Experimental Design

The experiments presented in this chapter are aimed at analyzing the performance of competing models when some specific forms of bias affect the

Figure 12.1. Contour plots of Bivariate Normal and Joe Copula when marginals are N(0,1).

responses to the second bid question in the double bound format. Each form of response effect implies a specific bivariate data generating process (DGP) for the two WTPs. The first DGP we consider introduces a mild form of elicitation effect, where the underlying WTP elicited after the second question is the same as the first, but because of some disturbance in the elicitation process they are not perfectly correlated:

where p is the correlation parameter, and uu and u2i are identically distributed random variables with mean zero, and variance a2. If the random errors are assumed to be distributed as a Normal, this specification gives rise to a BVN model, where parameters of the two equations are constrained to be equal, while the correlation parameter is unconstrained. In Cameron and Quiggin this was deemed as the best specification to fit their double bound data (a well known Australian study for the Kakadu area), which were mod eled by means of the univariate interval data (or double bound) model, and several alternative specifications of the bivariate probit model. The simulated data are constructed using the following specification: intercept parameter a =10, slope coefficient 3 =3, standard deviation a =5, and a BVN distribution, with correlation p =0.7. The corresponding value of t is given by t = (2/n) arcsin(p) =0.493. The variable x is generated from a uniform, with mean 3.95 and standard deviation 2.05.

The same BVN and interval data specifications used by Cameron and Quig-gin were analyzed again by Alberini (1995) on a slightly different sample from the same data, and in this case the preferred specifications was as in 3.1, but with different estimated variances for the two latent dependent variables. The underlying behavioral hypothesis could be that the cognitive process after the second elicitation question is more "disturbed" — and indeed Alberini (1995) finds that the estimated standard deviation of the second equation is substantially higher than the first. This more general structure has been analyzed again in Alberini, Carson and Kanninen (1997). Our Monte Carlo analysis considers both cases, which can be respectively referred to as restricted and unrestricted random effects models, with an equality restriction imposed on the standard deviation parameters of the former (experiment A). In the latter DGP, the standard deviation of the second equation is set at 7 (experiment B).

Another experiment (C) studies the performance of different models when the double bound elicitation method produces more serious forms of bias, leading to a downward shift of the second equation WTP: CGM indicate several possible causes of this effect, briefly reported in section 2 of this chapter. In its simplest form the bivariate model with shift is structured as follows:

0 < p < 1, 5< 0, i.e. the shift is simply a leftward translation of the WTP distribution. While more complex specifications may model the shift effect as dependent on some covariates, in our experiment we hold to this basic model, setting 5 =-2, the other data being constructed as in experiment B.

The Framed model proposed by DeShazo is relatively new, and, to our knowledge, has not been as yet studied by means of Monte Carlo methods. Here the structure of the DGP is somewhat more complex, since the model involves a mechanism of sample selection. Theoretically responses to both questions should be dictated by model B but because of framing effects a percentage of respondents belonging to the "yes,yes" class produce responses in the "yes,no" class. In DeShazo's proposed method to estimate such data affected by framing effects, follow-up responses from individuals who faced a downward sequence of bids enter the second equation, while for individuals facing an upward sequence only the first response is considered, as if it were a single bound elicitation. The bivariate model for descending sequences proposed by DeShazo is the following:

where Y2 is modeled for respondents in a descending sequence only (i.e. individuals who responded No to the first elicitation question). The parameter values are constructed as in experiment B above but we switch randomly a percentage of "yes,yes" responses to "yes,no". In order to evaluate the performance of the bivariate estimator versus its univariate counterpart, we also estimate a univariate Framed model: it is a censored double bound model, with the second bid included in the equation for respondents in the descending sequence only.

In another experiment (E) we study the performance of different estimators when the initial bids are poorly chosen: in particular, the initial bids of this experiment leave uncovered the left tail of the true WTP distribution, being placed at the 45, 65, and 85 percentile of the distribution. As discussed in section 2, this is a case where the double bound method is generally deemed superior to the single bound, since the follow-up question allows inspection of the previously excluded part of the distribution. All other aspects of the data generation process are as in experiment B.

Finally, in experiments F and G we analyze the case where the DGP departs from the Bivariate Normal distribution. WTP distributions are commonly specified as non normal: logistic or extreme value (for WTP or its log) or gamma distributions are typical choices in CV data analysis. Application of bivariate models to WTP data may induce a misspecification problem, if the standard bivariate probit is adopted. In this experiment we are especially interested in analyzing the performance of the bivariate probit estimator when the assumption of bivariate normality is wrong. We first simulate a bivariate distribution with normal marginals, but with a dependency structure different from linear correlation: i.e. the Joe copula distribution with normal marginals (JOE-N: experiment F). In experiment G, we use as a DGP a bivariate model with extreme value marginals, again linked by a Joe copula (JOE-E). The dependency parameter t is set at the value 0.499 which corresponds to a value of d equal to 2.85. All other settings are as in experiment B. The reader is referred to the Appendix for the algorithm used to generate the data by means of the JOE copula.

Given the above scenarios we analyze the performance of different estimators: the univariate single and double bound models; a univariate model for descending sequences (in experiment D only); the bivariate probit model;

the bivariate model based on the Joe Copula with normal marginals for experiments A-F; a bivariate Joe Copula with extreme value marginals for experiment G; and the bivariate probit with sample selection for descending sequences. The respective likelihoods for each experiment are given inTable 12.1, where a ii = and a2i = '^—T1.

Table 12.1. Likelihood of Estimated Models

LIKELIHOOD TERMS: LIKELIHOOD TERMS: MODELS Experiments (A) to (F) Experiment (G), where the cdf for extreme value is given by, G(a) = exp(-exp(-a))

SINGLE P(yes): 1-$(aH) P(yes): 1-G(au) P(no): $(aii) P(no): G(aH)

DOUBLE P(n,n) P(n,y) P(y,n) P(y,y)

$(a2i) P(n,n): G(a2i) $(aii)-$(a2i) P(n,y): G(aH)-G(a2i) $(a2i)-$(aii) P(y,n): G(a2i)-G(aii) 1-$(a2i) P(y,y): 1-G(a2i)

FRAMED1 P(n,n) (experiment P(n,y) D only) P(yes)

$(a2i) P(n,n): G(a2i) $(aii)-$(a2i) P(n,y): G(aii)-G(a2i) 1-$(aii) P(yes): l-G(aii)

BVN P(n,n) P(n,y) P(y,n) P(y,y)

&(aii,a2i,p) $(aii,-a2i,-p) $(-aii,a2i,-p) $(-aii,-a2i,p)

JOE P(n,n) COPULA P(n,y) P(y,n) P(y,y)

C($(aii), $(a2i),0) P(n,n): C(G(aii),G(a2i),0) $(aii)-C($(aii), $(a2i),0) P(n,y): G(aii)-C(G(aii), G(a2i), 0) &(a2i)-C($(aii), $(a2i),0) P(y,n): G(a2i)-C(G(aii), G(a2i),0) 1-P(n,n)-P(n,y)-P(y,n) P(y,y): 1-P(n,n)-P(n,y)-P(y,n)

FRAMED2 P(n,n) P(n,y) P(yes)

$(aii,a2i,p) P(n,n): C(G(aii), G(a2i), 0) $(aii,-aii,-p) P(n,y): G(aii)-C(G(aii),G(a2i),0) 1-$(aii) P(yes): l-G(aii)

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