Handling Unobserved Site Characteristics in Random Utility Models of Recreation Demand

Jennifer Murdock Economics Department University of Toronto [email protected] 150 St. George Street Toronto, ON M5S 3A7 CANADA tel: 416-946-0656 fax: 416-978-6713

1

A.1 The Participation Model A nested participation model is estimated to determine whether it has substantially different implications than the more compact site choice model. Graphically the participation model can be depicted as in Figure 1. At the top level of the decision tree the angler chooses whether to take a single day fishing trip, a multiple day fishing trip, or do something else entirely during each day of the sample period. The site choice model, which models the choice among fishing locations conditional on taking a fishing trip, would include only the leftmost branch of the figure.

Figure 1: Participation Model A nesting structure is introduced that allows the unobservable errors associated with the single day fishing locations to be more correlated with each other than not fishing or taking a multiple day trip, which makes it more likely that anglers substitute among single day fishing sites than with other alternatives. For the nested participation model the probability that an angler chooses to visit a single day fishing location, take a multiple day trip (MD), or not to fish (NF) are respectively Pit (j) = exp(sf Vij ) ∗

hP J

i(1/sf )−1

k=1 exp(sf Vik )

/Bijt , where Vij = Xj βi + Zi Xj γ + Dij ρ + ξj

Pit (M D) = exp(ViM D )/Bijt , where ViM D = αQi Pit (N F ) = exp(VitN F )/Bijt , where VitN F = φQi + ωWit and Bijt =

hP

i1/sf J k=1 exp(sf Vik )

+ exp(ViM D ) + exp(VitN F )

Qi are angler characteristics, and Wit are measures that vary over time and/or across individuals including the weather. Weather variables and individual characteristics affect the probability of taking a fishing trip rather that engaging in a non-fishing activity. The nesting parameter is sf . 2

When sf >= 1 the underlying probability density function is well defined (see McFadden (1978) and note that sf = 1/(1 − σf ) in his notation). When sf = 1 a simple non-nested model is obtained. See Morey [2] for a clear derivation of probabilities for nested logit models.

A.2 Contraction Mapping and Sample Gauss Code The contraction mapping is a computational tool that can assist the gradient methods typically used in maximizing a likelihood function. To understand how the contraction mapping works start by remembering that in any random utility model the inclusion of a dummy for a particular alternative means that the predicted number of times it is selected will be exactly equated with the actual number of times it is selected in the data.1 Hence, given estimates of the other parameters (such as ˆ they must equate predicted that on travel cost) we can solve for the alternative specific constants (δ): and actual trips. The contraction mapping returns the δˆ that gets the predicted and actual trips arbitrarily close. The contraction mapping is a convenient tool for solving what could be a nightmare system of equations to determine what vector δˆ equates actual and predicted trips given the data and other parameters. The contraction mapping shown in Equation 1 is specified in terms of “shares.” Simply dividing the number of trips to each site by the total number of trips to all sites yields the shares. ˆ Θ)) ˆ δˆnew = δˆold + ln(s) − ln(s(X, δ;

(1)

The new estimate of δˆ is a function of the previous guess and the difference between the actual and predicted share of trips, which depends on δˆ and the parameters of the model. This contraction mapping has the desirable properties that with each iteration the resulting δˆ is always closer to the ˆ the update (change) of δˆ is always smaller than the previous update, and true δ than the previous δ, it always converges to the same place regardless of the starting guess. The maximum likelihood routine uses standard gradient methods to search over the parameter ˆ which for each iteration is filled in by the contraction mapping. This method vector excluding δ, yields the exact same results as estimating δˆ along with the other parameters but is much faster and more stable.2 The contraction mapping is utilized for each iteration of the maximum likelihood search. The gradients iteratively return parameter estimates for all parameters other than the deltas. This parameter vector is then augmented by the delta’s calculated using the contraction mapping. The results are exactly the same whether the gradient method is used to search over the entire parameter space including the deltas or the much smaller parameter space excluding the deltas with estimates of the missing deltas filled in by the contraction mapping. 1

A proof that the maximum likelihood estimate of δ has this property is provided in this section. In fact, once estimation has converged the final parameter vector can be used as a starting value for maximum likelihood estimation without the contraction mapping. It will converge immediately and yield a variance-covariance matrix (and hence standard errors) for δˆ as well as the other parameters. 2

3

The following sample code in the Gauss programming language illustrates how the contraction mapping shown in Equation 1 works. Lines 01 through 04 are performed only once at the start of the program. Lines 07 through 14 are called for each iteration of maximizing the likelihood. The dimensions of the vectors are provided in the comments within the sample code. Sample Gauss code illustrating the use of the contraction mapping: 01 J = 569;

/* Number of alternative fishing sites */

02 tol_level = 0.01;

/* Tolerance for stopping the contraction mapping */

03 delta = rndu(J-1,1);

/* Starting delta (can be anything) (J-1*1) */

04 load ln_share;

/* Log each site’s share of total trips (J-1*1) */

05 06 /* Following code called from within log-likelihood procedure */ 07 tol = 10;

/* Any number > tol_level to ensure enter loop */

08 DO UNTIL (tol < tol_level); 09

call predict_share;

10

/* Returns s_share (J-1*1): predicted shares, which are a function of delta (among other things) */

11

delta_new = delta + ln_share - ln(s_share); /* Contraction mapping in action */

12

tol = maxc( abs( (delta_new - delta) ) );

13

delta = delta_new;

/* Calc sup-norm of differences */

/* Update current delta */

14 ENDO;

Two lines requires some additional explanation beyond that provided by the comments. Line 09 calls a procedure (not shown) that will calculate the predicted share of total trips to each site excluding one. This procedure uses the matrices containing the travel distances (or travel cost), the current parameter estimates from the log-likelihood procedure (such as ρ the parameter on travel distance), and the current delta vector.3 Predicted trip level probabilities are summed to compute predicted trips that can then be used to calculate predicted share. To predict the needed shares, the value of delta for the omitted site is normalized to zero without loss of generality. The tolerance calculation on Line 12 determines how much delta has been revised with each iteration of the contraction mapping. The contraction mapping has the properties that with each iteration the resulting delta is always closer to the truth than the previous delta, the update of delta is always smaller than the previous update, and it always converges to the same place regardless of the starting guess. The tolerance used in the estimation of the results presented in this paper is quite tight: the maximum change in any of the deltas is less than 0.01. This is tight given that the deltas themselves range from -5.7 to 10.4, have 10th and 90th percentiles of -3.4 and 3.2, and a standard deviation of 2.5. The contraction mapping obtains δˆ such that the predicted market shares are exactly equal to observed market shares. Here we prove that when alternative specific constants are included, the likelihood is maximized only when predicted number of trips equals actual number of trips. Begin by writing the log-likelihood function where i indexes individuals, t indexes fishing trips, and j indexes fishing sites. yijt is an indicator variable that equals 1 if site j is chosen by angler i on 3 For more general specifications that include interaction terms, the predicted market shares also depend on the matrices containing individual and site characteristics.

4

trip t and is otherwise 0. Pit (j) denotes the usual logit probabilities that each site is chosen and Vijt is the utility index, which guides anglers’ choices.

L=

T X N X J X

yijt ln(Pit (j))

i=1 t=1 j=1

where exp(Vijt ) Pit (j) = PJ k=1 exp(Vikt )

Vijt = δj + Xj σβ µi + Xij γ¯ + Xij σγ νi We can rewrite the log-likelihood by plugging in for Pit (j).

L=

N X T X J X

Ã

yijt Vijt − ln

à J X

i=1 t=1 j=1

!!

exp(Vikt )

k=1

WLOG examine the first order condition for maximizing the log-likelihood function with respect to δ1 , which is the alternative specific constant for fishing site 1. 



N X T J X X exp(Vi1t )  ∂L yi1t − = yijt PJ ∂δ1 k=1 exp(Vikt ) i=1 t=1 j=1

∂L = ∂δ1

N X T X i=1 t=1



J X



exp(Vi1t ) yijt  k=1 exp(Vikt ) j=1

yi1t − P J

Recognize that for each individual trip a single site is chosen so the last sum must be one and rewrite. Next, set this first order condition equal to zero to maximize the log-likelihood function.

N X T X ∂L = [yi1t − Pit (1)] = 0 ∂δ1 i=1 t=1

N X T X i=1 t=1

yi1t =

N X T X

Pit (1)

i=1 t=1

Actual Number of Trips to Site 1 = Predicted Number of Trips to Site 1 5

A.3 Addressing Sites that are Not Visited Some applications with very large numbers of alternatives and a relatively small sample of recreation trips, may have some locations that are not selected in the sample. Of course, if these alternatives are considered by recreationists they should be included in the model.4 However, the proposed method cannot be simply applied to data where some alternatives are not selected. The model will try to explain the zero visitation by saying that those site must have infinitely undesirable characteristics. The alternative of using the two-stage approach and simply estimating alternative specific constants on the chosen locations is not advisable as it will result in biased parameter estimates in the second stage (but no bias in first stage parameters). There are some “brute force” methods the researcher can employ to use the two-stage approach in cases where some alternatives are never selected, but these methods have some drawbacks. For example, an estimate of the market share of each of the not visited sites could be used and the zeros could be replaced by these small numbers.5 However, any error in the estimates (or likely, guesses) of the difficult to know market shares of the sites never visited in the sample will bias the second stage parameter estimates. Hence, in cases where not all alternatives are visited at least once the researcher will need to consider the benefits and costs of using either the two-stage or standard methods.6 Consider the properties of this “brute force” method with a Monte Carlo simulation. A simulation with the same design as the concentric circle simulations (discussed in Section ) but with 100 alternatives instead of 20 is conducted. With a total of only 2,000 simulated trips, on average about one quarter of the sites are never visited. These simulation show that by substituting a fractional trip (0.1 trips is used) for each site with zero trips the two-stage approach will produce substantially more efficient parameter estimates than the traditional approach and unbiased standard errors. For the two-stage approach with the fractional trips substituted there is a slight bias in the parameters estimated on site characteristics whereas the traditional approach produces unbiased parameter estimates.7 The degree of bias will be affected by the choice of the fractional number of trips substituted for locations with zero visits. However, in this simulated case the two-stage estimator seems clearly preferable using a mean-squared error criteria. But, one cannot generalize this result to suggest that the two-stage approach be utilized in all cases where some sites are not visited. There are benefits 4

The researcher must be wary of excluding considered sites as well as including sites that are not considered as an error in either direction will bias parameter estimates. 5 Mechanically one way this could be achieved is by including a full set of alternative specific constants for the visited sites in the first stage. Then throwing away those dummy parameters (but keeping the other first stage parameters such as that on travel cost) and using the contraction mapping to solve for the set of deltas that when combined with the parameters estimated in the first stage yields predicted market shares exactly equal to the adjusted market shares of the sample (adjusted such that no site has a strict zero market share). Then the deltas from the contraction mapping could be used in a second stage regression that includes both visited and not visited sites. 6 If the researcher proceeds with traditional approaches including sites never visited does not cause estimation problems or biased parameter estimates. But if there are unobserved characteristics and traditional methods are used the researcher will need to contend with biased standard errors, bias in the welfare measures, and potential bias in the travel cost parameter. 7 The parameter bias when the two-stage approach is employed without the inclusion of the fractional trips is considerable and makes this approach inadvisable.

6

and costs to consider with either the traditional or the adjusted two-stage approach when some alternatives are not selected by the sample and no macro level data is available that provides share (or total trip) estimates for all included locations.

A.4 Alternative Monte Carlo Simulations

Simulation Based on Concentric Circle Distribution of Sites and Anglers To conduct a simulation that uses a number of trips similar to that observed in the actual data it is necessary to limit the chance that some sites are never visited. To achieve this we limit the number of alternative locations to 20 as opposed to 185 in the original simulation. We also choose a spatial distribution of alternatives and residences such that in expectation each site has an equal probability of visitation. One design that achieves this is to have the sites located equidistant from each other on a circle and to also have residential centers on a concentric circle with the same spatial allocation.8 The specification shown in Equation 2 is the same as that in the paper except that β1 , β2 , ..., β8 are set to 0.8 and distance is now calculated as the crow flies between each simulated residential center and each simulated fishing location.9 From each residential center 200 simulated trips originate for a total of 4,000 trips. Uijt = β1 x1j + β2 x2j + β3 x3j + β4 x4j + β5 x5j + β6 x6j + β7 x7j + β8 x8j + ρDij + εijt

(2)

where β1 , β2 , ..., β8 = 0.8, ρ = −1 x1j , x2j , ..., x8j ∼ U nif orm(−1, 1) , Dij ∼ as crow flies, εijt ∼ Extreme Value with scale 1 Table I shows results similar to those we obtained in the original set of simulations. When there are any unobserved characteristics the 2S approach yields a more efficient estimate of the MRS. Also, the standard errors of the CL are severely biased downward to overstate parameter precision. Both methods produce an unbiased estimate of the true MRS (0.8). The efficiency gain from using the 2S approach is somewhat less than in the simulation reported in the paper because the spatial distribution and the number of sites can affect the magnitude of the results. However, the number of trips is not driving the difference in efficiency between 2S and CL models. This can be verified by repeating this second simulation using a very large number of trips, which shows the variation of both decreases, but that the difference is stable. Table I also shows that even with a much more realistic number of trips the bias in the CL standard errors remains large.

8

The radius of the residence and fishing location circles are set to 64 and 65 miles respectively, which along with the other assumptions sets the importance of travel distance similar to that in the real data. 9 With this specification distance explains about 83 percent of the variation in the utility index, site characteristics 8 percent, and epsilon 9 percent.

7

Table I: Monte Carlo Simulations Using Discrete Choice Model1 MRS: βd 1 /ρ

Observed

βˆ1

ρˆ

% Reject

Variables

Mod.

ave.

s.d.

s.e.

ave.

s.d.

s.e.

ave.

s.d.

s.e.

x1 -x8 ,D

CL

-0.80

0.065

0.061

0.80

0.067

0.062

-1.00

0.018

0.017

4

2S

-0.80

0.087

0.082

0.80

0.088

0.081

-1.00

0.019

0.018

5

CL

-0.80

0.373

0.055

0.76

0.358

0.054

-0.95

0.031

0.016

73

2S

-0.80

0.333

0.317

0.80

0.334

0.318

-1.00

0.019

0.018

5

CL

-0.80

0.529

0.050

0.69

0.465

0.044

-0.86

0.048

0.013

83

2S

-0.80

0.458

0.444

0.80

0.460

0.444

-1.00

0.019

0.018

4

CL

-0.80

0.590

0.048

0.64

0.478

0.039

-0.80

0.052

0.012

87

2S

-0.81

0.516

0.492

0.81

0.517

0.493

-1.00

0.019

0.018

4

x1 -x6 ,D

x1 -x3 ,D

x1 ,D

True

β1 2 ρ

1 Reported results based on 1000 independent draws of data. 2 Percent of times true marginal rate of substitution (MRS) between the first site characteristic and travel

distance is outside the 95 percent confidence interval.

The utility index used in the original simulation reported in the paper, actually somewhat downplays the role of site characteristics compared to the real data. This is to keep the size of the simulation within a reasonable range.10 Increasing β1 − β8 to 1.5 gets us very close to matching the variation in the actual data.11 As we increase the role of the site characteristics we increase the chances that some sites will have extremely bad draws of characteristics and have a near zero probability of visitation, which necessitates an even larger number of simulated trips. This simulation is run 50 times rather than 1000 times as above.12 The results indicate that as the site characteristics become more important in explaining choice, the efficiency gains from using the proposed two-stage approach become over twice as large.

A.5 Hedonic Wage Regression for Wage Prediction of those Not Working The socioeconomic and demographic variables available for use in a hedonic wage regression include age, gender, level of education, ethnicity, and household structure (number of adults and number of children). Information from the 1990 Census at the zip code level is also included.13 Table 10

The simulation reported in the paper takes about 8 days of computer time to run on a Pentium IV 2.8 Gigahertz machine with 2 Gigabytes of RAM. 11 Travel distance explains 84 percent, site characteristics 12 percent, and epsilon 4 percent of the variation in Uij . 12 This took 7 days of computer time and the program simulates 13.5 million choices with each iteration. 13 While a number of Census variables are included in initial regressions, only the median household income for each zip code is included in the final regression as it is the only one that adds to the overall fit.

8

Table II: Hedonic Wage Regression Independent Variables Parameter Est. Two adults 0.75 Three or more adults 1.94 One child 0.63 Two children 0.86 Three or more children 1.00 Male 1.54 Age 0.54 Age squared -0.0051 High school diploma 0.12 Some college 1.70 College degree 2.44 Some graduate work 4.39 Graduate degree 7.07 White 1.30 Zip code median HH income ($10,000) 1.33 Constant -9.90 Adjusted R-squared 0.23 Number of Observations 696

Std. Error 0.66 0.77 0.57 0.56 0.66 0.45 0.11 0.001 1.09 1.09 1.19 1.38 1.30 1.66 0.28 3.12

Note: Dependent variable is the wage rate after tax.

II reports the results of the hedonic wage regression, which is estimated using a sample of those working full or part time (at least 20 hours per week). The overall fit as measured by an adjusted R-squared statistic of 0.23 is comparable to other studies including hedonic wage regressions (see Feather and Shaw [1]). All of the included variables have the expected sign.

A.6 Fish Catch Rate Variables A WLS regression is estimated for each of the 996 origin sites and for each of the eight different species of fish. The dependent variable is the average catch rate for each site and the independent variables include a constant term and, in the case of inland lakes, a measure of abundance from the WDNR data. For inland lakes a single variable measuring fish abundance is constructed for each species by combining the dummy variables that measure whether the species is listed as abundant, common, present, or not present.14 Using a single measure of abundance rather than a set of fish abundance dummies prevents such dummies from becoming site specific constants in those regressions where only a single site has a particular abundance rating for a particular species. Including site specific constants defeats the purpose of increasing the number of observations by including other sites in the determination of catch. 14 The WDNR does not provide any numerical values for these categories. However, a single measure is constructed that is 0 for not present, 1 for present, 2 for common, and 8 for abundant. Using the full sample of inland lake sites, this restriction is not rejected for any of the species according to an F-test comparing this specification to the unrestricted case that includes a dummy variable for each abundance category.

9

Table III: Summary of Localized WLS Regressions % Regressions

% Regressions

with pos. coef.

with t-stat > 2

Average

Species

on abundance

on abundance

R-squared

Walleye

100.0

87.8

.18

Smallmouth

94.2

34.2

.04

Musky

99.4

73.3

.19

Northern

58.9

20.6

.02

Trout

90.0

48.9

.07

The WLS regression for each site includes those sites of the same type within a 50 mile radius, where four “types” are defined that include inland lakes, rivers, Green Bay, and Lake Michigan. This radius provides ample observations, which circumvents the small sample size problem, while including only those sites likely to be quite similar to the origin. On average this includes information on 95 different sites that together receive 678 trips taken by 155 different anglers. The weights vary according to distance in miles from the origin site, the number of trips to that site, and the physical similarity between the origin and destination site. Within the weight itself these components must be assigned relative importance; otherwise, their importance will be completely arbitrary and based on the units of measurement and variability. The selected weights assign equal importance to the number of trips, the distance from the origin, and physical similarity by rescaling them such that they all have the same range. A site twice as far from the origin as another site should have twice as many trips to receive equal weight. The formula used to construct the weights is given by weight =

p

1 1 1 trips ∗ √ ∗q ∗r √ √ mddist−1 dist mddist−1 1 + dacres mdacres 1 + ddepth mddepth

(3)

where trips is the number of trips to the site, dist is the distance in miles between the sites, dacres is the absolute value of the difference in the surface area of the lakes, and ddepth is the absolute value of the difference in the maximum depth of the lakes. The terms mdacres, mddepth, and mddist are the absolute value of the maximum difference in the surface area, depth, and distance respectively. Table III summarizes the fit of the localized WLS regressions for the inland lakes. For those species not reported in Table III and non-inland lakes, a simple weighted average is used as the catch measure. As expected, for all species the vast majority of the regressions produce a positive estimate of the coefficient on the fish abundance measure. Further, a substantial portion of these coefficients are

10

statistically significant as well as positive.15 Considering that there is only one included explanatory variable, the overall explanatory power of the models is acceptable as measured by the average Rsquared statistic. Because the fish abundance measures are restricted to a single variable, some of the predicted catch rates are negative. Zeros replace any negative predicted catch rates.

A.7 Full Estimation Results for Heterogeneous Tastes Specifications

Table IV: Results for Heterogeneous Tastes Specifications

Variables Panfish2

CL Interact

2S Interact1

RPL (1)

RPL (2)

2S RPL1

Est. (se)

Est. (se)

Est. (se)

Est. (se)

Est. (se)

0.16 (0.23)

0.37 (0.16)

-0.07 (0.12)

0.62 (0.53)

0.88 (0.26)

1.83 (0.10)

2.05 (0.10)

1.81 (0.11)

sd X male

0.46 (0.13)

0.56 (0.18)

0.33 (0.32)

0.37 (0.25)

X age

0.83 (0.36)

0.67 (0.58)

-2.23 (0.77)

-1.92 (0.50)

-0.77 (0.32)

-0.50 (0.47)

0.05 (0.88)

-0.96 (0.63)

-1.08 (0.56)

-0.39 (0.64)

-1.86 (0.49)

-1.93 (1.40)

0.88 (1.02)

5.10 (0.36)

4.48 (0.29)

5.15 (0.39)

X income Walleye sd X male

1.68 (0.32)

2.03 (0.44)

2.68 (0.70)

4.09 (0.87)

X age

3.12 (0.92)

2.93 (1.28)

0.34 (2.18)

-3.74 (1.27)

-0.53 (0.78)

-0.91 (1.02)

-2.01 (2.05)

-5.23 (1.67)

-2.18 (1.29)

-5.31 (0.83)

-3.46 (0.70)

-1.16 (2.55)

-10.45 (1.32)

8.34 (0.66)

7.98 (0.63)

9.18 (0.59)

X income Largemouth sd X male X age X income Smallmouth

0.09 (0.73)

0.16 (1.08)

-0.34 (1.69)

0.04 (1.57)

-7.49 (2.43)

-9.61 (3.88)

-12.25 (5.10)

-7.61 (3.49)

9.53 (1.72)

8.82 (2.55)

10.51 (3.34)

11.93 (3.55)

-0.86 (0.57)

0.67 (0.45)

-0.88 (0.38)

-1.96 (1.42)

-7.52 (0.71)

4.33 (0.37)

4.23 (0.40)

6.03 (0.36)

sd X male

0.87 (0.31)

1.07 (0.46)

0.84 (0.82)

1.69 (0.93)

X age

1.36 (0.91)

0.50 (1.26)

1.03 (2.67)

4.32 (1.87)

X income

1.72 (0.69)

1.27 (0.91)

1.43 (1.55)

-0.10 (1.72)

17.59 (4.27)

23.86 (3.82)

-5.96 (3.02)

-10.93 (8.31)

-12.01 (6.03)

21.85 (2.83)

21.11 (2.16)

32.45 (3.00)

Musky sd X male

-11.48 (2.42)

-12.80 (4.17)

-8.06 (5.02)

-14.28 (5.72)

X age

-12.46 (7.79)

-9.44 (11.85)

17.30 (15.34)

25.90 (12.53)

10.80 (6.57)

5.17 (10.67)

16.39 (11.52)

5.55 (13.29)

X income

15 For all species, none of the regressions produced any negative and significant coefficients, with a few exceptions for northern pike (less than 1 percent).

11

Table IV: continued

Variables Northern Pike

CL Interact

2S Interact1

RPL (1)

RPL (2)

2S RPL1

Est. (se)

Est. (se)

Est. (se)

Est. (se)

Est. (se)

4.69 (2.52)

2.07 (1.37)

-7.45 (1.25)

-1.05 (4.97)

-6.02 (2.02)

16.85 (1.08)

16.98 (1.13)

26.17 (1.19)

sd X male X age X income Trout

-1.40 (1.29)

1.03 (1.85)

2.10 (2.80)

2.48 (3.32)

2.17 (4.11)

2.62 (6.04)

-3.89 (8.97)

-13.88 (5.88)

-8.04 (2.73)

-8.66 (4.19)

-16.37 (7.38)

-24.86 (7.59)

-3.04 (1.17)

-0.87 (0.59)

-5.80 (0.82)

-8.38 (4.06)

-11.33 (0.94)

9.88 (0.40)

13.03 (0.71)

12.89 (0.54)

sd X male

2.28 (0.62)

1.80 (1.10)

3.38 (2.26)

4.27 (1.88)

X age

4.91 (1.62)

2.35 (3.75)

-3.13 (6.71)

-2.40 (3.81)

X income

3.58 (1.48)

5.08 (2.62)

-1.69 (5.53)

-3.26 (4.13)

8.75 (1.90)

4.84 (2.03)

-10.13 (2.08)

-9.09 (5.97)

-11.20 (3.20)

15.75 (1.35)

16.97 (1.10)

20.01 (0.82)

Salmon sd X male

3.85 (1.06)

6.62 (1.46)

3.03 (2.60)

6.25 (2.29)

-11.82 (2.96)

-6.64 (4.33)

-11.67 (10.39)

-20.97 (4.68)

-2.12 (2.35)

-5.29 (3.25)

7.16 (5.24)

-1.18 (4.53)

Acres < 50

-0.10 (0.12)

0.35 (0.24)

0.19 (0.11)

-0.37 (0.39)

X boat

-0.37 (0.15)

-0.25 (0.31)

-0.68 (0.14)

-0.26 (0.11)

0.05 (0.01)

0.19 (0.04)

0.11 (0.02)

0.08 (0.07)

0.07 (0.01)

0.08 (0.02)

0.09 (0.03)

0.21 (0.03)

-0.02 (0.09)

-0.43 (0.24)

0.01 (0.08)

-0.55 (0.38)

0.04 (0.09)

-0.06 (0.13)

-0.01 (0.11)

-0.15 (0.10)

0.78 (0.13)

0.91 (0.27)

0.60 (0.11)

1.05 (0.44)

0.07 (0.19)

0.01 (0.54)

0.09 (0.18)

0.02 (0.16)

-0.30 (0.10)

0.16 (0.24)

-0.22 (0.08)

0.04 (0.38)

0.46 (0.15)

0.66 (0.33)

0.40 (0.13)

0.49 (0.10)

-1.13 (0.31)

-0.06 (0.25)

-0.98 (0.32)

-0.20 (0.40)

-0.04 (0.17)

-0.24 (0.27)

-0.17 (0.18)

0.03 (0.19)

X age

1.35 (0.62)

0.72 (0.90)

1.31 (0.71)

2.28 (0.40)

X income

2.24 (0.32)

1.71 (0.56)

2.14 (0.27)

1.19 (0.35)

National Forest

-4.13 (0.63)

-1.90 (0.35)

-0.80 (0.59)

-2.00 (0.56)

X male

1.63 (0.41)

0.55 (0.74)

0.40 (0.43)

0.35 (0.40)

X age

5.24 (1.12)

4.30 (2.04)

2.83 (1.12)

4.42 (0.65)

X income

1.58 (1.13)

0.41 (2.18)

-2.82 (0.98)

-2.12 (1.02)

0.61 (0.33)

1.01 (0.27)

1.20 (0.27)

0.86 (0.43)

X age X income

Ln(Acres) X boat Motor Trolling X boat Boat Launch X boat Paved Launch X boat Quality Mgmt. X male

State Forest

12

-0.01 (0.09) 0.19 (0.02) 0.10 (0.06) 0.44 (0.07) -0.14 (0.06) 0.36 (0.06)

0.55 (0.15)

0.16 (0.04)

Table IV: continued CL Interact

2S Interact1

RPL (1)

RPL (2)

2S RPL1

Est. (se)

Est. (se)

Est. (se)

Est. (se)

Est. (se)

X male

-0.49 (0.19)

-0.58 (0.26)

-0.30 (0.18)

-0.60 (0.20)

X age

-0.53 (0.52)

-0.58 (0.73)

-1.91 (0.41)

-0.98 (0.33)

X income

-0.01 (0.54)

-0.98 (0.72)

0.06 (0.40)

-0.12 (0.41)

County Forest

-0.19 (0.45)

1.05 (0.31)

-0.64 (0.35)

-0.25 (0.50)

-0.35 (0.24)

-0.71 (0.36)

-0.66 (0.21)

-0.65 (0.22)

X age

0.04 (0.78)

-0.10 (1.29)

1.26 (0.65)

2.12 (0.43)

X income

2.66 (0.78)

1.77 (1.15)

1.89 (0.65)

1.44 (0.64)

Urban Area

-0.04 (0.22)

-0.33 (0.21)

0.27 (0.18)

-0.19 (0.33)

X male

-0.29 (0.12)

-0.30 (0.16)

-0.24 (0.11)

-0.23 (0.12)

X age

-0.82 (0.38)

-1.29 (0.49)

-1.48 (0.35)

-1.72 (0.23)

1.10 (0.29)

1.14 (0.38)

1.20 (0.24)

1.26 (0.22)

Residential Dev.

0.85 (0.37)

0.35 (0.21)

0.20 (0.30)

-0.12 (0.34)

X male

0.82 (0.20)

0.61 (0.58)

1.25 (0.15)

0.37 (0.17)

X age

-1.56 (0.65)

-1.71 (2.02)

-0.15 (0.52)

0.55 (0.33)

X income

-1.82 (0.52)

-0.97 (1.30)

-2.14 (0.45)

-1.06 (0.45)

Industrial Dev.

-0.37 (0.21)

-0.66 (0.17)

-0.49 (0.18)

-0.62 (0.28)

X male

-0.41 (0.12)

-0.52 (0.20)

-0.43 (0.10)

-0.30 (0.11)

X age

1.24 (0.36)

1.02 (0.59)

1.15 (0.32)

0.79 (0.21)

X income

1.26 (0.30)

1.35 (0.48)

1.45 (0.28)

0.95 (0.25)

-9.34 (0.29)

-11.24 (0.41)

-11.27 (0.30)

-12.69 (0.31)

-10.50 (0.44)

-10.22 (0.58)

-10.67 (0.43)

-10.45 (0.54)

X opp. cost

5.49 (2.06)

8.07 (2.96)

7.17 (2.22)

8.31 (2.37)

X not employ

0.66 (0.26)

0.39 (0.43)

0.54 (0.32)

0.19 (0.37)

X part-time

0.67 (0.34)

0.85 (0.59)

0.38 (0.35)

0.28 (0.33)

Variables

X male

X income

Miles (one-way) X waterfront

-0.12 (0.10)

-0.10 (0.04)

0.32 (0.08)

0.35 (0.05)

-12.34 (0.12)

1 The 568 alternative specific constants estimated in the first stage logit model are omitted for brevity. 2 Variable names in italics mark parameters estimated in the second stage regression. Non-italicized

variable names mark parameters estimated in the first stage logit model.

13

References [1] P. Feather, W.D. Shaw, Estimating the Cost of Leisure Time for Recreation Demand Models, J. Environ. Econom. Manage. 38(1) (1999) 49-65. [2] E.R. Morey, TWO RUMS unCLOAKED: Nested-Logit Models of Site Choice and Nested-Logit Models of Participation and Site Choice, in J.A. Herriges and C.L. Kling, eds., Valuing Recreation and the Environment, Massachusetts, Edward Elgar Publishing, 1999.

14