Repeated Dichotomous Choice Formats for Elicitation of Willingness to Pay: Simultaneous Estimation and Anchoring Effect

Abstract

Repeated dichotomous choice contingent valuation data are generated from responses to a succession of binary questions regarding alternative prices for an environmental good. In this paper we propose a simultaneous equation model that allows for endogeneity and error correlation across the responses at each stage of the bidding process. The model allows us to study the evolution of anchoring effects after the second dichotomous choice question. Estimation involves the Bayesian techniques of Gibbs sampling and data augmentation, and the application focuses on the preservation value of a natural area. The results for a data set involving up to four successive dichotomous choice questions show that restricted multiple-bounded models are rejected by the data with the general model. In addition, willingness to pay tends to stabilize after the second stage in the elicitation process for the general unrestricted model. When taking anchoring effects into consideration, it is revealed that individuals’ responses in the latter stages are influenced by the sequence of bid prices offered in earlier questions. Nevertheless, they do not have a significant effect on welfare estimates.

Appendix 1

The model outlined in Section 2 is estimated with a Bayesian approach. The principal difficulty is that the joint posterior distribution −π(θ Y ¹, Y ²,... Y ^m), where θ is the parameter vector to estimate, Y ¹ = (y ¹₁ ,... y ¹ _n ) and Y ² = (y ²₁ ,...,y ² _n ), – is difficult to evaluate by multiple integration methods. However, this task is more feasible with a Monte Carlo approach, based on data augmentation (DA) and a Gibbs’ sampling algorithm (GS), as developed by Tanner and Wong (1987) and Gelfand and Smith (1990), respectively.

GS allows the researcher to evaluate the joint posterior distribution by sampling directly from the conditional posterior distributions for θ. However, in the context of discrete choice models, direct application of the GS is not trivial, since the conditional posterior distributions are still difficult to evaluate. This can be addressed by combining GS and data augmentation (DA) methods. The DA technique simplifies computation by introducing the latent variables $\overline{\hbox{WTP}}_{i}^{1}, \overline{\hbox{WTP}}_{i}^{2}, \overline{\hbox{WTP}}_{i}^{3},\ldots\overline{\hbox{WTP}}_{i}^{m}$ into the model. As a result, the conditional posterior distributions are available in a tractable form.²⁰

For a general DCm process let $\theta =\{\overline{\hbox{WTP}}^{1}_{i}, \overline{\hbox{WTP}}_{i}^{2}, \overline{\hbox{WTP}}_{i}^{3},\ldots \overline{\hbox{WTP}}_{i}^{m}, \Sigma,\Pi,\mu_{1},\mu_{2},\ldots,\mu_{m}\}$ be the parameter vector to estimate, where $\overline{\hbox{WTP}}^{j}= (\hbox{WTP}_{1}^{j},\ldots,\hbox{WTP}_{n}^{j}) \forall j=1,\ldots m$, and Π is a matrix that collects parameters on the potential endogeneity of WTP (η _kr ∀ k > r. The steps of the GS algorithm are the following :

• Step 0. Determine the starting values for θ. Call these θ⁽⁰⁾. These values can be obtained by FIML estimation.

• Step 1. Generate a set of sample draws from the conditional distribution for WTP¹ for each individual in the sample (e.g. $\overline{\hbox{WTP}}^{1}$), assuming that the true values of all parameters are equal to these starting values. That is, π(WTP ¹⁽¹⁾ _i |WTP ²⁽⁰⁾ _i ,WTP ³⁽⁰⁾ _i ,...,WTP ^m(0) _i , Σ⁽⁰⁾, π⁽⁰⁾ , μ ⁽⁰⁾₁ , μ ⁽⁰⁾₂ ,..., μ ⁽⁰⁾ _m )

• Step 2. Generate sample values for $\overline{\hbox{WTP}}^2$ from the corresponding conditional distribution evaluated at the most recent values drawn for each individual in the sample, which are collected in $\overline{\hbox{WTP}}$.That is, π(WTP ²⁽¹⁾ _i |WTP ¹⁽¹⁾ _i ,WTP ³⁽⁰⁾ _i ,...,WTP ^m(0) _i , Σ⁽⁰⁾,π⁽⁰⁾, μ ⁽⁰⁾₁ ,μ ⁽⁰⁾₂ ,...,μ ⁽⁰⁾ _m )

• ...

• Step m. Generate sample values for $\overline{\hbox{WTP}}^m$ from the corresponding conditional distribution evaluated at the most recent values of the remaining parameters, that is, π(WTP ^m(1) _i |WTP ¹⁽¹⁾ _i ,WTP ²⁽¹⁾ _i , ..., WTP ^m-1(1) _i ,Σ⁽⁰⁾,π⁽⁰⁾, μ ⁽⁰⁾₁ ,μ ⁽⁰⁾₂ ,...,μ ⁽⁰⁾ _m )

• Step m + l. Once a set of draws-one for each individual in the sample and for all m WTP amounts – is available, sample Σ⁽¹⁾ from the corresponding conditional distribution evaluated at the most recent values of the rest of parameters.

• Step m + 2. Sample Π⁽¹⁾ from the corresponding conditional distribution evaluated at the most recent values of the rest of parameters.

• Step m + 3. Sample μ ⁽¹⁾₁ from the corresponding conditional distribution evaluated at the most recent values of the rest of parameters.

• ...

• Step 2m + 2. Sample μ ⁽¹⁾ _m from the corresponding conditional distribution evaluated at the most recent values of the rest of parameters.

• Step 2m + 3. Repeat steps 1 to 2m + 2 until convergence is reached.

The values generated by using this algorithm can be regarded as drawn from the joint distribution of θ. These series of simulated values are then used to estimate the posterior moments for the parameters, after the first d values – or burn-in period – in the chain are discarded (Data and program-codes used in this paper are available (only for academic and research purposes) at: www.personales.ulpgc.es/jarana.daea/soflware).