A Latent Class Approach to Modeling Endogenous Spatial Sorting in Zonal Recreation Demand Models

II. Unobserved Heterogeneity and Aggregation

Despite the current popularity of random utility modeling, there remain good reasons to model recreation demand using an aggregate count data specification, also known as a “zonal traVel cost model.” First, the discrete and nonnegative nature of the recreation decision lends itself to well-known count data frameworks such as the Poisson and negative binomial. Second, there is an abundance of readily available recreation permit data and aggregated census data that can substitute for individual surveys; the latter can be costly to collect and prone to selection bias (Weber and Berrens 2006). Third, compared to other approaches that rely on individual data, aggregate count data models are less prone to model specification bias and have performed well in Monte Carlo tests (Hellerstein 1995). And fourth, it is straightforward to obtain estimates of seasonal welfare changes including access value.

However. there are also potential problems associated with these models. First, it is not feasible to model the consumer’s complete utility maximization problem. Although the same is true of other modeling frameworks, this can lead to biased welfare estimates unless certain assumptions and restrictions are deemed valid. Second, reliance on aggregate rather than individual data can introduce aggregation bias into the analysis (Stoker 1993). And third, large sample populations likely exhibit substantial amounts of unobserved heterogeneity, which if neglected also can be a source of bias.

LaFrance (1990) notes three approaches to address the first problem: (1) aggregate across goods, (2) assume preferences are appropriately separable, or (3) estimate a theoretically consistent incomplete demand system. Shonkwiler (1999, 256) argues that aggregation is not desirable when the focus is on demand for specific goods, and LaFrance (1990) and Shonkwiler (1999, 256) outline several shortcomings of invoking arbitrary separability. Incomplete demand systems thus have emerged as an attractive option for addressing this problem, particularly when modeling demand for multiple goods that may be substitutes.

Stoker (1993) provides an excellent overview of aggregation issues. To be consistent with theory when working with aggregate data, the analyst should either aggregate individual demands exactly by integrating over the distribution of characteristics in the population (Hellerstein 1995; Moeltner 2003; Blundell and Stoker 2005, 352), or assume utility functions consistent with the Gorman form and estimate derived demand functions using the aggregate data (LaFrance, Beatty, and Pope 2008). Moeltner (2003) demonstrates how information about the distribution of characteristics in ZIP-code-level census data can be incorporated into a zonal recreation demand model, but finds only a 5% change in welfare estimates after doing so.

Individual heterogeneity can be addressed in a variety of ways, depending on its nature. As described by Moeltner (2003), it is straightforward to incorporate observable heterogeneity into the demand function by including the appropriate moments of regressors thought to influence demand (e.g., age, income, gender, etc.). Unobserved heterogeneity is often addressed with either fixed or random effects, or with latent class (Provencher, Baerenklau, and Bishop 2002; Boxall and Adamowicz 2002) or random parameters models (Train 1998). All of these approaches permit different parameter estimates to be associated with different individuals in the population, thus potentially improving measures of model fit and accuracy and avoiding certain types of bias.

Issues of heterogeneity and aggregation are thus related. When there is little heterogeneity or when the nature of it is well known in a population, aggregation is easier and the scope for aggregation bias is small. Moeltner’s (2003) analysis provides some evidence that neglecting to account for observable heterogeneity at the ZIPcode level does not substantially bias welfare estimates in an aggregate count data framework.¹ However, to date, there has been no examination of the consequences of neglecting to account for unobserved heterogeneity. This article addresses this gap and focuses on the relationships between unobserved heterogeneity, endogenous spatial sorting, and welfare estimation in zonal recreation demand models.

III. Modeling Unobserved Heterogeneity in an Aggregate Count Data Framework

Following Moeltner (2003), Englin, Holmes, and Niell (2006), and others, this analysis posits a semilogarithmic incomplete demand system at the individual level as the basis for the model (von Haefen 2002):

[1]

Here, x_ij is individual i’s demand for trips to site j∈{1,...,J}; α_j is a demand shifter that is a function of observable individual- and site-specific variables q_ij; p_ik is individual i's cost to access site k; each β is an estimable parameter; y_i is individual i’s income; and γ_j is an estimable parameter. One set of parameter restrictions that guarantees integrability of the demand system described by equation [1] is (LaFrance 1990; von Haefen 2002):

[2]

Imposing these restrictions on equation [1] gives

[3]

where a_j(q_ij)≡ln(α_j(q_ij)), and each β_j is restricted to be negative. As in previous studies, this analysis assumes a_j(q_ij) is linear in q: a_j(q_j)≡δ′q_ij, thus a(q_j)≡ exp(δ′q_j). LaFrance (1990) and von Haefen (2002) derive compensated welfare measures for this demand system, which are used later in the analysis to estimate recreation access values.

A statistical framework is needed to estimate the parameters β, γ, and δ. For reasons discussed below, this analysis specifies that individual demand for each site follows an independent Poisson distribution (Cameron and Trivedi 1986):

[4]

with mean and variance both equal to λ_ij, and λ_j ≡ E(x_ij) = exp(δ′q_ij + z β_jp_ij + γy_i), ∀j. Because zonal models use aggregate rather than individual data, this analysis and others before it assume q_ij=q_1j, p_ij = p_ij, y_i=y_I, ∀i ∈ I for consistent aggregation, where I indicates a specific zone and subscript I indicates an aggregated (typically per capita) variable for that zone. Furthermore, because the sum of N independent Poisson distributions is also Poisson with parameter , it follows that the density for the aggregate zonal demand, , is given by

[5]

where λ_ij ≡ E(x_Ij) = exp (δ′q_Ij + β_jp_Ij + γy_I), ∀j and n_I is the population of zone I.

As mentioned previously there are multiple ways to incorporate unobserved heterogeneity into equation [5]. The approach used here is to assume the population of agents within any zone can be characterized as a mixture of distinct but unobservable homogenous groups. This is known as a latent class or finite mixture model, and it has been used in several previous applications with individual data in the context of random utility (Swait 1994; Morduch and Stern 1997; Provencher, Baerenklau, and Bishop 2002; Scarpa and Thiene 2005). Wedel et al. (1993) and Bockenholt (1993) appear to provide the first instances of latent class count data models in the literature. These and subsequent applications—mainly in the marketing and health literatures—use individual data. The approach used here is similar, but the model structure and interpretation are somewhat different, and the aggregate nature of the data imposes some additional constraints that are discussed below. Although there is no theoretical reason to prefer a latent class model to a random parameters model, the latent class approach is intuitive and computationally easier than the random parameters approach. It also can be useful for identifying policy-relevant subpopulations of individuals.

In previous applications using individual data, the probability that an individual i belongs to group g ∈{1,...,G} often is assumed to be logistic:

[6]

where each ϕ_g is an estimable parameter vector (ϕ₁≡0 for identification), and z_i are individual characteristics thought to influence group membership.² However, when working with aggregate data it is not possible to allocate individuals to groups, even probabilistically, because z_i is not observed. Rather, the analogous concept is to interpret equation [6] as the expected proportion of individuals in zone I belonging to group g. To implement this it is again necessary to assume z_i = z_I, ∀i ∈ I; then the expected proportion is given by

[7]

Because the aggregate zonal demand remains a sum of independent Poisson distributions in the presence of multiple latent groups, the latent-class analog for equation [5] is parameterized by the expected aggregate demand across all classes of individuals in a given zone:

[8]

where subscript g refers to group-specific versions of previously defined variables. Incorporating [8] into [5] gives

[9]

with associated sample log-likelihood³

[10]

Note that because group-specific demands are not observed, the regression still relies on the observed aggregate zonal demands , which are generated by the mixture of groups in each zone. However each group is now characterized by an expected individual demand:

[11]

which can be aggregated up to give the expected demands by group and by zone: n_I π_I,g λ_Ij,g.⁴

These group-specific expected demands provide the basis for group-specific welfare estimates that can inform the distributional effects of policies affecting recreation demand. In this application, equivalent variation (EV) is used as the welfare measure. Conditional on group membership, EV for an individual in a particular zone is given by (Englin, Boxall, and Watson 1998)

[12]

where is the prepolicy set of access costs and the postpolicy set. It follows that aggregate EV for a group in a particular zone is n_I,gv_I,g. However, because groupspecific populations cannot be known with certainty, the expectation of this quantity must be used: E(n_I,gv_I,g) = n_I π_I,gv_Ig. These expectations may be aggregated across zones to provide estimates of group-specific welfare effects in the population.

Some further comments on the preceding model are warranted, particularly with regard to fixed and random effects and negative binomial (NB) specifications. A well-known criticism of the basic Poisson model is that the mean and variance are constrained to be equal. Because the Poisson is a member of the linear exponential family (LEF) of distributions, this constraint potentially affects the efficiency but not the consistency of the Poisson estimates under a distributional misspecification (i.e., if the true data-generating mechanism is not Poisson), provided the conditional mean function is specified correctly (Cameron and Trivedi 1986).⁵ However, typically it is possible to relax this constraint, obtain more efficient estimates, and incorporate a limited amount of unobserved heterogeneity by introducing random effects into the regression (Greene 1997, 939). When the random effects are assumed to follow a gamma distribution, this produces the NB model. Depending on the specific parameterization of the gamma function, different NB models are obtained including Cameron and Trivedi’s (1986) NB1 and NB2 specifications. A major advantage of using the NB2 model is that it is a LEF distribution; however, using the NB2 model with aggregate data is problematic because the sum of independent NB2 draws is not NB2. The sum of independent NB1 draws is NB1; however the NB1 is not a LEF distribution so it is not robust to misspecification. Furthermore, Cameron and Trivedfs (1986) quasi-generalized pseudo-maximum-likelihood estimator cannot be used to estimate either model with latent classes because group-specific demands are not observed, so consistent estimates of the nuisance parameters cannot be obtained. Other random effects estimation methods— quadrature, Monte Carlo, and quasi-Monte Carlo methods—all require evaluating the factorial of the total number of trips in equation [9]; but with aggregate data the value of the factorial can be quite large, resulting in numerical errors during estimation.

Another approach would be to incorporate fixed effects into the Poisson model, which normally is straightforward (Greene 1997, 940). However introducing fixed effects into a latent class Poisson model is problematic again because the group-specific demands are not observed. This prevents substituting sufficient statistics for the fixed effects to obtain a concentrated likelihood function that can be optimized over the slope parameters only. Greene (2001) proposes a “brute force” method that may be adaptable to a latent class Poisson model and avoids inverting a potentially large Hessian, but this approach has been left for future work.

The basic Poisson model thus provides a robust and theoretically sound framework for calculating consistent (though potentially inefficient) parameter estimates from aggregate data.⁶ Furthermore, in a related application using this dataset (Baerenklau et al. 2010), ML estimation of a NB1 model without latent classes predicted 3.4 times the actual number of trips and increased welfare estimates by 24% relative to a Poisson model, which correctly predicted the total number of trips. Other authors (e.g., von Haefen and Phaneuf 2003a; Englin, Holmes, and Niell 2006) have found similar results with NB models. Thus von Haefen and Phaneuf (2003b) have suggested the Poisson is preferable for policy purposes. For these reasons, this application relies on a Poisson framework with robust Eicker-White standard errors for hypothesis testing.

IV. Study Site and Data

The preceding framework is applied to backcountry recreation during 2005 in the San Jacinto Wilderness in the San Bernardino National Forest in southern California.⁷ The wilderness covers over 13,000 hectares and is located within a 2.5 hour drive of most of the Los Angeles, San Diego, and Palm Springs metropolitan areas (population: ~18 million) and attracts roughly 60,000 backcountry visitors each year. An additional 350,000 visitors ride the Palm Springs Aerial Tramway into the Mt. San Jacinto State Park but do not enter the backcountry. The Pacific Crest Trail traverses the wilderness form north to south, and elevations range from 1,800 to 3,300 meters.

Backcountry access is regulated by two U.S. Forest Service ranger stations and one state park office. Horses are allowed but bikes and motorized vehicles are prohibited. Day hiking is by far the most popular activity in the backcountry. Day hikers enter the backcountry via several vehicle-accessible trailheads located on the north, west, and south sides of the wilderness (regulated by a ranger station and the state park office, both located in Idyllwild), or by riding the tram and then hiking in from the east side (regulated by a ranger station located in Long Valley). Table 1 presents some statistics for the 10 trailheads that account for nearly all day-use visitors.

Backcountry visitors are required to obtain a permit in either Idyllwild or Long Valley, but the Forest Service estimates⁸ the compliance level is around 75%.⁹ Data needed to perform a standard count data travel cost analysis is available from permit receipts maintained by the Forest Service and state park offices. Each permit lists the date of the trip, the number of people in the group, the entry and exit points, and the home address of the group leader. Most wilderness areas maintain similar records, which helps explain the popularity and usefulness of travel cost models for estimating recreation value (Moeltner 2003). Because the focus of this article is the latent class approach, rather than estimating trail-specific access values or marginal values of trail characteristics, the estimation is simplified by combining all trails accessed through Idyllwild into one destination and all trails accessed through Long Valley into another. This is a reasonable simplification because all permitted hikers first visit one of these two destinations before proceeding to a trailhead, and only a relatively small additional cost is incurred to reach each trailhead after obtaining a permit.

Following conventional methods (e.g., Moeltner 2003; Weber and Berrens 2006), the hiking permit data is combined with the most recent census data (U.S. Department of Commerce 2000) to construct a dataset containing the number of backcountry trips taken from each of the 586 ZIP codes within a 2.5 hour drive of the wilderness and certain population characteristics of each ZIP code that are likely to help explain variation in recreation demand across ZIP codes (e.g., race, gender, age, education level, income).¹⁰ The price of a trip from each ZIP code is estimated to be the sum of driving costs and time costs. Driving costs are a function of distance (measured from the home address of the group leader using Google Maps¹¹), the average per mile cost of operating a typical car ($0.561/mile; AAA 2005), and the average number of passengers per vehicle (1.5; author’s dataset). Time costs are a function of travel time (also derived from Google Maps) and the opportunity cost of time, which is evaluated at one-third of the average hourly per capita income for each ZIP code (Hagerty and Moeltner 2005). For tramway users, the ticket price and one hour of wait and ride time are added to these amounts. When necessary, costs are adjusted to 2005 dollars using the U.S. Consumer Price Index.¹² The dataset also includes voting records on an environmental initiative from the 2000 election (California Secretary of State 2000) to help control for variation in environmental attitudes across ZIP codes.¹³ Table 2 summarizes the variables used in the statistical analysis; the following section discusses how these variables enter the group membership equation [7] and the demand equation [11].

V. Model Specification

In any latent class model, the analyst must decide which variables influence behavior within a group (the vector q in equation [11]) and which influence group membership (the vector z in equation [7]). In the context of random utility models with individual data, it seems intuitive to include characteristics of the good in q and characteristics of the individual in z; information on attitudinal variables seem particularly appropriate for inclusion in z, as shown by Boxall and Adamowicz (2002) and Morey, Thacher, and Breffle (2006). However, this is certainly not the only acceptable approach. Provencher, Baerenklau, and Bishop (2002) include some individual characteristics (employment status, senior citizen) as variables in the utility function, while others (age, experience) determine group membership. Morduch and Stern (1997) use the same set of variables in q and z. Gupta and Chintagunta (1994) and Bockenholt (1999) judge whether certain demographic characteristics are likely to influence group membership and whether they would be useful for identifying subpopulations, and include these in z. After finding no significant effects of any available socioeconomic variables on class membership, Scarpa and Thiene (2005) include only a constant in z and mostly site characteristics in q.

In the context of aggregate count data models of recreation demand, the situation is somewhat different. First, there is generally less available data. Characteristics of the good often can be reduced to a site-specific constant when, as is typical, demand is considered over a single season and access value is the primary focus. Demographic information tends to be limited to census data, and attitudes— except what can be gleaned from serendipitous voting data—are not observable. Second, parameters in the demand function, not the utility function, are being estimated, and theory tells us that both individual preferences and site characteristics affect demand. Price and income must be in each group-specific demand, but which other variables ought to shift demand within a group versus between groups is reasonably debatable.

Provencher and Moore (2006) observe that the delineation between q and z in previous applications has been driven by the specific goals of the analyst. For example, understanding how different segments of customers might be influenced by advertising. This article is concerned with endogenous spatial sorting of individuals relative to recreation sites. Therefore the previous discussion about candidate explanations for endogenous sorting is used to motivate the choice of q and z. Each demand equation includes eight variables: a site-specific constant, a site-specific travel cost variable, prop12, white, adultmale, elderly, college, and income (see Table 2).¹⁴ The group membership equation includes a constant and the same demographic variables as well as three additional variables thought to explain differing proportions of groups across ZIP codes assuming some type of sorting behavior is taking place in the population. The variable milesID measures the distance to Idyllwild, which is not only the west-side access point for the San Jacinto Wilderness, but also a mountain community that provides residents with nonrecreation amenity benefits.¹⁵ The variable milesSIM measures the distance to the nearest similar mountain community that provides comparable recreation opportunities and amenity benefits.¹⁶ Positive (negative) coefficients on these variables would indicate that the corresponding group tends to live further from (closer to) these communities. The variable Popdens is included to help further identify whether there exists an amenity value-seeking group in the population. A positive (negative) coefficient on this variable would indicate that the corresponding group tends to live in more (less) densely populated areas. These locational preferences of the latent classes can be matched with the corresponding recreational preferences (defined by the groupspecific demand and welfare estimates) to inform the question of spatial sorting.