Introduction
Currently, the valuation of ecosystem services (ES) is important for decision making on natural resources (^{Tallis & Polasky, 2009}). According to ^{De Groot, Wilson, and Boumans (2002)}, the ES are defined as "the ability of natural processes and components to provide goods and services that directly or indirectly meet human needs". The same authors indicate that the services can be regulation, habitat, provisioning and cultural services. In short, the importance of the ES lies in providing the sustenance for life on the planet.
Despite the importance mentioned, ES are difficult to quantify monetarily. The problem of environmental degradation is linked to socalled market failures. One of these are the negative externalities, which originate from the characteristics of public goods; that is, no exclusivity or rivalry in their consumption (^{Perman, Ma, McGilvray, & Common, 2003}). Therefore it is necessary to determine the value of ES from an economic perspective, so that they are known and the cost they represent when used is paid (^{Azqueta, 1994}). In this regard, the contingent valuation method (CVM) is the most significant direct method to estimate the total monetary value of the ES; i. e. the value of use, nonuse value and option value. In general terms, CVM is a technique that, through a survey, proposes a hypothetical market to determine monetary willingness to pay (WTP) when there is a change in welfare. There are two ways to ask for the WTP in the survey, the open format (ask directly the maximum amount to pay) and the dichotomous format, also called referendum (a payment is proposed and the respondent decides to accept or reject). Researchers who use the CVM with the dichotomous question face the complex task of estimating useful statistics from a sparse set of information; for this purpose, three procedures have been developed: parametric, nonparametric and semiparametric.
In the parametric estimates we first assume a functional form of the indirect utility model and then a cumulative distribution function of the errors (^{Hanemann & Kanninen, 1999}). A poor specification of these assumptions results in a biased estimate of welfare measures. In order to overcome the difficulties created by this problem, the use of nonparametric estimation methods has been proposed (^{Cerda & Vásquez, 2005}). This type of estimation does not require assumptions on the probability distribution of errors or the functional forms of utility models.
Based on the above, the objective of the study was to compare the parametric and nonparametric methods of estimation of the WTP through mean intervals, in order to give recommendations of use in the valuation of ecosystem services. In addition, in the nonparametric approach, a comparison of the mean and variance estimation of the WTP was proposed, using the formulas of ^{Boman, Bostedt, and Kriström (1999)} versus those of ^{Haab and McConnell (2002)}.
Materials and Methods
To provide support for the comparison of the parametric and nonparametric methods of estimation of the WTP, databases of two empirical cases were used: ^{Valdivia, García, López, Hernández, and Rojano (2011)} and ^{Valdivia, Abelino, López, and Zavala (2012)}. These authors used the contingent valuation method with the simple dichotomous format.
^{Valdivia et al. (2011)} estimated the WTP of the inhabitants of Axtla de Terrazas, San Luis Potosí, for protection, maintenance and improvement of the Axtla River. The authors used simple random sampling and in the year 2010 they applied 300 surveys to the family heads of households randomly selected from the municipal database. The results were analyzed using logistic regression techniques via maximum likelihood.
Furthermore, the study of ^{Valdivia et al. (2012)} had the objective of estimating the amount of the WTP of the inhabitants (household level) of Texcoco, Estado de México, to implement a recycling system. Data were collected in 2010 using simple random sampling with a confidence level of 95 %. The total of the sample was 402 households and the logit binomial econometric model with linear functional form in the income was used.
Estimation of the WTP using the parametric method
According to ^{Haab and McConnell (2002)}, responses in contingent valuation can be analyzed from a random utility model. Thus, there are two alternatives to choose the consumption of an ecosystem service. The indirect utility of the respondent j can be written as follows:
where i = 1 is the final state after generating the change in environmental quality; i = 0 is the unchanged state or status quo. The determinants of utility are: y _{ i } , the income of the jth respondent; z _{ j } , a mdimensional vector of the domestic attributes and characteristics of the choice; and ɛ _{ ij } , a component of preferences known for the respondent but not observed by the researcher. In this way, the respondent j will respond positively to the willingness to pay of t _{ j } when the utility with the valuation approach exceeds the status quo; that is, when the resulting utility is higher even though the income from the payment decreases.
The probability of a positive response is the probability that the respondent thinks he/she will be in a better scenario even with the required payment, so u _{ 1 } > u _{ 0 } . For the respondent j the probability is:
This probability statement provides an intuitive basis for the analysis of responses and can be used as a starting point for nonparametric approaches. However, the statement is very general for parametric estimations because two decision models are needed: first, the functional form of u(y _{ i } , z _{ j } , ɛ _{ ij } ) should be chosen, and second, the distribution of ɛ_{ ij } should be specified.
The two empirical cases of ^{Valdivia et al. (2011}, ^{2012)} assume a logistic probability distribution function:
If we consider a linear model in the income we obtain:
where α _{ q } is the change in utility and β is the change in marginal utility (^{Hanemann & Kanninen, 1999}). Thus, it is shown that the payment p* would leave the user indifferent, since the payment is equal to the change in utility between marginal utility. Thus, the mean of the WTP would be:
Estimation of the WTP using the nonparametric method
In the CVM with referendum format, different k1 payment offers are chosen A _{ 1 } , A _{ 2 } …, A _{ k = 1 } , and are administered or divided into k1 subsamples, where A _{ 1 } < A _{ 2 } …, < A _{ k = 1 } y k ≥ 2; we also obtain the answers of m _{ 1 } , m _{ 2 } …, m _{ k = 1 } individuals. For each subsample i(i = 1, 2…, k1) we obtain the quantity h _{ i } of persons accepting the supply A _{ 1 } and we can estimate the proportions p _{ i } = h _{ i } / m _{ i } . Thus, the responses of a typical discrete survey give a sequence of proportions p = (p _{ 1 } , p _{ 2 } …, p _{ k = 1 } ) (^{Boman et al., 1999}).
It is important to make clear that from the economic principles this sequence of proportions
is expected to be monotonically decreasing in A
_{
1
} ; that is, the proportion of "yes" responses should decrease
(or at least not increase) as A increases. However, it is
sometimes the case that p
_{
i+1
} > p
_{
i
} . At this point, the theorem of ^{Ayer,
Brunk, Ewing, Reid, and Silverman (1955)} in the isotonic regression
is useful. The theorem shows that when the sequence of proportions is smoothed
by substituting the proportions p
_{
i
} and p
_{
i+1
} with (h
_{
i
} + h
_{
i+1
} ) / (m
_{
i
} + m
_{
i+1
} ) and this procedure is repeated until a non increasing of estimates is
obtained, then this new sequence
Continuing with the method, we should determine an A
_{
0
} value to make
where A _{ 0 } ≤ 0
The above relationship suggests an estimation of mean WTP:
which is equivalent to:
Based on these principles, ^{Boman et al. (1999)} propose the formulas for mean and variance summarized in Table 1
Measure  Mean  Variance of mean 

Lower limit 


Intermediate 


Upper limit 


Regarding the formulas of variance of ^{Boman et al. (1999)}, ^{Vaughan and Rodríguez (2000)} argue that they are conceptually incorrect because they treat the supply and not the proportions of the subsamples as random variables. ^{Vaughan and Rodríguez (2000)} state that the offers are constant, since they are preselected by the designer of the referendum contingent valuation survey, and that the respondents react to the acceptance or rejection of the offer; those reactions are the random variables. On the other hand, the expected value of the nonparametric formulas is only a linear function of the random variables, but the formulas of the variance apparently were not explicitly derived with that fundamental property in mind. Based on the above, these authors propose the generalized formulas of Haab and McConnell as best alternative (summarized in Table 2).
Measure  Mean  Variance of mean 

Lower limit 


Intermediate 


Upper limit 


Source: ^{Vaughan and Rodríguez (2000}). N
_{
j
} represents the number of “no” answers and Y
_{
j
} the “yes” answers in each group j, F
_{
j
} = N
_{
j
} / (N
_{
j
}
+ Y
_{
j
} ). There are j = 1…M different payment offers specified in the survey; the supply j = M + 1 is the last level of supply that the researcher should assume, presumably carries F
_{
j
} to 1. The variance of each proportion F
_{
j
} is equal to
For the statistical comparison of welfare measures, confidence intervals were built. The intervals for the nonparametric method were shown in Table 1, while for the parametric method, the procedure developed by ^{Krinsky and Robb (1986)} was applied. This bootstrap process simulates the probability distribution from repeated random extractions of the parametrized multivariate normal distribution, based on estimates of coefficients and the covariance matrix of the estimated logit model (^{Mogas & Riera, 2003}). The Krinsky and Robb procedure is found in the WTPCIKR command (^{Abdullah & Jeanty, 2011}; ^{Jeanty, 2008}), a special module of the program Strata (^{StataCorp LLC, 2015}). A total of 5,000 iterations were applied with a confidence level of 95 %.
Results and Discussion
The data showed situations of proportions in which p _{ i+1 } > p _{ i } (Annexes 1 and 2), for which the theorem of ^{Ayer et al. (1955)} was applied to fit and find a nonparametric estimate of maximum likelihood of acceptance probability; in other words, the PAVA algorithm was used. After adjusting the proportions and obtaining the monotonically decreasing survival function, we had the conditions to calculate the estimators of interest (mean and variance). Thus, Tables 3 and 4 shows the results using the formulas of ^{Boman et al. (1999)} and those of ^{Haab and McConnell (2002)}.
Welfare measures  Parametric mean  Nonparametric mean  Variance of the nonparametric mean  

Boman et al. (1999)  Haab and McConnell (2002)  Boman et al. (1999)  Haab and McConnell (2002)  
Laspeyres (Lower limit)  55.07  53.05  52.00  2.66  5.86 
Intermediate  57.15  57.72  56.67  2.44  5.30 
Paasche (Upper limit)  59.23  62.38  61.33  2.37  5.91 
Source: Compiled by authors
Welfare measures  Parametric mean  Nonparametric mean  Variance of the nonparametric mean  

Boman et al. (1999)  Haab and McConnell (2002)  Boman et al. (1999)  Haab and McConnell (2002)  
Laspeyres (lower limit)  25.07  22.03  22.03  0.45  0.86 
Intermedia  27.18  25.11  25.11  0.38  0.88 
Paasche (upper limit)  29.86  28.19  28.19  0.36  0.63 
Source: Compiled by authors
Table 3 shows the parametric and nonparametric estimates, based on data from the economic valuation for the rehabilitation of the Axtla River, in San Luis Potosí, Mexico, by ^{Valdivia et al. (2011)}. The intermediate WTP estimated by the parametric method, with a linear functional form in the income (57.15, estimated and reported by ^{Valdivia et al., 2011}), does not shows significant difference compared to that obtained through the two nonparametric methods (57.72 and 56.67). Regarding the intervals (lower and upper limits), the nonparametric methods show a slightly wider range with respect to the intervals of the parametric method; however, it is also not possible to say that it is an important discrepancy. Regarding the variances within the nonparametric methods, greater variances were observed with the Haab and McConnell formulas.
To validate and corroborate these results in other studies with similar methodologies, Table 4 shows the parametric and nonparametric estimates, based on the economic valuation of urban waste recycling performed by ^{Valdivia et al. (2012)} in Texcoco, Estado de México. This study also obtained the WTP by the parametric method with a linear model in the income (previously reported in ^{Valdivia et al., 2012}). Similarly, the PAVA algorithm was also applied to fit the data and find the monotonically decreasing survival function.
When applying the nonparametric formulas of the WTP it was observed that the intervals of the amount to pay overlap with those of the parametric estimation, so that it can be asserted that there are no divergences. In this study, the intervals of the means obtained with the formulas of ^{Boman et al. (1999)} and those of ^{Haab and McConnell (2002)} do not differ, although the variances obtained by the method of Habb and McConnell are relatively higher.
According to the results, in the two study cases analyzed, it is indifferent to use the formulas of ^{Boman et al. (1999)}, and those of ^{Haab and McConnell (2002)} or the parametric method to estimate the intervals of the mean, because the difference is not significant. However, in terms of variance there are clear differences coinciding with the results obtained by ^{Vaughan and Rodríguez (2000)}.
The estimates of the mean of the WTP using the nonparametric estimation do not differ much from those obtained by the parametric approximation; however, despite of having some advantages, the disadvantage of the former is that they do not include explanatory variables. In this context, ^{Haab and McConnell (2002)} express that the parametric models allow the incorporation of characteristics of the respondents in the functions of willingness to pay. Understanding how the WTP responds to individual characteristics allows the researcher to obtain information on the validity and reliability of the CVM and to extrapolate sample responses to the general population. Moreover, a set of explanatory variables that adjusts to expectations makes the application of contingent valuation more convincing.
Similarly, ^{Riera, Brey, and Gándara (2008)} explain that nonparametric methods have limitations compared to semiparametric and parametric ones, such as the difficulty to include explanatory covariates, impose restrictions on estimation or interpret results. In spite of this, the authors indicate that, because of the simplicity of nonparametric methods, some studies use them, sometimes as a complement to the results obtained by the other methods or to judge the robustness of the parametric estimations in the case of changes in the functional form.
In relation to other empirical studies with results similar to those shown in this study, we find that of ^{Soncco and Armas (2008)}, where the values of welfare measures do not differ significantly between two types of approach (parametric and nonparametric). These authors claim that the results obtained using parametric procedures are robust to assumptions made on discrete probability distributions. These authors suggest that nonparametric methods be used as complements and validation tools.
Moreover, ^{Cerda and Vásquez (2005)} affirm that there is no significant difference when using the two methods and that the importance lies in the design of the survey. They explain that when the Kriström method (referring to the formulas of ^{Boman et al., 1999}) is compared with the linear parametric estimation, the differences are not significant based on the overlap of the confidence intervals. In contrast, there is a significant difference when the Kriström method is compared with a logarithmic parametric estimation or when the Haab and McConnell approach is compared with the parametric estimates. The authors concluded that it is better to invest time in the experimental design before using nonparametric estimation methods that lose the benefits of the underlying information in traditional econometric models, such as the incorporation of other explanatory variables into the valuation function.
^{Sánchez (2008)} compared parametric estimates using the linear and logarithmic functional form, and logit and probit regressions, against the nonparametric method of Kriström and found that there are no statistically significant differences when verifying that their confidence intervals overlap. This result applies in the comparison of the linear functional form, regardless of the type of regression; however, there are differences compared to the logarithmic model, in agreement with ^{Cerda and Vásquez (2005)}. Also, ^{Del Saz and García (2007)} used Kriström's nonparametric formula and compared it with parametric estimation through logit, probit and spike models, obtaining similar results with the two approaches.
Finally, ^{Villena and Lafuente (2013)} estimated measures of welfare change using parametric and nonparametric estimates, for which they built confidence intervals. In the case of the parametric model they used the linear and logarithmic functional form; while in the case of nonparametric estimation they used the formulas of Kriström and those of Habb and McConnell. The authors found that the estimated confidence intervals contain the true WTP and that the intervals overlap; therefore, they do not differ statistically.
Conclusions
The mean and, in particular, the interval of this estimator are important because they provide a rough idea of the range of income that comes from the welfare change approach of users of ecosystem services. In spite of the problems of specifying the functional forms of the utility in the parametric method and the relative ease of obtaining the data with the nonparametric methods, it is recommended to use the latter as a complement to the first one. This is because the parametric method has greater robustness, since it includes socioeconomic explanatory variables of the WTP. Within the nonparametric methods it is indifferent to use the formulas of ^{Boman et al. (1999)} or those of ^{Haab and McConnell (2002)} to estimate the mean, while the variance is significantly higher when using the Haab and McConnell method. Finally, the present study had the limitation of not comparing the parametric and nonparametric methods with the semiparametric method, which is considered a variant of the first method. Nevertheless, it is recommended to compare the three methods with the inclusion of a greater number of observations to increase the reliability of the results.