Estimating Recreational User Counts

Multiple-site recreation demand and welfare analyses typically estimate the number of trips to a particular site or set of sites. However, the number of individual visitors is often unknown. Although visitation estimates for each site may be available, the fact that individuals may visit more than one site per season or per trip, and may visit a site on more than one occasion, prevents summation of single-site attendance estimates into an aggregate estimate of the total number of visiting individuals. Hence, while the total number of site visits may be known, researchers do not know the total number of visiting individuals. The problem becomes more difficult when it is necessary to count several different types of recreational visitors (e.g., catch-and-release versus catch-and-keep anglers; day-trippers versus overnight visitors), each of whom may realize different benefits from a recreational experience, and may reveal different behavior and visitation rates.

Despite assumptions to the contrary, estimating visitor numbers is rarely a trivial matter (Smith). While the lack of accurate visitor counts over multiple sites may be of little interest in some cases, in others it may provide a significant barrier to estimation of aggregate benefits or economic impacts. For example, in the absence of accurate visitor or user counts, recreation demand models that estimate benefits per visitor may require ad hoc assumptions to estimate aggregate welfare impacts (e.g., Johnston et al., Garrod and Willis, pp. 289–310).¹ Even if welfare estimates may be legitimately estimated per average trip—and hence total benefits may be estimated as per trip surplus multiplied by the number of trips—policy makers may still wish to know how many different individuals realize these benefits, or are otherwise influenced by a particular policy. Moreover, as shown by Morey, interpretation of per-trip benefit measures is less straightforward than is typically assumed. Analogous cases appear in the tourism literature: ad hoc efforts are often used to estimate visitor numbers, allowing aggregate economic impacts or expenditures to be calculated (e.g., Crompton, Lee, and Shuster; Fesenmaier et al.).

Although estimates of recreational visitor numbers are often obtained using inappropriate or ad hoc methods (Smith), there are special cases in which the number of visitors is either known or may be estimated as a simple extension of the model. For example, if all visitors live in an identifiable geographic region, random surveys may allow one to estimate visitor numbers during any time period through simple multiplication of estimated participation rates by the population of potential users. Identification of the geographical extent of the market, however, is often a controversial issue (Cameron, Shaw, and Ragland); many recreation studies define markets based on data availability rather than on the true origin of recreational visitors. Such problems can be avoided if all visitors are from an identifiable pool, such as fishing license holders (e.g., Feather, Hellerstein, and Tomasi; Herriges, Kling, and Phaneuf). Similarly, if individuals visit only one site per season, and site visitor numbers are known, then simple summation will yield an estimate of total recreational users.

The above-mentioned instances, however, are special cases. In the general case, the geographic origin of visitors is unknown, preventing researchers from simply multiplying participation rates by a known population size. In other instances, research goals or constraints may require an on-site survey, rather than random state- or nationwide sampling (e.g., McKean, Johnson, and Walsh; Johnston et al.). In such cases, one generally lacks information concerning the total number of individual users of a particular set of sites (for an exception, see Cooper).

This article outlines a methodology for estimating the number of individual visitors to a predefined set of sites, typically in a single geographic region. Unlike special-case methods of estimating recreational visitor numbers, the model does not require that researchers specify the extent of the geographic market from which visitors originate. The model is designed for a set of multiple sites characterized by: (a) unrestricted recreation from a wide and at least partially unknown geographic market; (b) individuals who typically visit more than one site; (c) the availability of accurate visitation (i.e., gate) counts from each site. The third assumption concerns information regarding the number of visits, not the number of individual visitors. We assume that all recreational users of interest visit at least one of the sampled set of sites, but no single site attracts all users (Tyrrell and Johnston).

Where such conditions hold, the presented model provides statistically consistent estimates of the total number of recreational visitors to the sampled sites, based on efficient use of information embedded in site-level count and survey data. That is, the presented model provides a straightforward means to answer a simple, yet fundamental question: How many individuals of different groups, in total, visited a known set of recreational sites, during a specified season? The model is based on the idea that triangulation of evidence from site visitation and participation rates can resolve the user count problem. In estimating the total number of regional recreational users, the methodology also provides consistent estimates of visitation for specified visitor sub-groups.

Determining Recreational User Counts from Observable Data

For ease of exposition, we denote the number of individuals visiting one (or more) sites within a predefined set as the number of recreational visitors. Assume that there are j = 1, …, J different types of visitors and i = 1, …, I different sites of interest. We denote V_j as the total number of regional visitors of type j, and β_ij as the average number of visits, per season, to site i among visitors of type j (i.e., visitation rates). The total number of visits to site i is given by a_i such that

(1)

where V_j is unobservable, a_i is assumed observable from gate counts, ticket sales, direct observation, or other means, and β_ij is the visitation rate of particular sub-groups, observable through responses to random on-site surveys regarding visitation behavior. Such surveys would be conducted at each site in the relevant set, preferably in proportion to total attendance. If individuals visit each site a maximum of one time per season, then a_i is also equal to the number of visitors to site i.

Implicit in (1) is a particular time dimension, say a month, year, or weekend. Hence, V_j is the total number of visitors of type j to any of the I sites, over the time period chosen for analysis. For example, if a₁ represents total seasonal visits to a particular fishing site among all angler types, and β₁₁ represents the visitation rate of catch-and-release anglers, then V₁ would represent the unobserved number of catch-and-release anglers visiting any sampled site within the region. Similarly, if β₁₂ represents the visitation rate of catch-and-keep anglers, V₂ would represent the unobserved number of catch-and-keep anglers visiting any sampled site within the region.

It is important to note that, unlike the notation found in typical econometric models, the goal here is to produce consistent estimates of V_j, given data concerning a_i and β_ij. Hence, in this model the β_ij do not represent parameters to be estimated, but are rather true visitation rates that may be observed, albeit with unavoidable errors, from sample data, such as random on-site surveys of site visitors. As the visitation rate is only observable through some type of empirical sampling or surveying, it is composed of two components, the “true” visitation rate for the activity under consideration (β_ij) and a stochastic error (ɛ_ij). Hence one must distinguish between the true, fixed β_ij and the observable, stochastic, where the observed proportion. It is assumed that the visitor survey or other data collection is conducted appropriately, and that the estimates of β_ij are unbiased such that the expected value of ɛ_ij is zero (i.e., E(ɛ_ij) = 0).

Given (1) for all I sites of interest, one can summarize the relationship between numbers of visitors and activities measured in matrix notation by

(2)

which may be written in compact matrix notation as

(3)

where A is an I × 1 column vector of raw site attendance measures including all user groups, B is an I × J matrix of true participation rates of J visitor types who visit at least one of the I sites, V is a J × 1 column vector of the total numbers of regional visitors or recreational users of each group, and φ is an I × J matrix of stochastic sampling error terms, as shown in (2). Note that since A = BV by definition, φV = 0. However, only is observable.

The goal is to estimate V—representing the total numbers of the j different recreational user types visiting one or more of the I sites—using matrix equation (2). Note that represents the total visitor count for all sites combined, over the length of time chosen for analysis. As each V_j is unobservable, may not be estimated through simple summation of observable user counts.

If I ≥ J, and |B′B| ≠ 0, it is possible to estimate each V_j (and hence) from (2). That is, there must be more sites than user types, and B′B must be non-singular. In the special case that I = J, a unique algebraic solution to the problem exists, obtained by inverting B and multiplying the inverted matrix by both sides of (2). The more common, relevant, and desirable case, however, is the case in which I > J. The advantage of the I > J case is the additional information provided by a greater number of observed sites. Since in this case there are more sites than user types, there is more than enough information to solve the problem, preventing a single, simple algebraic solution. The remainder of this article is devoted to estimation of the number of regional recreational users in each of the J types, for the case in which I > J.

Estimators for V when I > J

Distributing V in (3) over [B + φ],

(4)

where

and

Thus, the stochastic nature of the I relationships is determined by the ɛ_ij, which are assumed to be stochastic with a mean of zero. If the relationships are linearly independent, one can choose a subset of J of these relationships (J is always ≤I) and solve uniquely for V in terms of B and A

(5)

where B_J is a J × J submatrix of B (i.e., a selection of J unique rows of B).

To show the unbiasedness of this initial estimator, one need only show (Judge et al., p. 69)

(6)

where the E outside parentheses is the expected value operator. Equation (8) holds because B and V are non-stochastic

(7)

Thus, any subset of J relationships can provide an unbiased (and therefore also consistent) estimator of V. However, although each subset provides an unbiased estimate, each one by itself does not incorporate the full range of information available from all I activities.

A Unique Estimator When I > J

Any subset of the J relationships provides an unbiased estimator of visitor group populations. However, because of errors in the observations of true visitation rates (β_ij), each subset will likely provide a different estimate of V; there is no simple way to choose one estimate over any of the others. To derive a unique estimator that incorporates information from all relationships at once, we first simplify (4) by defining ε = φV. Following the approach of standard linear statistical models, the simplest unique estimator minimizes the sum of squares of the difference between predicted and actual visitation for each site (ε′ε), leading to an estimator mathematically equivalent to ordinary least squares (OLS):

(8)

where an observation on the explanatory variables are the observed visitation rates of each of the visitor types for a particular site. That is, each site represents an observation in the statistical model. It is important to recognize that (8) does not represent typical use of an OLS estimator. Rather, it is a means to combine and reconcile sampling information on visitation from the I sites to generate estimates of the regional visiting population for each of the J visitor groups. This estimator, however, is analogous in mathematical structure to the standard OLS estimator, and may be derived through minimization of a sum of squared residuals.

The OLS estimator uses information from all sites and is unbiased because

(9)

However, as ε = φV, the variance of the estimator is given by

(10)

where

(11)

Equation (8) resembles the standard OLS variance when disturbances are non-spherical; it is well known that the OLS estimator is not efficient under these conditions. Given a known Ω, a best linear unbiased estimator would follow an approach analogous to generalized least squares. However, the elements of Ω are unknown since they depend on V as well as σ²_ij. Iterative procedures can be used for non-linear estimators that incorporate information about the variances and covariances of the ɛ_ij embedded in the observed. The small sample properties of these are unknown, but they are asymptotically more efficient. That is, given that the are observed through survey methods, one also has an estimate of the variances of the; this information is embedded in Ω. Incorporating this information into the estimator improves asymptotic efficiency.

Asymptotically More Efficient Estimators

The Generalized Least Squares Estimator

Let us assume that E(ɛ²_ij) = σ²_ij, E(ɛ_ijɛ_i′j) = σ_ii′j and E(ɛ_ijɛ_ij′) = E(ɛ_ijɛ_i′j′) = 0 where i ≠ i′ and j ≠ j′. The first two of these assumptions are standard in that they describe stochastic errors with constant variances and constant covariances between activities for each single type of visitor. The third assumption implies that there are no covariances of participation rate estimates at one site across J visitor types (errors in observations across visitor types are independent). It is important to note that we are not assuming that there is no competition for sites or that the rate of participation by one type at a site will be independent of the rate of participation of another group. These dependencies are captured in the estimated β's.

Given (3) and the assumptions noted above, the generalized least squares estimator for V is

(12)

where the asymptotic variance-covariance matrix of this estimator is given by

(13)

As V and Ω are unobserved, the true GLS models cannot be estimated (Judge et al.). However, a consistent feasible estimator can be constructed and estimated iteratively, based on an initial consistent estimate of V. A number of consistent initial estimates are available. One example is the simple OLS estimator derived above (8). This initial estimate of V is unbiased, and therefore consistent. The consistent initial estimate of V guarantees a consistent estimate of the variance-covariance matrix (i.e.,) for the Feasible GLS estimator (Greene, Judge et al.). Based on this initial estimate and the corresponding variance-covariance matrix, the Feasible GLS (FGLS) estimator is estimated iteratively, with each prior estimate of V and Ω providing initial estimates for the subsequent iteration. Generally, iterations of this type converge quickly to a constant solution for the number of visitors of each type. The Feasible GLS solution is consistent and asymptotically more efficient than the initial estimate of V, as it incorporates information from the variance-covariance matrix in each successive iteration.

The Weighted Least Squares Estimator

If one assumes that E(ɛ_ijɛ_i′j) = 0 where i ≠ i′ (i.e., the stochastic components associated with measurement of visitation rates at different sites are independent, even for the same visitor group) then (11) becomes

(14)

This simplifying assumption reduces the GLS estimator (12) to a weighted least squares (WLS) estimator

(15)

where is a J × 1 vector of the numbers of visitors of each type, is an I × 1 vector of weighted total visitor counts at each of the major attractions a_iρ_i, and X is an I × J matrix of weighted estimated visitation rates β_ijρ_i. In these definitions,

(16)

where is the square of a consistent estimate of the number of visitors of the jth type. As above mentioned, the estimator is solved iteratively, with the initial providing the initial consistent estimate. As implied by 12-13 above, the WLS estimator is a special case of the GLS estimator.

The Maximum Likelihood Estimator

One may also derive consistent and asymptotically efficient estimates of V using a maximum likelihood (ML) estimator. If we assume that the errors in each of the I attraction attendance equations are distributed multivariate normal, then

(17)

where Ω is defined as in (11). The Jacobian of the transformation from the V into these errors will be precisely B. The log-likelihood function for I observations is given by:

(18)

which is maximized by minimizing

(19)

The values of V that minimize this function are the ML estimators (V_ML). The asymptotic variance of the estimator is given by

(20)

Monte Carlo Simulation

While small sample properties may be calculated for the OLS estimator, the feasible GLS, weighted least squares, and ML estimators have unknown small sample properties. Although the OLS estimator is unbiased it may not be as efficient as alternative estimators. The feasible GLS, weighted least squares, and ML estimators are consistent, but may not perform well in small samples. This is an important consideration, as the sample size for all estimators is equal to the number of recreational sites in the sample, and is therefore likely to be relatively small. Accordingly, it is unclear a priori which estimator will provide the most appealing estimates.

To explore the performance of the alternative estimators, we present a Monte Carlo analysis of the four potential estimators (OLS, FGLS, WLS, and ML). The estimators are applied to four different recreation participation scenarios, denoted cases I–IV. Each scenario assumes two types of visitors and three recreational sites, yet is based on different assumed visitation behavior (and thus patterns of visits across sites). One thousand draws, each containing 100 observations are generated for each case. (This is equivalent to surveying 100 individuals (site users) to determine their site visitation behavior.)

The four scenarios are described in table 1. We refer to the two types of visitors as “catch-and-keep anglers” (j = K) and “catch-and-release anglers” (j = R), respectively. The 100 observations for each type are selected randomly from the illustrated distribution of participation rates (table 1). For this analysis, we make the simplifying assumption that total site counts add to exactly the number of site visitors (i.e., an angler will visit each site a maximum of one time per season). This implies that participation rates add to 1.00 and allows interpretation of participation rates as visitation probabilities. For example, in case II there is a 80% probability that a catch-and-keep angler will visit site 3 only, and a 10% chance that he will visit both sites 2 and 3 (table 1). Thus, the total probability of visiting site 3 on a trip is 90% and the total probability that he will visit site 2 is 10%. This may also be interpreted as an average catch-and-keep angler making 0.9 visits to site 3 during the season. Although this assumption simplifies modeling and discussion, it is not required, and has negligible implications for simulation results.

Table 1. Four Recreation Participation Scenarios

	Case I		Case II		Case III		Case IV
	Catch-and-Keep	Catch-and-Release	Catch-and-Keep	Catch-and-Release	Catch-and-Keep	Catch-and-Release	Catch-and-Keep	Catch-and-Release
Probabilities
P(1)	0.4	0.1	0.0	0.8	0.3	0.2	0.05	0.85
P(2)	0.1	0.05	0.0	0.0	0.4	0.6	0.1	0.1
P(3)	0.1	0.1	0.8	0.0	0.3	0.2	0.85	0.05
P(1,2)	0.05	0.1	0.0	0.1	0.0	0.0	0.0	0.0
P(1,3)	0.05	0.1	0.0	0.0	0.0	0.0	0.0	0.0
P(2,3)	0.1	0.1	0.1	0.0	0.0	0.0	0.0	0.0
P(1,2,3)	0.0	0.05	0.0	0.0	0.0	0.0	0.0	0.0
Participation rates
B1
P(1) + P(1,2) + P(1,3) + P(1,2,3)	0.5	0.35	0.0	0.9	0.3	0.2	0.05	0.85
B2
P(2) + P(1,2) + P(2,3) + P(1,2,3)	0.25	0.3	0.1	0.1	0.4	0.6	0.1	0.1
B3
P(3) + P(1,3) + P(2,3) + P(1,2,3)	0.25	0.35	0.9	0.0	0.3	0.2	0.85	0.05
∑P(·)	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
True visitor numbers and site visits
VK	500		500		500		500
VR	1000		1000		1000		1000
a 1 (true visits to site 1)	600		900		350		875
a 2 (true visits to site 2)	425		150		800		150
a3 (true visits to site 3)	475		450		350		475
Correlations
Corr(BH, BB)	0.50		−0.66		1.00		−0.59
Corr(VH, VB)	−0.97		−0.00		1.00		−1.00

The four scenarios were chosen to characterize different degrees of correlation among B_j and V_j for j = {K, R}. The last two rows of table 1 characterize these correlations. Corr(B_K, B_R) measures the correlation between participation rates of catch-and-release and catch-and-keep anglers across sites. Corr(V_K, V_R) measures the correlation between population estimates of the two groups (V_j) from the GLS (most efficient unbiased) estimator using the true underlying participation rates.

As noted above, the comparison of estimators is based on their average performance over 1,000 simulated data sets drawn for each of the four scenarios. The simulated datasets were designed to mimic data that might be obtained through on-site surveys, combined with site-specific visitation estimates. For each scenario defined in table 1, a simulated dataset was generated through 100 repetitions of seven successive draws from a binomial distribution (i.e., a random number between 0 and 1), with each draw corresponding to one of the seven visitation probabilities from table 1 (i.e., P(1), P(2),…, P(1,2,3)). If the random draw exceeded the associated probability, a ‘0’ was entered for that observation; otherwise a ‘1’ was entered. These entries represent yes-or-no responses to survey questions regarding visits to various sites or groups of sites. Again, to simplify the analysis we assume a maximum of one visit per site, per season. These visitation probabilities are used to calculate sample from the true B_j shown in table 1.

For each scenario the true probabilities (participation rates) are multiplied by the true numbers of visiting catch-and-keep and catch-and-release anglers, then aggregated as appropriate to generate a_i, or total visits per site (table 1). In actual analyses this number would be generated from gate counts, ticket sales, or similar data sources at each site.² For this analysis, we assume true values of V_R = 1,000 catch-and-release anglers and V_K = 500 catch-and-keep anglers; this is the total true angler (or visitor) population used to calculate simulated sample a_i. These are also the true values for V_R and V_K against which estimates are compared.

Estimated solutions for the FGLS and WLS estimators were calculated and found to converge to the fourth significant digit after three iterations. The log-likelihood function for the ML estimator was maximized using the generalized reduced gradient (GRG2) non-linear optimization algorithm (Lasdon et al.). The alternative estimators were tested in a large number of simulations, of which cases I–IV were selected to illustrate the estimators' performance under a wide range of conditions.

Results for the four estimators are summarized in tables 2 and 3. Results are characterized for (total catch-and-keep anglers), (total catch-and-release anglers), andTable 2 illustrates mean bias and variance of each estimator across the 1,000 samples. Table 3 illustrates mean squared error (MSE) and the ratio of sample-estimated variance to true variance.

Table 2. Monte Carlo Results: Bias and Variance

	Case I		Case II		Case III		Case IV
	Catch-and-Keep	Catch-and-Release	Catch-and-Keep	Catch-and-Release	Catch-and-Keep	Catch-and-Release	Catch-and-Keep	Catch-and-Release
True number of visitors	500	1000	500	1000	500	1000	500	1000
Bias of estimators (percent bias in parentheses)
OLS: Individual visitors	6.3 (1.26)	−5.4 (0.54)	−0.5 (0.1)	−0.7 (0.07)	−4.9 (0.98)	4.0 (0.4)	−0.6 (0.12)	−1.0 (0.1)
Total visitors	0.9 (0.06)		−1.2 (0.08)		−0.9 (0.06)		−1.6 (0.11)
WLS: Individual visitors	4.9 (0.98)	−0.6 (0.06)	0.0 (0.00)	0.2 (0.02)	9.7 (1.94)	−7.2 (0.72)	0.3 (0.06)	−0.3 (0.03)
Total visitors	4.3 (0.29)		0.2 (0.01)		2.5 (0.17)		−0.1 (0.01)
GLS: Individual visitors	4.1 (0.82)	0.0 (0.00)	0.0 (0.00)	0.1 (0.01)	9.3 (1.86)	−6.9 (0.69)	0.1 (0.02)	−0.1 (0.01)
Total visitors	4.2 (0.28)		0.1 (0.01)		2.4 (0.07)		0.0 (0.00)
ML: Individual visitors	125.9 (25.18)	−166.1 (16.61)	−1.8 (0.36)	−4.8 (0.48)	57.8 (11.56)	−85.4 (8.54)	−2.8 (0.56)	−10.6 (1.06)
Total visitors	−40.2 (2.68)		−6.6 (0.44)		−27.6 (1.84)		−13.4 (0.89)
Variance of estimators
BLUE: Individual visitors	95,878	105,445	277	1102	–	–	683	683
Total visitors	6979		1375		–		0
OLS: Individual visitors	121,860	132,602	279	1142	66,581	54,629	1141	1855
Total visitors	9480		1445		7759		2523
WLS: Individual visitors	122,932	133,790	277	1141	65,655	53,949	1145	1857
Total visitors	9592		1426		7712		2522
GLS: Individual visitors	121,704	132,364	278	1143	65,853	54,114	1158	1852
Total visitors	9620		1425		7736		2529
ML: Individual visitors	21,065	22,005	256	1032	19,585	15,694	1096	1581
Total visitors	6675		1294		5855		2326

• Note: Bias and variance are measured in visitor numbers; values in parentheses represent bias as a percentage of the true value.

Table 3. Monte Carlo Results: MSE and Sample-Estimated Variance

	Case I		Case II		Case III		Case IV
	Catch-and-Keep	Catch-and-Release	Catch-and-Keep	Catch-and-Release	Catch-and-Keep	Catch-and-Release	Catch-and-Keep	Catch-and-Release
MSE of Estimators
OLS: Individual visitors	121,900	132,632	279	1142	66,605	54,646	1141	1856
Total visitors	9481		1446		7760		2525
WLS: Individual visitors	122,957	133,790	277	1142	65,748	54,001	1145	1857
Total visitors	9610		1426		7719		2522
GLS: Individual visitors	132,365	132,365	278	1143	65,940	54,161	1158	1852
Total visitors	9638		1426		7742		2529
ML: Individual visitors	36,926	49,592	259	1055	22,920	22,985	1104	1694
Total visitors	8287		1338		6619		2506
Ratio of sample-estimated variance to true variance of estimators
OLS: Individual visitors	2.03	2.10	1.05	0.99	1.27	1.32	1.00	1.05
Total visitors	1.35		0.99		1.22		1.07
WLS: Individual visitors	2.06	2.13	1.02	0.99	1.27	1.31	1.00	1.05
Total visitors	1.34		0.99		1.21		1.07
GLS: Individual visitors	2.00	2.07	1.02	0.98	1.27	1.31	0.98	1.04
Total visitors	1.33		0.98		1.21		1.06
ML: Individual visitors	5.29	5.53	1.10	1.08	3.19	3.40	1.02	1.19
Total visitors	1.31		1.07		1.35		1.13

Frequency Distributions of Visitor Estimates

Figures 1–3 provide another depiction of estimator performance. These figures show the frequency distributions of visitor estimates from the 1,000 samples for case I. Figures are not included for cases II–IV. The frequency distribution of estimates for case III resembles that for case I. Cases II and IV result in estimates highly concentrated around the true values, with little difference among frequency distributions for any of the alternative estimators.

Figure 1 shows the case I distribution of estimates for the number of catch-and-keep anglers, where true V_K = 500. The ML estimate has a concentration of estimates in the 500–700 range with few estimates below 400. This concentration above the true value of 500 is reflected in the average bias reported for the ML estimates. The least squares estimates are spread more thinly across the entire figure with a substantial number of estimates below zero. The least squares estimates also show a relatively high concentration of estimates in the 500–700 range, but these overestimates are almost entirely balanced by the large number of low estimates, leading to the relatively low mean bias illustrated in table 2. The least squares estimators generate markedly similar frequency distributions. However, there is a substantial difference between the least squares distribution and that of the ML estimate of.

Figure 2 shows the complementary distributions of estimates of catch-and-release anglers for the same case I samples where true V_R = 1,000. The ML estimates are concentrated in the range 650 to 1,000 while the least squares estimates are again spread more evenly across a wider range of values. The ML concentration below the true value of 1,000 is reflected in the average negative bias of ML estimates. Despite the relatively wide distribution of the least squares estimates, there is minimal mean bias in any of these estimators. Figure 3 shows the distribution of estimates for, the sum of and. The distribution of more closely resembles a normal distribution. In contrast to the distributions of and, the (i.e., total visitor) distribution shows smaller tails, and a greater concentration of estimates around the true value; this is true for all estimators. The peak for the least squares estimators occurs just above the true value of 1,500, while the peak for the ML estimator occurs just below this value. As mentioned above, there is little difference between any of the least squares estimates, but a notable distinction between the ML estimator and the least squares estimators.

An Empirical Illustration

The visitor count model was applied to estimate the number of visitors to the principal Preservation Society of Newport County (PSNC) properties (mansion houses in Newport, Rhode Island), over the weekend of 7–8 September 2002. Data were drawn from surveys of visitors together with official gate counts from each site. Eight houses were incorporated in the analysis, including The Breakers, Breakers Stable, Marble House, The Elms, Chateau Sur Mer, Green Animals Topiary Garden, Kingscote, and Rosecliff. The Preservation Society maintains gate counts for each house, and also approximates the total number of visitors to all houses combined through a count of total ticket sales. Although this approximation over-counts visitors who purchase individual tickets at multiple houses (as opposed to purchasing a single multiple-house ticket), and does not distinguish visits by different groups (e.g., overnight visitors, day trippers), it does provide an approximate benchmark from which estimated visitor counts may be compared. Hence, the PSNC data provides an opportunity to both test and illustrate the proposed methodology.

Visitor surveys were conducted in proportion to the total gate count at each site, with an approximately equal number of visitors sampled during each day. Sampling was spread over the entire business day (i.e., the time that each house was open to the public). A total of 1162 complete and useable surveys were collected, with per-house samples ranging from 526 at the Breakers to 33 at Kingscote. The survey asked respondents to indicate (check off from a list) “which of the following mansions are you visiting today?” Answers provide data regarding individual sites visited by each respondent, and allow average visitation probabilities to be calculated for each site. Of the 1,162 visitors sampled, 781 reported visiting more than one PSNC property, preventing a simple summation of gate counts to estimate total visitor numbers. Respondents were also asked to indicate whether they were (a) Rhode Island residents and/or (b) staying overnight in Newport; allowing Newport day-trippers to be distinguished from non-resident overnight visitors. Visitation rates for each property and group are given by table 4.

Using visitor survey data and gate counts for each property, visitor counts for PSNC properties were estimated using the least squares and ML variants of the visitor count model. Results are reported in table 5. Compared to the PSNC ticket sales approximation of visitor numbers (3,922) for the same weekend, all least squares methods provide reasonable estimates (i.e., 3,642–3,764). The ML estimate (1,160), by comparison, appears unreasonably low. The difference between least squares estimates and the ticket sales approximation may be due to visitors who purchased multiple individual tickets at different houses (and were hence over-counted by the ticket sales approximation), rather than the standard and less expensive option of purchasing multiple-house tickets at a single booth. In this case, the least squares estimates might provide a more accurate illustration of actual visitor numbers, whereas the ticket sales approximation would provide upwardly biased results.

The relatively poor performance of the ML estimator is likely due to the characteristics of available data. The estimated residual variance component of the likelihood function (A − BV)′Ω⁻¹(A − BV) is close to zero for all the least squares estimators, yet is much larger for the ML estimator. It appears that the ML estimator sacrifices minimization of (A − BV)′Ω⁻¹(A − BV) in order to obtain a smaller estimated value for I ln (|Ω|), the remaining component of the likelihood function. The ML estimator's difficulty in minimizing I ln (|Ω|), and hence the larger value of (A − BV)′Ω⁻¹(A − BV), may be related to a violation of multivariate normality in the errors of site attendance equations.

The relatively large standard errors associated with estimates of overnight visitors and day trippers (table 5) is likely due to the highly correlated visitation patterns of the two groups; there is a 98.5% correlation between the visitation patterns of overnight visitors and those of residents/day trippers. This correlation reduces estimator efficiency, particularly with regard to estimates of the number of visitors in distinct groups. We expect that—on average—a greater divergence among the behavior of different visitor groups (i.e., less correlation) would reduce standard errors associated with estimates of visitor counts, particularly for specific visitor groups.

As found in the Monte Carlo analysis, the empirical application of the visitor count estimators suggests that the least squares estimators provide more accurate estimates of total visitor counts, despite the reduced variance of the ML estimator. However, unlike the reported Monte Carlo results, the empirical application illustrates a case in which the least squares estimators clearly outperform the ML estimator. Indeed, the ML estimator provides a visitor count estimate that appears to be less than half of the suspected true value, based on the best alternative approximation available (the PSNC ticket count).

Conclusions

The accuracy of visitor count estimates can have important implications for estimation of aggregate welfare estimates or other impacts of regional recreation. Without accurate visitor counts, even the most accurate estimates of per-visitor benefits cannot be extrapolated into meaningful aggregate benefit estimates. Errors in visitor counts will typically carry through to unknown and potentially substantial biases in aggregate estimates. The proposed recreational visitor count methods provide means to estimate visitor numbers in regions where direct counts of all visitors are intractable.

Data requirements for the proposed visitor count estimators are not trivial, but are in most cases obtainable. They include a sample of recreational sites of interest for which accurate visitation data may be obtained, and representative survey data including all visitor types. Visitor counts for particular sites may already be available for many recreational sites; for others it may be necessary to conduct on-site counts or compile existing attendance records, ticket sales or other data to obtain this information. Survey data need only cover a representative sample of visitors at each site; it is not necessary to obtain survey data for every visitor. The surveys must request at least two types of information: (a) information allowing researchers to distinguish among the different types of visitors, and (b) information regarding visitation to all sites.

The primary advantage of the proposed method is the ability to estimate total visitor numbers across multiple sites, where conditions prohibit a direct and accurate count. The alternatives often include estimates of visitor numbers based on ad hoc assumptions, attendance at a single site, or simple summation of attendance at a collection of sites (Smith). None of these alternatives will, in general, provide accurate results. In contrast, the presented methodology provides unbiased or at least consistent and asymptotically efficient estimates of regional visitors. Such estimates may be combined with the results of recreational demand models to provide more accurate estimates of the aggregate welfare implications of recreational activities.