OLS versus ML estimation of non-market resource values with payment card interval data

JOURNAL

OF ENVIRONMENTAL

ECONOMICS

AND MANAGEMENT

OLS versus ML Estimation Values with Payment

17, 230-246 (1989)

of Non-market Resource Card Interval Data1

TRUDY ANN CAMERON Department

of Economics, University of California, LAXAngeles, California 90024

AND DANIEL D. HU~PERT National Marine Fisheries Service, Southwest Fisheries Center, LA Jolla, California 92038

Received July 31, 1987; revised March 18,1988

Contingent valuation methods (CVM) have been shown to be potentially very useful for eliciting information about demands for non-market goods. We assess the sensitivity of “payment card” CVM results to the researcher’s choice of estimation method. Empirical payment card data are used in both (a) a naive ordinary least squares (OLS) procedure employing interval midpoints as proxies for the true dependent variable, and (b) an efficient maximum likelihood (ML) procedure which explicitly accommodates the intervals. Depending upon the design of the payment card, OLS can yield biased parameter estimates, misleading inferences regarding the effects of different variables on resource values, and biased estimates of the overah resource value. Q 1989 Academic Press, IX.

I. INTRODUCTION

Contingent valuation methods have now been recognized as potentially useful techniques for eliciting demands for non-market goods (see Cummings et al. [S], Mitchell and Carson [16], and references cited therein). These methods use hypothetical market scenarios posed to individual respondents to discern their likely behavior under different conditions. While we would prefer to elicit a respondent’s exact willingness to pay, WTP (or willingness to accept, WTA), for an increase (decrease) in consumption of a public good, it has been argued that respondents find it very difficult to name a specific sum. To avoid non-response, researchers have found it more fruitful to pose the valuation questions differently. For telephone or in-person surveys, one alternative is called “sequential bidding.” If a respondent indicates a willingness to pay the initial amount offered, a larger amount is proposed. The bidding continues until the interviewer reaches an amount that the respondent will not pay. However, it has been argued that the starting point (i.e., whether it lies above or below actual valuation) can influence the outcome. To

‘Without implicating them, we thank Michael Hanemann, John Loomis, Jane Murdoch, and two anonymous referees for helpful comments md suggestions. nLis paper represmtsa portion of UCLA Discussion Paper 448 (JuJy 198Q and subsumes Part of [53. 230 0095-06%/89 g3.00 CqqnQht

@I 1989 by Academic F’ms, Inc.

All rights of reproductjQll

in any Form resend.

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get around this problem, an alternative format is feasible. Rather than sequential bidding, the survey questionnaire can be designed to include an ordered set of threshold values, called a “payment card.” The respondent is then asked simply to peruse the range of values and to circle the highest (lowest) amount they would be willing to pay (accept). As with oral sequential bidding, it can then be inferred that the respondent’s true “point” valuation lies somewhere in the interval between the circled value and the next highest (lowest) option. Payment cards conserve effort because even a fairly detailed set of thresholds can be visually scanned quite quickly and there is no need for prompting by an interviewer. Payment cards also avoid the high rate of item non-response on open-ended (i.e., “name the amount”) valuation questions. Although payment cards were originally invented to avoid the starting point bias problem (Mitchell and Carson [17]), they are still subject to other forms of bias involving implied value cues. The range of values, anchoring effects, and size of intervals displayed on the card can affect responses. The range of values used on the payment card employed in our application was designed to cover the likely range of responses, based on pre-test surveys. This should mitigate starting point bias in this study. In exchange for the advantages of payment cards, however, one must confront the problem of having value responses in the form of intervals rather than “point” valuations. Whether the data are used simply to estimate average values or to estimate functional relationships between the values and the characteristics of respondents or the public good, a simplistic approach would be to assign to each interval a value equal to the interval midpoint. One could then use these midpoints as approximations to the true unobserved values in fitting a univariate distribution of values, or use them as the dependent variable in an ordinary least squares (OLS) regression. This “midpoint method” ignores the fact that expected values within the intervals are not necessarily equal to the interval midpoints. A biased average valuation or biased regression coefficients may result. There is, however, a considerable statistical literature on the efficient maximum likelihood (ML) estimation of regression models where the dependent variable is only measured on intervals of a continuous scale (see Hasselblad et al. [12], Burridge [3], Stewart [19], and Cameron [4]). These “interval regression” methods are appropriate for payment card data. The central task of this paper is to assess the potential distortion introduced into contingent valuation estimates due to the inappropriate application of OLS regression methods to payment card interval midpoints. Of course, a host of unresolved concerns regarding the overall accuracy and robustness of contingent valuation methods (CVM) are presently being debated. We abstract from these problems to focus on an important methodological issue which will always be present with payment card data. We illustrate the ML solution to the interval data problem by analyzing payment card data collected in a recent survey concerning values associated with changes in Pacific salmon and striped bass populations. Section II of this paper outlines the procedure for maximum likelihood estimation of the “payment card’ regression model. Section III briefly summarizes the data set. Section IV addresses the specification of the valuation (WTP) function. Section V compares “naive” results from OLS estimation using interval midpoints with those from the more-appropriate ML interval estimation method for a selection of alternative payment cards, and Section VI concludes.

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II. MAXIMUM LIKELIHOOD ESTIMATION “PAYMENT CARD” DATA

WITH

Since valuation ought to be non-negative, and since previous empirical studies (Cameron and James [6,7]) have indicated that the distribution of valuations is frequently skewed, we will propose that a lognormal conditional distribution for valuations should serve as a useful first approximation. If the respondent’s true valuation, Y, is known to lie within the interval defined by lower and upper thresholds tli and lui, then (log Y) will lie between (log t,;) and (log tui). It is generally presumed that E(log Y]xi) is some function g(xi, /3), for which a linearin-parameters form is computationally convenient. In the simplest case, we will have (log Y) = x,‘P + ui, where ui is distributed normally with mean 0 and standard deviation u. Then we can “standardize” each pair of interval thresholds for (log Y) and state that

Pr(r; G trli7t,i)) = Pr((log t,, - x$)/a

< zi < (log tui - x$)/a),

(I)

where zi is the standard normal random variable. The probability in (1) can be rewritten as the difference between two standard normal cumulative densities. Let zti and zui represent the lower and upper limits in (l), respectively. Then for a given observation, (1) can be written as @(z,~) - @(z,,), where Q, is the cumulative standard normal density function. The joint probability density function for n independent observations can then be interpreted as a likelihood function defined over the unknown parameters p and u (implicit in zui and zli). The log-likelihood function is then lOgL=

f i=l

lOg[+(Z”i)

-

@(zli)].

(2)

Appendix 1 details the formulas for the gradients and the Hessian matrix associated with this likelihood function.2 In this study, we use the computer package GQOPT (Goldfeld and Quandt [9]) to maximize the likelihood function explicitly with respect to the unknown parameters, p and u. However, for this particular interval data model, researchers who have access to the LIMDEP computer package (Greene [lo]) will probably find that the “GROUPED DATA” procedure described there can also be used to determine these parameters. Once the optimal values of p and u have been attained, it is a simple matter to reconstruct fitted values of (log Y). The conditional mean of (log Y) for any given vector of x variables will be ~$3. However, if we wish to retransform back to the estimated conditional distribution for the variable Y itself, exp(x$) will be the median of the distribution of Y rather than the mean3 These conditional medians can certainly be considered a valid measure of the central tendency of the unobserved dependent variable Y. However, we may also wish to compute the mean of ‘These formulas can substantially reduce computational costs for the optimization of the log-likelihood function. 3We are grateful to Michael Hanemann for reminding us of this point. For a description of the lognormal distribution, see Hastings and Peacock [13, p. 841.

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the untransformed Y variable, which is obtained by scaling the median by an estimated constant equal to exp(a*/2), where (I is an unbiased estimate of the true population error variance. Either the individual fitted conditional medians or the individual fitted conditional means can then be used to compute the weighted average of fitted valuations across the sample. Either of these quantities can readily be multiplied by the sample size to give an estimate of the total value of the resource to the members of the sample. Applied welfare economics does not exclusively require use of sample mean values. Compensation tests (based upon potential Pareto improvements) require that winners can compensate losers. As a practical matter, however, if the sample mean values and sample mean costs are unbiased estimates of population means, and if one is willing to make decisions based upon a crude compensation test (i.e., benefit-cost analysis as theory of public policy, Randall [X3]), a positive total net benefit (mean value exceeding mean cost) signals a socially desirable project. However, another simple rule for group decisions is the majority voting rule. Assuming each individual votes for projects that entail net benefits to him/her, a project (to be acceptable) would have a median value greater than its median individual cost. That is, since more than 50% of the people have individual values greater than the median, the median value represents a maximum cost consistent with majority approval of a project. Thus, both mean and median values are potentially of interest to those using economic information to examine public resource policies.

III. THE DATA

The data are drawn from a survey of saltwater anglers residing in northern and central California coastal counties centered on San Francisco Bay.4 This was a multiple-purpose survey using a two-stage design. A random telephone survey sampled the general population to identify angling households (defined as those containing at least one member who fished during the previous 12 months). The telephone sample size in each county is roughly proportional to the square root of the county population. Each angler contacted provides information about his/her recent angling activity, including number of fishing trips taken in the most recent 2-month period and whether the fishing trips were targeted on salmon and/or striped bass. A questionnaire was mailed to each telephone respondent who volunteered to cooperate. Approximately 79% of the anglers contacted agreed to complete a mail questionnaire. The mail questionnaire response rate varied among classes of anglers. Only 24% of anglers who had not gone fishing in the previous 2 months returned the questionnaire. Additional mail response rates were 42% of those taking one fishing trip in 2 months; 60% of those taking two trips; 72% of those taking more than two trips; and 75% for anglers having fished solely for salmon or striped bass in the previous 2 months. Clearly, the probability of appearing in the final data set varies with the angler’s county of residence (due to stratification in the telephone sample) and level of 4The survey was conducted by CIC Research, Inc., of San Diego under contract to National Marine Fisheries Service’s Southwest Fisheries Center.

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fishing activity (due to response rate) and both characteristics may be related to individual resource values. To compensate for potential sampling bias, we have computed statistical weights for each combination of county and fishing activity level; these weights equal the estimated proportion of the angling population falling into each category. In addition to the CVM questions, each respondent answered a series of questions concerning his/her angling activity (including details of the most recent three fishing trips), level of angling skill, boat ownership, expenditures on fishing gear, employment status, and household income. Additional details of the survey design and procedures are available in Thomson and Huppert [20]. The contingent value question analyzed in this paper concerns anglers’ WTP for enhancement of fish stocks. The questions were framed by an introductory note indicating that there are competing demands for water in rivers necessary for salmon and striped bass spawning, and that contributions to a special fund for hatcheries and habitat restoration could enhance the fish populations. The crucial valuation questions was: “What is the MOST you would be willing to pay each year

TABLE I Descriptive Statistics (n = 342)

Variable acronym

Description

MIDPT

Midpoint of interval

log(MIDPT)

log(midpoint of interval) No. salmon and striped bass fishing trips in past 12 months log($‘OOOhbld income using midpoint of reported interval) = 1 if all trips were salmon fishing trips = 1 if all trips were striped bass fishing trips = 1 if advanced fishing ability (“p, “Y on l-5) = 1 if respondent owns a boat Catch/trip of salmon on exclusively salmon trips

TRIPS

log(INC)

S-TRIP

B-TRIP

ABIL

OWNBOAT S-CATCHb

Unweighted mean (SD)

Weighteda mean (SD)

48.66 (80.49) 3.165 (1.273) 5.591 (6.835)

57.98 (132.96) 3.115 (1.371) 4.416 (5.449)

3.647 (0.6447)

3.602 (0.6544)

0.4298

0.4188

0.3743

0.3338

0.3363

0.2813

0.4035

0.3317

0.7544 (1.191)

0.7341 (1.046)

aWeights were derived from cross-tabulations of home county by frequency of trip for both the original sample and the estimation sample. Deletion criteria were number of trips unknown; income unknown; ability unknown; boat ownership unknown; and total number of saltwater angling trips in the last 12 months unknown. bSimple averages; values can be zero if no trips of the specified type were taken. Since data are retrospective over last three trips, we camrot simply use S-TRIP and B-TRIP to compute actual per-trip catch-some anglers will have reported mixed trip types.

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TABLE II Interval Selection Frequencies (n = 342) Interval

Raw frequency (W)

Weighted frequency (W)

$ o- 5 5- 10 lo- 15 15- 20 20- 25 25- 50 50- 15 75-100 loo-150 150-200 200-250 250-300 300-350 350-400 400-450 450-500 500-550 550-600 600-750 750 +

52 (15.2) 14 ( 4.1) 38 (11.1) 49 (14.3) 31 ( 9.1) 49 (14.3) 57 (16.7) 6( 1.8) 28 ( 8.2) 6 ( 1.8) 9 ( 2.6) 1 ( 0.3) 0 0 0 1 ( 0.3) 0 0 0 1 ( 0.3)

58.3 (18.3) 8.6 ( 2.7) 39.4 (12.4) 44.9 (14.1) 24.0 ( 7.6) 43.6 (13.7) 56.9 (17.9) 5.2 ( 1.6) 12.2 ( 3.8) 5.8 ( 1.8) 12.3 ( 3.9) 2.2 ( 0.7) 0 0 0 1.0 ( 0.3) 0 0 0 3.9 ( 1.2)

to support hatcheries and habitat restoration that would result in a doubling of current salmon and striped bass catch rates in San Francisco Bay and ocean area if without these efforts your expected catch in this area would remain at current levels? (Circle the amount).” The listed values were $0, $5, $10, $15, $20, $25, $50, $75, $100, $150, $200, $250, $300, $350, $400, $450, $500, $550, $600, and “$750 or more.” 5 Their response thus tells us the interval in which their valuation lies. For example, if $50 is circled we presume that their true value is greater than or equal to $50, but less than $75. Descriptive statistics for the usable portion of the sample are given in Table I. The payment card interval choice frequencies appear in Table II. Note that we do not attempt to compensate for the measurement error inherent in the income (INC) variable because of its categories; that issue is beyond the scope of this study. The pair of dummy variables (S-TRIP and B-TRIP) distinguishes between anglers who sought to catch only salmon or only striped bass (versus the omitted category-anglers who sought either species, without preference). Ability was originally rated on a subjective 5-point scale, with 1 being “novice” and 5 being “expert.” These categories were aggregated into “intermediate and below” and “better than intermediate.” The “intermediate and below” category was designated as the omitted category and the dummy variable (ABIL) captures the higher skill level. If the respondent indicated that they owned or operated a boat that could be used for

5Respondents who circled $0 were also asked: “Did you circle $0 because you feel this change has no value to you?” If a respondent replies “no” to this question, we assume that this answer indicates that they value the change by more than $0 but by less than $5.

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saltwater fishing, the dummy variable OWNBOAT takes on a value of 1. We will maintain the hypothesis that current catch rates will influence the respondents’ basic valuation of the fishery, and should therefore be expected to affect their valuation of a doubling of the catch rates. The survey was conducted by mail and asked for retrospective information on each respondent’s actual catch for their third-last, second-last, and most recent fishing trips (if at least one trip was take over the last 12 months). Recall that we are focusing in this study on fishing trips for which the target species was “exclusively salmon,” “exclusively striped bass,” or “either salmon or striped bass.” The catch data are in the form of “catch per trip.” Only S-CATCH, the salmon catch on exclusively salmon fishing trips, seems to matter in our specifications. Observed catch may not be fully exogenous (due to the fact that respondents’ fishing abilities and choice of site may affect both their average catch and their valuation), but we control to some extent for differing abilities with the subjective ABIL dummy variable.6

IV. THE FORM OF THE VALUATION

(WTP)

FUNCTION

CVM data are frequently used to estimate functional relationships between willingness to pay responses and other variables normally affecting demand. It is beyond the scope of this paper to develop the full theory of these willingness to pay functions, but a brief explanation is appropriate.’ The WTP for the enhancement of fish populations equals the change in money income that would leave the respondent as well off with the enhancement (and lower income) as without the enhancement (at the original income level), i.e., the Hicksian compensating variation for the enhancement in question. The fish population directly influences catch per angler trip, so it acts as a quality attribute of fishing trips, which are themselves unpriced activities yielding direct utility. Hence the Hicksian compensating variation (WTP) associated with increased fish population is equivalent to the increase in total surplus due to an upward shift in the demand curve for fishing.’ We have a direct expression of WTP and do not need to integrate between estimated demand curves to obtain estimated WTP. Nevertheless, the functional relationship between individual WTPs and fishing avidity, household income, and various other characteristics is useful for extrapolating values to the full population, or subpopulations, or to the same individual in changed circumstances. The main innovation of this paper is to estimate this functional relationship without invoking the simplifying assumption that each respondents’ WTP is exactly equal to the midpoint of the relevant interval. ‘Likewise, we must be aware that observed average catch rates for the last three fishing trips by a particular individual do not necessarily provide a good estimate of their expected catch. We would anticipate that this expected catch has a greater impact upon the demand for fishing days than do recent actual catch rates. 7Recent examples of WTP functions include Loomis [14], Brookshire et al. [2], and Brookshire and Coursey [l]. ‘If we were to assume that this change in value depended linear& upon the number of fishing days currently being consumed, we would be imposing a puralZe1 upward shift of the “demand” curve as a result of doubling the fish stock. Instead, we employ a log-linear transformation of the gain in surplus, which allows the implied shift in the demand curve to be non-parallel.

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As outlined in Section II, we assume that the valuation (WTP) function has the general form of log r;. = g( xi, p), where g( xi, 8) = ~$3 in the simple case.’ As in all empirical applications of CVM, it is the task of the investigator to decide upon which of the available variables (or which transformations of these variables) belong in the vector xi. To remain loyal to the neoclassical economic theory of constrained utility maximization, we would of course prefer to employ a specification for g(x,, p) derived from some utility function which has desirable properties (see McConnell [15]). Frequently, however, researchers are constrained by available data to working with specifications which can be considered approximations to these desired functional forms (e.g., Hanemann [ll]). Such is the case here. Preliminary OLS regression models using the midpoints of each of the valuation intervals must be relied upon to determine the most promising explanatory variables as well as to assess whether linear or log-linear specifications might be appropriate. V. COMPARISON OF OLS AND MLE ESTIMATES The first pair of columns in Table III gives the OLS midpoint and MLE interval estimates of the parameters of a log-linear valuation function. Since we have argued that the MLE estimates are a priori superior, we will concentrate on these parameter values in our initial overview of the empirical results. Consider the plausibility of our parameter estimates. Examination of the OLS midpoint estimates shows that the number of trips has a very small but significant effect on the log of valuation for a doubling of catch rates. This means that the “before” and “after” demand curves diverge slightly as the number of days increases. The doubling of the catch would also appear to be a “normal good” for which demand is inelastic with respect to income. Concerning the effect of target species in the OLS model, the omitted category covers respondents who sought either species without preference. If the angler sought only salmon on his last three trips, valuation is lower by about 48% (in the “brief’ model).‘O If only striped bass were sought, WTP is lower by about 59%. On the surface, it would appear that anglers who care only about one of these two species are only about harf as willing to pay for enhancement measures which would double the catch of both species. The ability rating enters significantly. Anglers with above-average ability have WTP about 58% higher. For serious anglers, it is possible that the catch rate figures more prominently among the “bundle” of utility-generating activities that make up a fishing day. Boat ownership enters significantly, contributing to a decrease of about 47% in WTP. This would seem to imply that the easier it is for an angler to increase his number of fishing trips, the less willing he is to pay for higher catch rates per trip. This is plausible. Among the average catch rate variables in the OLS models, only the salmon catch rate (on exclusively salmon fishing trips) significantly affects valuation. If, for a given respondent, the average salmon catch rate on past salmon fishing trips was higher by one fish, it would not be surprising that this individual would be more 9We consider a Box-Cox generalization of linear and log-linear functional specifications in an appendix. “This “percentage” terminology is used loosely, for expositional purposes. The changes are, of course, infinitesimal.

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TABLE III Original versus Coarser Intervals: OLS Midpoint Estimates and MLE Interval Estimates Original intervals

Double intervals

Quadruple intervals

Variable

OLS

MLE

OLS

MLE

OLS

MLE

Constant

2.251 (4.434) 0.02908 (1.989) 0.3425 (3.1%) - 0.4559 ( - 2.465) - 0.5816 - 2.985) 0.5609 (3.301) - 0.4412 - 2.871) -0.1690 - 2.111) 1.286

2.207 (5.151) 0.03008 (2.023) 0.3537 (3.177) - 0.4753 ( - 2.511) - 0.5884 (-2.956) 0.5824 (3.355) - 0.4770 (- 2.982) -0.1798 (-2.175) 1.292 (22.13)

2.551 (6.955) 0.03097 (2.386) 0.2850 (2.992) - 0.3532 ( - 2.152) - 0.5283 - 3.055) 0.4909 (3.255) - 0.3845 - 2.819) - 0.1578 - 2.221) 1.142

2.458 (6.373) 0.03218 (2.414) 0.3011 (3.002) - 0.3770 (- 2.218) - 0.5457 (-3.048) 0.5224 (3.348) - 0.4235 (- 2.939) - 0.1703

2.586 (7.405) 0.02640 (2.141) 0.2510 (2.774) -0.1025 ( - 0.6574) - 0.2576 (- 1.569) 0.4571 (3.190) - 0.3299 (-2.546) - 0.1274 (- 1.888) 1.084

2.124 (4.877) 0.02907 (1.985) 0.3050 (2.730) - 0.1424 (- 0.7422) - 0.2956 - 1.467) 0.5935 (3.415) - 0.4237 - 2.586) - 0.1734 - 2.019) 1.171 (15.88)

TRIPS log(INC) S-TRIP B-TRIP ABIL OWNBOAT S-CATCH (I

(- 2.290) 1.143 (21.55)

Average median WTP

$26.03 (16.88)

$25.76 (17.60)

$30.04 (16.79)

$28.59 (17.21)

$32.34 (13.67)

$24.70 (13.50)

Average mean WTP

$59.51 (38.59)

$61.80 (42.22)

$57.66 (32.23)

$56.71 (34.14)

$58.20 (24.60)

$50.69 (27.70)

satisfied with current catch rates and therefore less willing to pay for a doubling of overall catch rates (by approximately 18%). With intervals as fine as those specified on the actual payment card used in our survey, Table III shows that the OLS midpoint parameter estimates and the MLE interval parameter estimates are very close. The same is true for the average median and mean WTP values for the doubling of catch rates. This is greatly reassuring and suggests that the payment card used for this survey was well-designed. In other studies, however, a great variety of payment cards have been used. We will now make use of our data to simulate the results which might have be obtained had different payment cards been used for this study.

Coarser Intervals The finer the intervals listed on the payment for a respondent to decide exactly which interval pay. Consequently, questionnaire designers must card against the danger of item non-response. upon the valuation function parameter point

card, the more difficult it becomes contains their actual willingness to trade off resolution in the payment Here, we will consider the effects estimates and the marginal mean

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valuations of imposing coarser and coarser groupings. We rely only on the presumption that individuals implicitly do know their true underlying valuation and therefore will make consistent responses across different payment card formats. (Violations of this assumption would of course confound the estimates even further, beyond the purely statistical distortions illustrated here.) In our first set of simulations, we counterfactually reduce the number of intervals on the payment card by first collecting the existing intervals in pairs. The question might have listed only the values $0, $10, $20, $50, $100, $200, $300, $400, $500, and $600 or more. It is reasonable to assume that someone who circled $15 before would now circle $10. (It would be inconsistent to circle $20 if this amount was not willingly paid before.) Based on these counterfactual “double intervals,” the resulting point estimates of the parameters are displayed in the second pair of columns in Table III; the third pair of columns gives the corresponding estimates when the original intervals are collected in quadruples. Broader intervals make it easier for the respondent to select the interval wherein his true valuation lies, but there will be a loss of information for estimation purposes. Cameron [4] demonstrates using Monte Carlo techniques that as interval widths increase in a log-linear normal specification, it is unfortunately not possible to systematically sign the bias introduced into the parameter estimates by the OLS midpoint method-even in large samples, these biases oscillate. The MLE interval estimates, however, are much more robust to interval width. This tendency is observed as one scans across the columns of Table III. Examining the pairs of OLS and MLE parameter estimates for each set of intervals reveals that the absolute value of the OLS slope estimates decreases systematically with increasing interval width. This damping effect on the apparent influence of explanatory variables is consistent with the simple regression Monte Carlo findings in Cameron [4]. The MLE slope estimates in the quadruple interval case are uniformly closer to the estimates obtained using the true intervals. In the two cases where the parameter estimates differ substantially from the “true interval” values (S-TRIP and B-TRIP), the aggregation of the intervals has sufficiently obscured the consequences of these dummy variables that statistically significant parameter estimates cannot be obtained either by OLS or by MLE. The precision of these estimates is low in either case. The average median and average mean values of fitted willingness to pay appear at the bottom of Table III. For the OLS midpoint estimates, the average medians appear to increase systematically with interval width, from $26 for the true intervals to $32 for the coarser set. (By changing the arbitrary value used for the “midpoint” of the uppermost interval, however, a very wide range of value estimates can be obtained at will.) In contrast, the corresponding estimates for the MLE midpoint method seem more stable, with $26 for the true intervals and $25 for the quadruple intervals. Another fundamental statistical problem with the OLS midpoint method comes into play when the average medians are converted to average means by multiplying by exp(a*/2). By assuming that the true value within each interval is exactly the midpoint of that interval rather than simply some random value within the interval, the OLS method systematically overstates the information in the data. The OLS estimates for u are therefore smaller than the true standard deviation of the conditional distribution of the dependent variable, and the extent of the underestimation worsens as the intervals widen. Consequently, the OLS median values are

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being scaled up by a factor which is too small to begin with (even for the “true” intervals), and increasingly too small as the intervals widen. This problem is substantially corrected in the MLE interval estimation method. Thus, while it may appear that the OLS mean WTP estimates are fairly stable as the payment card intervals widen, and that the MLE mean WTP estimates decrease by about one-sixth, the stability of the OLS means is merely an accident of the u estimates decreasing just sufficiently to offset the increase in the fitted median values. For an unscrupulous researcher, therefore, it seems that the apparent average median social value of this resource could be increased by offering respondents a smaller selection of relatively coarser payment card intervals and using the midpoint technique. The OLS average mean values are simply wrong, because of their strong dependence upon the estimated value of cr.

Fewer Upper Intervals

In our second set of counterfactual experiments, the existing intervals are retained for the lower levels, but the single open-ended upper interval for valuation changes from “ > 750” (which contained only one observation and produced the results given in the first column of Table III) to “ > 100” (which contains 46 observations, or 13.45% of the sample), then to “ > 50” (containing 109 observations, or 31.87% of the sample), to the extreme but possible case of “ > 20” (containing 189 observations, or 55.26% of the sample). This last situation is clearly undesirable and could probably be avoided with some judicious pre-testing of the payment card design, but it could easily happen if the pre-test sample is not representative. Again, one of the biggest problems with the OLS midpoint method is the arbitrary selection of a value to represent the midpoint of the open-ended upper interval. This choice can have a dramatic effect upon the parameter estimates and upon the implied value of the resource. In the three counterfactual payment cards analyzed in Table IV, we used a value 50% larger than the lower bound of the upper interval as that interval’s midpoint. There is no standard algorithm for the choice. For the last case (“ > 20”) however, we illustrate succinctly the effect of one extreme alternative choice, $250, for this last midpoint. (If all one knew was that 189 respondents had valuations greater than $20, what number should be chosen?) The bottom of Table IV also gives the resulting marginal averages of the fitted median and mean valuations. Again, a decision not to discriminate so finely among the upper intervals results in considerable damping of most of the OLS coefficients and of the overall average fitted median and mean values computed from the OLS estimates. From an average fitted median of $26 for the true intervals, the value drops to only $16 for our last OLS simulation, that is, unless a different value is chosen to represent the open-ended upper interval, in which case a very much larger average fitted median valuation of $53 can be attained. The fitted means will involve the same underestimates of u that were present in the last section, but here, since the medians also shrink as we move from left to right in Table IV, the total effect on the calculated means is not offset as before. The average mean falls from $59 under the “true” intervals to $24 under the last scenario. But if we adopt $250 as the upper interval’s midpoint in this last case, both the average median and (I become substantially larger and the computed average mean reaches $280! In this light, the variability in the average means from the MLE

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DATA

TABLE IV Original versus Fewer Upper Intervals: OLS Midpoint Estimates and MLE Interval Estimates Highest interval

Highest interval

(’ 100)

(’ 50)

Highest interval (’ 20) OLS “$250”

MLE

Variable

OLS

MLE

OLS

MLE

OLS “$30”

Constant

2.424 (6.186) 0.03314 (2.395) 0.2756 (2.714) - 0.4180 (- 2.390) - 0.5939 (- 3.223) 0.4198 (2.612)

2.316 (5.360) 0.04091 (2.626) 0.3309 (2.918) - 0.4980 (- 2.577) - 0.7132 ( - 3.473) 0.5101 (2.883)

2.561 (6.847) 0.02932 (2.221) 0.2106 (2.173) - 0.3641 (-2.181) -0.5160 (- 2.934) 0.3138 (2.045)

2.452 (4.777) 0.05392 (2.668) 0.3194 (2.358) - 0.5224 (- 2.247) - 0.8062 (- 3.287) 0.4425 (2.086)

2.406 (8.263) 0.01526 (1.484) 0.1496 (1.983) -0.2852 (-2.194) - 0.2774 (- 2.025) 0.2210 (1.850)

2.985 (5.071) 0.03905 (1.879) 0.3025 (1.983) - 0.2907 (-1.106) - 0.4336 (- 1.566) 0.4691 (1.942)

2.285 (4.319) 0.04237 (1.914) 0.3305 (2.387) - 0.4237 (- 1.798) - 0.5224 (- 2.038) 0.4346 (1.850)

- 0.3688 (-2.537) - 0.1556 (- 2.054) 1.217

((-

-0.3358 ( - 2.420) -0.1105 (- 1.529) 1.161

-0.5113 (-2.653) -0.1710 (- 1.724)

- 0.2704 (-2.503) - 0.06623 (-1.176) 0.9042

((-

- 0.5425 (- 2.549) - 0.1256 (-1.271) 1.447 (11.57)

TRIPS log(INC) S-TRIP B-TRIP ABIL OWN BOAT SCATCH IJ

0.4452 2.751) 0.1961 2.338) 1.290 (19.98)

1.474 (16.02)

Average median WTP

$24.07 (13.41)

$26.03 (18.78)

$22.03 (9.613)

$29.09 (23.30)

$15.75 (4.274)

Average mean WTP

$50.48 (28.12)

$62.28 (44.93)

$43.22 (18.86)

$90.87 (72.78)

$23.70 (6.432)

0.5399 2.472) 0.1299 1.142) 1.828

$52.57 (30.90)

$279.5 (164.3)

$27.13 (17.38)

$81.31 (52.09)

estimates (due almost entirely to the increase in the estimate of u as the upper interval widens) seems relatively innocuous. Again, it seems that an unscrupulous researcher could readily influence the estimated total value of the resource by appropriately tailoring the upper intervals of the payment card, making a judicious choice of the arbitrary “midpoint” for that interval, and then selecting either the medians or the means in order to achieve the desired effect.

VI. SUMMARY

AND CAVEATS

It seems that the implications of a payment card contingent valuation study can be influenced to a considerable extent by the design of the payment card and by the estimation technique employed to fit the valuation function. However, it is important to qualify our specific empirical findings.

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First, the fisheries enhancement program being proposed can only be interpreted as an increase in the stock of fish aoailable for catching. The question is posed in terms of catch rates overall, not in terms of the indiuidual ‘s catch rate. There is only a loose correlation between fish stocks and individual catch rates (and in many instances, the individual’s catch is currently zero). Hence, in this study we do not pursue translation of “ WTP for a doubling of overall catch rates” into “ WTP for an additional fish”-a measure which has been sought in other studies.” The ultimate objective in a project like this one is a measure of the social value of the enhancement project (i.e., the social value of having twice as many fish out there). This figure could presumably be compared to the project’s overall cost to assess its advisability. Second, we cannot know the actual values of the “true” underlying population parameters for the relationships which actually produced the real data employed in this study-i.e., the true “data generating process.” Therefore, we are at a loss to compare our preferred ML interval estimates with these true parameters to assess the sign or extent of the innate bias in the payment card format. Findings reported in Cameron [4] suggest that there will always be some bias and that its sign will be indeterminate, but that the ML interval technique in log-linear models is unambiguously more reliable than OLS used on interval midpoints. Neither can we say unequivocally whether the log-linear specification we are estimating is the “true” functional relationship between willingness to pay and the other variables. Evidence examined in Appendix 2 suggests that the log-linear specification may be too restrictive-a Box-Cox transformation employed with the ML interval method (which frees up an additional parameter) rejects the restriction imposed by the log-linear form. Since our main points here are methodological, however, we have opted to adhere to the log-linear model and its relatively easier interpretation. One would certainly expect the discrepancies highlighted in this paper to carry through for a wide variety of specifications of the relationship between valuation and the vector of explanatory variables. The findings described in this paper have demonstrated that the implications of any payment card study can be influenced to a considerable extent by the valuation question intervals chosen by the survey designer and by the estimation method employed by the data analyst. We have illustrated this sensitivity with just one survey and just one (ad hoc) specification for the underlying valuation model, but we emphasize that interval choice and estimation method biases will be a fundamental issue in all payment card studies, independent of other problems with the technique. It may be a disservice to the profession to point out that the resulting total value of a non-market resource (the derivatives of that value with respect to explanatory variables) can be manipulated by intentional decisions regarding the payment card and estimating procedures. However, armed with the knowledge that intentional manipulations can distort the apparent results of a CVM study, policymakers

“In auxiliary experiments, we have segregated respondents according to their target species. For each of these groups, we have transformed the midpoint of the valuation interval by dividing through by the respondent’s average catch, if the catch was non-zero. This makes the very strong assumption that if overall catch rates double, so will this respondent’s catch. ‘Ibis procedure generates a new dependent variable: the value of increasing expected catch by one fish. In these models, however, this value still depends significantly on the number of fish caught.

ML ESTIMATION

FOR PAYMENT

CARD

243

DATA

charged with the critical assessment of payment card studies will have a better sense of the reliability of the value estimates these studies produce. The analysis described here leads us to three recommendations. Without these caveats, it is extremely difficult to judge the results of a payment card study. First, researchers considering using payment cards in a contingent valuation survey must be prepared to conduct careful prstesting of the survey instrument. This pre-testing will identify the approximate marginal distribution of values in the population and can avert the use of vastly inappropriate payment card intervals for the full sample. A description of the thoroughness of pre-testing should be provided for evaluators of payment card CVM research. At a minimum, however, we recommend most strongly that a detailed description of the final payment card and observed choice frequencies should be included in any reporting of payment card value estimates. Second, if OLS midpoint estimation methods are to be used, the researcher should be able to argue that the payment card intervals are sufficiently fine to minimize the bias from using interval midpoints as proxies for the true values of the dependent variable. In particular, the arbitrary but critical selection of a “midpoint” value for any open-ended intervals must be justified and sensitivity analysis with respect to this arbitrary choice should be conducted. If a log-linear OLS midpoint model is being employed and fitted valuations are being reported as means rather than medians, the researcher must be careful to acknowledge that the means are distorted to an unknown degree by the systematic underestimation of the regression standard error in the midpoint model. Since most researchers can gain access to Greene’s LIMDEP software package quite readily, we must conclude by advocating the use of the preferred maximum likelihood method whenever payment card valuation data are being examined. APPENDIX 1 Derivatives of the Log-Likelihood

Function for the ML Interval Model

To simplify the formulas for the gradient vector and the Hessian matrix, it is useful to define the following abbreviations: Pi = @(z”i) - @(zJ Pi = +(‘ui>

- 9tzli)

Qi = zui+(zui) - zli+(zli) 4i = zi9(zli) Ri

=

Pi =

Zfi+(Zli)

-

z,“i+(zui>

-

Zii+(Zui)

Pi/‘,

oi = Q/Pi.

The elements of the gradient vector are given by

a1% L/w,

= C-

a log L/h

= c

PixiJu,

- 6+/a

r=

1 )...)

k

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CAMERON AND HUPPERT

and the elements of the Hessian are r,s = ~2log L/ah ah, = C [~,(l - ~~~~~~~~~~~~~~

1 ,***> k

r=l,...,k a2logL/ap, aa= C[p,+ gilPi - P~c.+]x~~/u~,

a2i0gL/aa2 =C

[2@,+ RJP,- t&02.

APPENDIX 2

ML Interval Estimates under a Box - Cox Transformation

of Valuation

Although a log-linear specification is adopted in the body of this paper (since our emphasis is primarily upon the different estimates achieved by alternative estimation methods), we did explore alternative transformations of the underlying continuous dependent variable. Specifically, we have experimented with a Box-Cox transformation, p

=

(r;:X- 1)/X = x$

(with the transformation parameter X estimated simultaneously with the /3 vector and (I by ML). The results for this specification, by the ML interval method, are displayed in Table A.2.1. The estimated Box-Cox transformation parameter X is 0.1114. While the X is indeed statistically significantly different from zero (which would imply a log transformation), we elect to utilize the log transformation primarily because of its simplicity. Of course, since the Box-Cox model has an additional parameter free to be determined by the evidence in the sample, these estimates reflect closer conformity to the relationship between the variables implied by the data. These estimates can therefore be considered “superior” to any of those appearing in the body of the paper, as far as ad hoc models are concerned. In the body of the paper, we took care to distinguish between the fitted conditional median values for WTP and the fitted conditional mean WTP estimates. In the simple log-linear specification, the latter could be obtained from the former by transforming by exp( u 2/2). In the Box-Cox example, the correction is not this simple. Specifically, the fitted value of ~$3 gives the mean of the transformed variable, Y@) = (Y” - 1)/A. The mean of Y itself will be given by the expectation of a function [XY (*) + l](‘/‘), of the N(x$, a2) random variable, Y(‘). Since this integral seems prohibitively difficult to evaluate, we will report only the exponentiated value of the mean of Y@). Recall that the average of the fitted median WTP values in the log-linear ML interval model was $25.76 (with a standard deviation of $17.60). In the “brief” Box-Cox model, the corresponding marginal mean is $27.42 (with a standard deviation of $18.74). These differences are relatively modest. From a policy-making point of view, they may or may not be “significant.”

ML ESTIMATION FOR PAYMENT CARD DATA

245

TABLE A.2.1 ML Interval Estimates with Box-Cox Transformed Dependent Variable Interval estimates (asymptotic r ratio)

Variable

constant TRIPS log(WC) S-TRIP B-TRIP ABIL OWNBOAT S-CATCH 0

x Average median WTP

2.385 (3.750) 0.04349 (1.897) 0.5487 (2.848) - 0.7004 (- 2.321) - 0.8626 (- 2.667) 0.9123 (2.965) - 0.7209 (- 2.699) - 0.2756 (- 2.095) 1.906 (5.943) 0.1114 (2.466) $27.42 (18.74)

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10. W. H. Greene, “LIMDEP (trademark) User’s Manual” (1986). 11. W. M. Hanemann, Welfare evaluations in contingent valuation experiments with discrete responses, Amer. J. Agr. Econom. 66, 332-341 (1984). 12. V. Hasselblad, A. G. Stead, and W. Galke, Analysis of coarsely grouped data from the lognormal distribution, J. Amer. Statist. Assoc. 75, 771-778 (1980). 13. N. A. J. Hastings and J. B. Peacock, “Statistical Distributions,” Wiley, New York (1975). 14. J. B. Loomis, Balancing public trust resources of Mono Lake and Los Angeles’ water right: An economic approach, Water Resour. Res. 23,1449-1456 (1987). 15. K. E. McConnell, Models for referendum data, mimeo, Department of Agricultural and Resource Economics, University of Maryland, College Park, Maryland (1987). 16. R. C. Mitchell and R. T. Carson, “Using Surveys to Value Public Goods: The Contingent Valuation Method,” Resources for the Future, Washington, D.C. (1989). 17. R. C. Mitchell and R. T. Carson, Some comments on the state of the arts assessment of the contingent valuation method draft report, in “Valuing Environmental Goods: An Assessment of the Contingent Valuation Method” (R. G. Cummings et a/., Eds.), Rowman and Allanheld, Totowa, NJ (1986). 18. A. Randall, Total economic value as a basis for policy, Tram. Amer. Fish Sot. 116, 325-335 (1987). 19. M. B. Stewart, On least squares estimation when the dependent variable is grouped, Rev. Econom. Stud. 50, 737-153 (1983). 20. C. J. Thomson and D. D. Huppert, “Results of the Bay Area Sportfish Economic Study (BASES),” U.S. Department of Commerce, NOAA Technical Memorandum, National Marine Fisheries Service (1987).