser C. Analyse microećonome´trique par usages de la consummation re ´sidentielle d’eau a ` l’�ile de

La Reúnion, unpublished master thesis in econometrics, Department of Econometrics, University of Geneva.

2006.

Vonesh E. F, Chinchilli V. H. Linear and Nonlinear Models for the Analysis of Repeated Measurements, Marcel

Dekker, New York. 1997.

Eco

no

met

ric

Mo

del

ing

and

An

alys

iso

fR

esid

enti

alW

ater

Dem

and

Bas

edo

nU

nb

alan

ced

Pan

elD

ata


End of Table 5

Ô. Êàðëåâàðî, Ê. Øëåññåð, Ì. Å. Áèíå, Ñ. Äþðàí, Ì. Ïîëü

Ýêîíîìåòðè÷åñêîå ìîäåëèðîâàíèåè àíàëèç ïîòðåáíîñòè â âîäå â æèëèùíîì ñåêòîðå

íà îñíîâå íåñáàëàíñèðîâàííîé ïàíåëè äàííûõ

Èññëåäîâàíèå, ïðîâåäåííîå â 1997 ãîäó íà îñòðîâå Ðåþíüîí, âûÿâèëî ôàêò ÷ðåçìåð-

íîãî ïîòðåáëåíèÿ âîäû, êîòîðîå â ñðåäíåì ñîñòàâëÿëî 246 ë â äåíü íà æèòåëÿ, â òî

âðåìÿ êàê â êîíòèíåíòàëüíîé Ôðàíöèè ýòîò ïîêàçàòåëü áûë íå âûøå 145 ë.

×òîáû îáúÿñíèòü ýòó öèôðó, ðåãèîíàëüíûé Äåïàðòàìåíò îõðàíû îêðóæàþùåé

ñðåäû (DIREN) ïîðó÷èë Öåíòðó ýêîíîìè÷åñêèõ è ñîöèàëüíûõ èññëåäîâàíèé Óíèâåðñèòå-

òà Ðåþíüîí (CERESUR) ïðîâåñòè ðàáîòó, âêëþ÷àþùóþ â ñåáÿ:

(i) èçìåðåíèå äîìàøíåãî âîäîïîòðåáëåíèÿ stricto sensu (â ñòðîãîì ñìûñëå, — ëàò.)

ñ öåëüþ îïðåäåëåíèÿ êîëè÷åñòâåííîé çíà÷èìîñòè ÷ðåçìåðíîãî ïîòðåáëåíèÿ âîäû;

(ii) îáúÿñíåíèå îïðåäåëÿþùèõ ôàêòîðîâ âîäîïîòðåáëåíèÿ â æèëèùíîì ñåêòîðå,

â ÷àñòíîñòè ðîëü ðàçëè÷íûõ âèäîâ ïîëüçîâàíèÿ âîäîé, èìåþùåãîñÿ îáîðóäîâàíèÿ è ïî-

âåäåíèÿ èíäèâèäóóìîâ, ñ öåëüþ ðàçðàáîòêè ìåð ïî ðàöèîíàëüíîìó èñïîëüçîâàíèþ âîä-

íûõ ðåñóðñîâ.

Äëÿ äîñòèæåíèÿ ïîñòàâëåííûõ öåëåé â 2004 ãîäó áûëî ðàçðàáîòàíî è ïðîâåäåíî ñòðà-òèôèöèðîâàííîå ðàíäîìèçèðîâàííîå èññëåäîâàíèå íà îñíîâàíèè âûáîðêè èç 2000äîìîõîçÿéñòâ. Ýòî ïåðâîå èññëåäîâàíèå, ïðîâåäåííîå ïî òåëåôîíó, áûëî äîïîëíåíî

ïî÷òîâîé ðàññûëêîé 1000 äîìîõîçÿéñòâàì, âûêàçàâøèì æåëàíèå äîáðîâîëüíîãî ó÷àñòèÿäëÿ îïðåäåëåíèÿ îáúåìà âîäîïîòðåáëåíèÿ, óêàçàííîãî â òðåõ ïîñëåäíèõ ñ÷åòàõ çà âîäó ýòèõäîìîõîçÿéñòâ. Ê ñîæàëåíèþ, äàííàÿ ïî÷òîâàÿ ðàññûëêà äàëà òîëüêî 173 íàäåæíûõ îòâåòà.

Íàøå èññëåäîâàíèå ïðîäîëæàåò ïåðâûé àíàëèç äàííûõ, ñîáðàííûõ â ðåçóëüòàòå îïðîñà,ïðîâåäåííîãî ñïåöèàëèñòàìè CERESUR, îòâåòñòâåííûìè çà èññëåäîâàíèå DIREN [Binet et al.(2005)]. Åãî öåëüþ ÿâëÿåòñÿ óãëóáëåííûé àíàëèç äàííîé èíôîðìàöèè ïóòåì ðàçðàáîòêè ìèê-ðîýêîíîìåòðè÷åñêîé ìîäåëè, ïîçâîëÿþùåé âûÿâèòü îñíîâíûå êëèìàòè÷åñêèå, äåìîãðàôè-÷åñêèå, ýêîíîìè÷åñêèå è òåõíîëîãè÷åñêèå ôàêòîðû áûòîâîãî âîäîïîòðåáëåíèÿ è îïðåäå-ëèòü äîëþ êàæäîãî âèäà ïîòðåáëåíèÿ âîäû â îáùåì îáúåìå.

Ñ ýòîé öåëüþ ìû ôîðìóëèðóåì ïðè÷èííî-ñëåäñòâåííóþ ýêîíîìåòðè÷åñêóþ ìîäåëü, îïè-ñûâàþùóþ åæåäíåâíîå ïîòðåáëåíèå âîäû ïàíåëüþ äîìîõîçÿéñòâ íà îñíîâàíèè ðàçëè÷íûõâèäîâ èñïîëüçîâàíèÿ âîäû â äîìàøíèõ óñëîâèÿõ (Ðàçäåë 2). Â ñîîòâåòñòâèè ñ ýòèì ïîäõîäîìîáùåå ïîòðåáëåíèå âîäû â äîìîõîçÿéñòâå ÿâëÿåòñÿ ðåçóëüòàòîì ðàçëè÷íûõ íåçàâèñèìûõâèäîâ åå èñïîëüçîâàíèÿ. Â äàëüíåéøåì ìû ïðåäñòàâèì âîäîïîòðåáëåíèå ïî åãî âèäàì êàêïðîèçâåäåíèå êîëè÷åñòâà ïîëüçîâàòåëåé âîäû íà ïîòðåáëåíèå îäíèì ïîëüçîâàòåëåì. ×èñ-ëåííîñòü ïîëüçîâàòåëåé ÿâëÿåòñÿ èçìåðÿåìîé âåëè÷èíîé, â òî âðåìÿ êàê âîäîïîòðåáëåíèåíà îäíîãî ïîëüçîâàòåëÿ ìîäåëèðóåòñÿ êàê ëèíåéíàÿ ðåãðåññèÿ íåêîòîðûõ îáúÿñíÿþùèõ ïå-ðåìåííûõ. Âëèÿíèå íåíàáëþäàåìûõ ôàêòîðîâ ãåòåðîãåííîñòè íà ïîâåäåíèå äîìîõîçÿéñòâ


ó÷èòûâàåòñÿ ïîñðåäñòâîì ñòðóêòóðû êîìïîíåíòû ïîãðåøíîñòè âîçìóùåíèÿ ýòîé ðåãðåñ-ñèâíîé ìîäåëè. Èñïîëüçóÿ ñáàëàíñèðîâàííóþ âî âðåìåíè ïàíåëü äàííûõ, ìû ïîëó÷èëè, àã-ðåãèðóÿ âî âðåìåíè, îáùóþ ìîäåëü ëèíåéíîé ðåãðåññèè, êîòîðàÿ îáúÿñíÿåò ñóùåñòâóþ-ùóþ, äîñòóïíóþ äëÿ ïðîâåäåíèÿ ýìïèðè÷åñêîãî àíàëèçà íåñáàëàíñèðîâàííóþ âî âðåìåíèïàíåëü äàííûõ (Ðàçäåë 3). Ñ öåëüþ îïðåäåëåíèÿ ïàðàìåòðîâ ýòîé ãåòåðîñêåäàñòè÷íîéè ñåðèàëüíî-êîððåëèðîâàííîé ìîäåëè ëèíåéíîé ðåãðåññèè, ìû ïðèìåíÿåì ýôôåêòèâíóþäâóøàãîâóþ îáîáùåííóþ îöåíêó ïî ìåòîäó íàèìåíüøèõ êâàäðàòîâ (Ðàçäåë 4).

Èñïîëüçóÿ ýòó ìîäåëü, ìû àíàëèçèðóåì äàííûå îáñëåäîâàíèÿ DIREN ïóòåì ðàññìîòðå-íèÿ äâóõ øèðîêèõ êàòåãîðèé áûòîâîãî âîäîïîëüçîâàíèÿ: «îñíîâíîå âîäîïîòðåáëåíèå»è «âîäîïîòðåáëåíèå äëÿ äîñóãà» (Ðàçäåë 5).Îïðåäåëÿåì ðàçìåð ïîïóëÿöèè âîäîïîëüçîâàòå-ëåé äëÿ ïåðâîé êàòåãîðèè ïîñðåäñòâîì øêàëû ýêâèâàëåíòíîñòè, ãäå âçðîñëûé ÷åëîâåê âçÿòçà åäèíèöó èçìåðåíèÿ, à ðåáåíîê îöåíèâàåòñÿ êàê ÷àñòü âçðîñëîãî. Âîäîïîòðåáëåíèå íà êà-æäûé ýêâèâàëåíò âçðîñëîãî ÷åëîâåêà ìîäåëèðóåòñÿ êàê ëèíåéíàÿ ôóíêöèÿ äâóõ îáúÿñíÿþ-ùèõ ïåðåìåííûõ: äîëÿ ðàáîòàþùèõ âçðîñëûõ ëþäåé â äîìîõîçÿéñòâå, ñîñòîÿùåì èç ýêâèâà-ëåíòîâ âçðîñëûõ ëþäåé, è ÷èñëà êîìíàò, ïðèõîäÿùèõñÿ íà êàæäûé ýêâèâàëåíò âçðîñëîãî÷åëîâåêà. Ìû îæèäàåì íåãàòèâíîå âëèÿíèå ïåðâîé ïåðåìåííîé íà âîäîïîòðåáëåíèå äîìî-õîçÿéñòâà è ïîçèòèâíîå âëèÿíèå âòîðîé. Áûòîâîå âîäîïîòðåáëåíèå «äëÿ äîñóãà» âêëþ÷à-åò â ñåáÿ èñïîëüçîâàíèå âîäû äëÿ óõîäà çà ñàäîì è äëÿ áàññåéíà, â ñëó÷àå èõ íàëè÷èÿ. Ðàç-ìåð ýòèõ «äîïîëíèòåëüíûõ óäîáñòâ» èçìåðÿåòñÿ êàê ôóíêöèÿ äîõîäà ñåìåéíîé åäèíèöûâ òûñ. åâðî, ïðèõîäÿùåãîñÿ íà ýêâèâàëåíò âçðîñëîãî ÷åëîâåêà. Ìû ìîäåëèðóåì âîäîïî-òðåáëåíèå, ïðèõîäÿùååñÿ íà òûñ. åâðî/ýêâèâàëåíò âçðîñëîãî ÷åëîâåêà êàê ëèíåéíóþ ôóíê-öèþ äâóõ îáúÿñíÿþùèõ ïåðåìåííûõ: èíòåíñèâíîñòü äîæäåé (êîãäà ìû ó÷èòûâàåì óõîä çà ÷à-ñòíûì ñàäîì) è ôèêòèâíóþ ïåðåìåííóþ, êîòîðàÿ âûðàæàåò íàëè÷èå èëè îòñóòñòâèå áàññåé-íà â äîìå. Ìû îæèäàåì, ÷òî ïåðâàÿ ïåðåìåííàÿ îêàæåò íåãàòèâíîå, à âòîðàÿ — ïîçèòèâíîåâëèÿíèå íà îáúåì áûòîâîãî âîäîïîòðåáëåíèÿ. Îöåíêè â òàáë. 2 ñîîòâåòñòâóþò îæèäàåìûìâëèÿíèÿì è ïîçâîëÿþò èçìåðÿòü èõ êîëè÷åñòâåííîå çíà÷åíèå äëÿ äîìîõîçÿéñòâà ëþáîãîðàçìåðà.

È íàêîíåö, ìû ðàçðàáîòàëè ýôôåêòèâíûé ìåòîä, ïîçâîëÿþùèé ñèñòåìíî ðàçäåëÿòü îáú-åì âîäîïîòðåáëåíèÿ, óêàçàííûé â ñ÷åòàõ äîìîõîçÿéñòâ çà âîäó â ñîîòâåòñòâèè ñ ýòèìè äâóìÿêàòåãîðèÿìè èñïîëüçîâàíèÿ âîäû. Ñèñòåìíîñòü äîñòèãàåòñÿ ïóòåì äîáàâëåíèÿ ýòèõ îöåíîêâîäîïîòðåáëåíèÿ ïî âèäàì èñïîëüçîâàíèÿ ê íàáëþäàåìîìó îáùåìó îáúåìó (Ðàçäåë 6). Ê ñî-æàëåíèþ, äàííûé ìåòîä íå ïîçâîëÿåò èçáåæàòü îòðèöàòåëüíûõ îöåíîê âîäîïîòðåáëåíèÿ ïîâèäó èñïîëüçîâàíèÿ. Ïðèìåíåíèå äàííîãî ìåòîäà äëÿ íàøåé âûáîðêè, ïðèâåäåííîéâ òàáë. 3, äàåò îöåíêó íåîáõîäèìîãî âîäîïîòðåáëåíèÿ â ïðåäåëàõ îò 210 äî 215 ë íà ÷åëîâå-êà â äåíü êàê ñðåäíåå çíà÷åíèå è îò 175 äî 177 ë íà ÷åëîâåêà â äåíü êàê ìåäèàííîå, â òî âðå-ìÿ êàê èñïîëüçîâàíèå âîäû «äëÿ äîñóãà» îöåíèâàåòñÿ â ïðåäåëàõ îò 52 äî 59 ë êàê ñðåäíååè îò 15 äî 18 ë êàê ìåäèàííîå. Çàìåòíî áîëåå íèçêîå ìåäèàííîå çíà÷åíèå îòíîñèòåëüíîñðåäíåãî òàêæå ñâèäåòåëüñòâóåò î òîì, ÷òî âûñîêîå ñðåäíåå çíà÷åíèå ÷ðåçìåðíîãî âîäîïî-òðåáëåíèÿ æèòåëÿìè Ðåþíüîíà âûçâàíî äîñòàòî÷íî íåáîëüøîé ãðóïïîé äîìîõîçÿéñòâ, èñ-ïîëüçóþùèõ áîëüøèå îáúåìû âîäû.

Äëÿ ðåàëèçàöèè öåëåé íàøåãî èññëåäîâàíèÿ åñòü íåîáõîäèìîñòü ðàñøèðåíèÿ áàçû äàí-íûõ ïóòåì ñáîðà áîëåå ïîäðîáíîé è íàäåæíîé èíôîðìàöèè, êîòîðàÿ ïîçâîëèëà áû ïðîâåñ-òè áîëåå ÷åòêóþ ðàçáèâêó áûòîâîãî âîäîïîòðåáëåíèÿ ïî âèäàì èñïîëüçîâàíèÿ. Èìåííî ïîýòîìó îñíîâíîìó íàïðàâëåíèþ è äâèæåòñÿ íàø èññëåäîâàòåëüñêèé ïðîåêò.


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