A multivariate econometric approach for domestic water demand modeling: An application to Kathmandu, Nepal

Transcription

1 Water Resour Manage (2007) 21: DOI /s ORIGINAL ARTICLE A multivariate econometric approach for domestic water demand modeling: An application to Kathmandu, Nepal M. S. Babel A. Das Gupta P. Pradhan Received: 11 July 2005 / Accepted: 8 May 2006 C Science + Business Media B.V Abstract Domestic water use/demand is a complex function of socio-economic characteristics, climatic factors and public water policies and strategies. This study therefore develops a model based on the multivariate econometric approach which considers these parameters to forecast and manage the domestic water use/demand. The model applies statistical tools to select suitable demand function and most relevant explanatory variables that have strong relationship with water use/demand. The model applicability is demonstrated with an example of domestic water use in Kathmandu Valley, Nepal. The results indicate that the number of connections, water pricing, public education level, and average annual rainfall are significant variables of domestic water use/demand. The paper further analyzes the effect of length of data series on accuracy of model results. The developed model is used to forecast the water use/demand in the future in the study area. Keywords Multivariate econometric model. Kathmandu Valley. Water demand forecast. Demand management 1. Introduction Water demand forecasts are required for proper planning, development and management of water resources. Mathematical models can be used to analyze the factors influencing water demand, understand their effects, predict the future demand and develop management plans accordingly. Demand management aims to reduce the wastage of water due to overuse and leakage. Demand management measures in the domestic sector include water conservation measures (leakage detection in distribution system, reduction of illegal household connections, in-house retrofitting, outdoor water saving measures etc.); water pricing (metering and tariff structures); information and education (awareness raising, public involvement and inschool education); and legal measures (rules and regulations that form the basis of policy, M. S. Babel ( ) A. D. Gupta P. Pradhan Water Engineering and Management, Asian Institute of Technology, P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand msbabel@ait.ac.th

2 574 Water Resour Manage (2007) 21: regulations on the sale of water and municipal bylaws that promote water conservation). The water demand management models can be helpful in assessing the effect of various conservation measures and take suitable decisions on the development of policies and strategies for demand management and implementation of demand management options. Various widely used techniques to forecast domestic water demand are (Pradhan, 2003): (a) Time extrapolation: It is assumed that future water use follows trends in the past and the water use over time is extrapolated into the future by graphical or mathematical means and the change in demand over time may be assumed to follow a linear, logistic, exponential or some other function; (b) Single coefficient requirement methods: The per capita requirements approach estimates future water use as the product of projected service area population and a projected value of per capita water use; (c) Multiple coefficient requirement models: In these models various parameters excluding the price of water, or any other economic factors, as explanatory variables are considered; (d) Multiple coefficient demand models: They are similar to the requirement models above but include the price of water charged to the users, and/or other related economic variables, such as, income level of the consumers; and (e) Disaggregated water use forecast models: They specify water use for each sector, season or region separately, utilizing the best available explanatory variables which are unique to a given type of water use sector and generally yields a more accurate composite forecast. The extrapolation of past water use may not represent changing socio-economic status, technological changes, improvements in water use efficiency and government policy decisions related to water use and conservation that can affect the use of water by people. Similarly, the inherent simplified assumption in the single coefficient requirement model is that the demand is constant for each person regardless of the lifestyle, income and level of education is not true. The multiple coefficient requirement models imply that water use is an absolute requirement unaffected by economic choice. Disaggregated water use forecast models require a large number of data for different sectors which are normally not available in developing countries. The domestic water use/demand depends on socio-economic characteristics (population, household size, income, education level, etc.), climatic factors (temperature, rainfall, etc.) and public water policies and strategies. Hence, the application of multivariate coefficient demand model, that can integrate the effects of all or several of these factors, is gaining popularity. The water use/demand models proposed by Katzman (1977), Dandy et al. (1997), Malla and Gopalakrishnan (1997), Billings and Agthe (1998), Lahlou and Colyer (2000) and Babel et al. (2003) are some of the examples of multivariate and econometric water demand models. Katzman (1977) emphasized the importance of income and price elasticities for the projection of demand. Dandy et al. (1997) found in Adelaide, Australia that water consumption above the free allowance; being sensitive to price, responds less to social and climatic factors than consumption below the free allowance. Malla and Gopalakrishnan (1997) used the generalized least squares (GLS) procedure to model the residential demand for water as a function of price, income of household and rainfall in Hawaii and the results indicate that raising price appeared feasible in private multi-units housing. Billings and Agthe (1998) developed and compared state-space and regression forecast models using factors such as monthly temperature and precipitation, marginal price of water, block rate subsidy and real income per capita. Lahlou and Colyer (2000) analyzed approaches for sustainable water management using multiple regression demand models. The results indicated that residential/commercial water demand is weakly responsive to price changes compared to institutional water demand and that considerable savings can be attained through a comprehensive water demand management program. Recently, Babel et al. (2003) commended the suitability of multivariate econometric approach for forecasting and managing water demand in urban areas.

3 Water Resour Manage (2007) 21: The present study therefore aims to develop a multiple coefficient water demand forecast and management model for the domestic sector considering various socio-economic, climatic and policies related factors. The applicability of the model is illustrated using an example of domestic water use in Kathmandu Valley, Nepal. 2. Modeling methodology A simple user-friendly model is developed to forecast the water demand and to analyze the effect of management measures in domestic sector. The model can accommodate either monthly or annual data series in the analysis. The user interface is developed using the Visual Basic 6.0 programming with Microsoft Excel as the backhand for major calculations and the Statistical Package for the Social Sciences (SPSS) is used to carry out multiple regression analysis. The interface can be installed and run in computers with Microsoft Office setup, preferably Office 98 or Office The model is menu-driven and the help menu is provided to guide the user through the process of model development and application. The step-by-step modeling procedure is presented and explained below Model variables and correlation analysis The multivariate water demand model can use as many variables as required that can directly or indirectly affect the water demand in a site-specific condition. They can be socio-economic, climatic and public policies and strategies related factors. The most commonly used variables based on the literature are: number of connections or population, household size, number of households, income or factors representing the standards of living, price of water, educational level and climatic factors such as temperature and rainfall. The model carries out correlation analysis and determines the partial correlation coefficients of the independent and dependent variables and displays the results in the form of a matrix. This analysis indicates degree of correlation between dependent variable and independent variables and among the independent variables. Higher the correlation more is the relative importance of the independent variable in predicting the dependent variable. In case the independent variables have high correlation coefficients (0.95 or more) with each other, one of them is to be selected to avoid the effect of multi-collinearity. The multi-collinearity can distort the standard error of estimate and may therefore lead to incorrect conclusions as to which independent variables are statistically significant. This can be done either on judgmental ground or on the basis of comparison of model fit with one eliminated versus the other(s) eliminated. A common rule of thumb is that the correlations among the independent variables between 0.70 to 0.70 do not cause difficulties (Mason et al., 1999). After dropping one or more independent variables that are strongly correlated the regression is recomputed Functional relationships and multiple regression analysis The general form of the multivariate model is: Y = f (X 1, X 2, X 3,...,X n ) (1) where, Y is the total water use/demand (dependent variable); and X 1 to X n are different relevant factors affecting the water use/demand (independent variables).

4 576 Water Resour Manage (2007) 21: As there is no prior basis for choosing a functional relationship, the model is provided with the options to analyze water use/demand using three popular functional forms as follows: Linear model: Semi-log model: Log-log model: Y = b 0 + b 1 (X 1 ) + b 2 (X 2 ) + b 3 (X 3 ) + +b n (X n ) (2) lny = b 0 + b 1 (X 1 ) + b 2 (X 2 ) + b 3 (X 3 ) + +b n (X n ) (3) lny = b 0 + b 1 ln (X 1 ) + b 2 ln (X 2 ) + b 3 ln (X 3 ) + +b n ln (X n ) (4) where, b 0 is the model intercept and b 1, b 2, b 3,...,b n are the coefficients of corresponding independent variables. Linear demand functions are often chosen because of their ease of estimation. However, they do not yield constant elasticities at all points of the demand function. The log-log functional form provides direct estimates of the respective elasticities of the independent variables with respect to the dependent variable. The semi-log function is often used to compare the results with the linear and log-log functions (Garcia et al., 2001). The values of the intercept and coefficients of all three functional forms are obtained through multiple regression analysis employing SPSS software package with time series data of the dependent and independent variables as inputs Evaluation of demand function The model performs two kinds of tests to evaluate the fitness of the water demand equation, global test and individual test Global test The ability of the independent variables X 1, X 2, X 3,...,X n to explain the behavior of the dependent variable Y is checked by the global test. Basically, it investigates if it is possible for all independent variable to have zero net regression coefficients. In other words, could the amount of explained variation R 2 occurred by chance? If r 1, r 2, r 3,..., r n are sample net regression coefficients, the corresponding coefficients in the population are β 1,β 2,β 3,...,β n. It is tested whether the net regression coefficients in the population are zero. The null hypothesis is: H 0 : β 1 = β 2 = β 3...= β n = 0 and the alternate hypothesis is: H 1 : β 1 β 2 β 3... β n 0. If the null hypothesis is accepted, it implies the regression coefficients are all zero and logically, the independent variables are of no use in estimating the dependent variable. In such a case one has to search for some other independent variables or take a different approach to predict the dependent variable. In general to check for the null hypothesis that the multiple regression coefficients are all zero, F distribution at a specified level of significance (generally 0.05) is used. The F value

5 Water Resour Manage (2007) 21: can be computed by: F = (SSR/k)/(SSE/(n (k + 1))) (5) where, n = number of samples used; k = number of independent variables used; SSR = sum of square for regression; and SSE = residual sum of square or sum of square for error. The critical value of F is given in tabular form in the standard statistical books. If the computed F value is greater than the critical F value, the null hypothesis is rejected and the alternative hypothesis is accepted Individual test It is required to check if any of the independent variables coefficients is zero or not. If a b i is zero, it implies that the particular independent variable is of no value in explaining any variation in the dependent variable. If there are variables for which null hypothesis cannot be rejected then, it can be eliminated from the regression equation. Generally the hypotheses are tested at the 0.05 level of significance. The null hypothesis is: H 0 : β 1 = 0 and the alternative hypothesis is: H 1 : β 1 0. The test statistics is the student t distribution with n (k + 1) degrees of freedom. The critical t value can be taken from the standard statistics books and H 0 is rejected if the t computed is out of range of the two tail s critical values. If any one of the independent variables is found to accept the null hypothesis by t-test, it is removed from the regression equation assuming that it is not a significant predictor of the dependent variable. The analysis is then again carried on with the remaining variables. If two or more independent variables are found to accept the null hypothesis, the one with the smallest t value is removed first and the regression analysis if re-run and preceded as before checking the remaining variables (Mason et al., 1999). In addition, the p value (i.e., probability value), also known as the observed or exact level of significance is calculated and if it is less than a specified significance level (generally 0.05), the null hypothesis is rejected and the alternative hypothesis is accepted. Technically, the p value is defined as the lowest significance level at which a null hypothesis can be rejected. The user s judgment is required in deciding the acceptance of a particular variable at a certain level of significance. The final selection of the functional form is done based on the coefficient of determination (R 2 ), adjusted R 2 and the standard error and the significance of the t-test of each variable Partial residual plots Large R 2 and significant F-test and t-test are highly desirable, but they don t guarantee that the data has been fitted well. Even if the regression analysis results are same, the nature of data may be entirely different. This may be due to outliers and/or curvature in the plot of the residual versus an explanatory variable included in the analysis. The outliers are identified by the model using the limits defined by the first and third quartiles (Q 1 and Q 3 ) and the median of the data set. Any value above and below [Q 3 ± 1.5(Q 3 Q 1 )] is considered as an outlier (Mason et al., 1999). However, once outliers are identified they must be checked with the data monitoring authority for their accuracy. The model performs partial plot analysis to determine the need of transformation of the variables and thus to confirm the chosen functional form using the evaluation criteria explained earlier. The curvature in a plot of residuals versus an explanatory variable included

6 578 Water Resour Manage (2007) 21: in the model indicates that further transformation of the variables should be tried. The relationship should be linear for the best results. However, if the residuals are plotted directly against the explanatory variables, other variables will influence the plots. To avoid this, partial residual plots (also called adjusted variable plots) are constructed. The widely used general form of useful transformation is natural logarithm (Helsen and Hirsch, 1991). The partial plot (e j versus X j ) describes the relationship between the dependent variable Y and the jth explanatory variable after the effects of all other explanatory variables have been removed. The partial residual is given by The adjusted explanatory variable is given by ej = Y Y ( j) (6) X j = X X ( j) (7) Where, Y( j) is the predicted value of Y from a regression equation with X j left out of the analysis; Y is the actual value of the dependent variable; j is the number to denote a particular explanatory variable ( j = 1 to number of explanatory variables); X( j) is the predicted value of X j from a regression against all other explanatory variable (so X j is treated as response variable to obtain the X( j) ); and X = actual value of the independent variable X j. 3. Model application: An illustrative example of Kathmandu Valley 3.1. Study area The domestic water use/demand data of Kathmandu Valley, Nepal is used to illustrate the model applicability. Kathmandu Valley, situated in the mid-western development region of Nepal with a total geographical area of about 900 km 2, includes three administrative districts of Kathmandu, Lalitpur, and Bhaktapur as indicated in Figure 1. Nepal Water Supply Corporation (NWSC) is the government agency responsible for providing adequate drinking water supply and sewage services that ensures health security in Kathmandu Valley and other urban centers in the country. Both the surface water and groundwater sources are tapped to supply water in Kathmandu Valley to a population of about 1.65 million in Database for modeling Factors affecting water use/demand are grouped into three categories, water utility management policy factors, socio-economic factors, and climatic factors. The dependent variable (Y) is the total domestic water use (10 3 m 3 per day). The nine independent variables representing the utility management policies, socio-economic, and climatic factors are included and they are denoted as: number of connections [X 1 ]; water tariff rate after minimum allowance of water supply in Nepalese Rupees (NRs)/m 3 [X 2 ]; population [X 3 ]; per capita GDP at the current price in NRs [X 4 ]; ratio of the total population to the university students [X 5 ]; number of households [X 6 ]; average household size [X 7 ]; average annual temperature in 0 C[X 8 ]; and annual rainfall in mm [X 9 ]. These parameters are selected based on the availability and reliability of data and to best represent the water use/demand characteristics.

7 Water Resour Manage (2007) 21: Fig. 1 Location map of the study area and the three districts of Kathmandu Valley

8 580 Water Resour Manage (2007) 21: Table 1 Database for domestic water modeling in Kathmandu Valley Year Y X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X , ,003, , , , ,037, , , , ,071, , , , ,105, , , , ,159, , , , ,213, , , , ,267, , , , ,321, , , , ,375, , , , ,429, , , , ,483, , , , ,537, , , , ,591, , , , ,645, , ,407 Y = domestic water use (10 3 m 3 per day); X 1 = number of connections; X 2 = water tariff rate after minimum allowance of water supply in NRs/m 3 ; X 3 = population; X 4 = per capita GDP at the current price in NRs; X 5 = ratio of the total population to the university students; X 6 = number of households; X 7 = average household size; X 8 = average annual temperature in 0 C; and X 9 = annual rainfall in mm The yearly time-series data on the above-mentioned variables, for the period , as presented in Table 1, were obtained from NWSC, Central Bureau of Statistics (CBS), Department of Hydrology and Meteorology (DHM), and Tribhuvan University (TU) head office in Kathmandu. Water tariff is applicable for water use above 10 m 3 /month and is charged according to the block tariff structure. The pricing of water depends on the type of user, the industrial sector is charged about 1.5 folds of the domestic sector. 4. Results and discussion 4.1. Correlation matrix The correlation analysis between the assumed explanatory variables indicated that X 1, X 3, X 4, X 6 and X 7 are highly correlated. This is reflected by the correlation coefficients being higher than ±0.95 (Table 2). To avoid the effects of multi-collinearity, only one of the five variables is used in further analysis. Hence, only X 1 is chosen due to the reliability of its historical record Water demand function The domestic water use equation for Kathmandu Valley is developed using ten years of data ( ). The data of 1998 to 2001 is used for evaluating the performance of the developed model. Multiple regression analysis is carried out to find out the relationship of water use with the remaining explanatory variables. F-test and t-test are applied to check the suitability of the functional relationship and the use of the explanatory variables in the model with the available dataset. The procedure adopted here is similar to backward elimination method, a standard stepwise procedure to eliminate the insignificant variables from the analysis. The steps in the analysis, using three functional forms, are as follows:

9 Water Resour Manage (2007) 21: Table 2 Correlation matrix X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X X X X X X X X X Table 3 Regression results (Linear model) No. of observations R 2 Adj. R 2 Std. error F-value Sig. F Coefficient Std. error t-stat p-value Intercept X X X X Linear model: (1) The multiple regression analysis is carried out with the remaining independent variables X 1, X 2, X 5, X 8 andx 9. The variable X 8, with the smallest absolute t-value, is then removed as it accepts the null hypothesis of the individual test (t-stat = and p-value = 0.256). (2) The regression analysis is repeated with X 1, X 2, X 5 andx 9 as the explanatory variables. All these variables are found to be significant at a significance level of 0.05 except X 9, which is significant at only As it is close to the assumed significance level of 0.05, it is also accepted as a factor affecting the water use. The linear model, with these four variables, explains 95% variation in the water use, as shown by adjusted-r 2 (Table 3). Semi-log model: (1) The multiple regression analysis is carried out with X 1, X 2, X 5, X 8 and X 9. As in the linear model, it is required to remove X 8 from the analysis, as it accepts the individual null hypothesis (t-stat = and p-value = 0.452). (2) The analysis is repeated using the explanatory variables X 1, X 2, X 5 and X 9 and it is found that X 9 is significant only at 0.065, the adjusted-r 2 being as shown in Table 4. (3) Further analysis is carried out to see if the removal of X 9 yields better results. In that case it is observed that X 5 is significant only at 0.088, with t-stat of If X 5 is also not considered in the analysis then only two variables X 1 and X 2 are found to be significant at the level The adjusted-r 2 in this case is The comparison of the adjusted-r 2 and the standard error indicates that the model with the

10 582 Water Resour Manage (2007) 21: Table 4 Regression results (Semi-log model) No. of observations R 2 Adj. R 2 Std. error F-value Sig. F Coefficient Std. error t-stat p-value Intercept X X X X Table 5 Regression results (Log-log model) No. of observations R 2 Adj. R 2 Std. error F-value Sig. F Coefficient Std. error t-stat p-value Intercept X X X X four explanatory variables (X 1, X 2, X 5 and X 9 ) performs better than that with the two independent variables (X 1 andx 2 ). The model coefficients and analysis results using X 1, X 2, X 5 and X 9 are presented in Table 4. Log-log model: (1) Using the five explanatory variables (X 1, X 2, X 5, X 8 andx 9 ), the results suggest that as for the linear and semi-log models X 8 is not a significant predictor of Y, as it accepts the null hypothesis with the t-stat of and p-value of (2) Further analysis with X 1, X 2, X 5 and X 9 reveals that these variables are significant at 0.05 significance level and these four variables together in the model can explain 96.5% variation in water use (Table 5). Hence, the final domestic water demand model equations are as follows: Linear model: Y = (X 1 ) (X 2 ) (X 5 ) (X 9 ) (8) Semi-log model: lny = (X 1 ) (X 2 ) (X 5 ) (X 9 ) (9)

11 Water Resour Manage (2007) 21: Log-log model: lny = ln(x 1 ) ln(x 2 ) ln(x 5 ) ln(x 9 ) (10) The above analysis with ten years of data, for the period , shows that the water use is dependent on four factors namely; number of connections, water tariff, education level (ratio of population to university students) and rainfall. The effect of the variables that are not considered in the analysis, but are somehow affecting the water use, is accounted for by the value of the model intercept. The above three models are validated with the known values of explanatory variables for the period All the three models under-predicted the water use in 2001 and the error of 8.74, 7.85 and 6.26% with respect to the observed water use is found by the linear, semi-log and log-log model respectively. However, with the four independent variables the log-log model could represent the water use better than the other two models. The log-log equation is therefore considered the best among the three functional forms, based on the highest R 2, lowest standard error (Tables 3 5) and the least percentage error in predicting the water use for the period 1998 to 2001, to represent to water use in the Kathmandu Valley. The partial residual plots of each of the explanatory variables with original data are obtained as shown, for example, for the number of connections in Figure 2a. The non-linearity of the partial residual plots suggests the necessity of transformation of the explanatory variables and dependent variable. The plots, as shown in Figure 2b, make it clear that the log-log transformations of both independent and dependent variables provide better results. This supports the analysis which indicates suitability of log-log form of demand equation for the study area Significant water demand variables The number of connections (X 1 ) is highly correlated with the population. The water use, therefore, will increase with the number of connections unless the per capita use is drastically reduced due to application of various water efficient technologies and conservation measures. The possibility of such reduction is very rare as the per capita water consumption generally increases with living standard, which is expected to improve in the future. The elasticity of water use with respect to the number of connections is indicating that a 1% increase in the number of connections will result in 1.055% increase in the water use. Another significant variable identified is the water tariff rate after the minimum allowed use of 10 m 3 /month (X 2 ). The price elasticity of water is found to be This implies that water use will decrease by 1.67% if the water charge is increased by 10%. This low price elasticity of water may be because of it being charged at a flat rate for a multitude of uses. Moreover, as the increases in the water charges are very low, their effect on water usage is very minor. The average water tariff in excess of 10 m 3 in was NRs 2/m 3 (US$ 0.028/m 3 ); in it rose to NRs 6/m 3 (US$ 0.084/m 3 ); it was increased to NRs 10/m 3 (US$ 0.14/m 3 ) and NRs 12/m 3 (US$ 0.168/m 3 ) in 2001 and 2002 respectively. It can be inferred that the government doubled the water charges within two years in an effort to control the water use in the valley. The results suggest that water pricing could be one of the demand management measures for the valley.

12 584 Water Resour Manage (2007) 21: Fig. 2 Partial residual plot of number of connections (a) before transformation and (b) after transformation of the variables The analysis also shows that the water use is increasing with the ratio of population to the number of university students (X 5 ). The elasticity of water use in the valley with respect to X 5 is about 0.5. This can be interpreted as the decrease in water use with the overall increase in the education level of the people. It is reasonable to assume that education brings increased awareness among the people for conservation and judicious use of water. The relation of rainfall to the water use is found to be negative, indicating the decrease in pipe supplied water use with respect to increase in total rainfall. This result is obvious because rainfall decreases the need for outdoor water use in households for gardening or lawns and people in Kathmandu Valley are beginning to harvest rainwater for various purposes other than drinking. The rainfall elasticity of water use is 0.21, that is, a 10% increase in rainfall will decrease the water use by about 2.1% Effect of length of data series An analysis is carried out to evaluate the effect of length of data series used in model development on the accuracy of the forecast. Figure 3 displays the observed and predicted

13 Water Resour Manage (2007) 21: Fig. 3 Observed and predicted water use using log-log model with different datasets (by the log-log model) water use amounts obtained using different datasets. It can be clearly seen that the accuracy of the forecast decreases as the gap between the forecast year and the data year increases. The percentage error in the prediction of the water use by log-log models with different length of datasets can be visualized from Figure 4. This indicates, as expected, that incorporating recent data in the analysis provides better results. Therefore, for improved forecast, updating the multivariate econometric models as developed in the present study, by inclusion of recent data is crucial and should be carried out regularly by the water utility authorities. Fig. 4 Error in predicted water use using log-log model with different datasets

14 586 Water Resour Manage (2007) 21: Forecast of independent variables Number of connections, water tariff after the minimum allowance, ratio of population to the number of university students and rainfall are the four variables found to be capable of explaining the water use variation in Kathmandu Valley. Therefore to predict the water use in Kathmandu Valley, the forecast of these four variables is mandatory. Since the number of connections and the ratio of population to the number of university students depend on population, this variable is forecasted based on the historical data. Curve fitting the time series data of population is tried with various trend-lines, and the 2-degree polynomial equation is found to fit the data best. Hence, the future population in Kathmandu Valley is estimated using the developed equation. It is, however, assumed that the city expansion is possible and the availability of land is not a constraint. Also, at present, most of the buildings in Kathmandu Valley are less than three storeys, giving the possibility of vertical expansion to accommodate the increased population in the future. The current development scenario in Nepal that is focused and centralized in Kathmandu Valley also supports this assumption. The number of connections is a policy decision to be made by NWSC. It is reasonable to assume that the NWSC will have to increase the number of connections according to the increase in population. Hence, the number of connections is projected taking the constant ratio of population to the number of connections as as per the data of The water tariff is also a policy decision to be made by the concerned authority. As observed from the past records, the average increase in water tariff after the minimum allowance for domestic sector in Kathmandu Valley is around NRs 2.6/m 3 (US$ /m 3 ) of water each year. Hence, water use in the valley is predicted with an annual average increase in tariff of NRs 2.6/m 3 (US$ /m 3 ) of water. The number of students in Kathmandu Valley is forecasted using trend line extrapolation, which seems to be satisfactory as indicated by the R 2 value of Using the forecasted values of population and number of university students, the ratio of the population to the number of university students is estimated. The average absolute variation in the total rainfall in each year from the mean annual rainfall (using the data of the period ) is about 13.5%. Hence, the average annual rainfall of 1475 mm is assumed to be constant during the forecast period Water use/demand forecast Figure 5 compares the observed water use ( ) with the calculated ( ) and forecasted ( ) water use based on the log-log model and the known values of the explanatory variables. The model performed well in estimating the water use in However, it is suggested that such multivariate equations should be updated, by incorporating newly emerging explanatory variables and recent available data, on a regular basis so as to improve the predictions. The developed model is used to forecast the water use/demand in the valley in 2005, 2010 and 2015 by predicting the four explanatory variables as explained above and the results are presented in Table 6. The domestic water demand in Kathmandu in 2015 is estimated at (10 3 m 3 /d) compared to (10 3 m 3 /d) in 2001, an increase of about 20%. Data in Table 1 indicates that the actual per capita water consumption from 1988 to 2001 varies from 38 to 45 L/capita/day. In future it is expected that due to economic growth the per capita use will increase. However, using a value of 45 L/capita/day and the projected population (projected number of connections multiplied by 15.33) as given in Table 6 the

15 Water Resour Manage (2007) 21: Table 6 Projected explanatory variables and water demand Explanatory variable No. of Water tariff Rainfall Water demand Year connections (NRs) Population/students (mm) (10 3 m 3 /day) , , , Fig. 5 Observed and calculated water use water demand in 2005, 2010 and 2015 based on the classical engineering approach would be 86.16, , and (10 3 m 3 per day) which is much higher than the corresponding values of 66.23, and (10 3 m 3 per day) estimated based on the modeling approach of the present study. This clearly indicates that the traditional approach overestimates the water demand and this multivariate approach can help make realistic forecast which can help water utility authority to manage their resources properly and judiciously. 5. Conclusions The study presents a methodology for developing a multivariate demand model for domestic sector considering various factors representing socio-economic characteristics, climatic parameters and public water policies and strategies. The model can accommodate any number of variables given that they are correlated and have logical relationships with the water use or demand. The model carries out a correlation analysis to identify highly correlated independent variables and to eliminate some of them to avoid the effects of multi-collinearity.

16 588 Water Resour Manage (2007) 21: Three functional forms of multivariate equation are incorporated in the model. The multiple regression analysis is carried out using SPSS software package to estimate the values of the intercept and coefficients of the chosen functional forms of demand equation. The model performs two statistical tests, F-test or p-value and t-test to evaluate the fitness of the water demand equation. The final selection of the functional form is done based on the coefficient of determination (R 2 ), adjusted R 2 and the standard error. Partial residual plot analysis is also carried out to without and with logarithmic transformation of independent and/or dependent variables to confirm the selected the functional form of the demand equation. The developed model is user-friendly and the user interface can be installed and run in computers with Microsoft Office set-up, preferably Office 98 or Office The model is menu-driven and the help menu is provided to guide the user through the process of model development and application. The modeling process is illustrated using the water use data of Kathmandu Valley, Nepal. The analysis indicates that the log-log model, using the four determinants of water use, namely; number of connections, water tariff after the minimum allowance, education level and total annual rainfall, is the best among the most widely used functional forms for the domestic sector in Kathmandu Valley. The developed model equation is: lny = ln(x 1 ) ln(x 2 ) ln(x 5 ) ln(x 9 ) The elasticity of water use with respect to the number of connections is indicating that the water use in the valley will increase by 1.055% with a 1% increase in the number of connections. The price elasticity of water use is estimated to be 0.167, which implies that with a 10% increase in the water price, the use will decrease by only 1.67%. Although increasing water price can reduce the water use, it is not significant as the past water tariffs have been low. Public education and awareness also has paramount effects on water use as indicated by its elasticity of about 0.5. It implies that there can be a reduction in water use of 0.5% with an increase in the ratio of total population to the university students of 1%. The analysis concludes that the price of water, public education and information and awareness would be the most effective factors in managing water demand in Kathmandu Valley. Since water demand in the domestic sector is a complex function of different factors which may vary spatially and temporally, planning and management of water supply systems should be a continuous process with due consideration to ever-changing socio-economic conditions, technological changes, improvements in water use efficiency and government policy decisions related to water use and conservation that can affect the use of water by the people. It is, therefore, recommended that NWSC adopt and update multivariate water demand models on a regular basis, as developed in the present study, so as to provide more accurate and reliable demand estimates for the future planning and management of water supply systems. The model, if developed incorporating long and recent datasets, can give better results. However, the main constraint in the application of the present model in developing countries is the availability of a variety of required data. It is therefore suggested that the concerned authorities realize the importance of and give emphasis to the collection of data and information essential to make management modeling a tool for the analysis and development of demand management policies and strategies, which are necessary to address the challenge of water scarcity being faced by many countries.

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