Joint Estimation of Revealed and Stated Preference Data from the Alaska Saltwater Sportfishing Economic Survey1 John C. Whitehead Department of Economics Appalachian State University Boone, North Carolina Daniel K. Lew Alaska Fisheries Science Center National Marine Fisheries Service Seattle, Washington March 27, 2015 Abstract. We develop econometric models to jointly estimate revealed preference (RP) and stated preference (SP) models of recreational fishing behavior and preferences using survey data from Alaska. The RP data are from site choice survey questions and the SP data are from a choice experiment. Models using only the RP data are likely to estimate the effect of cost on site selection well but catch per day estimates may not reflect the benefits of the trip as perceived by anglers. The SP models are likely to estimate the effects of trip characteristics well but may less attention may be paid to the cost variable. The combination and joint estimation of revealed and stated preference data seeks to exploit the contrasting strengths of RP and SP data. We find that there are significant gains in econometric efficiency and differences between RP and SP willingness to pay estimates are mitigated by joint estimation. The nested logit “trick” model fails to account for the panel nature of the data and is less preferred to scaled, random parameter and generalized mixed logit models that account for panel data and scale differences. We find scale differences across data sources in only one candidate model, the error components mixed logit. While this model outperformed the standard mixed logit, willingness to pay estimates do not differ across these two models after adjusting for scale differences. 1 A previous version of this paper was presented at the Society for Benefit-Cost Analysis annual meeting in Washington, DC, March 2015. The authors thank Trudy Cameron for numerous comments. Funding for this research was provided by the Alaska Fisheries Science Center, National Marine Fisheries Service. Opinions expressed are those of the authors and do not reflect those of NMFS, NOAA, or the U.S. Department of Commerce. 1 Introduction Under the Halibut Catch Sharing Plan recently adopted by the North Pacific Fisheries Management Council (Council) and being implemented by NMFS, the Council will begin annual evaluations of charter sector-specific regulations to manage recreational harvest of Pacific halibut. The goal of the overall research project is to better understand the demand and value of saltwater recreational fishing in Alaska, with a primary emphasis on Pacific halibut and its primary economic substitutes and complements, Pacific salmon. In this paper we develop econometric models to jointly estimate revealed preference (RP) and stated preference (SP) models of recreational fishing behavior and preferences using survey data collected by NMFS (Lew, Lee and Larson 2010). The RP data is site choice behavior and the SP data is from a choice experiment. The RP and SP data have been separately analyzed by Lew and Larson (2011, 2012, 2014) and Larson and Lew (2013). The RP models in these publications use travel and time costs measured by distance to various fishing sites and income data reported in the survey. The benefits of the trips are measured by catch per day estimated from creel surveys and from estimates provided by the anglers themselves. The SP models use a total trip cost measure that does not make explicit travel distance and time cost. Expected catch per day and other attributes are included in the choice experiment. In this context, the strengths of the RP data are also the weaknesses of the SP data, and vice versa, so that the RP and SP data should be considered complementary. The combination and joint estimation of RP and SP data seeks to exploit the contrasting strengths of RP and SP data while minimizing their weaknesses (Whitehead et al. 2008, Whitehead, Haab and Huang 2011). 2 Data combination can be used to mitigate problems in both RP and SP data. In short, the RP models are more likely to provide an unbiased estimate the effect of cost on site selection but catch estimates may not reflect the benefits of the trip as perceived by anglers.2 The SP models are more likely to estimate unbiased coefficients for trip attributes (including catch) but hypothetical bias may lead to attribute non-attendance on cost. If less attention is paid to the cost variable it will be biased downward (in absolute value) leading to upwardly biased willingness to pay estimates. Site/species-specific revealed preference catch estimates do not vary across angler and, therefore, may be highly correlated. Multicollinearity may hinder estimation of unbiased catch coefficients. Estimation of individual/site specific catch rates is feasible but the data often does not support estimation (Haab et al. 2012). The SP choice experiment is designed to collect data on hypothetical behavior with individual variation in catch rates. Also, the revealed preference data are limited to analyzing behavior in response to a limited range of catch per day variation across site. The SP choice experiment can be designed to assess variation in attributes beyond the observed range. There is much evidence of hypothetical bias of choice experiment data in the transportation literature (e.g., Hensher 2010, Fifer, Rose and Greaves 2014). Metcalf et al. (2012) find that total willingness to pay is higher with choice experiment data relative to comparable contingent valuation data. They scale the willingness to pay estimates from choice experiment data down by the choice experiment-contingent valuation ratio of total willingness to pay. But, 2 The RP cost will be unbiased under the heroic assumption that the cost variable is not measured with error. 3 mostly, the environmental economics literature has been silent on this issue. Hypothetical choices may not reflect budget and time constraints on behavior or data may suffer from attributed non-attendance. Combining SP data with RP data grounds hypothetical choices with real choice behavior. While there have been a number of RP/SP joint estimation applications in other literatures and the SP choice experiment literature in environmental economics has exploded, we know of only four applications in environmental economics that combine RP and SP choice experiment data. Adamowicz, Louviere, and Williams (1994) present the first application of RP/SP joint estimation using choice experiments in the environmental economics literature. They find that the RP data are limited due to collinearity and the SP data are prone to hypothetical bias. Adamowicz, et al. (1997) present an application to moose hunting with a focus on a comparison of RP-SP models with objective or subjective characteristics of the choices. They conclude that the RP-SP model with subjective characteristics outperforms the other models, suggesting that RP models are limited by the lack of subjective measurement of choice attributes. Using the Adamowicz et al. data, von Haefen and Phaneuf (2008) extend the model with observed and unobserved preference heterogeneity. They reject the hypothesis of consistency between the RP and SP (i.e., equality of coefficients across data sources) and illustrate how sensitivity analysis can be used to account for the inconsistency. Abildtrup, Olsen and Stenger (2015) estimate forest recreation choices with a mixed logit error components model. They find significant hypothetical bias in the choice experiment data. In the next section we describe the models available for RP and SP data combination. These include the nested logit “trick,” the mixed logit error components model and the 4 generalized mixed logit model. Then we describe the data. Since the survey was not specifically designed for joint estimation we use manipulate the RP and SP data in order to create a parallel data set. Then we present the results including sensitivity analysis for some of the choices made in the data manipulation process. Finally, we offer some conclusions and directions for future research. Models Random utility theory is the basis for recreational fishing models involving joint estimation of RP and SP discrete choices, , (1) where ′ is the utility angler i receives from fishing alternative j, i = 1, …, I, j = 1, …, J, is the systematic portion of utility, is a vector of parameters, is a vector of variables specific to the choice (e.g., travel cost, catch), and ε is the random error. Given the unobserved elements of utility, we consider the probability of individual i choosing alternative j is ;∀ ∈ (2) . The multinomial logit model assumes the error terms in (2) are independent and identically distributed (iid) extreme value (also known as Gumbel) variates and takes the form: (3) , ∑ 5 where μ is a scale parameter (that cannot be identified in a typical discrete choice setting and is thus implicitly set equal to one). In the multinomial logit model, the model parameters describing site choice, , are assumed to be constant across individuals, indicating homogeneous response to site characteristics. In contrast, the mixed logit model adds additional flexibility to the site choice model by allowing for preference heterogeneity (Train 2003). For MXL, we assume that individual angler preferences differ randomly according to a specified distribution, f(ß), such that the unconditional site choice probability takes the form: (4) . ∑ In application, (4) does not have a closed form solution so estimation of the parameters requires simulation of the integral. See Train (2003) for a detailed description of the mixed logit model. A number of discrete choice models have been developed in order to appropriately combine discrete choice RP and SP data. A naïve approach simply stacks the data and employs standard logit models. When RP or SP data are estimated separately the scale parameter in the multinomial logit is arbitrarily set equal to one. When RP and SP data are stacked and estimated jointly, it is common for the error terms that result from the different data to have unequal variance leading to unequal scale parameters (Swait and Louviere 1993). It is typical, but not automatic, for the SP data to have a higher variance due, perhaps, to the unfamiliarity of the choice task. The difference in the scale parameter will cause the MNL coefficients to differ across RP and SP data sets. In this case the RP coefficient estimates will be larger than the SP coefficient estimates indicating that the characteristics of the SP choices have an unduly large effect on each choice, relative to the RP data. 6 The relative scale factor in a stacked data set can be estimated as described by Hensher and Bradley (1993), Hensher, Rose and Greene (2005) and Whitehead (2011). The so-called nested logit “trick” involves creating a choice situation for each RP (SP) observation that counterfactually also includes the SP (RP) alternatives. The model is then estimated as a nested logit with the inclusive value for the RP branch set equal to one. The SP alternatives are estimated in individual branches but with the inclusive value for each branch constrained to be equal. The variance of the RP nest is constrained equal to one while the variance of the SP alternatives is allowed to diverge from one. The nested logit model scales the SP coefficients so that the RP and SP data can be combined and jointly estimated. Each variance of the error term is an inverse function of the individual scale factor (Swait and Louiviere 1993). When the scale parameter for the RP data is set equal to one the nested logit model will estimate the relative scale factor for the SP data as the inclusive value for the SP data. The SP data is then appropriately scaled and the coefficients can be estimated jointly. The nested logit “trick” assumes that the stacked RP and SP observations from each angler are independent (Hensher, Rose and Greene 2008). Mixed logit discrete choice models acknowledge the panel nature of the data, allow correlation across choices for each angler and are capable of estimating scale parameters. Brownstone, Bunch and Train (2000) and Greene and Hensher (2007) develop a mixed logit error components model where scale is estimated as the standard deviation of alternative specific SP random coefficients where the mean effect is set equal to zero. The scale parameter is estimated by included alternative specific constants for the SP alternatives with zero mean and free variance. The scale parameter is inversely proportional 7 to the estimated variance. Hensher (2008) and Börjesson (2008) provide examples of its estimation in the RP/SP context. A scaled multinomial logit model is one in which individual scale parameters are estimated. In effect, the model estimates a different coefficient vector for each individual in the sample (Hensher 2012). The generalized mixed logit model can be used so that individual scale and preference heterogeneity can be accounted for (Fiebeg, et al. 2009). First, consider the basic ∆ mixed logit model which involves individual specific coefficient estimates, where ~ is a vector of individual characteristics, Δ is a vector of parameters, Γ , is the error term, 0,1 and Γ is the lower triangular Cholesky matrix. A mixed logit model with unobserved preference heterogeneity results when Δ is equal to zero. The basic conditional logit model results when Δ and Γ are both equal to zero. Greene and Hensher (2010) show that the mixed logit parameter estimates can be individually scaled: ∆ (5) where ′ , 1 is the mean variance, Γ are parameters, characteristics, is the coefficient on the unobserved scale heterogeneity, are individual , and 0 1 is a weighting varialbe. The scaled multinomial logit model in heterogeneous scale with a data source shift can be estimated when Δ and Γ are both equal to zero and ′ , where SP is equal to one for the stated preference data source and zero otherwise (Hensher 2012). The independently estimated RP and SP models and jointly estimated RP/SP models can be used to estimate a willingness-to-pay for changes in the non-cost attributes. Haab and McConnell (2002) show that the marginal willingness-to-pay for a change in catch for each 8 fishing trip is , where is the functional form of the change in the catch rate being evaluated which can be nonlinear (i.e., diminishing in catch). Since the coefficient on travel cost is in the denominator of the willingness to pay function, it plays a large role in value determination. Hypothetical bias in SP cost coefficients will lead to bias in willingness to pay for catch. All of these models are estimated with NLOGIT software (Greene, 2012, Chang and Lusk 2011). In the next section we estimate each of these models but efforts to estimate the SP scale parameter with a scaled MNL and generalized mixed logit models failed (these results are available upon request). Data A 2007 survey of Alaska saltwater anglers collected information on saltwater fishing participation, effort, and preferences of resident and non-resident anglers over the 2006 fishing season. The survey was administered to three groups of anglers for which separate survey instruments were developed: non-residents, residents of Southeast Alaska, and all other Alaska residents. Lew, Lee and Larson (2010) describe the development, content, and structure of the three survey versions, their implementation, and a summary of the data. In this study we focus on the southeast Alaska resident data. The southeast Alaska RP data were analyzed separately in Larson and Lew (2013). Using a repeated mixed logit model the authors show how multiple sources of catch data, creel and survey, could be used in the same model and weighted to provide improved value estimates. The southeast Alaska SP data were 9 analyzed separately in Lew and Larson (2014). The authors focused on the relationship between catch and keep and catch and release rates. In this study we focus on a sample of southeast Alaska resident anglers who took a trip to a southeast Alaska fishing site and also answered each of four stated preference questions. We use a sample size of n = 204. There is a preponderance of nonusers in the n = 398 sample and the no RP trip option makes combining data less informative (e.g., the no trip option dummy variable is strongly positive in the RP data). Only considering users in this exercise puts an emphasis on the gains from combining data from an intercept sample. This type of data combination will be most informative to other recreation demand choice experiment models that can be combined with surveys of on-site users. In order to combine data, it is recommended to have the same number of observations so that neither data source dominates in estimation. There are four SP observations (i.e., trip preferences) and 2245 RP observations (i.e., trips) in the data. There are at least two approaches to generating four RP observations: (1) we could use the 10 site data and stack the same choices four times ignoring of the total number of trips taken by the angler or (2) randomly select two or three other sites to combine with the observed choice. Both of these were attempted with option (2) eventually preferred. Reducing the choice set in this way is not expected to affect willingness to pay estimates (Whitehead and Haab 1999). First, we consider whether a model with a choice set of two or three alternatives is preferred. We compare a repeated RP model with 10 sites and all trips, a typical trip RP model with 10 sites, a typical trip RP model with 2 sites and a typical trip RP model with 3 sites. The 10 typical trip is equal to one if the site has the maximum number of trips: max 1 for and 0 otherwise. There are four ties in the data. In this case the typical trip site was chosen randomly by the computer. The distribution of typical trips is very similar to the distribution of total trips (Table 1). In order to develop an RP data set we randomly choose one and two non-chosen sites for inclusion in the model. We estimate single period multinomial logit models and find similar results. The travel cost coefficients are negative and statistically significant in both models. Importantly, the magnitude of the travel cost coefficient is similar across models. The pattern of catch coefficients is also similar with only king salmon being statistically significant. Given these similarities we begin this exercise with the 2 site model. The SP site choice data is adapted from a contingent ranking exercise where respondents are asked for their most preferred and least preferred of three alternatives. The first two alternatives (A, B) are fishing trips that vary over fishing mode, fishing days, target species, daily bag limit, actual catch and size, and cost per trip. Catch levels are truncated where the actual catch exceeds the daily bag limit. Alternative C is the no trip alternative. We assume that respondents would choose the fishing site that they rate as their most preferred. We delete cases whose choices are ambiguous. Less than 10% of the anglers prefer no trip across the four choice occasions. Anglers choose equally across alternatives A and B. Results The multinomial logit RP site selection models are presented in Table 2. We present three models, the repeated trip model including all 2245 trips across 10 sites, the typical trip model 11 across 10 sites and the typical trip model across two sites. The travel cost coefficient is negative and statistically significant in each model with the z-statistic falling as the number of trips and alternatives fall. The coefficient for king salmon is positive and statistically significant in each model with the same pattern of efficiency. The coefficient on halibut is not statistically significant in any model. The coefficient on silver salmon is only statistically significant in the total trips model. The pattern of willingness to pay estimates is similar across models with king salmon valued highest. Notably, willingness to pay for halibut is not statistically significant in any of the revealed preference models. Given that our primary interest is in improving SP models by including RP data in the estimation, we begin our investigation with the simplest data available and consider a two RP alternative data situation. In order to develop RP data for combining with the SP data we randomly select a non-chosen fishing site four times. These four alternatives are each paired with the chosen site and stacked to create a RP panel. The SP panel consists of SP choices for respondents who answered each of the four questions. The RP and SP multinomial logit (MNL) results are in Table 3. The RP model has statistically significant travel cost, king and silver salmon catch coefficients. The halibut coefficient remains statistically insignificant. The SP model has a negative travel cost coefficient but it is an order of magnitude lower (in absolute value) and estimated much less efficiently relative the RP coefficient. This suggests that anglers pay less attention to the trip cost in the hypothetical questions relative to their actual behavior. In contrast, the catch coefficients are each statistically significant in the SP model and their magnitudes are lower. The silver salmon coefficient and the no trip option coefficient is negative. 12 Next, the data are stacked and jointly estimated. Several gains from joint estimation are observed. Each of the coefficients are estimated more efficiently due to the simple fact that there are more observations. One might think that the jointly estimated willingness to pay estimates will fall in between independently estimated estimates. But, if the SP data breaks collinearity across alternatives in the RP data then this naïve expectation will not be observed. The willingness to pay estimate for halibut falls in between the RP and SP estimates and is statistically significant. Willingness to pay for halibut and king salmon are not statistically different. The king salmon willingness to pay estimate falls below both RP and SP estimates. Silver salmon willingness to pay is in between RP and SP estimates and is not statistically significant. The nested logit “trick” results are similar to the naïve stacked model but there are subtle differences. Each of the coefficient estimates are statistically significant. The SP data dominates in estimation of in estimation of the each of the coefficients. The travel cost coefficient is increased in absolute value but the catch coefficients are lower than both the RP and SP model coefficients. Both of these effects lead to lower willingness to pay estimates. The silver salmon coefficient is negative and statistically significant with the SP data dominant. Willingness to pay estimates for halibut and king salmon are lower when compared to the naïve stacked model. The SP scale parameter is greater than and statistically different from one which indicates greater variance in the SP data. The magnitude of the scale parameter is very similar to the scale parameters from two case studies in Whitehead (2011). In contrast, the scale parameter from a nested logit “trick” model with RP data including all 10 of the RP sites stacked four times is an order of magnitude greater (these results are available upon request). This suggests that there is 13 extreme heteroskedasicity in SP data when compared to RP data after stacking of uniform RP choices with additional alternatives, perhaps artificially introducing less randomness in the RP data. The choice of only 2 alternatives in the RP data is strengthened by this result but sensitivity analysis is needed. For example, increasing non-chosen alternatives greater to be greater than 2, but less than 10, may increase the efficiency and accuracy of the RP data while not introducing artificial non-randomness across choices. We next apply the scaled, mixed and generalized mixed logit models to the SP data and then apply the same models to the RP-SP with and without scale effects to determine gains from joint estimation. The mixed logit is specified with triangular distributions for the catch variables (Hensher and Greene 2003). In this case the limits of the distribution are equal to the mean and the standard errors of the limits are equal to the standard errors of the coefficients. The generalized mixed logit model is estimated with the gamma weighting parameter set to one. When the model is estimated with gamma freely estimated it is greater than one which violates the theory (Fiebig et al. 2010). In these models the no trip option is parameterized by interacting the no trip alternative specific constant with income and fishing experience. The travel cost coefficient is negative and statistically significant in each model, although it is only weakly significant in the scaled MNL model. Halibut and king salmon coefficients are positive and statistically significant in each model. The silver salmon coefficient is statistically significant in the generalized mixed logit model. Angler participation increases with income and fishing experience in the random parameters and generalized mixed logit model. The tau coefficient on the scale parameter is statistically significant in the scaled MNL and generalized mixed logit models, indicating heterogeneity in the scale parameter. The mean of the scale 14 parameter (i.e., sigma) is equal to one in the scaled and generalized mixed logit models but each individual scale parameter is not positive as required by theory (Fiebig et al. 2010). Willingness to pay estimates are larger than in all previous models. The generalized mixed logit willingness to pay estimates are more than twice as great as those from the scaled MNL and mixed logit models. The value of halibut catch is greater than the value of king salmon catch but the differences are not statistically significant. The value of silver salmon catch is only statistically significant in the generalized mixed logit model. The jointly estimated models are in Table 6. Again, several gains from joint estimation are observed. Coefficients are estimated more efficiently and the silver salmon coefficient is positive and statistically significant in each model. The travel cost coefficient is negative and (strongly) statistically significant in each model. Other results are similar to the SP models. Angler participation increases with income and fishing experience and the tau coefficient on the scale parameter is statistically significant. The mean of the scale parameter is equal to one in the scaled and generalized mixed logit models but each individual scale parameter is not positive, making these results questionable. The error components model contains scale parameters for each SP alternative. The two trip alternatives have standard deviation estimates that are not statistically different from one, indicating no heteroskedasticity. The standard deviation for the no trip option is large and statistically different from one. This indicates heteroskedasticity in the no trip option, but since there is not a no trip option in the RP data, this may not be an issue. 15 In contrast to the SP data results, the willingness to pay for catch is consistent across three of the four models. Willingness to pay for halibut and king salmon is between $41 and $49 in the scaled and mixed logit models. Willingness to pay for silver salmon is statistically lower but consistently positive across each model. The willingness to pay estimates in the generalized mixed logit model are larger than the others and follow the same pattern across species as in the RP data. Conclusions This study finds that there are significant gains to jointly estimating RP recreation trip and SP choice experiment data. There are significant gains in econometric efficiency and differences between RP and SP willingness to pay estimates are mitigated by joint estimation. The only model where the SP scale parameter made a significant difference was in the nested logit “trick” model. This model fails to account for the panel nature of the data and is less preferred to scaled and random parameter models that account for panel data and scale differences. With these data we found scale differences across data sources in only one candidate model, the error components mixed logit. While this model outperformed the standard random parameter model, there is little statistical evidence of scale differences and willingness to pay estimates did not differ across these two models. These results should not be directly compared to the Lew and Larson papers since our interpretation of the data differs. The Alaska sportfishing survey was not designed to collect data for RP and SP joint estimation. In order to appropriately combine the data we made a number of data decisions that could lead to differences in separate RP and SP analyses. Nevertheless, the 16 data handling decisions made here are not invalid; i.e., the separate RP and SP analyses in this study are different, but valid, approaches to obtaining willingness to pay values. These results raise questions about much of the SP choice experiment literature in environmental economics. Even with the attention accorded to the Adamowicz, Louviere, and Williams (1994) seminal contribution (over 1200 cites on Google Scholar), the choice experiment literature has proceeded with little to no attention paid to its similarities to RP data3. This is in stark contrast to the contingent valuation and contingent behavior literatures which have spent considerable time comparing and jointly estimating RP and SP data (Carson et al. 1996, Whitehead, Haab and Huang 2011). The lack of attention to this issue may be due to the fact that many choice experiments are not designed as behavioral surveys. With the southeast Alaska data we assumed that a ranking of the “best” trip would be a chosen trip. Many other choice experiments are designed to elicit non-behavioral preferences and elicit nonuse values for which there is no behavioral trace. Our results suggest that choice experiment survey design should pay more attention to their behavioral counterparts and that choice experiments should receive more scrutiny than seems to be the case4. While these results are promising they should only be considered as preliminary. In our models we have simplified the data to (1) focus on variables that are contained in both the RP See Araña and León (2013) for one counter-example in the environmental economics literature. See Krucien, Gafni, and Pelletier‐Fleury (2014), Lancsar and Swait (2014) and Fifer, Simon, Rose, and Greaves (2014) for discussion in the health and transportation literatures. 4 Also, Crastes and Mahieu (2014) find that choice experiments take 62% less time to publication compared to contingent valuation papers. 3 17 and SP data (travel cost and catch) and (2) assess the simplest RP choice scenarios. Considering (1), it is not necessary for the RP and SP coefficients vectors to contain the same elements. In fact, it is typical for RP data sets to have some variables that do not vary or are unobserved across individuals. In this case, the SP data must be used to identify the effect of the variable on the choices (e.g., fish size). Also, SP choice experiments may be difficult to design some attributes that are relatively easy to estimate with RP data (e.g., time cost). Future work with these data will include additional attributes from the SP survey and time cost variables from the RP data (Larson and Lew, 2014). Considering (2), future work with these data should investigate the most appropriate number of RP alternative in the panel data and include the full sample size (including nonusers from the RP data, n = 398). 18 References Abildtrup, Jens, Søren Bøye Olsen, and Anne Stenger. "Combining RP and SP data while accounting for large choice sets and travel mode–an application to forest recreation." Journal of Environmental Economics and Policy ahead-of-print (2014): 1-25. Adamowicz, W., Louviere, J. and Williams, M. (1994) Combining revealed and stated preference methods for valuing environmental amenities, Journal of Environmental Economics and Management, 26: 271-292. Adamowicz, W., Swait, J., Boxall, P., Louviere, J., and Williams, M. 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Whitehead, John C., and Timothy C. Haab. "Southeast marine recreational fishery statistical survey: distance and catch based choice sets." Marine Resource Economics 14, no. 4 (1999): 283-298. Whitehead, John, Tim Haab, and Ju-chin Huang, eds. Preference Data for Environmental Valuation: Combining Revealed and Stated Approaches. Vol. 31. Routledge, 2011. Whitehead, John C., Subhrendu K. Pattanayak, George L. Van Houtven, and Brett R. Gelso. "Combining revealed and stated preference data to estimate the nonmarket value of ecological services: an assessment of the state of the science." Journal of Economic Surveys 22, no. 5 (2008): 872-908. 23 Table 1. Revealed Preference Trips Total Typical Site Sum % Sum % 1 46 1.9% 4 2.0% 2 76 3.1% 8 3.9% 3 997 41.1% 87 42.6% 4 9 0.4% 2 1.0% 5 407 16.8% 30 14.7% 6 120 4.9% 13 6.4% 7 250 10.3% 18 8.8% 8 397 16.4% 31 15.2% 9 78 3.2% 9 4.4% 10 45 1.9% 2 1.0% 24 Table 2. Revealed Preference Conditional Logit Models RP - Total Trips RP - Typical Trip RP - Typical Trip Coeff. S.E. z Coeff. S.E. z Coeff. S.E. z Travel cost -0.05 0.00 -43.14 -0.05 0.00 -12.46 -0.06 0.01 -5.08 Halibut -0.12 0.18 -0.65 0.47 0.61 0.76 0.80 1.39 0.57 King salmon 7.74 0.30 25.79 7.93 1.05 7.52 7.91 2.75 2.88 Silver salmon 1.35 0.14 9.57 0.61 0.49 1.25 0.83 1.12 0.74 Cases 204 204 204 Trips 2245 204 204 Alternatives 10 10 2 Time Periods 1 1 1 Willingness to pay for one additional fish WTP S.E. z WTP S.E. z WTP S.E. z -2.50 3.89 -0.64 9.10 11.65 0.78 12.40 21.35 0.58 Halibut 167.96 5.86 28.68 154.32 18.26 8.45 123.14 39.54 3.11 King salmon 29.35 3.40 8.65 11.95 10.06 1.19 12.95 17.68 0.73 Silver salmon 25 Travel cost Halibut King salmon Silver salmon No trip Scale Cases Time Periods Log-Likelihood AIC Halibut King salmon Silver salmon Table 3. Multinomial Logit Models RP - MNL SP - MNL RPSP – Naïve Coeff. S.E. z Coeff. S.E. z Coeff. S.E. z -0.059 0.006 -10.730 -0.005 0.001 -3.480 -0.020 0.001 -20.250 0.969 0.748 1.300 0.409 0.071 5.720 0.659 0.077 8.590 7.428 1.306 5.690 0.340 0.067 5.040 0.594 0.076 7.830 1.142 0.553 2.060 -0.078 0.039 -2.000 0.037 0.040 0.910 -1.543 0.169 -9.140 -2.497 0.158 -15.810 204 4 -92.69 193.4 WTP S.E. 16.31 12.33 124.99 19.78 19.21 9.37 z 1.32 6.32 2.05 204 204 4 8 -726.35 -978.45 1462.7 1966.9 Willingness to pay for one additional fish WTP S.E. z WTP S.E. z 89.77 26.62 3.37 32.78 3.59 9.12 74.62 23.49 3.18 29.56 3.73 7.93 -17.04 10.56 -1.61 1.82 1.99 0.92 26 RPSP -"NL Trick" Coeff. S.E. z -0.015 0.001 -18.330 0.332 0.067 4.930 0.202 0.061 3.290 -0.083 0.039 -2.120 -3.213 0.266 -12.090 1.359 0.091 14.880 204 8 -2211.39 4434.80 WTP 22.80 13.85 -5.73 S.E. 4.41 4.19 2.70 z 5.17 3.31 -2.12 Table 4. Stated Preference Data Models Scaled Multinomial Logit Random Parameters Logit Generalized Mixed Logit Coeff. S.E. z Coeff. S.E. z Coeff. S.E. z Halibut 0.873 0.395 2.210 0.529 0.084 6.320 1.869 0.310 6.040 King salmon 0.753 0.360 2.090 0.444 0.080 5.510 1.832 0.383 4.790 Silver salmon -0.046 0.070 -0.660 -0.046 0.040 -1.140 0.190 0.099 1.910 Travel cost -0.008 0.005 -1.700 -0.004 0.001 -3.020 -0.007 0.002 -3.840 No trip x income -0.051 0.044 -1.150 -0.012 0.002 -4.810 -0.015 0.003 -5.260 No trip x experience -0.081 0.083 -0.980 -0.017 0.007 -2.340 -0.014 0.008 -1.780 Halibut_SD 0.529 0.084 6.320 1.869 0.310 6.040 King salmon_SD 0.444 0.080 5.510 1.832 0.383 4.790 Silver salmon_SD 0.046 0.040 1.140 0.190 0.099 1.910 Tau 1.413 0.422 3.350 1.609 0.075 21.400 Gamma 1.000 fixed Scale SD 1.005 0.909 Sigma SD 2.529 2.051 Cases 204 204 204 Time Periods 4 4 4 Willingness to pay for one additional fish WTP S.E. z WTP S.E. z WTP S.E. z Halibut 111.57 34.45 3.23 132.60 42.17 3.14 285.17 77.17 3.70 King salmon 96.26 32.50 2.96 111.22 36.17 3.07 279.48 75.47 3.70 Silver salmon -5.87 8.45 -0.69 -11.51 11.72 -0.98 28.94 14.51 1.99 27 Halibut King salmon Silver salmon Travel cost No trip x income No trip x experience Halibut_SD King salmon_SD Silver salmon_SD SP Alt1 SD SP Alt2 SD SP No trip SD Tau Gamma Sigma mean Sigma SD Cases Periods LL1 LL0 Chi-squared AIC McFadden's R2 Table 4. Revealed and Stated Preference Data Models Scaled Multinomial Random Parameters Random Parameters Logit Logit Logit ECM Coeff. S.E. z Coeff. S.E. z Coeff. S.E. z 1.233 0.220 5.610 0.937 0.097 9.710 1.009 0.113 8.920 1.189 0.237 5.020 1.034 0.120 8.590 1.091 0.138 7.940 0.113 0.071 1.600 0.161 0.046 3.480 0.149 0.051 2.930 -0.029 0.005 -6.30 -0.021 0.001 -19.17 -0.024 0.001 -18.100 -0.037 0.009 -4.17 -0.020 0.003 -7.410 -0.080 0.018 -4.410 -0.054 0.019 -2.890 0.820 0.141 5.810 1.00 0.98 204 8 -990.30 -1792.94 1605.28 1994.60 0.45 -0.029 0.937 1.034 0.161 0.008 0.097 0.120 0.046 -3.730 9.710 8.590 3.480 0.043 0.113 0.138 0.051 0.362 0.509 1.368 -2.430 8.920 7.940 2.930 1.130 0.590 5.120 -0.030 2.038 2.764 0.517 1.251 1.000 204 204 8 8 -973.88 -862.56 -1792.94 -1792.94 1638.11 1860.75 1959.80 1743.10 0.46 0.52 Willingness to pay for one additional fish 28 -0.105 1.009 1.091 0.149 0.408 0.302 7.002 Generalized Mixed Logit Coeff. S.E. z 2.038 0.255 8.000 2.764 0.334 8.270 0.517 0.094 5.510 -0.026 0.001 -17.72 -0.027 0.003 -8.35 0.009 0.255 0.334 0.094 -3.370 8.000 8.270 5.510 0.062 20.240 fixed 0.956 1.500 204 8 -922.60 -1792.94 1740.66 1859.20 0.49 Halibut King salmon Silver salmon WTP 43.24 41.70 3.97 S.E. 4.24 4.68 2.33 z 10.20 8.91 1.70 WTP 43.84 48.37 7.52 S.E. 4.04 5.09 2.09 z 10.84 9.50 3.60 29 WTP 41.59 44.98 13.92 S.E. 4.13 5.04 1.14 z 10.07 8.93 12.16 WTP 78.06 105.87 19.80 S.E. 8.56 11.07 3.28 z 9.11 9.56 6.04
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