LONG RUN WATER CONSUMPTION Cointegration

Econometric Methods: Single Equation Models < C -Mathematical and Quantitative Methods, Q25 -Water < Q2 -Renewable Resources and Conservation|Environmental Management < Q -Agricultural and Natural Resource

Keywords: BOUNDS TESTING, ARDL APPROACH TO COINTEGRATION, LONG RUN WATER

Long-run water demand estimation. Habits, adjustment dynamics and structural breaks

Antonio Musolesi, Mario Nosvelli

Introduction

While there is a large body of literature on econometric estimation of a water demand equation, which use data on highly disaggregated spatial units over short time periods, only a few studies have focused on European cities, and over the long run.

Many recent works generally converge in concluding that there is a signi…cant, but small, negative e¤ect of price on water demand in many European countries (Nauges and Thomas, 2003;Martínez Espiñeira, 2002[START_REF] Credoc | L'Eau et les Usages Domestiques. Comportements de consommation de l'eau dans les ménages[END_REF]. Furthermore, instantaneous price elasticities obtained in a static framework, in general are smaller than long-run elasticities based on dynamic models. This could be due either to the use of durable equipment, such as washing machines, dish washers, showers (Arbuès et al., 2003) or on slow adaptations in consumer behaviour (Martínez Espiñeira and Nauges, 2004). This paper examines a water demand equation for Milan, for the second half of the twentieth century using annual data and assesses the impact of price on consumption -in both the short-and the long-term -and measures consumption habits. While the use of cross section or (short) panel data for highly disaggregated spatial units allows many degrees of freedom, low collinearity, building complex behavioural models, the use of a long time series has the comparative advantage to be more appropriate for: i) analysing how structural changes occurred in the economy during some decades have a¤ected consumption, ii) comparing long-run and short-run consumption's behaviours, iii) measuring long-run (persistent) habits. The procedure developed by [START_REF] Pesaran | An autoregressive distributed lag modelling approach to cointegration analysis[END_REF] and [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] is adopted. The focus is on testing and estimating a long-run level relationship starting from a dynamic ARDL model which also allows for short run dynamics. Before presenting the results of estimation, the methodology is brie ‡y recalled and adapted to the context of a water demand equation.

Let fz t g 1 t=1 denote a (k + 1)-vector random process partitioned as z t = (c t ;

x 0 t ) 0
where c is a scalar variable representing the natural logarithm of water consumption in year t, x is a k vector of explanatory variables (also expressed in logarithmic form) like the price of water and some other variables such as income, the productive structure, demographic and climatic factors, etc. The data-generating process for fz t g 1 t=1 is assumed to be a VAR model of order p:

(L) (z t ) = " t ; t = 1; 2; :::

where L is the lag operator, is an unknown (k + 1)-vector of intercept coe¢cients, the (k + 1; k + 1) matrix lag polynomial (L) = I k+1 P p i=1 i L i with f i g p i=1 (k + 1; k + 1) matrices of unknown coe¢ cients. Under few standard assumptions, [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] 1 obtain an ARDL conditional ECM model de…ned as:

c t = a 0 + c t 1 + 0 x t 1 + p 1 X i=1 i c t i + p X i=0 ' i x t i + t (2)
It follows from (2) that if 6 = 0 and 6 = 0 there exists a conditional level relationship between c t and x t the long run domestic water demand equation -de…ned by:

c t = 0 + 0 x t + v t (3)
where 0 a 0 = ; = is the k-vector of long run response parameters andfv t g is a zero mean stationary process. Equation (2) can be reformulated in ECM form as:

c t = a 0 + & (c t 1 0 x t 1 ) + p 1 X i=1 i c t i + p X i=0 ' i x t i + t (4)
where & represents the speed of adjustment to the long run equilibrium. For & = 0 there is no adjustment to the long run equilibrium. If & < 0 then the long-run relationship between the levels of c t and x t is stable since there is adjustment to the long-run equilibrium. For example, for & = 1 the disequilibrium in the previous year is completely corrected in the current year. According to [START_REF] Pesaran | An autoregressive distributed lag modelling approach to cointegration analysis[END_REF] and [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF], the ARDL modelling approach to cointegration requires the following two steps. In the …rst step, the absence of any long run relationship between c t and x t is tested through exclusion of the lagged level variables c t 1 and x t 1 in (2) :

Critical values of the test statistics are tabulated in [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF]. In the second step of the analysis -following estimation of the ARDL model de…ned by (2), which gives the short run impacts -we obtain the long run parameters of (3) from the coe¢ cients of the lagged variables of (2); …nally, by re-parameterising (2), the ECM model ( 4) is estimated and the speed of adjustment to the equilibrium is determined.

This econometric framework is very useful for our research objectives. Firstly, it provides statistical information on consumption habits and on their modi…cation over- 2 Probably, the main limitation of our data is that they do not allow the decomposition of total water consumption with respect to di¤erent typologies of users (residential, industrial, commercial) except for few years at the end of the period. In a micro-econometric setting with disaggregated data, water demand for water of each typology of user is estimated in a speci…c way. While residential water demand is obtained by maximising consumers' utility, industrial water is introduced as an input of production (generally it is considered as a substitute of labour and complementary to capital; [START_REF] Renzetti | Estimating the Structure of Industrial Water Demands: The Case of Canadian Manufacturing[END_REF] and it is often estimated in a cost function framework. In spite of such a limitation, we believe that our data allow us to provide interesting new information about long run structural dynamics of total water consumption in a relevant policy scenario, which is that of a big European city. Moreover, we did some non reported robustness checks using an alternative de…nition of the dependent variable relative to consumption since we add total employment to total population within the denominator. This alternative de…nition could be relevant for studying industrial water , 2007). Since the foundation of Milan's water network, the price of water in Milan has followed a nonlinear tari¤ structure. The choice of which price to include in the demand function is controversial. Economic theory suggests the use of marginal prices, while some studies on water consumption point out that users react to average prices rather than marginal prices [START_REF] Credoc | L'Eau et les Usages Domestiques. Comportements de consommation de l'eau dans les ménages[END_REF]. An approach commonly adopted in the face of a multiple-block price structure is to include in the demand function both the marginal price and a di¤erence variable in order to take account of the intramarginal structure [START_REF] Nordin | A Proposed Modi…cation of Taylor's Demand Analysis: Comment[END_REF].

In our study, it is not possible to calculate actual average and marginal prices,

given the lack of data relative to disaggregated water consumption over the period considered. Indeed, the use of ex post average prices produces a problem of simultaneity, and predetermined information is required in order to identify the demand equation [START_REF] Taylor | The Demand for Electricity: A Survey[END_REF]. Despite the continuing debate among researchers on the most appropriate price variable to adopt, recent surveys show that results in terms of sign and intensity of price elasticity do not depend hugely on the type of price variable selected (Arbues et al., 2003). In a …rst stage, we employ the value (in real terms) of the …rst block tari¤ as a proxy for water price. This block accounts for the largest share of consumption 3 and represents the core of the entire tari¤ plan, which applies to all typologies of users.

For these reasons, the majority of users should perceive the e¤ective water price to be the price of the lower block, rather than the ex-post average or marginal prices that also account for the other blocks, representing more speci…c typologies of consumers such as small and large …rms. In a later stage, our analysis focuses on the price structure and its evolution.

RAINFALL. The variable rain t represents rainfall (per year), measured in millimetres per year (Osservatorio Metereologico di Milano Duomo, 2002).

PRODUCTION ACTIVITY. Production activity prod t is included in the model to take account of non-domestic consumption. Here, we adopt di¤erent proxies for the productive structure:

Description of the time series

The time series plots of the …ve variables included in the VAR are given in …gure 1. The number of employees in services (SERV) was fairly stable until the 1970s when a strong tertiarisation process occurred, resulting in a major increase in service employees that has continued. Thus, the share of service employment in the population (SERV_EMP) shows a U shaped trend: it decreased up to the beginning of the 1970s and increased thereafter.

Total employment (TOT=SERV+IND) shows fairly linear growth over the entire time period. Up to the 1970s it was driven by industry and after that time received a boost from services employment. The share of total employment in the population (TOT_EMP) follows a U shaped trend.

Preliminary results

4.1 Detecting the order of integration: are the time series really nonstationary?

Before testing for the existence of a long run relationship and estimating the model, we conducted a preliminary statistical analysis in order to detect the order of integration of the variables. Since the Augmented Dickey-Fuller (ADF) test was introduced, a number of alternative tests have been proposed that improve the size and power of the ADF. Elliott et al.'s (1996) quasi-di¤erencing variant through local GLS detrending of the data (DFGLS) yields the greatest advantages with respect to alternative approaches.

14 propose the so called KPSS test in which the unit root is the null hypothesis to be tested, and the way in which classical hypothesis testing is carried out ensures that the null hypothesis is accepted unless there is strong evidence against it.

When we apply the KPSS test (column iv, table 1) we cannot cross-validate the outcomes of ADF and DFGLS unit-root tests since for many of the time series under consideration the KPSS test strongly supports the stationarity null hypothesis.

The failure of the ADF and DFGLS tests to reject the unit root null hypothesis could be due to breaks or non-linearities in the trend function. Based on …gure 1 there is reason to suspect that there may be a break in the trend functions for C, TOT_EMP and SERV_EMP. We therefore applied the [START_REF] Zivot | Further evidence of the great crash, the oil-price shock and the unit-root hypothesis[END_REF] 15 The idea that a time series can be stationary around a breaking deterministic linear trend, as in [START_REF] Zivot | Further evidence of the great crash, the oil-price shock and the unit-root hypothesis[END_REF] and [START_REF] Perron | The great crash, the oil price shock, and the unit root hypothesis[END_REF], was generalised by [START_REF] Bierens | Testing the unit root with drift hypothesis against nonlinear trend stationarity, with an application to the US price level and interest rate[END_REF], who adopted the notion of integration around a deterministic nonlinear trend. The notion of non-linear trend stationarity can usefully be applied to the C, PRICE, TOT_EMP, SERV_EMP series, which present some breaks. It can also be applied to the income (M) series, which is clearly non-linear.

Here, we focus on the [START_REF] Bierens | Testing the unit root with drift hypothesis against nonlinear trend stationarity, with an application to the US price level and interest rate[END_REF] (5) [START_REF] Bierens | Testing the unit root with drift hypothesis against nonlinear trend stationarity, with an application to the US price level and interest rate[END_REF] We conducted the non-linear ADF test on the basis of the auxiliary regression (1) with a lag length p chosen using the AIC and the Chebishev time polynomial order m, settled in accordance with the degree of non-linearity of the series. 5 . In order to To sum up, based on the evidence from the di¤erent unit root testing procedures, we can conclude that P, TOT_EMP, SERV_EMP, M and RAIN are stationary time series, and C is …rst order integrated.

Testing for the existence of a long-run relationship

As in the cointegration analysis, after checking the order of integration of the variables and before estimating the model, we need to test for the existence of a long-run [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF]. However, the asymptotic distribution of the F-statistic is not standard, irrespective of whether the regressors are I(0) or I(1). [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] give two sets of critical values: one assumes that all regressors are purely I(1), and the other is computed assuming that all regressors are purely I(0), thereby providing for all possible classi…cations of the regressors as purely I(0), purely I(1) or mutually cointegrated.

To determine whether a deterministic trend is required we estimated the ARDL model with and without a linear trend, for p=1,2,3 and p chosen using the AIC. We found the time trend to be insigni…cant in both equations. We therefore chose not to introduce a linear trend. It should be noted also that a visual inspection of the time series reveals the absence of common trends in the variables.

With respect to the intercept, we concentrate on the unrestricted intercept model 2001) at the conventional signi…cance level: the hypothesis that there is no level water demand equation is rejected irrespective of the lag order and the proxy used for the productive structure.

(
5 Estimation results

Benchmark results

The set of speci…cations presented below takes account of the inclusion of di¤erent variables relative to the productive structure, which we suppose to be a crucial determinant of water consumption over the long-run. In a …rst speci…cation, we include a variable that represents the average intensity of productive activity and is equal to the share of total workers in the population (TOT_EMP). In a second estimation we focus on services intensity -measured as number of service workers in total population (SERV_EMP) -since the service sector historically has increased in Milan as in other su¢ ciently small such that the "unrestricted ECM" is not over-parameterised. The orders of the ARDL (p1, p2, p3, p4, p5) were chosen using the Akaike Information Criterion (AIC) subject to a maximum lag of 3, which seems a reasonable point of departure giving the (annual) frequency of the data and the number of observations available for estimation. For example, for both the speci…cations with TOT_EMP and with SERV_EMP as proxies for productive activity, the AIC results in an ARDL

(1,0,3,2,0) speci…cation. The "basic" ARDL then becomes:

c t = a 0 + c t 1 + 1 price t + 2 rain t + 3 rain t 1 + 4 rain t 2 + (6)
+ 5 rain t 3 + 6 tot_emp t + 7 tot_emp t 1 + 7 tot_emp t 3 + + 8 m t + t while the conditional ECM can be written as:

c t = a 0 + & (c t 1 0 x t 1 ) + ' 1 price t + ' 2 rain t + ' 3 rain t 1 + (7) +' 4 rain t 2 + ' 5 tot_emp t + ' 6 tot_emp t 1 + ' 7 m t + t
and long run level water demand is: The results from estimating the demand equation are presented in tables 5 and 6.

c t = 0 + 1 price t + 2 rain t + 3 tot_emp t + 4 m t + v t (
All speci…cations pass the set of diagnostic checks for serial correlation, non-normal errors, functional form and heteroskedasticity.

The estimated coe¢ cients and standard errors obtained from estimating the model with all explanatory variables, are presented in columns vi to viii, while the results obtained excluding income from the list of the explanatory variables as suggested by a general-to-speci…c modelling strategy 8 are in columns ii to v.

SHORT RUN ESTIMATES. Among the results obtained from the di¤erent econometric speci…cations, four are relevant.

The …rst, very clear result, concerns the variable for lagged water consumption.

In each econometric speci…cation it presents a positive coe¢ cient around 0.7, always highly signi…cant. It clearly reveals that persistent habits are crucial for water consumption.

The theoretical bases for estimating a dynamic water demand are described in the literature. Tversky and Khaneman (1991) point out that current consumer utility depends, not only on present consumption levels, but also on past consumption. Nauges and Thomas (2003) show that a dynamic model of residential water consumption can 8 Note, that since price and income present some degree of collinearity (see also Martinez Espiñeira, 2007) we also estimated the model including income, but excluding price as a regressor. The results (which are not reported here) con…rm that income is never signi…cant and presents an estimated be derived from a structural optimisation programme solved by the communities.

TABLE 5 ABOUT HERE

These theoretical …ndings imply that ignoring habits in the estimation of a consumption function could lead to problems of mis-speci…cation and biased results.

Empirical studies show that water consumption habits depend either on the use of durable equipment, such as washing machines, dishwashers, showers (Martínez Espiñeira, 2002;Arbuès et al., 2003) or on slow adaptations in consumer behaviour (Martínez Espiñeira and Nauges, 2004).

Among the studies that focus on the role of habits 9 , we would draw attention to work that estimates autoregressive models, for example, Nauges and Thomas (2003) who …nd lower values for the estimated autoregressive component. However, direct comparisons are not easy, given the panel data structure of these studies.

A second and related point concerns the parameter associated with the ECM, which, as previously discussed, represents the speed of adjustment following a disturbance, for the restoration of equilibrium in the dynamic model. This is always signi…cant and is further evidence of a stable long-term relationship. In our estimates, the estimated ECM coe¢ cient is about -0.2, which corresponds to a correction of 9 See, among others, [START_REF] Carver | Short run and long run e¤ects of price on municipal water use[END_REF] , 2004). There are also other explanations, which can be summarised as follows. First, consumers' lack of awareness of the cost of water may cause income elasticity to be close to zero [START_REF] Dalhuisen | Price and income elasticities of residential water demand: a meta analysis[END_REF]. Second, interesting results regarding income e¤ects typically emerge from estimating di¤erent levels of demand for di¤erent levels of income/wealth using micro data (Hanke and de Maré, 1982;[START_REF] Agthe | Equity, price elasticity, and household income under increasing block rates for water[END_REF]. Third, income e¤ects on water demand can only be indirect since they depend, at least partially, on the characteristics of the water-using capital stock (Martínez Espiñeira, 2007).

We now turn of our preferred estimates excluding income and presented in columns ii to v. Long-run price coe¢ cients are signi…cant in every speci…cation, and with a negative sign, which con…rms previous …ndings. The clearest result in terms of magnitude and signi…cance, emerges from the speci…cation where we introduce employment in the service sector (SERV_EMP) as a proxy for productive activity (the estimated coe¢ cient of price is -0.145 with a p value equal to 0.02). In general, in our estimates, the intensity of long-run price elasticities is lower than in other European studies using 26 dynamic models in the range (-0.50 to -0.30).

TABLE 6 ABOUT HERE

The underlying and relevant evidence arising from comparison of the short and long-term results, is that water users are more likely to adjust their consumption following a permanent increase in tari¤s than in response to a one year shift. This could derive from the speci…c typology of the good we are considering, since water is essential for life and a large component of its consumption is not reducible at least in the short-run (Arbues et al., 2003;Haneman, 2006). One major e¤ect on long-run water consumption is due to the renewal of water using capital stock, which refers to both domestic and industrial durable equipment (Nauges and Thomas, 2003;Martinez-Espineira, 2007). The slow rate of change in the stock of appliances, and the role of well entrenched habits, will contribute to a process of slow adaptation to price changes.

The next result concerns rainfall, which is signi…cant in almost all cases. The estimated coe¢ cients have negative signs, in line with the literature [START_REF] Garcia | Estimation of residential water demand: a state-of-the-art review[END_REF]Martinez Espiñeira, 2002[START_REF] Credoc | L'Eau et les Usages Domestiques. Comportements de consommation de l'eau dans les ménages[END_REF]. This mostly re ‡ects the e¤ect of climate on water consumption due to increases in temperature and evapotranspiration rates [START_REF] Renzetti | The economics of water demand[END_REF]. In this respect, outside water use, mainly for irrigation and street cleaning, plays a signi…cant role, which is included in our dependent variable. A third result relates to the group of variables used to analyse the impact of the productive structure. The variable that measures the average intensity of total productive activity (TOT_EMP) shows a positive and signi…cant long-run elasticity, which is estimated at 0.16. The role played by water in industrial processes is crucial, despite the increasing e¢ ciency of water consumption induced by technological innovation [START_REF] Renzetti | The economics of water demand[END_REF] and the growing relative importance of services. This latter aspect is highlighted in the model that includes intensity of the service sector (SERV_EMP) as an explanatory variable: the estimated coe¢ cient is signi…cant and positive (0.11), but is lower than that for total productive activity referred to above.

Our analysis estimates the impact on water demand of the currently prevailing sector, through the inclusion of the variables IND_EMP5074 and SERV_EMP7501.

The estimated coe¢ cients are highly signi…cant and their magnitudes are higher with respect to the estimated parameters obtained through the inclusion of total productive sectors, and the service sector alone. This shows very clearly that changes in the productive structure are a relevant explanatory factor in the evolution of long-term water consumption.

The cross validation exercise provides con…rmation of these results. It estimates the impact of services in the …rst period, and of manufacturing in the second period.

The estimated coe¢ cients are close to zero and highly non-signi…cant. In the previous section we focused on the price relative to the …rst block tari¤, which includes all typologies of consumers: households, and small and large …rms.

In this section we focus on the modi…cation to the price structure over time, and its impact on water consumption. When the Milan water network was …rst established, prices were based on a three-block structure. In the 1970s, this structure was substantially modi…ed, and up to 1975 water users experienced decreasing block rates; however, since 1976, the price structure has been based on increasing block rates (…gure 3). the "European range" 10 . Second, the inter-block price di¤erences are higher in the second half of the period because of the sharp increase of the price of the third block following the 1976 changes to the price structure.

In order to investigate how these changes in price structure a¤ected consumption, we re-estimated the demand equation and simultaneously include the two prices variables relative to the two periods. We built the …rst price variable by making the price interact with a step dummy DU5075 that takes the value 1 if year< 1975; we built the second by multiplying the price by a step dummy DU7601 that takes the value 1 if year 1976. In a …rst set of estimations we include the prices for the …rst block (P1) for the two periods 1950-1975 and 1975-2001 (P15075 and P17601). In the next estimations we replace these variables with the prices for the third block (P3) for the same two periods (P35075 and P37601). As before, we estimate three di¤erent speci…cations according to the three proxies for productive structure.

The results of our estimations show that both short and long-run price elasticities, relative to the …rst block tari¤ (table 7, columns ii, iii and iv), are higher (about 1.35 times) for the period 1950-75, which is characterised by a higher …rst block price, than 1976-2001 under a decreasing block structure. These results provide clear evidence that water users reacted to the mid-1970s'modi…cations to the water price structure. We now turn to discussion of the estimates obtained using the price variables relative to the third block (P35075 and P37601, in columns v, vi and vii of table 7).

As expected, the results are reversed: consumption elasticities with respect to the price corresponding to the third block are higher (about 1.7 times) in the last period 1976-2001, under an increasing block tari¤, than in the …rst period. The e¤ect on price elasticities of changing the price structure, is greater for the third than the …rst block. Moreover, the third block short and long-run price elasticities for the period 1950-75, under decreasing price rates, are often not signi…cant.

TABLE 7 ABOUT HERE

To better understand these results we need to consider that the …rst block accounts for all types of users, both domestic and …rms, whereas the third block involves only large consumers, typically industrial …rms.

According to our …ndings, even though the price change stimulated reactions from both small and big consumers, its most incontrovertible e¤ect was on …rm behaviour, since variation in price elasticity is most evident for large scale consumers. This could be due to the fact that demand for industrial water also depends both on the degree of substitution from other inputs and on the level of other prices. We can assume that in the …rst period, 1950-75, water was a very convenient input with respect to some other factors of production. 6 Concluding remarks and some comments on water policy

This paper estimated a water demand function on a 50 year sample for the city of Milan. Allowing for trend break stationarity or non-linear trend stationarity, we …nd that the evidence does not support the unit root hypothesis for many time series.

Accordingly, the standard cointegration analysis is not appropriate and therefore we adopted an alternative estimation and testing procedure. We focused especially on the so-called bounds testing approach proposed by [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] which can be applied irrespective of the level of integration of the variables and which provides a useful modelling strategy given that dynamics are important in the estimation of a water demand equation (Nauges and Thomas, 2003).

We estimated the e¤ect on water consumption per user, of di¤erent factors: price, climate, income and productive structure, over the period 1950-2001.

The …rst relevant result is related to the estimated parameter of lagged water consumption, which is positive and highly signi…cant, showing the existence of well entrenched habits. Alternatively, we can say that after a disturbance, water users adjust their levels of consumption towards a long run equilibrium, only slowly. A second important …nding is evidence that long run price elasticity is higher than short run elasticity. This could be due to the intrinsic economic characteristics of water -since it is essential for life and in part, not substitutable -and to the long time needed to change water using equipment. We …nd that long run price elasticities are signi…cant even though they are smaller than those obtained in other studies on dynamic water demand.

The tari¤ scheme introduced in the mid 1970s promoted substantial changes in the pricing structure, from a decreasing to an increasing block structure. These changes have led to a decrease in the price elasticity for the …rst block and an increase in the elasticity for the third block. Moreover, the third block price elasticity was not signi…cantly di¤erent from zero before the change and become signi…cant thereafter.

The third block is related to large scale consumption typically attributable to …rms, which seems to show a signi…cant reaction to changing price schemes.

These results on price elasticities present quite clear implications for policy. Price policies can have an e¤ect on water consumption over the long term. Water price policies could be fundamental to changes in behaviour since, for all types of consumers, households and …rms, they signal the need to adopt new consumption habits in the face of an increasingly scarce and essential good such as water.

However, since our results show that habits play an important role in water consumption, other policies will need to be combined with price policies in order for con-33 Notes.

: …rst-di¤erence operator. *: rejection of null hypothesis at 5% level. DC: deterministic component. c is a constant while t is a linear time trend. ADF: Augmented Dickey-Fuller test. DFGLS: quasi-di¤erencing variant the ADF proposed by Elliott et al. (1996) through local GLS detrending of the data.

ADF,DFGLS: the order of the auto-regressive component has been chosen with the AIC. KPSS uses the automatic bandwidth selection procedure proposed by [START_REF] Newey | Automatic lag selection in covariance matrix estimation[END_REF].

1

Page The ARDL estimated coe…icients of p15075,...,p37501 are from the "basic" ARDL speci…cation The long run coe¢ cient are estimated from the level relation. ... Non reported ARDL estimates for saving space. "Total": the demand equation is estimated using tot_emp as proxy for the productive activity. "Service" serv_emp is used. "Ind&Serv" means that both ind_emp5074 and serv_emp7501 are used. "Cross val.": inclusion of serv_emp5074 and ind_emp7501. *,**,*** signi…cant respectively at 10% 5% and 1% level.

1

  [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF]make a non-standard assumption that: The roots of I k+1 P p i=1 i L i = 0 are either outside the unit circle jzj = 1 or satisfy z = 1: This enables the elements of z t to be I(1

  time: i) in the ARDL speci…cation, the parameter associated to the auto-regressive component indicates the persistence of the series and thus provides a measure of habits; ii) the parameter associated to the ECM term gives useful information on adjustment dynamics; iii) long-run estimated parameters indicate how a modi…cation in the price, personal income, etc modify long-run habits in consumption. Second, it allows comparing long-run and short-run agents'behaviours by comparing long-run and short-run estimated parameters. Finally, it is a suitable framework for estimating a model irrespective of the degree of integration of the variables of interest.We use an annual time series data set which covers the period 1950-2001 for the city of Milan. All variables have been transformed in logarithmic form.3.1 Variables and sourcesWATER CONSUMPTION. The annual data on daily average water consumption per-user -which is the sum of residential, commercial and industrial consumptions 2was collected by the municipal water network Acquedotto del Comune di Milano (c t , in litres). The explanatory variables used to estimate the demand equation are:

:

  considers the null unit root hypothesis with drift against the alternative of non-linear trend stationarity, and develops several test statistics: b t (m) is the t-statistic of the estimated coe¢ cient b ; Where under the null = 0, while under the alternative < 0. The most important fractiles of the null distribution of b t (m) and b A (m) for m = 1; : : : ; 20 are tabulated in Bierens (1997) on the basis of 10000 replications of a Gaussian random walk with sample size 500. Since these two tests do not take account of all the available information, Bierens proposes two other tests b F (m) and b T (m) for the joint hypothesis that and the last m components of the parameter vector are zero. The b F (m) is a conventional F-test whereas b T (m) is a 2 test based on non-conventional testing principles.

  sise distortion we simulate the critical values using wild bootstrap 6 based on 2000 replications. TABLE 3 ABOUT HERE Looking at the asymptotic critical values we can reject the null unit root hypothesis for all variables excepting consumption -C. The results also show evidence of substantial sise distortion and the simulations (F (m) and T (m)) still indicate the rejection of the unit root hypothesis for PRICE, TOT_EMP, SERV_EMP and M at the 5% signi…cance level.

7

  See Pesaran and Shin (1999) and[START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] for a more detailed discussion of the economies. In the third and fourth estimations we analyse and control for the role of the prevailing sector in di¤erent periods during the 50 years considered. This last approach is aimed at studying the role of structural change in the productive structure and its e¤ect on long-run water demand. In the third speci…cation we introduce two variables relative to productive activity. The …rst is built by multiplying the industry intensity (IND_EMP) with a step dummy DU5074 that takes the value 1 if year< 1975; the second is obtained by multiplying the service intensity variable (SERV_EMP) by a step dummy DU7501 that takes the value 1 if year 1975. We label these two variables IND_EMP5074 and SERV_EMP7501 respectively. The idea behind this modelling is that, given the trends in relative sise observed for the di¤erent macro-sectors, industrial production will push non-domestic water consumption in the …rst half of the sample, while in the second half it will be services that have the main in ‡uence. The fourth speci…cation cross-validates the third speci…cation by estimating the impact of services in the …rst half of the time period (SERV_EMP5074), and the impact of manufacturing in the second half (IND_EMP7501). We expect the values and signi…cance of the estimated coe¢ cients to be lower.The speci…cation de…ned in eq. 2 is based on the assumption that the disturbances are serially uncorrelated. It is therefore important that the lag order p is selected appropriately. There is of course a delicate balance between choosing a p that is su¢ ciently large to allow serially uncorrelated residuals and, at the same time

  of the price structure over time and its impact on water consumption

FIGURE 3 Page

 3 FIGURE 3 ABOUT HERE

  10 E.g., in 2001, the price of the third block was 0.41 euros per m3, while in many other European cities it was 1-2 euros(UNEP, 2004

Finally

  second half of the time period, might derive from technological innovation, which should increase water utilisation e¢ ciency and the capacity to save water.

Figure 1 .Page

 1 Figure 1. The time series (logarithms)

Figure

  Figure 2. Unit Root tests with break

-

  .273*** -.256*** -.258*** -.303*** -.230*** -.249*** -.207**The estimated coe…icients of c(-1),...,ind_emp7501(-3) are from the "basic" ARDL de…ned in equation8The coe¢ cient of the ecm term is estimated from the conditional ecm de…ned by equation 9. "Total": the demand equation is estimated using tot_emp as proxy for the productive activity. "Service": serv_emp is included. "Ind&Serv" means that both ind_emp5074 and serv_emp7501 are used. "Cross val.": inclusion of serv_emp5074 and ind_emp7501. *,**,*** signi…cant respectively at 10% 5% and 1% level.

7

  Page

10 Page 11 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

	tot_emp log (total employment=population) ;
	ind_emp log (employment in manufacturing industry=population) ;
	serv_emp log (employment in services=population) ;
	which measure respectively, the intensity of total employment, manufacturing em-
	ployment and services employment (Crenos, 2007).
	F o INCOME. Another variable suggested by economic theory is personal income,
	r even if in empirical studies on water consumption, it is often non-signi…cant. Time
	P series of personal income have not been gathered in Italy at the urban level. In a
	log-linear speci…cation, the parameters that are of interest are elasticities, which are e e r invariant to a change of scale. Under the quite reasonable assumption that during
	the …fty the period covered by our analysis, the evolution of personal income has been similar R for Milan and Italy, then per-capita income for Milan can be proxied by per capita e w e i v income for Italy, m t ( Heston et al., 2006) 4 .
	years considered in this analysis and depicted in …gure 1 concern total water consumption and not
	only residential/domestic water consumption. A recent comparative report on water consumption in
	Europe shows that Milan, with around 350 litres per day, presents a very high per capita domestic
	consumption compared to other European cities (Ambiente Italia, 2006).

3

In the year 2001, the most relevant part of the water bill regards the …rst block price rate which considers per capita consumption within 350 litres per day. The average water supplied for domestic consumption by Milan's water network is around 353 litres per day, obtained by 172,7 million m3 supplied in the year to the 1.339.933 citizens. Statistics about water consumption relative to

11 Page 12 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

	FIGURE 1 ABOUT HERE
	WATER CONSUMPTION. Daily per user water consumption increased up to
	the 1960s, when it stabilised at around 500 litres. There are some exogenous events,
	not accounted for here, that might have had some short run e¤ects on consumption.
	These are primarily supply side factors, mostly linked to modi…cations to the struc-
	ture and management of water systems. The early 1960s also saw the emergence of F o pollution problems in Milan's water system. This led to the closure in 1963 of 37 out
	r of 55 wells. In the 1974 another discovery of serious water pollution, mainly due to
	P the intensive industrialisation, was accompanied by a lowering of the water table due
	e to over-exploitation (Bianchi, 1989). The Milan water network solved the pollution e r problems in the 1980s by temporary closure of the polluted wells. These closures did
	not a¤ect water supplies, which were guaranteed by more intensive utilisation of the R unpolluted wells. However, wells closures could reasonably impact on operating costs e v and, indirectly, on water price. i e PRICE (…rst block). Up to 1974 the nominal price of water was set by a w government committee, with no reference to management costs. In 1984, a law estab-
	4 Obviously there are many other variables involved in water demand, such as the demographic lished that any increase in the price of water should not be higher than the forecast
	structure of the city. Unfortunately, yearly data on this and other variables are not available; many in ‡ation rate. The real price of water shows a generally slowly decreasing trend, with
	time series that would have provided interesting data are collected by the national census only every dramatic shocks in the 1970s and 1980s primarily due to the rate of in ‡ation in Italy,
	ten years. which in these periods went up to 20%.

12 Page 13 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

	RAINFALL. The rainfall series shows a very slightly decreasing trend with rel-
	atively constant variability over time: about 80% of the data fall within the interval
	750-1,250 mm. There are some particular climatic anomalies in three years in the
	period: 1587 mm in 1950, 1583 mm in 1958 and 643 mm in 1990.
	INCOME. Real per-capita income for Italy shows a general increase around a
	logarithmic trend with some shocks, corresponding to the oil crises.
	F o r PRODUCTION ACTIVITY. The variables for productive activity, IND_EMP, F o SERV_EMP and TOT_EMP were plotted. The process of heavy industrialisation r
	P (IND is the number of workers in industry) that occurred after the Second World P
	e e r War was coupled with a signi…cant increase in population (POPU) up to the end of the 1960s. From the early 1970s a process of de-industrialisation was accompanied by e e r
	R e v i e w a decrease in population, which also occurred in many other Italian cities mainly do to the declining birth rates and increased emigration from downtown to peripheries outside of the municipal territory. Consequently, the share of industrial workers in the population (IND_EMP) re-mained fairly stable over the entire period. However, we can see that during the 1960s R e v i e w
	this share shows a generally decreasing trend, while in the 1990s it increases due to
	the stabilisation of employee numbers in the industrial sector which was coupled with
	a decrease in total population.
	13

  The ADF and the DFGLS tests (table 1) both provide strong evidence favouring the unit root hypothesis for many time series. The ADF test indicates that all the variables, except RAIN, are non-stationary, while the DFGLS test shows both RAIN and IND_EMP to be stationary.

	Submitted Manuscript
	F o
	r
	P
	e e r
	R e v i e w

Page 15 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

TABLE 1 ABOUT HERE However, since Neslon and Plosser (1982) seminal work, many econometricians have shown that, for many economic time series, standard unit root tests fail to reject the null hypothesis of a unit root. For example, Kwiatkowski et al. (1992) argue that the standard unit root tests are not very powerful against relevant alternatives. They

  test, which allows for a break in the trend, with the result that two (TOT_EMP and SERV_EMP) out of three of the series are trend break stationary. Similarly, we can apply the Clemente et al. (1998) Innovative Outlier test to the price (PRICE) series. This test allows for two structural breaks -which occur gradually -in the mean of the series. Under the innovative outlier hypothesis, the price of water can now be considered stationary around a mean that changes in 1974 and 1985. (Results are presented in table 2 and …gure 2).

	F o
	r
	TABLE 2 ABOUT HERE P r e e FIGURE 2 ABOUT HERE
	R e v i e w

Page 16 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  On the other hand, if m is too high it may also reduce the power of the test by estimating super ‡uous parameters. We conducted the test using di¤erent values for m; the results were similar and are

	Submitted Manuscript
	relationship (long-run water demand). The results from the previous section render
	cointegration tests and estimation inappropriate. We therefore test for the existence
	of a level relation between water consumption and its determinants by computing
	the F-statistic to test the signi…cance of the lagged levels of the variables in the
	error correction speci…cation of the underlying "conditional ECM" model de…ned
	by equation 2, as suggested by
	F o
	r
	P
	e e r
	R e v i e w
	available upon request.
	6 Draw the model errors from normal distribution with zero mean and variance from the squared
	OLS residuals.

18 Page 19 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  model III) which does not incorporate the constraint a 0 = ( ; 0 ) .

	Submitted Manuscript	Page 20 of 50
	III do not incorporate such restriction 7 .
	TABLE 4 ABOUT HERE
	Table 4 presents the F-statistics for testing the existence of a level water demand
	equation. In all cases the F-statistics fall outside the critical bounds tabulated in
	o F Pesaran et al. (
	r
	P
	e e r
	R e v i e w
	It should
	be noted that the Data Generating Processes for the models with restricted and
	unrestricted intercept are identical while estimation and hypothesis testing for model
	19

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Page 34 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  Nelson CR, Plosser CI. 1982. Trends and random walks in macro-economic time series. Journal of Monetary Economics 10 : 139-162. DOI: 10.1016/0304-

	Page 35 of 50 Page 39 of 50					Submitted Manuscript Submitted Manuscript Submitted Manuscript Submitted Manuscript	Page 38 of 50
		6.3					-1.5
	sumption levels to be moderated. Such policies could include information campaigns. Hakkio CS. Rush M. 1991. Cointegration: how short is the long-run? Journal of Kwiatkowski D, Phillips PCB, Schmidt P, Shin Y, 1992. Testing the null hy-
	As in other European countries, these campaigns should be aimed at informing house-holds about both the prices/costs of water supply and about the most e¢ cient ways International Money Finance 10 : 571-581. DOI: 10.1016/S0261-5606(99)00028-5 pothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?. Journal of Econometrics. 54 : 6.1 6.2 -3 -2 log water consumption -2.5 log price (first block)
	to use and save water. 159-178. DOI: 10.1016/0304-4076(92)90104-Y 6 -3.5 Haneman WM. 2006. The economic conception of water, in Water Crisis: myth 1950 1960 1970 1980 1990 2000 1950 1960 1970 1980 1990 2000
				year			year
		or reality? (Eds.) P.P. Rogers, M.R. Llamas, L. Martinez-Cortina, Taylor & Martínez Espiñeira R, Nauges C. 2004. Is all domestic water consump-7.4 10
	log rain	F o F o F o F o Francis plc., London. Hanke SH, De Maré L. 1982. Residential water demand: a pooled time-series F o 7.2 tion sensitive to price control?. Applied Economics. 36 : 1697-1703. DOI: 10.1023/A:1014547616408 F o F o 9.5 F o 6.8 7 9 log income
	.2 .3	r P e e r r P e e r r P e e r r P e e r cross-section study of Malmö, Sweden. Journal of the American Water Resources Association. 18. 621-625. DOI: 10.1111/j.1752-1688.1982.tb00044.x Heston I, Summers R, Aten B. 2006. Penn World Table Version 6.2. Center for r P e e r r r r 6.6 Martínez Espiñeira R. 2002. Residential Water Demand in the Northwest of Spain. Environmental and Resource Economics. 21 : 161-187. DOI: 10.1023/A:1014547616408 P e e r P e e r 6.4 1950 1960 1970 1980 1990 2000 8.5 1950 1960 1970 1980 1990 2000 P year year e .4 r e .6
	.1 0 -.1 .4	R e v i e w R e v i e w R e v i e w R e v i e w International Comparisons of Production, Income and Prices at the University of Pennsylvania, September, http://pwt.econ.upenn.edu/ ISTAT. 2007. Il valore della moneta dal 1861 al 2006. ISTAT. 18 november 200 Klein B, Kenney D, Lowrey J, Goemans C. 2007. Factors in ‡uencing res-R e v i e w .2 Martínez Espiñeira R. 2007. An Estimation of Residential Water Demand Using Co-Integration and Error Correction Techniques. Journal of Applied Economics. 10 : 161-184 Naimzada A.K., Tramontana F. 2008. Un modello dinamico del consumatore con razionalità limitata e analisi globale. Economia Politica. April. 1 : 59-94. R e v i e w R e v i e w 0 R -.2 e 1950 1960 1970 1980 1990 2000 -.4 1950 1960 1970 1980 1990 2000 v year year i ind_emp ind popu popu serv_emp serv e w
	.2	idential water demand: a review of the literature, University of Col-
		DOI: 10.1428/26456
	0	orado, Western Water Assessment, Working Paper, 29 february 2008,
	-.2	http://wwa.colorado.edu/resources/ water_demand_and_conservation/ liter-Nauges C, Thomas A. 2003. Long-run study of residential water con-
		1950	1960	1970	1980	1990	2000
		year sumption. Environmental and Resource Economics. 26 : 25-43. DOI: popu tot_emp ature_review_version_1_12_07.pd tot
		10.1023/A:1025673318692
							34 37 38
	Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 1 :

 1 Unit root tests

	Variable	DC ADF	DFGLS KPSS
	c	c, t -3.101	-1.704	0.115
	c	c	-3.668*	-0.852	0.39
	c	c	-6.157 *	-6.096*	0.326
	price	c, t -1.105	-1.173	0.131
	price	c	-5.143 *	-5.187*	0.109
	rain	c, t -6.808 *	-5.538*	0.063
	rain	c	-11.717 * -8.551*	0.083
	m	c, t -1.030	-0.171	0.250
	m	c, t -6.780 *	-4.620*	0.146
	tot_emp	c,t	-1.355	-1.135	0.238*
	tot_emp	c, t -5.329*	-2.813	0.103
	ind_emp	c	-1.637	-2.139	0.245
	ind_emp	c	-5.5718*	-5.637*	0.236
	serv_emp	c, t -2.5413	-1.819	0.233*
	serv_emp c, t -2.017	-1.840	0.121

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Table 2 .

 2 Testing for unit root with breaks

				Submitted Manuscript
	Variable	DC ZA	CMR	Type of break	time of the break
	c	c, t -3.058		t	1993
	price	c		-5.538* c1,c2 (IO)	1974, 1985
	tot_emp	c,t	-5.015*		t	1967
	serv_emp c,t	-4.866*		t	1963
	*: rejection of null hypothesis at 5% level.
	DC: deterministic component. c is a constant while t is a linear time trend.
	ZA is the Zivot and Andrews (1992) test.
	CMR: Clemente et al. (1998) test allowing two changes in the mean.
	t indicates that the break occurs in the deterministic linear trend while c means a break in
	the mean.
	IO: Innovative Oulier version of the test allowing gradual changes.

2 Page 46 of 50 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 3 .

 3 Test of the unit root with drift hypothesis against nonlinear trend stationarity[START_REF] Bierens | Testing the unit root with drift hypothesis against nonlinear trend stationarity, with an application to the US price level and interest rate[END_REF].

	Variable	p m	Test type test statistics	Fractiles	simulated
					0.05	0.10	p-values
	c	0 10 b t (m)	-5.7862	-6.67	-6.29	0.2780
			b A (m)	-38.9295	-80.30	-73.70	0.6820
			b F (m)	5.6622	2.15	2.36	0.8510
			b T (m)	1654.7126	280.57 359.51 0.9370
	price	1 10 b t (m)	-6.947	-6.67	-6.29	0.0630
			b A (m)	-135.86	-80.30	-73.70	0.0610
			b F (m)	8.7316	2.15	2.36	0.9530
			b T (m)	2597.6870	280.57 359.51 0.9050
	m	1 5	b t (m)	-4.2173	-5.16	-4.83	0.2685
			b A (m)	-37.2462	-48.70	-43.40	0.4015
			b F (m)	6.6236	1.83	2.08	0.9680
			b T (m)	1417.7731	18.60	30.15	0.9955
	tot_emp	1 5	b t (m)	-5.7052	-5.16	-4.83	0.0050
			b A (m)	-71.6307	-48.70	-43.40	0.0587
			b F (m)	10.8567	1.83	2.08	0.9997
			b T (m)	2737.2529	18.60	30.15	0.9997
	serv_emp 1 5	b t (m)	-4.161	-5.16	-4.83	0.3824
			b A (m)	-54.589	-48.70	-43.40	0.1784
			b F (m)	9.887	1.83	2.08	0.9948
			b T (m)	7165.041	18.60	30.15	0.9999
	Nonlinear ADF test with lag length p chosen using the AIC.
	Simulated critical values are computed using wild bootstrap based on 2000 replications.

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Table 4 .

 4 F statistics for testing the existence of level water demand equation

	Speci…cation	total employment service employment Ind&Serv
	p
	1	3.9608**	3.9690**	3.3750**
	2	4.6034***	4.7792***	2.6761*
	3	3.8464**	3.8808**	4.0127***
	AIC	3.9827**	4.1866**	3.3300**
	"Total employment": the demand equation is estimated using tot_emp as proxy for the productive activity.
	"Service employment": serv_emp is used as a proxy for the productive activity.
	"Ind&Serv": both ind_emp5074 and serv_emp7501 are used.
	*,**,*** signi…cant respectively at 10% 5% and 1% level.

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Table 5 .

 5 Short run estimates

	Without income	With income

Table 6 .

 6 Long run estimates The long run coe¢ cient are estimated from the level relation de…ned by equation 10. "Total": the demand equation is estimated using tot_emp as proxy for the productive activity. "Service": serv_emp is used. "Ind&Serv" means that both ind_emp5074 and serv_emp7501 are used. "Cross val.": inclusion of serv_emp5074 and ind_emp7501. *,**,*** signi…cant respectively at 10% 5% and 1% level. Standard error in parentheses.

			Without income			With income
		Total	Service	Ind&Serv Cross val. Total	Service	Ind&Serv
	price	-.067*	-.145**	-.113*	.019	-.104	-.152*	-.153
		(.040)	(.060)	(.066)	(.054)	(.092)	(.091)	(.113)
	rain	-.294**	-.349***	-.137	-.208	-.378*	-.363*	-.242
		(.120)	(.130)	(.123)	(.147)	(.228)	(.194)	(.244)
	m					-.052	-.008	-.093
						(.108)	(.082)	(.164)
	tot_emp	.159*				.241
		(.092)				(.203)
	lserv_emp		.106*				.114
			(.056)				(.096)
	ind_emp5074			.198**				.276
				(.095)				(.187)
	serv_emp7501			.221**				.296
				(.104)				(.191)
	ind_emp7501				-.037
					(.207)
	serv_emp5074				-.028
					(.202)
	CON ST AN T	9.114*** 9.081*** 8.342***	7.505***	10.591* 9.294*** 10.366***
		(1.022)	(.969)	(.890)	(2.157)	(3.332)	(2.340)	(3.804)

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Table 7 .

 7 Estimating price elasticities

	ARDL	FIRST BLOCK		THIRD BLOCK
		Total	Service	Ind&Serv	Total	Service	Ind&Serv
	Regressor
	p15075	-.0432*	-.060***	-.047*
		(.024)	(.023)	(.025)
	p17601	-.032*	-.048***	-.035**
		(.016)	(.016)	(.016)
	p35075				-.024	-.030*	-.022
					(.016)	(.016)	(.019)
	p37601				-.046*	-.057**	-.038
					(.024)	(.025)	(.026)
	:::::
	Constant	2.55***	2.34***	2.29***	2.35***	2.07***	2.02***
		(.556)	(.496)	(.570)	(.550)	(.509)	(.573)
	Serial corr. 2 (1) .061[.80] .499[.48]	.282[59]	.510[.47]	.089[.76] 1.302[.25]
	Func. form 2 (1) .675[.41] .238[62]	1.461[.22]	1.207[.27] .490[.48] 1.920[.16]
	Normality 2 (2)	.320[.85] .376[.82]	.040[.98]	1.117[.57] .717[.69] .210[.90]
	Hetero. 2 (1)	.843[.35] .323[56]	.692[.40]	1.271[.26] .403[.52] .641[.42]
	Long-run estim.
	p15075	-.165	-.244**	-.173*
		(.1087)	(.117)	(.094)
	p17601	-.122*	-.197**	-.127*
		(.071)	(.084)	(.066)
	p35075				-.089	-.153*	-.085
					(.064)	(.086)	(.074)
	p371601				-.167*	-.235*	-.146
					(.098)	(.137)	(.110)
	tot_emp	.306*			.118
		(.177)			(.116)
	serv_emp		.202*			.409
			(.108)			(.410)
	ind_emp5074			.251**			.114
				(.109)			(.093)
	serv_emp7501			.264**			.123
				(.112)			(.091)
	rain	-.286**	-.339**	-.111	-.261**	-.204	-.137
		(.125)	(.134)	(.119)	(.117)	(.135)	(.131)
	CON ST AN T	9.77**	9.48***	8.41***	8.58***	7.14***	7.81***
		(1.294)	(1.107)	(.851)	(.965)	(1.005)	(.859)

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

In choosing m we are faced with a tradeo¤: if m is too low -i.e. the nonlinerar trend is more non-linear than the Chebishev polynomial approximation -this might reduce the power of the test.