Abstract

Time series referring to water prices at different regions all over the world are examined in this paper by using fractionally integrated methods. We look at series corresponding to the following regions: Asia Pacific and Russia, Europe, United States and Latin America as well as global data. The results indicate large degrees of persistence, with the values of the differencing parameter being close to one in all cases and higher under the assumption of uncorrelated errors. If autocorrelation is permitted, a small degree of mean reversion is found in all except the Latin American series. The possibility of structural breaks is also investigated and the results indicate the presence of multiple breaks in the data: three in the case of Latin America and global data; four in Europe and USA and five for the Asian Pacific and Russia. Nevertheless, we do not observe a significant change in the degree of persistence across subsamples and once more mean reversion is found if autocorrelation is permitted.

Highlights

  • This research paper is to understand water prices and their behaviour with a daily time frequency, from January 2010 to January 2020.

  • We examine the time trends, persistence and seasonality in a unified treatment based on long memory and fractional integration in water prices data in United States, Latin America, Europe, Asia Pacific and Russia, and taking also into account global prices around the world.

Introduction

Water, that is the source of life, is considered to be the ‘blue gold’ or ‘the oil of the 21st century’ (Coy, 2002), because of its scarcity nowadays. Population and industrial activities continue to grow in the world, increasing the need for clean water. Looking at the numbers provided by the U.S. Geological Survey (USGS) and knowing that 70% of the earth's surface is made up of water, it should be noted that 97% is saltwater that cannot be used for consumption, irrigation or industrial uses. Of the remaining 3%, only 1% of water resources are available for human consumption.

In the report published in 2002, the American Water Resources Association (AWRA, 2002) mentioned that the water industry was rapidly transforming into a global business where the annual revenues were estimated at US$300 billion, predicting that they would double because of scarcity and the industry maturing. As was the case with the gas, electricity and telecommunication industries, the new economy of water is being consolidated through privatization and market competition. The concept of ‘water economy’ is due to the agreement between the water and socio-economic side based on water sources, users, physical infrastructure that connects water and consumers and the different institutions that assign property rights, rules and water allocation standards (see Tsur, 2020).

In 2019, the rapid industrialization, pollution, the growing agricultural use of water, and the growth of the world population may bring about further scarcity in the water supply that is fit for human consumption. There is a considerable body of literature on urban and industrial water demand (see, among others, Baumann et al. (1997), Renzetti (2002, 2015); Olmstead et al. (2007), House-Peters & Chang (2011), Baerenklau et al. (2014) and Smith & Zhao (2015)). Literature related to agricultural demand includes Just et al. (1983), Moore et al. (1994), Howitt (1995), Mundlak (2001), Tsur et al. (2004), Scheierling et al. (2006), Schoengold et al. (2006) among many others. According to the report of the United Nations (2019), today there are 7.76 billion people living in the world and it is expected that in 2050 there will be 9.73 billion people.

Figure 1, illustrated by the United Nations (UN), represents the Sustainable Development Goals (SDGs) regions from 1952 to 2020, using the medium-variant projection with 95% prediction intervals from 2020 to 2100.

Fig. 1.

Sustainable Development Goals regions. Source: United Nations (2019). (*excluding Australia and New Zealand).

Fig. 1.

Sustainable Development Goals regions. Source: United Nations (2019). (*excluding Australia and New Zealand).

Areas that have experienced a lack of water for consumption include China, Egypt, India, Israel, Pakistan, Mexico, parts of Africa and the United States (Colorado, California, Las Vegas and the East Coast), among others. In the United States the chemical compound methyl tertiary butyl ether (MTBE), which is an additive in unleaded gasoline, can be found in well water from California to Maryland. These facts are not limited to the West, since Russia, China and elsewhere show high levels of pollution, thus limiting the amount of fresh water available for human use. Research papers such as Diamond (2005), Bazilian et al. (2011), De Amorim et al. (2018) and Mahlknecht et al. (2020), among others, analyze the link between these issues, the risks and how they are interconnected.

Everything mentioned above leads to another important factor that balances the supply and demand of a good or service, and that is the price. Related to water tariff (see Pinto & Marques, 2015) and according to OECD (1999), the procedures and elements which determine a customer's bill is the tariff. The charge is any part of that bill and the rate is any unit price.

Until very recently, water has been subsidized and supplied at low or no cost because the foundation of these policies was that water is a basic service of life. Berg & Tschirhart (1995) were the ones who started the research line about public services in general. In many places, these policies have created an infrastructure crisis and have left many of the poorest sectors of society without access to good clean water. Middle East, the South of Europe, the United States West Coast, Cape Town, among other regions mentioned Garrone et al. (2019) are examples of these issues. In another research line, Hanemann (1993) highlights that there is a need for a distinct approach due to the unique features of water related industries where higher capital intensity and technical characteristics, institutional status and environmental importance, among others, are connected to the public/private economic nature of these services and their social value.

According to Renwick & Green (2000), Rogers et al. (2002), Martínez-Espiñeira & Nauges (2004), Grafton et al. (2011, 2015), Mansur & Olmstead (2012), Asci et al. (2017), Beecher & Gould (2018), Rougé et al. (2018), among others, the new pricing policies are moving towards real cost pricing to address financial sustainability and economic efficiency, equity, fairness and resource conservation and also with these objectives try to avoid the water scarcity.

Following Rogers et al. (2002) and Carvalho et al. (2012), there are many different price setting systems and structures for water and related services around the world because municipalities run water utilities in some countries and private companies in others. This fact occurs for several reasons, one of them is the faulty rate setting practices causing arbitrariness in price-setting strategies (Hoque & Wichelns, 2013). Therefore, the recovery of financial costs alone along with the recovery of environmental and social costs is a difficult milestone (Zetland & Gasson, 2013).

Based on the above comments, one of the novelties of this research paper is to understand water prices and their behaviour with a daily time frequency. This paper contributes to the literature in the analysis of water prices from a time series viewpoint by focusing on some its features such as seasonality, time trends and persistence. For this purpose, we examine the time series properties of water prices from January 2010 to January 2020 using daily data. Another contribution of this work is that for the first time we jointly examine the three features (time trends, persistence and seasonality) in a unified treatment based on long memory and fractional integration, which has not been jointly studied so far in water prices data by regions such as the United States, Latin America, Europe, Asia Pacific and Russia, also taking into account global prices around the world. The dataset was obtained from the Thomson Reuters Eikon database.

The rest of the paper is structured as follows. The next section briefly reviews the literature on water prices, followed by the methodology applied in the paper. The next section describes the data and presents the main empirical results, while the final section concludes the paper.

About water prices

In the literature, a limited number of studies has considered the behavior of water prices. There is research related with the impact of water tariffs on customers surveyed by numerous authors, including Worthington & Hoffman (2008) and Monteiro & Roseta-Palma (2011). Also, there are meta-analysis studies that use meta-regression analysis (MRA) to identify the price elasticity of water demand (see Espey et al., 1997; Dalhuisen et al., 2003; Marzano et al., 2018), income elasticities (Dalhuisen et al., 2003) and, more recently, the most common analysis in the literature has focussed on household size elasticity (see Sebri, 2014) and elasticity and scarcity in water from a residential and local point of view. In this context, for example, Krause et al. (2003) argue that price elasticity is very sensitive to water scarcity.

There are others studies based on water economics (see, e.g. Dinar & Schwabe, 2015) with the objective of proposing a pricing mechanism that implements the optimal water policy inside the context of a comprehensive water economy. The National Research Council (2005), Milennium Ecosystem Assessment (2005) and the references therein contribute by including within the regulatory mechanism all environmental externalities in relation to water allocation policy. There is a growing literature that studies the pricing mechanisms, taking into account the environmental allocation of water (see Weber et al. (2016), Tsur & Zemel (2018), Grant & Langpap (2019), Jack & Jayachandran (2019) and Tsur (2019), among others).

This empirical paper focuses its attention on transitory shocks (associated with trend stationary processes) and permanent shocks (related to difference stationary processes), focusing on the degree of persistence observed in the series and using fractional integration. The contributions of the paper are two-fold. First, to our knowledge this is the first paper that proposes to study the time series properties of the price of water in different regions such as the United States, Latin America, Europe, Asia Pacific and Russia using daily data. Second, in this paper we use some developed methods based on the concepts of long run dependence and long memory using fractional integration. Fractional integration is more general than the standard methods that exclusively use integer orders of differentiation (i.e. AR(I)MA models), allowing for a higher degree of flexibility in the dynamic specification of the model. Moreover, the possibility of structural breaks is also investigated, still within the framework of fractional integration (see the following section for more specific details).

Methodology

We use techniques based on long memory and fractional integration. For this purpose we define an integrated process of order 0 or I(0) as a second order stationary process with the infinite sum of the autocovariances (γj) assumed to be finite. Alternatively, in the frequency domain, the spectral density function is defined as the Fourier transform of the autocovariances, i.e.:
formula
(1)
and an I(0) process can be defined as a process with a density function that is positive and finite at the zero frequency. These are broad definitions that include not only the white noise model but also many others, including, for example, weakly autocorrelated structures such as the one produced by the stationary and invertible ARMA form. At the other extreme, we have nonstationary processes, defined as having unit roots or integrated of order 1, i.e. I(1) processes, which, in their simplest form, is the random walk model:
formula
(2)
where L is the lag operator (Lxt = xt–1) and ut is I(0). In this context, if ut is weakly autocorrelated, for example, if it follows an ARMA(p, q) process, xt is then said to be an ARIMA(p, 1, q) process. However, both the stationary I(0) and the nonstationary I(1) are specific cases within a more flexible and general type of model known as fractional integration or I(d) where ‘d’ can be any real value and thus, potentially fractional. We can extend the model in (2) to the fractional case such that:
formula
(3)
where ut is I(0) and ‘d’ can be 0 (short memory), a value between 0 and 1 (fractional integration), 1 (a unit root model), or even above 1 (see Robinson (2003), Doukham et al. (2003) and Gil-Alana & Hualde (2009) for a review of these models and long memory in general).
Processes such as those given by Equation (3) with d > 0 belong to a category of long memory, so named because of the large degree of association between observations far distant in time. They are characterized because the sum of their autocorrelation is infinite, or, alternatively, in the frequency domain, because the spectral density function has a singularity or pole at the smallest (zero) frequency, i.e.:
formula
(4)
Granger (1980) was the first to propose these models, noting that many aggregated data presented an extremely large value in the estimated spectrum at the smallest frequency, (i.e. as in Equation (4)), consistent with first differentiation, but once the series were differenced, the estimated spectrum displayed a value close to zero at the zero frequency, i.e.:
formula
(5)
which was an indication of over-differentiation. These models became popular in the 1990s through the works of Sowell (1992), Baillie (1996), Gil-Alana & Robinson (1997), Robinson & Zaffaroni (1997), Breidt et al. (1998), Silverberg & Verspagen (1999) and others, and they have also been employed more recently in the analysis of other commodity prices such as Gil-Alana & Gupta (2014), analyzing historical oil prices; Gil-Alana et al. (2015), examining precious metals prices such as gold, silver, rhodium, palladium and platinum; Monge et al. (2017a, 2017b) with crude oil prices, Monge & Gil-Alana (2019) studying the behavior of lithium and cobalt and Bouri et al. (2019) analyzing the behavior of cryptocurrencies, among many others.
Note that, for any real value d, the polynomial in the left hand side in Equation (3) can be expressed in terms of its binomial expansion, such that:
formula
(6)
and thus, Equation (3) can be expressed as:
formula
(7)
implying that higher the value of ‘d’ is, the higher the level of dependence between the observations is. Therefore, the parameter ‘d’ can be taken as a measure of the degree of persistence in the data.

We estimate the differencing parameter ‘d’ by using a version of a testing procedure developed by Robinson (1994) which is based on the frequency domain representation of the Whittle function. Among the advantages of this method, one is that it is valid for all range of values of ‘d’, not being restricted to the stationary region (d < 0.5); also, it allows the incorporation of deterministic terms such as an intercept or time trends and its limiting distribution is standard N(0,1). One limitation of the I(d) approach is that some authors argue that this may be a spurious phenomenon caused by the presence of structural breaks in the data that have not been taken into account (Ohanissian et al., 2008). Thus, in addition to the presented methodology, the presence of structural breaks is also taken into account in the paper by using Bai & Perron's (2003) methodology along with its extension to the fractional case in Gil-Alana (2008).

Data and empirical results

The data examined in this research paper, obtained from Thomson Reuters Eikon database, correspond to:

  • Thomson Reuters Global Water & Related Utilities;

  • Thomson Reuters United States Water & Related Utilities;

  • Thomson Reuters Latin America Water & Related Utilities;

  • Thomson Reuters Asia Pacific & Russia Water & Related Utilities;

  • Thomson Reuters Europe Water & Related Utilities.

These indices collect the entire water process until it reaches the final consumers, through the performance of the listed companies. We use daily prices from January 12, 2010 until January 10, 2020.

Table 1 shows the average, minimum and maximum data of water prices by regions and globally. According to the results presented in the table, we see that the price of water on average and the maximum is in the United States. The minimum price is in Europe.

Table 1.

Summary of time series.

Global (US$)United States (US$)Latin America (US$)Asia Pacific and Russia (US$)Europe (US$)
Mean 167.15 624.6 109.14 132.55 147.52 
Min. 96.62 253.1 91.24 93.78 80.65 
Max. 240.35 1,282 122.45 173.47 196.58 
Global (US$)United States (US$)Latin America (US$)Asia Pacific and Russia (US$)Europe (US$)
Mean 167.15 624.6 109.14 132.55 147.52 
Min. 96.62 253.1 91.24 93.78 80.65 
Max. 240.35 1,282 122.45 173.47 196.58 

Figure 2 plots the indices that represent the prices of water at the World, Europe, United States, Latin America and Asia Pacific and Russia.

Fig. 2.

Time series plots.

Fig. 2.

Time series plots.

Assuming yt below is each of the time series we observe, we start by considering the following regression model:
formula
(8)
testing the null hypothesis:
formula
(9)

In Equation (8) for do-values from 0, 0.01, … until 1.99 and 2, under three different modelling assumptions: (1) β0 = β1 = 0 a priori in Equation (8); (2) β1 = 0 a priori, and (3) with β0 and β1 unknown and freely estimated from the data. In Table 2 we suppose ut is a white noise process and report the estimated values of d along with the 95% confidence intervals of the non-rejection values with the test of Robinson (1994). We marked in bold the selected models according to the deterministic terms. We see that the time trend is only required in the case of United States prices (a significant time trend is usually related with the technological change); for the remaining cases, the intercept seems to be sufficient. Focusing on persistence by means of d, we see that the estimated values of d are close to 1 in all cases. However, we also observe some slight differences across the regions. Thus, for example, the I(1) null hypothesis cannot be rejected in the cases of Asia Pacific and Russia, Europe and Latin America; this hypothesis is rejected in favor of mean reversion (i.e. d < 1) for the United States, and it is rejected in favor of alternatives with d > 1 for the global series. For the United States, the time trend is found to be significantly positive (Table 3), but the fact that it is found to be significantly smaller than 1 indicates to us that shocks in the series will be mean reverting though taking long periods of time to disappear completely.

Table 2.

Estimates of d under the assumption of white noise errors.

SeriesNo termsAninterceptA linear time trend
Asia Pacific and Russia 0.99 (0.96, 1.02) 1.00 (0.94, 1.03) 1.00 (0.94, 1.03) 
Europe 1.00 (0.97, 1.03) 0.98 (0.95, 1.02) 0.98 (0.95, 1.02) 
USA 0.97 (0.94, 1.01) 0.95 (0.92, 0.98) 0.95 (0.92, 0.98) 
Latin America 0.98 (0.89, 1.10) 1.03 (0.93, 1.16) 1.03 (0.93, 1.16) 
Global 1.01 (0.98, 1.04) 1.04 (1.00, 1.07) 1.04 (1.00, 1.07) 
SeriesNo termsAninterceptA linear time trend
Asia Pacific and Russia 0.99 (0.96, 1.02) 1.00 (0.94, 1.03) 1.00 (0.94, 1.03) 
Europe 1.00 (0.97, 1.03) 0.98 (0.95, 1.02) 0.98 (0.95, 1.02) 
USA 0.97 (0.94, 1.01) 0.95 (0.92, 0.98) 0.95 (0.92, 0.98) 
Latin America 0.98 (0.89, 1.10) 1.03 (0.93, 1.16) 1.03 (0.93, 1.16) 
Global 1.01 (0.98, 1.04) 1.04 (1.00, 1.07) 1.04 (1.00, 1.07) 

The selected models in terms of the deterministic components are given in bold; 95% bands of the non-rejection values of d using Robinson (1994) are in parentheses.

Table 3.

Estimates of d under the assumption of white noise errors.

SeriesNo termsAninterceptA linear time trend
Asia Pacific and Russia 1.00 (0.94, 1.03) 120.40 (85.26) – 
Europe 0.98 (0.95, 1.02) 91.62 (53.70) – 
USA 0.95 (0.92, 0.98) 266.71 (40.31) 0.371 (4.18) 
Latin America 1.03 (0.93, 1.16) 99.96 (55.68) – 
Global 1.04 (1.00, 1.07) 108.10 (22.52) – 
SeriesNo termsAninterceptA linear time trend
Asia Pacific and Russia 1.00 (0.94, 1.03) 120.40 (85.26) – 
Europe 0.98 (0.95, 1.02) 91.62 (53.70) – 
USA 0.95 (0.92, 0.98) 266.71 (40.31) 0.371 (4.18) 
Latin America 1.03 (0.93, 1.16) 99.96 (55.68) – 
Global 1.04 (1.00, 1.07) 108.10 (22.52) – 

In the third and fourth columns, t-values are given in parentheses.

Tables 4 and 5 refer to the cases where ut in Equation (8) is autocorrelated. Here, we employ a non-parametric approach developed by Bloomfield (1973) that approximates ARMA structures with very few parameters (see Gil-Alana (2004) for the application of this model in the context of fractional integration). The first thing we observe in these two tables is that along with the United States, the time trend is now also required in the cases of Europe and the global case in the three cases with a significant positive coefficient. More importantly, evidence of a small degree of mean reversion is found in four out of the five series examined, with values of d statistically significantly below 1. In fact, only for Latin America, can the I(1) hypothesis not be rejected. Nevertheless, the estimates of d are very large in all cases, ranging between 0.88 (Latin America) and 0.91 (Asia Pacific, Europe and Russia) and thus implying large degrees of persistence in the data.

Table 4.

Estimates of d under the assumption of autocorrelated errors.

SeriesNo termsAn interceptA linear time trend
Asia Pacific and Russia 0.97 (0.92, 1.00) 0.91 (0.87, 0.95) 0.91 (0.87, 0.95) 
Europe 0.97 (0.92, 1.00) 0.91 (0.86, 0.96) 0.91 (0.86, 0.96) 
USA 0.95 (0.90, 0.99) 0.90 (0.87, 0.95) 0.90 (0.86, 0.94) 
Latin America 0.92 (0.79, 1.10) 0.88 (0.72, 1.10) 0.88 (0.71, 1.10) 
Global 0.97 (0.93, 1.02) 0.91 (0.87, 0.96) 0.91 (0.87, 0.96) 
SeriesNo termsAn interceptA linear time trend
Asia Pacific and Russia 0.97 (0.92, 1.00) 0.91 (0.87, 0.95) 0.91 (0.87, 0.95) 
Europe 0.97 (0.92, 1.00) 0.91 (0.86, 0.96) 0.91 (0.86, 0.96) 
USA 0.95 (0.90, 0.99) 0.90 (0.87, 0.95) 0.90 (0.86, 0.94) 
Latin America 0.92 (0.79, 1.10) 0.88 (0.72, 1.10) 0.88 (0.71, 1.10) 
Global 0.97 (0.93, 1.02) 0.91 (0.87, 0.96) 0.91 (0.87, 0.96) 

The selected models in terms of the deterministic components are given in bold; 95% bands of the non-rejection values of d using Robinson (1994) are given in parentheses.

Table 5.

Estimates of d under the assumption of autocorrelated errors.

SeriesNo termsAn interceptA linear time trend
Asia Pacific and Russia 0.91 (0.87, 0.95) 120.22 (86.04) – 
Europe 0.91 (0.86, 0.96) 91.63 (54.21) 0.036 (2.15) 
USA 0.90 (0.86, 0.94) 266.54 (40.63) 0.367 (5.99) 
Latin America 0.88 (0.72, 1.10) 100.21 (56.92) – 
Global 0.91 (0.87, 0.96) 107.97 (87.20) 0.048 (3.88) 
SeriesNo termsAn interceptA linear time trend
Asia Pacific and Russia 0.91 (0.87, 0.95) 120.22 (86.04) – 
Europe 0.91 (0.86, 0.96) 91.63 (54.21) 0.036 (2.15) 
USA 0.90 (0.86, 0.94) 266.54 (40.63) 0.367 (5.99) 
Latin America 0.88 (0.72, 1.10) 100.21 (56.92) – 
Global 0.91 (0.87, 0.96) 107.97 (87.20) 0.048 (3.88) 

In the third and fourth columns, t-values are given in parentheses.

The large degree of persistence observed in the results presented across Tables 25 can be a consequence of the presence of structural breaks which have not been taken into account. In fact, this is a classical argument that has been put forward by numerous authors in recent years (Diebold & Inoue, 2001; Granger & Hyung, 2004; Mikosch & Starica, 2004; Ohanissian et al., 2008, etc.). Thus, we next perform Bai & Perron's (2003) approach for detecting breaks in the data. Identical results were obtained when using Gil-Alana's (2008) approach, which extends the previous method to the fractional case. We observe three breaks for Latin America; four in the case of Europe and the United States and five for Asia Pacific and Russia and the global case. The break dates are reported in the third column of Table 6.

Table 6.

Structural breaks using Bai & Perron (2003).

SeriesNo. of breaksBreak dates
Asia Pacific and Russia 08.08.2011; 05.02.2013; 29.08.2014; 29.02.2016; 06.02.2018 
Europe 13.07.2011; 11.09.2013; 97.04.2015; 05.10.2016 
USA 05.06.2012; 13.03.2014; 28.01.2016; 16.07.2018 
Latin America 19.06.2019; 15.08.2019; 02.12.2019 
Global 14.07.2011; 10.01.2013; 29.07.2014; 08.04.2016; 16.07.2018 
SeriesNo. of breaksBreak dates
Asia Pacific and Russia 08.08.2011; 05.02.2013; 29.08.2014; 29.02.2016; 06.02.2018 
Europe 13.07.2011; 11.09.2013; 97.04.2015; 05.10.2016 
USA 05.06.2012; 13.03.2014; 28.01.2016; 16.07.2018 
Latin America 19.06.2019; 15.08.2019; 02.12.2019 
Global 14.07.2011; 10.01.2013; 29.07.2014; 08.04.2016; 16.07.2018 

In Tables 710 we display the estimates of d for each sub-sample and each series, once more for the three cases of: (1) no regressors, (2) with a constant, and (3) with a constant and a linear time trend. Tables 7 and 8 refer to white noise errors, while the results in Tables 9 and 10 correspond to the weakly autocorrelated (Bloomfield, 1973) case.

Table 7.

Estimates of d for each subsample: White noise case.

RegionSubsampleNo regressorAn interceptA time trend
Asia Pacific and Russia 1st period 0.96 (0.90, 1.04) 0.98 (0.92, 1.06) 0.98 (0.92, 1.06) 
2nd period 0.99 (0.93, 1.07) 0.98 (0.91, 1.05) 0.98 (0.91, 1.05) 
3rd period 0.99 (0.93, 1.07) 1.00 (0.92, 1.09) 1.00 (0.92, 1.09) 
4rd period 0.99 (0.93, 1.07) 1.01 (0.93, 1.11) 1.01 (0.93, 1.11) 
5th period 0.99 (0.93, 1.06) 0.97 (0.90, 1.06) 0.97 (0.90, 1.06) 
6th period 0.98 (0.93, 1.05) 0.93 (0.87, 1.01) 0.93 (0.87, 1.01) 
Europe 1st period 0.98 (0.91, 1.05) 0.98 (0.91, 1.08) 0.98 (0.91, 1.08) 
2nd period 1.00 (0.94, 1.08) 0.97 (0.91, 1.05) 0.97 (0.91, 1.05) 
3rd period 1.01 (0.94, 1.08) 0.99 (0.93, 1.07) 0.99 (0.93, 1.07) 
4rd period 0.99 (0.92, 1.06) 0.95 (0.84, 1.07) 0.95 (0.84, 1.07) 
5th period 0.98 (0.93, 1.03) 0.96 (0.92, 1.02) 0.96 (0.92, 1.02) 
USA 1st period 0.98 (0.92, 1.04) 0.79 (0.74, 0.85) 0.78 (0.73, 0.85) 
2nd period 1.01 (0.95, 1.09) 0.94 (0.88, 1.01) 0.94 (0.88, 1.01) 
3rd period 1.01 (0.95, 1.08) 0.90 (0.84, 0.97) 0.90 (0.84, 0.97) 
4rd period 1.00 (0.94, 1.06) 0.97 (0.92, 1.04) 0.97 (0.92, 1.04) 
5th period 0.99 (0.92, 1.06) 0.94 (0.87, 1.04) 0.94 (0.87, 1.04) 
Latin America 1st period 0.95 (0.78, 1.19) 1.11 (0.95, 1.34) 1.11 (0.95, 1.34) 
2nd period 0.87 (0.66, 1.16) 0.83 (0.65, 1.18) 0.82 (0.61, 1.18) 
3rd period 0.96 (0.82, 1.16) 0.89 (0.70, 1.18) 0.89 (0.70, 1.18) 
4rd period 0.87 (0.59, 1.24) 0.98 (0.75, 1.28) 0.99 (0.78, 1.26) 
Global 1st period 0.98 (0.92, 1.07) 1.04 (0.96, 1.14) 1.04 (0.96, 1.14) 
2nd period 0.99 (0.93, 1.07) 0.98 (0.90, 1.08) 0.98 (0.90, 1.08) 
3rd period 0.99 (0.93, 1.07) 1.22 (1.04, 1.19) 1.11 (1.04, 1.20) 
4rd period 0.98 (0.92, 1.06) 1.03 (0.95, 1.11) 1.02 (0.95, 1.11) 
5th period 1.00 (0.95, 1.07) 1.04 (0.97, 1.12) 1.04 (0.97, 1.12) 
6th period 0.99 (0.92, 1.07) 0.99 (0.91, 1.08) 0.99 (0.91, 1.08) 
RegionSubsampleNo regressorAn interceptA time trend
Asia Pacific and Russia 1st period 0.96 (0.90, 1.04) 0.98 (0.92, 1.06) 0.98 (0.92, 1.06) 
2nd period 0.99 (0.93, 1.07) 0.98 (0.91, 1.05) 0.98 (0.91, 1.05) 
3rd period 0.99 (0.93, 1.07) 1.00 (0.92, 1.09) 1.00 (0.92, 1.09) 
4rd period 0.99 (0.93, 1.07) 1.01 (0.93, 1.11) 1.01 (0.93, 1.11) 
5th period 0.99 (0.93, 1.06) 0.97 (0.90, 1.06) 0.97 (0.90, 1.06) 
6th period 0.98 (0.93, 1.05) 0.93 (0.87, 1.01) 0.93 (0.87, 1.01) 
Europe 1st period 0.98 (0.91, 1.05) 0.98 (0.91, 1.08) 0.98 (0.91, 1.08) 
2nd period 1.00 (0.94, 1.08) 0.97 (0.91, 1.05) 0.97 (0.91, 1.05) 
3rd period 1.01 (0.94, 1.08) 0.99 (0.93, 1.07) 0.99 (0.93, 1.07) 
4rd period 0.99 (0.92, 1.06) 0.95 (0.84, 1.07) 0.95 (0.84, 1.07) 
5th period 0.98 (0.93, 1.03) 0.96 (0.92, 1.02) 0.96 (0.92, 1.02) 
USA 1st period 0.98 (0.92, 1.04) 0.79 (0.74, 0.85) 0.78 (0.73, 0.85) 
2nd period 1.01 (0.95, 1.09) 0.94 (0.88, 1.01) 0.94 (0.88, 1.01) 
3rd period 1.01 (0.95, 1.08) 0.90 (0.84, 0.97) 0.90 (0.84, 0.97) 
4rd period 1.00 (0.94, 1.06) 0.97 (0.92, 1.04) 0.97 (0.92, 1.04) 
5th period 0.99 (0.92, 1.06) 0.94 (0.87, 1.04) 0.94 (0.87, 1.04) 
Latin America 1st period 0.95 (0.78, 1.19) 1.11 (0.95, 1.34) 1.11 (0.95, 1.34) 
2nd period 0.87 (0.66, 1.16) 0.83 (0.65, 1.18) 0.82 (0.61, 1.18) 
3rd period 0.96 (0.82, 1.16) 0.89 (0.70, 1.18) 0.89 (0.70, 1.18) 
4rd period 0.87 (0.59, 1.24) 0.98 (0.75, 1.28) 0.99 (0.78, 1.26) 
Global 1st period 0.98 (0.92, 1.07) 1.04 (0.96, 1.14) 1.04 (0.96, 1.14) 
2nd period 0.99 (0.93, 1.07) 0.98 (0.90, 1.08) 0.98 (0.90, 1.08) 
3rd period 0.99 (0.93, 1.07) 1.22 (1.04, 1.19) 1.11 (1.04, 1.20) 
4rd period 0.98 (0.92, 1.06) 1.03 (0.95, 1.11) 1.02 (0.95, 1.11) 
5th period 1.00 (0.95, 1.07) 1.04 (0.97, 1.12) 1.04 (0.97, 1.12) 
6th period 0.99 (0.92, 1.07) 0.99 (0.91, 1.08) 0.99 (0.91, 1.08) 

The selected models in terms of the deterministic components are given in bold; 95% bands of the non-rejection values of d using Robinson (1994) are given in parentheses.

Table 8.

Coefficient estimates for each subsample: White noise case.

RegionSubsampleNo regressorAn interceptA time trend
Asia Pacific and Russia 1st period 0.98 (0.92, 1.06) 120.364 (96.30) – 
2nd period 0.98 (0.91, 1.05) 108.388 (98.10) – 
3rd period 1.00 (0.92, 1.09) 120.940 (87.91) – 
4rd period 1.01 (0.93, 1.11) 155.215 (74.76) – 
5th period 0.97 (0.90, 1.06) 124.067 (101.8) – 
6th period 0.93 (0.87, 1.01) 141.981 (112.4) – 
Europe 1st period 0.98 (0.91, 1.08) 91.618 (73.58) – 
2nd period 0.97 (0.91, 1.05) 114.860 (85.20) – 
3rd period 0.99 (0.93, 1.07) 142.285 (88.40) – 
4rd period 0.95 (0.84, 1.07) 173.368 (72.02) – 
5th period 0.96 (0.92, 1.02) 170.162 (97.78) – 
USA 1st period 0.78 (0.73, 0.85) 266.460 (76.87) 0.1844 (4.69) 
2nd period 0.94 (0.88, 1.01) 383.537 (94.01) 0.2684 (1.98) 
3rd period 0.90 (0.84, 0.97) 516.346 (90.82) 0.3332 (2.29) 
4rd period 0.97 (0.92, 1.04) 700.443 (85.87) – 
5th period 0.94 (0.87, 1.04) 917.532 (91.79) 0.8334 (2.28) 
Latin America 1st period 1.11 (0.95, 1.34) 99.900 (51.73) – 
2nd period 0.83 (0.65, 1.18) 110.280 (65.50) – 
3rd period 0.89 (0.70, 1.18) 109.507 (64.91) – 
4rd period 0.98 (0.75, 1.28) 108.998 (73.41) – 
Global 1st period 1.04 (0.96, 1.14) 108.098 (11.65) – 
2nd period 0.98 (0.90, 1.08) 128.913 (11.65) – 
3rd period 1.22 (1.04, 1.19) 147.126 (11.65) – 
4rd period 1.03 (0.95, 1.11) 178.453 (11.65) – 
5th period 1.04 (0.97, 1.12) 186.444 (11.65) – 
6th period 0.99 (0.91, 1.08) 186.442 (134.7) 0.1291 (1.94) 
RegionSubsampleNo regressorAn interceptA time trend
Asia Pacific and Russia 1st period 0.98 (0.92, 1.06) 120.364 (96.30) – 
2nd period 0.98 (0.91, 1.05) 108.388 (98.10) – 
3rd period 1.00 (0.92, 1.09) 120.940 (87.91) – 
4rd period 1.01 (0.93, 1.11) 155.215 (74.76) – 
5th period 0.97 (0.90, 1.06) 124.067 (101.8) – 
6th period 0.93 (0.87, 1.01) 141.981 (112.4) – 
Europe 1st period 0.98 (0.91, 1.08) 91.618 (73.58) – 
2nd period 0.97 (0.91, 1.05) 114.860 (85.20) – 
3rd period 0.99 (0.93, 1.07) 142.285 (88.40) – 
4rd period 0.95 (0.84, 1.07) 173.368 (72.02) – 
5th period 0.96 (0.92, 1.02) 170.162 (97.78) – 
USA 1st period 0.78 (0.73, 0.85) 266.460 (76.87) 0.1844 (4.69) 
2nd period 0.94 (0.88, 1.01) 383.537 (94.01) 0.2684 (1.98) 
3rd period 0.90 (0.84, 0.97) 516.346 (90.82) 0.3332 (2.29) 
4rd period 0.97 (0.92, 1.04) 700.443 (85.87) – 
5th period 0.94 (0.87, 1.04) 917.532 (91.79) 0.8334 (2.28) 
Latin America 1st period 1.11 (0.95, 1.34) 99.900 (51.73) – 
2nd period 0.83 (0.65, 1.18) 110.280 (65.50) – 
3rd period 0.89 (0.70, 1.18) 109.507 (64.91) – 
4rd period 0.98 (0.75, 1.28) 108.998 (73.41) – 
Global 1st period 1.04 (0.96, 1.14) 108.098 (11.65) – 
2nd period 0.98 (0.90, 1.08) 128.913 (11.65) – 
3rd period 1.22 (1.04, 1.19) 147.126 (11.65) – 
4rd period 1.03 (0.95, 1.11) 178.453 (11.65) – 
5th period 1.04 (0.97, 1.12) 186.444 (11.65) – 
6th period 0.99 (0.91, 1.08) 186.442 (134.7) 0.1291 (1.94) 
Table 9.

Estimates of d for each subsample: Bloomfield case.

RegionSubsampleNo regressorAn interceptA time trend
Asia Pacific and Russia 1st period 0.99 (0.88, 1.11) 1.07 (0.94, 1.22) 1.07 (0.94, 1.22) 
2nd period 1.02 (0.91, 1.15) 0.99 (0.88, 1.13) 0.99 (0.87, 1.13) 
3rd period 0.98 (0.89, 1.09) 0.84 (0.76, 0.95) 0.84 (0.76, 0.95) 
4rd period 0.99 (0.88, 1.11) 0.85 (0.76, 0.96) 0.84 (0.75, 0.96) 
5th period 0.97 (0.89, 1.09) 0.84 (0.74, 0.94) 0.84 (0.76, 0.94) 
6th period 0.96 (0.87, 1.08) 0.87 (0.79, 0.97) 0.87 (0.80, 0.97) 
Europe 1st period 0.97 (0.86, 1.09) 0.85 (0.75, 0.98) 0.84 (0.73, 0.98) 
2nd period 0.99 (0.91, 1.09) 0.86 (0.76, 0.98) 0.86 (0.76, 0.98) 
3rd period 1.00 (0.90, 1.13) 1.04 (0.92, 1.16) 1.04 (0.92, 1.17) 
4rd period 0.98 (0.87, 1.09) 0.62 (0.53, 0.79) 0.62 (0.52, 0.79) 
5th period 0.95 (0.89, 1.04) 0.95 (0.89, 1.04) 0.95 (0.89, 1.04) 
USA 1st period 0.97 (0.89, 1.07) 0.79 (0.72, 0.92) 0.80 (0.69, 0.92) 
2nd period 0.97 (0.80, 1.08) 0.91 (0.82, 1.03) 0.92 (0.83, 1.03) 
3rd period 1.01 (0.92, 1.14) 0.94 (0.84, 1.07) 0.94 (0.84, 1.07) 
4rd period 0.99 (0.91, 1.09) 0.94 (0.85, 1.06) 0.94 (0.86, 1.06) 
5th period 0.96 (0.87, 1.11) 0.79 (0.72, 0.88) 0.76 (0.67, 0.86) 
Latin America 1st period 0.83 (0.54, 1.21) 1.06 (0.71, 1.51) 1.07 (0.71, 1.51) 
2nd period 0.70 (0.21, 1.23) 0.67 (0.37, 1.20) 0.68 (0.33, 1.20) 
3rd period 0.88 (0.61, 1.25) 0.38 (0.06, 0.77) 0.37 (0.08, 0.77) 
4rd period 0.47 (0.19, 1.24) 0.96 (0.16, 1.97) 0.97 (0.03, 1.66) 
Global 1st period 0.97 (0.87, 1.10) 0.89 (0.80, 1.02) 0.90 (0.79, 1.02) 
2nd period 0.97 (0.87, 1.11) 0.85 (0.76, 0.97) 0.84 (0.74, 0.97) 
3rd period 0.97 (0.88, 1.10) 0.99 (0.91, 1.10) 0.99 (0.91, 1.10) 
4rd period 0.97 (0.86, 1.08) 0.89 (0.77, 1.04) 0.89 (0.77, 1.04) 
5th period 0.99 (0.91, 1.08) 0.89 (0.81, 1.00) 0.89 (0.81, 1.00) 
6th period 0.97 (0.87, 1.10) 0.84 (0.76, 0.94) 0.81 (0.71, 0.93) 
RegionSubsampleNo regressorAn interceptA time trend
Asia Pacific and Russia 1st period 0.99 (0.88, 1.11) 1.07 (0.94, 1.22) 1.07 (0.94, 1.22) 
2nd period 1.02 (0.91, 1.15) 0.99 (0.88, 1.13) 0.99 (0.87, 1.13) 
3rd period 0.98 (0.89, 1.09) 0.84 (0.76, 0.95) 0.84 (0.76, 0.95) 
4rd period 0.99 (0.88, 1.11) 0.85 (0.76, 0.96) 0.84 (0.75, 0.96) 
5th period 0.97 (0.89, 1.09) 0.84 (0.74, 0.94) 0.84 (0.76, 0.94) 
6th period 0.96 (0.87, 1.08) 0.87 (0.79, 0.97) 0.87 (0.80, 0.97) 
Europe 1st period 0.97 (0.86, 1.09) 0.85 (0.75, 0.98) 0.84 (0.73, 0.98) 
2nd period 0.99 (0.91, 1.09) 0.86 (0.76, 0.98) 0.86 (0.76, 0.98) 
3rd period 1.00 (0.90, 1.13) 1.04 (0.92, 1.16) 1.04 (0.92, 1.17) 
4rd period 0.98 (0.87, 1.09) 0.62 (0.53, 0.79) 0.62 (0.52, 0.79) 
5th period 0.95 (0.89, 1.04) 0.95 (0.89, 1.04) 0.95 (0.89, 1.04) 
USA 1st period 0.97 (0.89, 1.07) 0.79 (0.72, 0.92) 0.80 (0.69, 0.92) 
2nd period 0.97 (0.80, 1.08) 0.91 (0.82, 1.03) 0.92 (0.83, 1.03) 
3rd period 1.01 (0.92, 1.14) 0.94 (0.84, 1.07) 0.94 (0.84, 1.07) 
4rd period 0.99 (0.91, 1.09) 0.94 (0.85, 1.06) 0.94 (0.86, 1.06) 
5th period 0.96 (0.87, 1.11) 0.79 (0.72, 0.88) 0.76 (0.67, 0.86) 
Latin America 1st period 0.83 (0.54, 1.21) 1.06 (0.71, 1.51) 1.07 (0.71, 1.51) 
2nd period 0.70 (0.21, 1.23) 0.67 (0.37, 1.20) 0.68 (0.33, 1.20) 
3rd period 0.88 (0.61, 1.25) 0.38 (0.06, 0.77) 0.37 (0.08, 0.77) 
4rd period 0.47 (0.19, 1.24) 0.96 (0.16, 1.97) 0.97 (0.03, 1.66) 
Global 1st period 0.97 (0.87, 1.10) 0.89 (0.80, 1.02) 0.90 (0.79, 1.02) 
2nd period 0.97 (0.87, 1.11) 0.85 (0.76, 0.97) 0.84 (0.74, 0.97) 
3rd period 0.97 (0.88, 1.10) 0.99 (0.91, 1.10) 0.99 (0.91, 1.10) 
4rd period 0.97 (0.86, 1.08) 0.89 (0.77, 1.04) 0.89 (0.77, 1.04) 
5th period 0.99 (0.91, 1.08) 0.89 (0.81, 1.00) 0.89 (0.81, 1.00) 
6th period 0.97 (0.87, 1.10) 0.84 (0.76, 0.94) 0.81 (0.71, 0.93) 

The selected models in terms of the deterministic components are given in bold; 95% bands of the non-rejection values of d using Robinson (1994) are given in parentheses.

Table 10.

Coefficient estimates for each subsample: Bloomfield case.

RegionSubsampleNo regressorAn interceptA time trend
Asia Pacific and Russia 1st period 1.07 (0.94, 1.22) 120.512 (96.87) – 
2nd period 0.99 (0.88, 1.13) 108.394 (98.09) – 
3rd period 0.84 (0.76, 0.95) 121.022 (91.01) 0.0780 (2.82) 
4rd period 0.84 (0.75, 0.96) 155.551 (77.63) –0.0821 (–1.92) 
5th period 0.84 (0.76, 0.94) 124.499 (105.1) 0.0453 (2.12) 
6th period 0.87 (0.79, 0.97) 141.901 (113.7) – 
Europe 1st period 0.84 (0.73, 0.98) 91.599 (75.82) 0.0592 (2.30) 
2nd period 0.86 (0.76, 0.98) 114.716 (86.89) 0.0433 (1.74) 
3rd period 1.04 (0.92, 1.16) 142.124 (88.44) – 
4rd period 0.62 (0.53, 0.79) 176.034 (95.79) – 
5th period 0.95 (0.89, 1.04) 170.084 (97.81) – 
USA 1st period 0.80 (0.69, 0.92) 266.596 (76.08) 0.1834 (4.16) 
2nd period 0.92 (0.83, 1.03) 383.608 (94.29) 0.2661 (2.21) 
3rd period 0.94 (0.84, 1.07) 516.625 (90.43) 0.3467 (1.89) 
4rd period 0.94 (0.85, 1.06) 700.580 (86.12) – 
5th period 0.76 (0.67, 0.86) 915.420 (98.74) 0.8667 (6.50) 
Latin America 1st period 1.06 (0.71, 1.51) 99.941 (51.47) – 
2nd period 0.67 (0.37, 1.20) 110.877 (73.24) – 
3rd period 0.38 (0.06, 0.77) 108.683 (146.0) – 
4rd period 0.96 (0.16, 1.97) 109.034 (73.47) – 
Global 1st period 0.90 (0.79, 1.02) 107.947 (120.7) 0.0560 (2.12) 
2nd period 0.84 (0.74, 0.97) 128.587 (130.6) 0.0463 (2.19) 
3rd period 0.99 (0.91, 1.10) 147.223 (141.8) – 
4rd period 0.89 (0.77, 1.04) 177.987 (117.3) – 
5th period 0.89 (0.81, 1.00) 186.771 (138.6) – 
6th period 0.81 (0.71, 0.93) 186.050 (141.1) 0.1331 (5.49) 
RegionSubsampleNo regressorAn interceptA time trend
Asia Pacific and Russia 1st period 1.07 (0.94, 1.22) 120.512 (96.87) – 
2nd period 0.99 (0.88, 1.13) 108.394 (98.09) – 
3rd period 0.84 (0.76, 0.95) 121.022 (91.01) 0.0780 (2.82) 
4rd period 0.84 (0.75, 0.96) 155.551 (77.63) –0.0821 (–1.92) 
5th period 0.84 (0.76, 0.94) 124.499 (105.1) 0.0453 (2.12) 
6th period 0.87 (0.79, 0.97) 141.901 (113.7) – 
Europe 1st period 0.84 (0.73, 0.98) 91.599 (75.82) 0.0592 (2.30) 
2nd period 0.86 (0.76, 0.98) 114.716 (86.89) 0.0433 (1.74) 
3rd period 1.04 (0.92, 1.16) 142.124 (88.44) – 
4rd period 0.62 (0.53, 0.79) 176.034 (95.79) – 
5th period 0.95 (0.89, 1.04) 170.084 (97.81) – 
USA 1st period 0.80 (0.69, 0.92) 266.596 (76.08) 0.1834 (4.16) 
2nd period 0.92 (0.83, 1.03) 383.608 (94.29) 0.2661 (2.21) 
3rd period 0.94 (0.84, 1.07) 516.625 (90.43) 0.3467 (1.89) 
4rd period 0.94 (0.85, 1.06) 700.580 (86.12) – 
5th period 0.76 (0.67, 0.86) 915.420 (98.74) 0.8667 (6.50) 
Latin America 1st period 1.06 (0.71, 1.51) 99.941 (51.47) – 
2nd period 0.67 (0.37, 1.20) 110.877 (73.24) – 
3rd period 0.38 (0.06, 0.77) 108.683 (146.0) – 
4rd period 0.96 (0.16, 1.97) 109.034 (73.47) – 
Global 1st period 0.90 (0.79, 1.02) 107.947 (120.7) 0.0560 (2.12) 
2nd period 0.84 (0.74, 0.97) 128.587 (130.6) 0.0463 (2.19) 
3rd period 0.99 (0.91, 1.10) 147.223 (141.8) – 
4rd period 0.89 (0.77, 1.04) 177.987 (117.3) – 
5th period 0.89 (0.81, 1.00) 186.771 (138.6) – 
6th period 0.81 (0.71, 0.93) 186.050 (141.1) 0.1331 (5.49) 

Starting with the case of white noise errors, we see that significant time trends are found in four out of the five sub-periods in the United States and for the last subsample in the global series. Mean reversion is detected only for the United States case in the first and third subsamples, and in general we do not observe any systematic (increase/decrease) pattern in the values of d across the subsamples.

With autocorrelation, the time trend coefficient is statistically significant in many more cases than in the case of white noise errors, being positive in all except one single case (Asia Pacific and Russia, fourth subsample), and mean reversion takes now place in numerous subsamples: the last four subsamples in Asia Pacific and Russia: the first, second and fourth in Europe; in the first and fifth in the United States; in the third subsample for Latin America, and in the second and sixth subsamples for the global series. Nevertheless, as with the white noise case, the values of d are relatively high in most of the cases, the exceptions being the second and third subsamples in Latin America (0.67 and 0.38 respectively) and the fourth in the European case (0.62).

Concluding comments

In this paper we have examined five time series referring to water prices in different regions all over the world, applying statistical methods based on long memory and fractional differentiation. As far as we are concerned there are no previous econometric works related to the price of water and using these methodologies.

We employ daily data of the water prices of the following four regions, Asia Pacific and Russia, Europe, the United States and Latin America, as well as global data. Our results indicate first that if the errors are uncorrelated, water prices do not show mean reversion in any of the series examined except for the United States data, where a positive time trend is also observed. That means that in the event of an exogenous shock increasing the price of water in the US, the series will revert by itself in the long run to its original trend, though it can take a very long time to recover completely. However, if autocorrelation is permitted, a small degree of mean reversion (i.e. an estimate of the differencing parameter significantly smaller than one) is detected in four of the series (Asia Pacific and Russia, Europe, the United States and global) while the I(1) hypothesis cannot be rejected for the Latin American data. Nevertheless, the estimated values for the degree of integration are large in all cases, ranging from 0.88 to 0.91, thus implying large degrees of persistence, with long lasting effects from shocks.

Following the literature and knowing that the large degree of persistence observed in our results can be a consequence of the presence of structural breaks that have not been taken into account, we used Bai & Perron's (2003) and Gil-Alana's (2008) approaches for detecting breaks in the data. In that case, we observe three breaks for Latin America, four for Europe and United States and five for Asia Pacific and Russia and the global case. Estimating the differencing parameter for each subsample and each series, if the errors are white noise, mean reversion is only observed in the case of the United States. On the other hand, allowing for autocorrelation, the time trend is significant in a number of cases and mean reversion is found in many of the subsamples implying in these cases that shocks will have a transitory nature, returning to their original long trend projections after some periods of time.

An interesting conclusion in this work is that water prices are highly persistent. Thus, in the event of an exogenous shock, for example, unexpectedly increasing the price of water, strong policy measures should be adopted to recover the original levels/trends in the data, since otherwise the series will take a very long time to recover by themselves.

This paper can be extended in several directions. First, non-linear trends can also be considered, noting that fractional integration and non-linear structures are isssues which are very much related (Diebold & Inoue, 2001). In this context, the non-linear approach developed in Cuestas & Gil-Alana (2016) and based on the Chebyshev polynomials in time in the context of fractional integration can be implemented with this dataset. Moreover, in a multivariate context, fractional cointegration can also be examined to look at potential long run equilibrium relationships among the variables used in this work, and here, the fractional cointegration VAR (FCVAR) approach developed by Johansen & Nielsen (2010, 2012) can be performed. Work in all these directions is now in progress.

Acknowledgements

Prof. Luis A. Gil-Alana gratefully acknowledges financial support from the Ministerio de Economía y Competitividad (ECO2017-85503-R). Prof. Luis A. Gil-Alana and Assoc. Prof. Manuel Monge also acknowledge support from an internal project from the Universidad Francisco de Vitoria. Comments from the Editor and two anonymous reviewers are gratefully acknowledged.

Data availability statement

All relevant data are included in the paper or its Supplementary Information.

References

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