Electronic Supplementary Material #1
Statistical methods to calculate CIs for WTP estimates
The Delta Method
The Delta method uses the outcomes of the estimation procedure in order to compute standard errors
for any function of the outputted estimates, including the marginal WTP values. For example, Hole
[1] applied the Delta method to compute the standard errors for marginal WTP values for models
assuming a linear in the attribute indirect utility specification. Ai and Norton [2] and Greene [3] used
the method to compute the standard errors for estimated interaction effects. In this document, we
demonstrate how the method may be applied to calculate the standard errors for marginal WTP
estimates involving complex non-linear in the attributes indirect utility specifications.
Suppose that ˆ denotes the vector of estimated parameters and  the corresponding
variance-covariance matrix (note that the standard errors are the square roots of the diagonal of this
matrix). Let the marginal WTP of attribute k for general indirect utility functions V be given by
function f,
dV
dxk
WTPk  
 f k (  | x).
dV
dxc
(1)
Then the delta method states that the asymptotic probability distribution of WTP is a normal
distribution with mean f k ( ˆ | x) and variance T , where  is the vector of partial derivatives of
this function f, evaluated at ˆ , and where superscript T denotes the vector transpose. Mathematically,
this can be written as
 df k ( ˆ | x) 


 d 1 
.
WTPk  N f k ( ˆ | x), T  ,     f k ( ˆ | x)  


 df k ( ˆ | x) 


 d K 


(2)
The (asymptotic approximation of the) confidence interval of the WTP can then be directly computed
using the square root of the variance. It is important to note that both the mean and the variance will in
general contain values for the attribute levels, x. Since these levels are not fixed across choice
situations (or even across respondents), we assume that these functions are evaluated in the average
value over the entire data set, denoted by x .
As an example, take the indirect utility function given in Equation (3)
V
 1 xk  2 xk xc  3 xc2 
(3)
1
The mean WTP (see Equation (4))




d
1 xk   2 xk xc  3 xc2
xc
dxk
1   2 xc
WTPk  


.
d
xk
 2 xk  2 3 xc
1 xk   2 xk xc  3 xc2
dxc
(4)
is given by ( ˆ1  ˆ2 xc ) /( ˆ2 xk  2ˆ3 xc ), while the variance can be computed as
 dV

 d 1
 dV
var(WTPk )  
 d 2
 dV

 d 3
T

 dV



 d 1

 dV



 d 2

 dV


   ˆ
 d 3








   ˆ
T




1


 ˆ

ˆ
1
 ( 1   2 xc ) xk  x 

c

( ˆ2 xk  2ˆ3 xc ) 2  ˆ2 xk  2ˆ3 xc


 2 xc ( ˆ1  ˆ2 xc ) 


ˆ
ˆ
  2 xk  23 xc 
(5)




1


ˆ
ˆ
ˆ
ˆ
 var( ˆ1 )

cov( 1 ,  2 ) cov( 1 , 3 ) 


 ( ˆ  ˆ x ) x
2 c
k
 cov( ˆ2 , ˆ1 )
var( ˆ2 )
cov( ˆ2 , ˆ3 )   1
 xc  .


  ˆ2 xk  2ˆ3 xc

 cov( ˆ3 , ˆ1 ) cov( ˆ3 , ˆ2 )
var( ˆ3 )  

  2 x ( ˆ  ˆ x ) 
c
1
2 c


ˆ x  2ˆ x

2 k
3 c


Although the mathematics may seem complicated, it is a very quick way of analytically computing
the confidence intervals using some derivatives and matrix algebra, without having to rely on
simulations. A downside of this method may be that the confidence interval is always symmetric
around the mean, as it relies on an asymptotic normal distribution.
The procedure involves the following steps after model estimation, from which the parameter
estimates ˆ and the variance-covariance matrix  are obtained.
1. Determine the WTP function f k (  | x ) by taking the derivatives towards the levels of
attributes k and c.
2. Calculate the mean of the WTP by   f k ( ˆ | x ).
3. Determine the vector of first order derivatives of the WTP function towards the parameter
estimates,    f k (ˆ | x ).
4. Calculate the variance of the WTP by  2  T .
5. The lower and upper bounds of the 95 percent confidence interval of the WTP is given by
  1.96 and   1.96 , respectively.
In the two following methods, the analyst has to rely on Monte Carlo simulations.
2
Krinsky and Robb Procedure
The Krinsky and Robb procedure [4, 5] represents a method for simulating the asymptotic properties
of the maximum likelihood estimated parameters. Originally applied to calculate variances
surrounding elasticities, the method is not specific to calculating confidence intervals around WTP
measures, but rather is a general method that can be applied to any transformation of estimated
parameter estimates. The procedure involves a Monte Carlo simulation taking simulated draws from
the multivariate parameter distribution, and as such accounts not just of the standard errors of the
estimated parameters, but also for the parameter covariances. Once the Monte Carlo simulation of the
multivariate normal parameter distribution has been performed, the simulated draws can be used to
construct confidence intervals around marginal WTP values.
The procedure involves a number of steps following model estimation. After obtaining the
parameter estimates ˆ , and the models estimated covariance matrix,   var( ˆ ), the following steps
are applied.
1. Calculate the Cholesky decomposition, C, of the covariance matrix, such that
CC '  var    .
2. Draw R (where R ≥ 5000) k dimensional vectors of independent standard normal random
variables, zk.
3. For each k dimensional vector, generate a new vector from a multivariate distribution by
calculating  m    C ' zk .
4. For each new parameter vector, calculate the marginal WTP value based on the relevant
attribute transformations and their respective derivatives.
5. Calculate the statistics of interest, including mean, median and percentiles. The percentile
values are used as the confidence level bounds. For example, the 0.025 and 0.975
percentiles correspond to the 95 percent confidence interval.
Although the method assumes that the parameter estimates are multivariate normally distributed, the
use of percentiles to construct the confidence level bounds does not predispose that the resulting
calculated estimates, in this case marginal WTP values, are symmetrical. For more detailed
information about the procedure, see Haab and McConnell [6].
Bootstrapping Procedure
The Bootstrap method is similar to the Krinsky and Robb procedure in that it is a Monte Carlo method
involving simulation. Unlike the Krinsky and Robb procedure however, bootstrapping involves
randomly drawing respondents from the sample population and estimating the parameter estimates of
interest. Thus, the method differs from the Krinsky and Robb procedure in that a) it involves the use
data from actual respondents as opposed to simulated draws taken from parameters estimated on the
full sample population, and b) it makes no assumptions about the underlying distribution of the
parameter estimates. The steps of the procedure are outlined below.
1. Sample with replacement, n respondents R times from the sample population.
2. For each of the R samples, estimate the model and store the parameter estimates,  .
3. Calculate the marginal WTP value based on the relevant attribute transformations and their
respective derivatives for each of the R samples of n respondents.
4. Compute the statistics of interest, including mean, median and percentiles. As with the
Krinsky and Robb procedure, the percentile values are used as the confidence level bounds.
For more background information about the method, see Hole [1].
3
References
1.
Hole AR. A comparison of approaches to estimating confidence intervals for willingness to
pay measures. Health Econ. 2007 Aug;16(8):827-40.
2.
Ai C, Norton E. Interaction terms in logit and probit models. Economics Letters.
2003;80(1):123-9.
3.
Greene WH. Testing hypotheses about interaction terms in nonlinear models. Economics
Letters. 2010;107(2):291-6.
4.
Krinsky I, Robb AL. On Approximating the Statistical Properties of Elasticities. The Review
of Economics and Statistics. 1986;68(4):715-9.
5.
Krinsky I, Robb AL. On approximating the statistical properties of elasticities: a correction.
Review of Economics and Statistics. 1990;72(1):189-90.
6.
Haab TC, McConnell KE. Valuing environmental and natural resources: The econometrics of
non-market valuation: Edward Elger Publishing; 2003.
4