Theoretical Framework

Much of the theory in this section is based on Lopez-Feldman‟s (2012) paper which describes the statistical process in STATA© for double bounded estimation. Consequently, it is also Lopez-Feldman‟s (2012) programming code that was used to run the double bounded model within STATA©. Response data from Contingent Valuation studies are analysed using the random utility model (McFadden, 1974; Hanemann, 1984). Individuals (which will be denoted by i) are assumed to be utility maximisers therefore, the model assumes that when respondents are faced with two choices or alternatives, they will always choose the option which yields the highest utility value. As dichotomous choice (referenda) are of the closed- ended CV type, a binary choice between two states is presented, one where the respondent states „no‟ (yi =0) and the other where the respondent states „yes‟ (yi = 1), to a predetermined

offer amount (ti,) and varies at random across individuals. WTP can therefore be modelled

using the following linear function:

( ) (1)

Where is a vector of explanatory variables as well as their corresponding parameters and is an error term which incorporates individual and question error (Lopez-Feldman, 2012, Watson and Ryan, 2007). In the above case, explanatory variables would include age, gender, education, knowledge, familiarity, occupation, interest, residential background, scenery, income, electricity paid for monthly and wind turbine attractiveness, while the error term represents the random differences in each individual‟s preferences for a particular good, which in this case is scenery. The random component of utility is not observable therefore the model relies on probabilities for estimation (Aravena-Novielli, Hutchinson and Longo, 2010). If the individual‟s WTP is higher than the bid amount, ti, (WTP > ti) then the expected

response would be a „yes‟ answer. Accordingly, the probability of a yes response given the values of the explanatory variables is expressed:

( | ) ( ) ( ) ( )

following from Lopez-Feldman (2012), assuming that ( ), indicating a normal distribution, it then proceeds that:

( | ) ( )

( )

( | ) ( ) (2)

where ( ) and ( ) denotes that the model uses the standard normal cumulative distribution. Equation (2) is closely related to the probit model, but distinguished by the presence of the variable in addition to the other explanatory variables (Lopez-Feldman, 2012). Estimation of this model can be done by maximum likelihood to solve for and or alternatively using probit74. However, equation (2) represents the single bounded model, in which only one response to a bid ( ) is allowed for. A follow-up bid, as used in the double bounded dichotomous choice procedure, would require some adjustment to the model specified.

The double bounded model, which was originally proposed by Hanemann (1985) and Carson (1985) and later implemented by Hanemann et al. (1991), involves a second or follow-up bid after the first offer amount. As explained earlier, a „yes‟ response to the first bid would result in a higher follow-up amount being offered and a „no‟ response would result in lower follow- up amount being offered, resulting in a „sharpening‟ or truncation of the single bounded estimates. Secondly, if a „no‟ response is followed by a „yes‟ response75, then a clear bound can be placed on WTP (Hanemann et al., 1991). It is crucial to note that the first bid is determined exogenously76, but the follow-up bid is endogenous as it depends on the answer given to the first bid77 (Cameron and Quiggin, 1994). The potential problem arising from assuming a single valuation function for both bids is that the distributions for the first and

74 _{Lopez-Feldman (2012) covers the probit estimation steps in detail and is beyond the scope of this section as} the interest lies more specifically with the double bounded model rather than the single bounded model. Further, maximum likelihood is adopted as the method of estimation and not the probit.

75_{This is also true if a „yes‟ response is followed by a „no‟ response.} 76_{The first bid amount is determined randomly by the interviewer}

77_{If the response was „no‟ in the first case, bids would be restricted to lower amounts than the initial bid. The} same applies if the response is „yes‟, although this time around, amounts would have to be higher that the initial bid level.

second bids respectively, may be different. However, Arana and Leon (2007) assert that as respondents become accustomed to a second bid offer, they will “settle into a new cognitive regime where the underlying value stabilises”.

Looking at the responses for the double bounded model more closely and naming the first bid amount and the second , we find that the individual will fit into one of four classes:

1. The individual answers „yes‟ to the first bid and then „no‟ to the second bid. In this case we have < and we can further infer that . Willingness to Pay falls between bid 1 and bid 2.

2. The individual answers „yes‟ to both the first and second question. In this case we have . Willingness to Pay is higher than the second bid up to infinity. 3. The individual answers „no‟ to the first question and then yes to the second question.

In this case we have and we can further infer that . Willingness to Pay has a distinct bound between the first high bid and the second low bid.

4. The individual answers no to both the first and second bid. In this case we have . Willingness to Pay lies between the second lowest bid and zero.

Cases 1 and 3 have well defined intervals for Willingness to Pay, while cases 2 and 4 are similar to the single bounded model except that the values will be closer to the true value of WTP due to truncation (Hanemann et al., 1991). Defining and as the first and second responses to the closed bids78, the probability that an individual will answer „yes‟ on the first bid and „no‟ on the second bid is expressed as ( | ) ( ). Where ( ) is basically simplified as stating that the probability is conditional on the values of the explanatory variables. Together with this in mind, and the assumption that ( ) and ( ), Lopez-Feldman (2012) states that the probability of each of these cases is given by:79

1. For cases with „yes‟ as the first response and „no‟ as the second ( and = 0):

78

Dichotomous variables

( ) ( )

( ) ( ) ( ) (3)

2. For cases with „yes‟ as the first and second response ( and = 1):

( ) ( )

( ) ( ) (4)

3. For cases with a „no‟ as the first response and „yes‟ as the second ( and = 1):

( ) ( )

( ) ( ) ( ) (5)

4. For cases with „no‟ as the first and second response ( and = 0):

( ) ( )

( ) ( ) (6)

Equations (3) to (6) cannot be estimated using the probit model and as a result a likelihood function needs to be constructed in order to obtain estimates for and using maximum likelihood estimation (Aravena-Novielli et al., 2010; Lopez-Feldman, 2012). To find the parameters of the model, the function that needs to be maximised is given:

∑ * ( ( ) ( )) ( ( )) ( ( ) ( )) ( ( ))+ (8)

where , are indicator variables which either takes the value of 0 or 1, depending on the responses of the individual. Therefore, out of four possible cases, only one of the indicator variables will be assigned a 1 with the rest being assigned 0. In this approach,

̂ ̂ are directly estimated and from this information WTP can be estimated (Lopez- Feldman, 2012).