MODELING OVERNIGHT RECREATION TRIP CHOICE

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The data were originally collected to focus on the Penobscot. River, a water where the majority of Maine's Atlantic salmon angling effort takes place, and where.
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MODELING OVERNIGHT RECREATION TRIP CHOICE: APPLICATION OF A REPEATED NESTED MULTINOMIAL LOGIT MODEL* [Suggested Running Head: Overnight Recreation Trip Choice] [3rd Revision, March 1998]

by W. Douglass Shaw, Associate Professor Department of Applied Economics and Statistics/204 University of Nevada Reno, NV 89557-0105 E-mail: [email protected] and Michael T. Ozog Quantitative Research Group 1712 Westchester Lane Fort Collins, CO 80525 E-mail: [email protected]

* Shaw is corresponding author. An earlier version of this paper was presented at the 1996 meetings of the American Agricultural Economics Assn. in San Antonio, Texas and we thank Jon Conrad for giving us the opportunity to present it there. Andie Litt provided us with the data. John Duffield, Joe Herriges, Cathy Kling, Kenneth Train, and two anonymous reviewers gave us constructive comments on previous versions of this manuscript. We thank Frank Lupi for engaging in discussions with us, and Betsy Fadali for her careful read of this paper. The usual caveat pertains. This research is partially supported by the Nevada Agricultural Experiment Station via funding from the U.S.D.A.

MODELING OVERNIGHT RECREATION TRIP CHOICE: APPLICATION OF A REPEATED NESTED MULTINOMIAL LOGIT MODEL

Abstract In this paper we apply the repeated nested multinomial logit model, a version of a random utility model (RUM), to estimate the choice of an overnight versus single day recreation trip, along with the other usual choice of which of the sites to visit, and less typically, the choice of whether to participate (in our application - to fish) at all. We also find statistically significant income effects in the empirical results. The application is to Atlantic Salmon fishing and the data set is for Maine resident angler's fishing trips to rivers in Maine and Canada.

JEL Classification Numbers: C25, D61, Q26 Keywords: Repeated Nested Multinomial Logit, RUM, Recreation Demand, Salmon Fishing, Trip length decisions

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MODELING OVERNIGHT RECREATION TRIP LENGTH CHOICE: APPLICATION OF A REPEATED NESTED LOGIT MODEL 1. Introduction In this paper we apply the repeated nested multinomial logit model, a version of a random utility model (RUM), to estimate the choice of an overnight versus single day recreation trip, along with the other usual choice of which of the sites to visit, and less typically, the choice of whether to participate (in our application - to fish) at all. A sample of recreators can include individuals who are more likely to take a multiple day visit rather than a single or part day visit. However, even when they appear in the original sample of those who return the survey, such multiple day trip takers are often omitted from the estimating sample for purposes of predicting demand and calculating consumer's surplus. Omitting them from the sample understates the total value (i.e. aggregated across the population who may visit the sites) for recreational resources, presuming these individuals have a positive consumer's surplus. This is especially true when a recreation destination offers special opportunities (unique geological wonders, beach outings, catch of saltwater species, or salmon and other anadromous species of fish which are not readily available in land-locked locations) that draw visitors from far away. (An example at the extreme is the European tourist who visits Arizona's Grand Canyon or California's Yosemite National Park.) Unlike much RUM analysis, which assumes away incomes effects, we are allowing for income effects to influence recreation behavior in the model below, which may be of particular importance when overnight trips are allowed. The data set for the application comes from a sample of anglers licensed in Maine for 1988 who fish for Atlantic Salmon. The data were originally collected to focus on the Penobscot River, a water where the majority of Maine's Atlantic salmon angling effort takes place, and where potential environmental damage from building a new dam was being examined (see Morey et al. 1993; or

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RCG/Hagler, Bailly). Because damages may affect those who take different length trips quite differently, our paper contributes to the recreation demand literature by exploring these differences. Specifically, changes in an individual's demand are estimated in response to changes in catch rates that could be linked to changing environmental quality. 2.

Including Trips of Varying Length in a Recreation Demand Model Because taking a recreation trip takes time away from work or alternative activities, pioneers of travel

cost modeling (see Shaw or Larson for many of the important references) include the opportunity cost of travel time in the usual travel cost proxy for the trip price, which otherwise includes only the out of pocket money costs (petroleum, road tolls) incurred in travel. These recreation demand modelers built on more general concepts about the allocation of time (Becker, De Serpa). Shaw reminds us that the time spent doing recreation activities always has a value at least equal to the value from doing the next best alternative activity, in opportunity cost terms. Though De Serpa distinguishes between the value of time "saved" (in reducing commuting time by taking an alternate route, for example) and the value of time in consumption of an activity, the usual approach in recreation modeling is to ignore possible differences in values in different uses of time. The per unit time cost is typically multiplied by the total amount of trip time is added to the out of pocket travel costs, to obtain the "full" travel cost. In micro-theoretic recreation demand models, it is the relative prices of the trips to different sites that matter in the site choice decision. When adding time cost to obtain full travel cost, the individual's cost per unit of time is often assumed constant no matter what site is visited, but the difference in his or her travel time across destinations may generate a richer variation than those differences generated by out-of-pocket costs alone. If a trip can also differ in its length, then we must consider whether and how the total trip price is impacted by trip length. At the minimum, one must consider the marginal cost of staying overnight

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(exogenous per night lodging costs). A typical modern modeling approach (fully discussed in Section 3) is the RUM, and this approach accommodates key features of the microeconomic theory. Strangely, this additional cost has not been carefully addressed in the new literature which endogenizes the trip length (discussed briefly below). The Random Utility Model An interpretation of the RUM model framework assumes that the individual knows the characteristics of possible alternatives, weighs each against the other, and chooses the alternative which provides maximum expected utility. Consider a focus of the modeling on a small set of alternatives out of a larger possible total set of alternatives (for example, recreation site visits out of the larger possible set of all activities that could be done with one's leisure time). A narrow focus such as this can be made consistent in the RUM with an assumption about weak separability between groups of goods, including activities which are time intensive (Morey 1994; or Anderson, de Palma, and Thisse). A carefully derived RUM fixes the time period for the trip decision (often called the "choice occasion") and thus also determines the budget available for recreation trips for that period. By assuming trip decisions are made each week for example, trips longer than one day can be introduced. Formally, the RUM is used to examine the difference between the conditional indirect utility function (V), evaluated at the jth versus the kth alternatives (i.e. is Vj >, =, or < Vk ?). The model commonly used in recreation demand modeling is the conditional multinomial logit model (McFadden 1978), which assumes that alternative sites are chosen, conditioned on a fixed number of total trips, i.e. a quantity which does not change when attributes of alternatives change. [See Smith 1997 or Anderson, de Palma, and Thisse]. Note that if the indirect utility function is linear in arguments (the usual assumption in almost all applications of the conditional multinomial logit that we know of), and the values the variable takes for two alternatives

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being evaluated is constant, then a variable of potential interest essentially drops out of the comparison, i.e. it has no importance in determining an individual's choice between these two alternatives. For example, suppose V( ) for a visit to site j can be written as a function of a site characteristic (aj), money income (Y), and the site price (pj), or: V j =  +  1 a j +  2 (Y - p j )

(1)

where 2 is equal to the constant marginal utility of money. Let a visit to site k (where j  k) have a corresponding Vk, written identically in function to equation (1), and note that money income drops out in comparing the two alternatives. This linearity assumption is no accident, as obtaining a closed form for the welfare measure from the discrete choice models is impossible otherwise (see Small and Rosen and discussion below). McFadden (1998) introduces a nonlinear form of the difference Y-pj, say f(Y-pj), to allow for income effects, suggesting the preferred form must be invertible to be consistent with the usual regularity conditions for utility functions (i.e. the indirect utility function is required to be quasi-convex in income and prices): -1 V j = V j [ f ( Y - p j ), OA j ]

(2)

where OAj is a vector of all other attributes of alternative j. In summary, two ways to assure the influence of an explanatory variable are (i) to make certain that the levels of key explanatory variables differ across the alternatives, or (ii) to impose a nonlinear form for the utility function which allows interactions with other variables, which unfortunately creates welfare measure related problems (laid out in McFadden 1998).

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If there are to be variables which determine the length of a trip among competing sites using the RUM, they must enter the conditional indirect utility functions for specific trip durations such that they do not drop out, as above. Otherwise, there is no point in using the RUM to evaluate any decisions involving onsite or travel time in such models. Recreation trip length has been made an endogenous choice in the context of a RUM in four other unpublished studies we know of [Desvousges et al.; Jones and Sung; Roach; Chen et al.], and one published study (Kaoru). Our model differs to an extent from all of the above in that we confine the examination to a RUM specification (unlike Jones and Sung), we model individual site choice behavior (unlike Chen et al. who aggregate to county level "sites"), we allow for specific variables such as income which intuitively have much to do with a decision to stay overnight (unlike all of the previous studies), and allow for different nest structures (unlike Desvousges et al.), examining implications of each. 3.

A Repeated Nested Multinomial Logit (NL) Model The NL is now a fairly common way to estimate recreation demand (e.g., Morey et al. 1993; Parsons

and Needelman). We focus on our trip length choice application. The data we have available tell us about the sites each individual angler chose during the course of a fishing season, though we do not literally know when the individual took each trip or what he did on every trip. One can still construe our type of data to be the chosen trips to alternative recreation sites on a number of observed choice occasions and hence the sample can be generated by a series of repeated discrete choices. We show the specifications for the utility functions below. There are eight recreation sites (j), and we want to estimate the probability of taking one day trips and trips greater than one day to the eight sites, as well as taking no trips to any of the sites (NP = no participation), yielding 17 alternatives.1 When trip duration is being considered, it doesn't seem very

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attractive to assume the Independence of Irrelevant Alternatives (IIA) assumption holds, though it is a possible structure to the model. We use the NL which assumes that the IIA property holds only for those alternatives within a group at a specific level of the nest. We model the demand for trips to fishing sites (j), allowing for the decision to participate or not (denoted by superscript f and nf ), and allowing for the decision of trip length, m. We assume individual I makes all three decisions simultaneously. The NP Alternative Note that in any possible nesting structure we let NP be an alternative. If an angler chooses not to fish on a choice occasion the utility received is Unf, but if the angler chooses to fish, utility is Uf. (Subscripts for period t are suppressed here but are implicit throughout because the analysis assumes that a decision is made on each trip choice occasion.) The angler therefore chooses to fish whenever Uf > Unf. In a given period (during a given occasion to take a trip), if the angler chooses not to fish, utility is: nf nf nf nf .5 U = V +  = ( c )+  0 PPY +  1 PPY + 

(3)

where c is a vector of observable individual characteristics that affect the decision to recreate in a given period,  is a vector of parameters corresponding to c, PPY is the budget available in period t (and equals Y/T, with T being the total number of periods), and nf is an unobservable component associated with not recreating. Let the function  of the characteristics be:

 [ ci ]  ao + a1 chi + a2 yrsi + a3 agei + a4 agei.5

(4)

where a0 is a constant term, ch = 1 if the individual reports having children in the household ( = 0 otherwise), age is the individual's age in years (allowed to have a nonlinear influence), yrs = the number of years the individual has been salmon fishing; In modeling on-site time as an endogenous choice, it may be important to include the NP decision in the

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model. Otherwise, there is no allowance for any activities besides the recreational activity on the part of the individual - he must reallocate among sites in response to a price or quality change, and cannot do something else with his time. A RUM where all individuals in the sample participate in recreation implicitly assumes that recreation is weakly separable from all other activities, so the opportunity cost of time is only in terms of fishing at some other fishing site. Such an assumption is implausible here because in choosing to devote time to fish during some period, especially when facing overnight trips, the individual likely considers the full spectrum of substitute activities. The Fishing Alternative If the angler decides to engage in salmon fishing during a given period the utility received from a visit to site j from a trip of length m is given by: .5 f .5 U m j =  o ( PPY - pm j ) +  oo ( PPY - pmj ) +  1 a j +  2 a j +  j +  m j

(5)

where PPY is again individual I's budget for the period, pmj is the cost of a trip to site j for trip length m for the individual, aj is the average number of salmon caught per trip at site j, mj is a random component known to the individual during the period, which varies across all subscripts; site to site, individual to individual, etc. Note that equation (5) features a nonlinearity in the per-choice occasion budget (using the ½ exponent follows Morey et al. and is similar to what one sees in flexible functional form literature [see Morey 1986, as example], which allows for income effects in the model. The budget per choice occasion may determine the length of a trip at each site. Assuming no income effects, income can still determine whether a trip will be taken, but not the length of the trip. It therefore seems of interest to model onsite time as an endogenous choice with income effects allowed. The cost (pmj) of visiting the site is per trip, which differs for each individual, depending on their

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trip length. Per trip cost consists of different components including out of pocket transportation costs; average on-site costs such as guide fees; and costs of time in travel. We used the same calculations as Morey et al. except for our treatment of lodging costs, which relate to how long an individual stays on a trip. [Out of pocket travel costs are calculated using the standard method of multiplying round trip travel miles by a per-mile operating cost of operating an automobile. Travel time is part of the full cost of a trip, is multiplied by the individual's per-hour value of time. Details are provided in a report (RCG/Hagler, Bailly 1989).] For those who take one (multiple) day trips this price variable does not (does) include the costs associated with lodging or staying overnight. Finally, the salmon catch rate at a specific site is assumed to be exogenous and constant for given trip lengths; it is the site characteristic that attracts the angler to the site, holding the other factors constant. Number of Choice Occasions and Income Effects Using the repeated model one may assume the season can be divided into T periods or choice occasions, each of equal length. Morey et al. briefly allude to this, and assume that an angler can not take more than one trip (q) per choice occasion, though he can take fewer (none), implying that Q  T, where Q = qt. (Though Morey et al. do not explicitly discuss this, such a relationship could be part of a constraint in a constrained maximum utility framework.) The number of choice occasions in the season is especially important in models which allow for the NP alternative and income effects because T determines the amount of the budget in each period, or per-period income. If T is assumed to be the same for every individual in the sample, then assuming a large value of T leads to many anglers having at least one choice occasion for which they take no trips. Assuming that a smaller value of T holds may require trimming those from the sample who would violate the constraint (Q  T) or truncating their actual trips to be equal to number of choice occasions, T. As will be seen below, we essentially estimate T empirically.

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Nest Structure Another key issue in NL modeling is the order of the nest (Kling and Thomson) - in our case, where to place the trip duration decision. Whether the trip duration decision should be at the bottom of a complicated nest (where Devousges et al. have placed it) or somewhere else, is the investigator's decision, which can be influenced by the logic of assumed correlations, or by empirical tests (see Herriges and Kling, 1996 or 1997a). The nest structure has implications for the correlation patterns among alternatives, though these could well be due to similarities unobserved by the researcher rather than because of a sequential decision process on the part of the individual (this is originally noted by Train, McFadden and Ben-Akiva). We specify two structures to the nest, testing to see which is preferred. In our first structure (see the left hand side of Figure 1), we assume that the angler simultaneously chooses whether to participate (the highest level-1), whether to stay overnight at a site (level 2), and which site to visit (level 3). By choosing this ordering the unobserved components are shared across site visits of the same trip length. Put another way, this structure assumes that the IIA applies to sites within a given trip length. We refer to this structure below as Model A. Next, we examine an alternative nesting structure. In our second model we flip the 2nd and 3rd levels of the nest in Model A, somewhat mimicking the structure used by Desvouges et al. This resulting model (B) is illustrated in the right hand side of Figure 1. As Kling and Thomson point out, the NMNL model's flexibility of different nest structures allows consideration of different variables to explain choices of alternatives in different groups. In choosing Model B we could rationalize that the unobservables are shared across trip lengths for the same site. For example, Kaoru estimates a three-level nested model with trip length (one, two, three, and greater than three days) at the top of the nest. His is not a repeated model and he does not allow for non-participation. He states that "recreationists are likely to decide trip length first,

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given a certain amount of time available for recreation." However, this is somewhat confusing because as we point out, the recreationist isn't necessarily making the decision in any order in the decision process. In our structure the IIA holds across trip lengths, given the site choice. We estimate both models A and B below and compare the results. 4.

The Data and Expected Consumer's Surplus for Changes in Catch Rates

4.1

The Data

The data used to estimate the model were obtained in a combined mail and telephone survey of a sample of recreational anglers who had a resident, non-resident, or complimentary license to fish for Atlantic salmon in Maine in 1988. The final sample of respondents include 168 resident anglers. Details on the original data set, and steps taken to create the variables used in estimation are in a consulting report [RCG 1989]. Anglers fish in Maine and at several sites in Canada. The survey was originally mailed out at the end of September and very few trips to Maine sites occur after this time, so for the Maine sites the individuals in the sample effectively report at least some information on their trips for the entire season. The survey questionnaire focuses on use of the Penobscot River. The angler is asked for details about fishing this river, and two other rivers on given trips, and also asked to report his total days in 1988 in which he fished for Atlantic salmon on rivers other than the Penobscot. The two other rivers questions were open-ended, allowing the respondent to name any river where he fished most frequently for Atlantic salmon. Only a few rivers were frequently mentioned out of many possible ones in Maine and Canada, resulting in some, but not complete data on approximately 18 rivers and streams in Maine and 6 rivers in several Canadian provinces. The survey questionnaire mixes the term "days" with the term "trips" in questions that seek information about total seasonal participation and only asks for Atlantic salmon, as opposed to all, fishing trips. Individuals were actually only asked their trips up to the date they received the

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survey for three rivers. This confines our modeling approach here. Even if the term "trips" had been used in soliciting the seasonal total across all sites, we would have to assume that the only important influence relating total trips to marginal trip decisions are specifically the trips taken for Atlantic salmon fishing. Aggregation of 13 rivers into eight river groups captures almost 90 percent of the Atlantic Salmon fishing trips to this region by license holders. Rivers were aggregated into eight river groups by the authors of the original study because some rivers had too few reported visits to generate reliable data. We recognize the possible bias that this aggregation may have caused (see Kaoru, Smith, and Liu or Parsons and Needelman), but do not have access to the original data (as it comes off the coded survey questionnaires) to explore the effects of regrouping the sites for this bias. However, we note that the authors who put together the original data state that sites within each group were much alike, and the groups themselves were quite heterogeneous, precisely the conditions that lead to a small bias in aggregation. The result is eight recreation "sites" for the model, five of which are in Maine and three of which are in the Canadian provinces of Nova Scotia, New Brunswick, and Quebec. Catch rates vary a good deal over these eight sites. The survey allows exploration of the trip length decision to a reasonable extent. Respondents are asked the length of their typical trip to the Penobscot River, and they are asked in separate questions the number of trips, and the number of days to date spent at the two other rivers. Most resident anglers in the sample take a one day trip to one of the sites, but 39 of our sample of 168 generally take multiple day trips somewhere. Fourteen anglers take no salmon fishing trips to the 8 sites, but eleven resident anglers take more than 50 trips in total across the 8 sites, and the mean (median) number of total trips taken is about 17 (6). The most trips are taken to Penobscot River, as 86 anglers in the sample took a one day Penobscot River trip (mean trips for this group is 10.7) and 14 anglers took a multiple day Penobscot trip (mean 1.1); only four people in the sample took trips to Nova Scotia rivers, and

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nine took trips to rivers in Quebec. All the anglers in the sample who went to the Canadian rivers took one or more multiple day trips and took no one day trips. To summarize, this sample of Maine resident anglers take more one day trips to sites close to home, leaving the multiple day trips for visits to sites further away. This feature is probably typical of many samples of recreators, at least those obtained using mail surveys. We model the choice of whether the angler takes a one day, or more than one day trip to each of the eight sites. 4.2

Expected Consumer's Surplus for Enhanced Catch Rates

The expected per period compensating (PPCV) and equivalent variation (PPEV) measures of exact consumer's surplus for the three level NMNL are derived and shown in Morey et al. (1993) [and generally for the NL in McFadden (1996).] The usual framework assumes away income effects, consistent with Small and Rosen, so a closed form solution for the measures is possible. Under these assumptions, the PPCV equals the PPEV. As stated above, in contrast we allow for income effects and thus, there is no closed form solution for either welfare measure. In this case general, the PPCV will not be generally equal to the PPEV. Morey (1994) suggests use of a numerical optimization procedure to solve for the PPCV and PPEV generated when there are income effects. Using the PPCV as an example, Morey minimizes: M = [V(ppy, yrs,age, P0 ,catch 0 ) - V(ppy - PPCV, yrs,age, P1 ,catch 1 ) ] 2

(6)

A nonlinear optimization procedure (such as can be found in any numerical optimization package) can simply be used to search for the global value of PPCV which minimizes the expression above for each individual in the sample. McFadden (1996) demonstrates that use of the Morey method generates a bias in the PPCV and recommends the use of a different method of simulating the CV (using a Markov Chain approach he calls a GEV sampler). However, the bias appears to be small. First, McFadden's numerical

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computations indicate the bias is about 10 percent for a 40 percent quality change. The degree of difference between the GEV sampler method and the Morey method is better illustrated in simulations performed as part of new paper by Herriges and Kling (1998), who are the first to test the GEV sampler. They show that the computational method suggested by McFadden is very tedious and that "...the consistent welfare estimates provided by the GEV sampler are not substantially different from the simpler linear and representative consumer approximations... This in turn suggests that, at least for [their] data set, the extra computational burden associated with the GEV sampler many not be worth the trouble" (p. 23). We derive the per season CV (EV) using Morey's method to obtain the PPCV and by then multiplying PPCV (PPEV) by T. Given the assumptions of the repeated model, we have a straight-forward way of calculating the welfare measure for the entire season. We can calculate these for all anglers, those who take only one day, and those who take multiple day trips. 5. The Likelihood Function, Estimation and Results 5.1

The log likelihood function

Given our model and the usual assumption about the distributions of the error term [i.e. that the error term for the nested logit follows the generalized extreme value distribution or GEV (McFadden, 1997)], the log of the likelihood function for our sample of 168 resident anglers is 168 8

_ =   y ji

ln 

ji

(7)

i=1 j=0

where yji is the number of trips individual I took to site j (if j = 0 it is the number of times the individual chooses not to fish) and  is the probability of fishing (not fishing) at site j for individual I, conditional on higher level decisions. The probabilities are different in Models A and B because of the structure or order

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of the nest; to review - Model A (B) has site choice (trip length) conditional on participation and trip length (site choice). The form of the specific probabilities for the three level NL model are shown in Morey (1994). 5.2 Log Likelihood Results We estimate our model using Full Information Maximum Likelihood (FIML).2 Results for models A and B are reported in Table I. Model B's lower likelihood suggests a better fit of the variation in trip patterns and all parameters are asymptotically significant except the no-fishing constant term. All parameters in Model A are also highly significantly different from zero except for the linear term for the catch rate. Of particular importance here, note the parameter on the nonlinear term involving the budget is significantly different from zero in both models. This indicates the importance of income effects in our model. For variables that appear linearly in the conditional indirect utility functions (which have easily interpretable signs), there are few surprises. We defined some dummy variables to help better explain overnight visits to the Penobscot River; specifically, we added a retirement dummy variable (= 1 if retired) in the indirect utility function for Penobscot River overnight trips. Its estimated negative sign is unexpected, as we anticipated that retired anglers might take overnight trips to Penobscot more "easily" than employed anglers. Having children in the household increases the probability of not fishing, as we expected could be true, and more years of fishing experience has a negative influence on the probability of not fishing. Model B overall also fits the data reasonably well, though the interesting phenomenon is that the signs on the two catch rate variables are reversed from Model A. What is of interest here is whether the probability of a site visit increases with an increase in the catch rate. While the probabilities are individual-

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specific, our intuition says the relationship should be positive, because catching more salmon should be a "good" thing for most anglers. However, the formula for the probability of a site visit varies considerably depending on the nesting structure (for example the inclusive value from a chosen trip length enters the site visit probability formula differently in Model B than in Model A because of the order of the nest), so there is no reason to expect that the overall influence will be similar in the Models A and B. Because of the highly nonlinear way in which the catch rate enters the probability formula it is difficult to tell by simple inspection of the parameters whether the overall effect of catch rates is positive or negative. (Salmon catch rates are typically quite small, but the exponential form accentuates the importance of high catch rates). Simulations (using the estimated parameters in the probability formulas, fixed catch rates, and fixed values of the other variables that enter into the formulas) shows that the overall influence is positive in both models when the catch rate becomes large enough, but it is negative in Model B for small catch rates. Model B's overall negative influence in this range is consistent with the estimated influence in Morey et al. The difference in the two models is akin to a form of specification error. In Model A and Model B, our scale parameters are different than one, indicating that the IIA does not hold between all alternatives.3 Model A yields results consistent with the Daly-Zachary/McFadden global conditions; Model B does not. Our Models (A and B) have the same variables and hence, the same number of independent parameters, but quite different nest structures. For comparison of nest structures Kling and Thompson suggest using Pollak and Wales' likelihood dominance criterion (LDC), which is in fact a test for model selection, to test preference for different nest structures. The test indicates a preference for Model B, however Model A might still be preferred because of consistency with the Daly-Zachary/McFadden condition. Estimate of T

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As alluded to above, the sample trip data may be used to help determine the appropriate number of choice occasions for the model. Otherwise, we have no good feel for where to begin in setting a value of T for the problem. For example, using Morey et al.'s value of T = 50, our parameter estimates, particularly the scale or dis-similarity parameters, took on many different values at convergence (from different starting values), an unsettling. We think the reason for this is due to the difference in our model's structure from Morey et al. To find an appropriate value of T for our problem we fix T at several different values (including the median trips value) and estimate to obtain convergence and values for all of the other parameters. In this manner we can search for the value of T which yields the smallest likelihood at convergence. T = 10 resulted in convergence with about the same parameter estimates for Models A and B each time we estimated the models. 5.3 Welfare Estimates The PPCV and PPEV measures are estimated for each individual for the purpose of illustrating possible differences across those who take trips of different lengths. The scenario we examine is a doubling of salmon catch rates at the Penobscot River. This scenario is chosen because of controversial management plans there (see Morey et al.). The individual is assumed to face the original salmon catch rates, and the PPCV indicates the angler's maximum willingness to pay to bring about the doubling of the catch rates. For the PPEV, the angler faces the doubled catch rates first, and the PPEV indicates the amount of money which would have to be taken away to make him as well off as if he faces the original catch rates. Recall that because the model predicts total seasonal participation, the seasonal CVs and EVs can be obtained by multiplying the per-period CV by the number of periods in the season. Results for the CVs from Model A are reported in Table II for each group of individuals: those who stay

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overnight (n = 39), those who do not (n = 129), and for the full sample of 168 anglers. Those who stay one day have lower average CVs than those who take multiple day trips. This is perhaps somewhat surprising because overnighters must pay a higher actual trip cost involving lodging fees, thus extracting some of the surplus. It is tempting to rigorously compare our welfare measures to those calculated by Morey et al. because of the similarity in the sample, and we note here that ours are on average, lower. We caution against too much comparison. In our Model A there are 10 periods, meaning that the PPCVs are multiplied by 10 and in Morey et al. (1993) each PPCV is multiplied by 50 periods. Our model structure is quite different from theirs because we model trip length while they model region choice assuming all those who go to Canadian fishing sites take four day trips. In our model income effects influence the trip length decision and we also re-calculated the travel prices so the prices are not the same for the two studies, which obviously can have an influence on resulting welfare measures. 6.

Summary and Conclusions The model estimated in this paper offers a unified approach to modeling the choice of varying length

trips as well as no participation in the recreation activity, and inclusion of income effects (and testing for their presence). Our nested logit approach allows examination of two different places in a nest for the trip length decision. Income is a significant variable in our model which includes those who take overnight trips, and thus, despite the consequences for estimating welfare measures, they are important for this application. For our sample, the welfare measures for those who take trips longer than one day are, on average, a bit larger than for those who only take one day trips - perhaps a surprising result. A test also indicates that putting trip length at the bottom of the nest is preferable to putting it in the middle. Our empirical result

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may confirm suspicions about the assumption that trips of unequal length exhibit the IIA property, which would be true in the model structure chosen by Desvouges, et al. Our repeated model is one approach to handling overnight trips and it may be appropriate for some types of recreation and for some types of people. It does restrictively assume there is independence between the periods. One alternative approach is to link the decisions in the periods together, requiring yet another assumption, such as diminishing marginal utility from each subsequent trip taken to a particular site (Adamowicz et al. 1990). Another alternative is to link the site choice model to a total-trips model, presuming one has the data on the total number of trips taken (e.g. Shaw and Jakus 1996), but problems remain with this approach (Smith 1997; Shonkwiler and Shaw 1997). This linked approach abandons the concept of using the repeated model to obtain seasonal welfare measures, as the aggregate demand function is used for that purpose in the linked models.

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NOTES 1. Few NL's include the participation or nonparticipation decision, i.e. modeling the choice of whether the individual recreates at all or not at all. Morey et al. 1993 do so and our model also allows this. 2. Some authors (e.g. Parsons and Needelman) suggest breaking the estimation steps, which is what they mean by "sequential" estimation. Sequential estimation unfortunately yields coefficients that are not asymptotically efficient and calculation of efficient standard errors requires correction on the variancecovariance matrix (Murphy and Topel). 3. Care must be exercised when comparing results on the scale parameters to standard conditions found in many different nested logit papers. As Herriges and Kling (1997) note, there are two terms frequently used, with opposite meanings: the dissimilarity and similarity parameters. Morey et al. (1993) added a third term, which relates to the others. All terms refer to a group parameter which when estimated reveals two things: whether the model is consistent with utility maximization; and whether the nested logit as opposed to the non-nested conditional logit, is worth undertaking. In our model we use Morey's (1994) "s" parameter which is equal to 1/, using  as the definition for the dissimilarity parameter from Kling and Herriges. Under the Daly, Zachary and McFadden global conditions, Morey's s should be greater than or equal to one.

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REFERENCES Adamowicz, W.; S. Jennings; A. Coyne. (1990) A sequential choice model of recreation behavior. Western J. of Agricultural Econ. (July): 91-99. Anderson, S.P.; A. de Palma; J. Thisse. (1996) Discrete Choice Theory of Product Differentiation. Cambridge, Mass. U.S.: MIT Press. Becker, G. (1965) A Theory of the allocation of time. Economic Journal 75: 493-517. Ben-Akiva, M. and S.R. Lerman.(1985) Discrete choice analysis: theory and application to travel demand. Cambridge: MIT Press. Bockstael, N.E.; W.M. Hanemann; C.L. Kling. (1987) Estimating the Value of Water Quality Improvements in a Recreational Demand Framework. Water Resources Research 23:951-60. Carson, R.; W.M. Hanemann; T. Wegge (1987) Southcentral Alaska Sport Fishing Economics Study. Report prepared by Jones and Stokes Associates (Sacramento, CA) for the Alaska Dept. of Fish and Game. Chen, H.Z.; J.P. Hoehn; F. Lupi; and T. Tomasi (1995) A repeated nested logit model of fishing participation, site choice and trip duration. Proceedings of the W-133 Regional Research Group Conference (Monterey, CA): Benefits and Costs Transfer in Natural Resource Planning, Eighth Interim Report. (Complied by Doug Larson, U. of California, Davis). Daly, A. and S. Zachary. (1979) Improved multiple choice models. In: Identifying and measuring the determinants of mode choice, D. Hensher and Q. Dalvi (eds.). Teakfield, London. 335-57. De Serpa, A.C. (1971) A theory of the economics of time. Economic Journal 81: 233-46.

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Desvousges, W.H.; S.M. Waters; and K. Train. (1996) Addendum to Report on Potential EconomicLosses Associated with Recreation Services in the Upper Clark Fork River Basin. (Vol.IV Submitted to U.S. District Court, Dist. of Montana). Triangle Economic Research, Durham NC 27713. Herriges, J. and C. Kling. (1996) Testing the consistency of nested logit models with utility maximization. Econ. Let. 50:33-39. Herriges, J. and C. Kling. (1997) The performance of nested logit models when welfare estimation is the goal. American J. of Agricultural Econ. 79: 745-67. Herriges, J. and C. Kling. (1998) Nonlinear income effects in random utility models. Forthcoming, Rev. of Econ. and Stat. Jones, C.A. and Y.D. Sung. (1991) "Valuation of Environmental Quality at Michigan Recreational Fishing Sites: Methodological Issues and Policy Applications." Draft Final Report prepared for the U.S. EPA, Contract No: CR-816247-01-2. Kaoru, Y. (1995) Measuring marine recreation benefits of water quality improvements by the nested random utility model. Resource and Energy Econ. 17: 119-36. Kaoru, Y.; V.K. Smith; J. Liu. (1995) Using random utility models to estimate the recreational value of estuarine resources. American J. of Agricultural Econ. 77: 141-51. Kling, C. and J. Herriges. (1995) An empirical investigation of the consistency of nested logit models with utility maximization. American J. of Agricultural Econ. 77 875-84. Kling, C. and C.J. Thomson. (1996) The implications of model specification for welfare estimation in nested logit models. American J. of Agricultural Econ. 78 No. 1 (Feb.) Larson, D. (1993) Joint recreation choices and implied values of time. Land Econ. 69, No. 3: 270-86. McConnell, K.E. (1992) On-site time in the demand for recreation. American J. of Agricultural Econ. 74:

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919-25. McFadden, D. (1978) Modeling the choice of residential location. A chapter in Spatial interaction theory and residential location. A. Karlquist et al. (eds), North Holland, Amsterdam: 75-96. McFadden, D. (1998) Computing willingness to pay in random utility models. Forthcoming, special issue: Resource and Energy Economics. Morey, E.R. (1986). An introduction to checking, testing, and imposing curvature properties: the true function and the estimated function. Canadian J. of Economics XIX: 207-35. Morey, E.R. (1994) Two Rums uncloaked. Discussion paper, Dept. of Economics, University of Colorado and Proceedings of the W-133 Regional Research Conference, Tucson, AZ. Morey,E.R.; M. Watson; R.D. Rowe. (1993) A Repeated Nested-Logit Model of Atlantic Salmon Fishing," American J. of Agricultural Econ. 75: 578-592. Murphy, K.M. and R.H. Topel (1985) Estimation and inference in two-step econometric models. J. of Business and Economic Statistics 3: 370-79. Parsons, G. and M.S. Needelman. (1992) Site aggregation in a random utility model of recreation. Land Econ. 68: 418-433. Pollak, R. and T. Wales. (1991) The likelihood dominance criterion. J. of Econometrics 47: 227-242. RCG/Hagler, Bailly Inc. (1989) Fishing for Atlantic Salmon in Maine: An investigation into angler activity and management options. Final consulting report prepared for Bangor Hydroelectric Company, Bangor, Maine.

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Roach, B. (1995) Valuing travel time by isolating the components of recreation benefits. Proceedings of the W-133 Regional Research group conference (Monterey, CA): Benefits and Costs Transfer in Natural Resource Planning. Complied by Doug Larson, U. of California-Davis. Shaw, W.D. (1992) Searching for the opportunity cost of an individual's time. Land Econ. 68: 107-115. Shaw, W.D. and P.Jakus. (1996) Travel cost models of the demand for rock climbing. Agricultural and Resource Econ. Review 25, No. 2: 133-142. Shonkwiler, J.S. and W.D. Shaw (1997) Aggregation of conditional multinomial logit site-specific demands. Discussion paper, Dept. of Applied Econ. and Stat., University of Nevada, Reno. Small, K.A. and H.S. Rosen. (1981) Applied welfare economics with discrete choice models. Econometrica 49: 105-130. Smith, V.K. 1997. Combining discrete choice and count data models: a comment. Discussion paper, School of Environmental Studies, Duke University (Durham, NC) Train, K.E.; D. McFadden; M. Ben-Akiva. (1987) The demand for local telephone service: a fully discrete model of residential calling patterns and service choices. Rand J. of Econ. 18: 109-23.

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Insert figure 1

Table I: Results of Maximum Likelihood Estimationa Model A Participation Trip Length Site Choice

Model B Participation Site Choice Trip Length

-0.279 (0.230)

22.140*** (1.490)

(catch rate).5

1.710*** (0.484)

-38.901*** (2.850)

(PPY - Price)

0.003*** (0.000)

.009*** (0.001)

(PPY - Price).5

0.126*** (0.018)

0.672*** (0.084)

Penobscot River trips - constant term

0.643*** (0.107)

2.056*** (0.130)

Retirement dummy variable for overnight trips to Penobscot River

-0.130*** (0.028)

-1.869*** (0.432)

Yrs - years spent fishing

-0.269*** (0.014)

-0.298*** (0.015)

Age - in years

1.636*** (0.259)

2.030*** (0.279)

Age.5

-1.489*** (0.326)

-1.956*** (0.348)

Constant for non-fishing alternative

3.289*** (1.038)

-1.690 (1.167)

Have children dummy variable for non-fishing choice

0.653*** (0.117)

0.748*** (0.123)

Bottom level scale parameter (s)

2.256*** (0.344)

0.378*** (0.023)

Middle level scale parameter (t)

10.434*** (1.612)

0.688*** (0.035)

-5269.7

-4489.7

Variable catch rate

Log likelihood value

a All results obtained using a FIML routine. Standard errors in parenthesis. *** in both models indicates asymptotic significance at the one percent level.

Table II Estimated Compensating Variations for 100 percent Penobscot River Catch Rate Improvementa Model Ab

All 168 individuals

n = 39 who took overnight trips

n = 129 who took one day trips

mean CV

$268

$285

$263

minimum CV

$51

$85

$51

maximum CV

$821

$821

$615

median CV

$258

$259

$257

a All CV's involve income effects and are therefore estimated using a numerical optimization procedure. Note that CVs and EVs for Model B could not be estimated because of problems using the OPTMUM Gauss numerical search method to find the PPCV given the values of the estimated Model B parameters. Specifically, we found that the catch rate parameters cause the algorithm we used to search in the wrong quadrant of expected maximum utility space, resulting in nonsensical estimates of the PPCV and instability over given starting values.

b Model A has trip length in the middle of the nest.

TECHNICAL APPENDIX: PROBABILITY EQUATIONS The form for the probabilities varies depending on the nest structure for the alternatives. We can write the probability for not fishing at all, for Model A (see Table 1) as: eV NF

Probnf =

J

V NF

e

+[

8where each Imj is the inclusive value term for each alternative site and

M

 ( I

t/s

mj

) ]

1/t

j=1 m=1

trip length that can be chosen. The inclusive value is the sum of all of the possible alternatives at a given level of the nest and when using FIML, the parameters are estimated simultaneously with all other parameters. Let m = od be a one day trip and m = o indicate an overnight trip. The probability the individual will choose site j of a one day trip length is: J

e

sV j

+ [  ( I j )t/s ] 1/t-1 ( I od )t/s-1 j=1

Prob j =

J

V NF

e

+[

9

M

( I

t/s

mj

) ]

1/t

j=1 m=1

The probability that the individual chooses site j of a more than one day, or overnight, trip length, is: J

e

sV j

+ [  ( I j )t/s ] 1/t-1 ( I o )t/s -1 j=1

Prob j =

J

V NF

e

+[

10

M

( I

t/s

mj

) ]

1/t

j=1 m=1

The Model B structure flips the site choice and trip length decision in the nest and therefore the probability equations for both the "not fishing", and other alternatives, differ accordingly. For example, the inclusive value term for the middle level group appears in the numerator for each probability formula for the chosen site in the Model A probability equation above. However, because the structure for Model B has the middle level group as the site choice rather than the trip length, the appropriate inclusive value term would be the sum of appropriate alternatives across possible sites rather than over the two possible trip lengths.