After specifying the likelihood function and the priors, the next step in the Bayesian analysis is then to combine these two components to generate posterior distributions for all model parameters. With the use of independent Normal-Gamma prior in our model, the posteriors are in the form of full conditional distribution. The full conditional posterior distribution of β is to follow a multivariate normal distribution:
) , (
~ , ,
| y h MVN β V
β Ω where V =(V−1+hX'Ω−1X)−1 ) ' (V 1 hX 1y
V − + Ω−
= β
β
The full conditional posterior distribution of h is to follow a Gamma distribution:
) , (
~ , ,
|y G s 2 v
h β Ω − where v= N+v
v
s v x y x
s y
1 2
2 =( − β)'Ω− ( − β)+
The full conditional posterior distribution of each λi is to follow a Gamma distribution:
) 1 1 ,
(
~ , , ,
| 2 +
+ +
λ λ
λ ελ
β
λ v
v h G v v h y
i i
The full conditional posterior distribution of vλis to follow )
exp(
2) ( 2 )
( , , ,
| λ 2 λ λ
λ β λ v λ v ηv
h y
v N
Nv
− Γ
∝ − where ∑
=
− +
+
= N
i
i
v 1 i
1) ]
2 [ln(
1
1 λ λ
η
λ
ii). Model Space Selection
Table 2.1 captures the model space that exhaustively list all competing models considered in this essay. The selection of this set of models is guided by the following idea. The estimation results in essay 1 showed that a number of variables in the meta- equation are not statistically significant at any level. This fact implicitly suggests that those variables might be irrelevant to the wetland values. Thus, the inclusion of them might bring undesired effect into our analysis and thus will undermine the model
performance in some way. Facing this challenge, dropping all insignificant variables in our meta-equation might be a possible solution to this situation in terms of econometric view.
Model
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Intercept Income
Acres X X X X X X
Share X X
Freshwatermarsh Saltwatermarsh Prairiepothole
Watersupply X X X
Quality
Flood X X X X X X X X X
RecFish
ComFish X X X X
Bird X X X X X X X X
Amenity X X X X X
Habitat X X X X X X X
Publish X X X X X X X X X X X X
EA
PFMPNFI X
CVM X X X X X X X X X X
HP X X X X X X X X X X X X X
TCM X X X X X X X X X X X
R1 R2 R3
X: a symbol denotes the variable(s) dropped from the corresponding model Table 2.1: The specification of 14 selected models in the model space
In fact, the extent to which insignificant variables should be removed from the meta-equation is an arbitrary choice to researchers. In order to manage this uncertainty, we apply backward elimination model selection technique12 to one by one remove insignificant variables from the model that has a complete set of variables. Next, we retain every model from each of the subsequent 13 backward steps as a candidate model in our model space, which results in a total of 14 competing models. Among these 14 models, the one from step 0 is labeled as the FULL mode or MODEL 0 and the one from step 13 is labeled as the REDUCED model or MODEL 13.
In this study, these 14 models play the role as follows. First, these competing models allow us to examine our study hypothesis under different model specifications.
Second, through the comparison across these competing models, we might be able to provide suggestions on whether or not to remove insignificant variables from the BT meta-equation. Third, we can provide a thorough BT prediction, which is free from the trade-off among competing models. The detail specifications for these 14 models are summarized in Table 2.1 and the variable descriptions can be found in essay 1.
iii). Prior Refinement
In this study, we use the estimated coefficients from Brander et al.’s meta- equation, which was estimated based on a 215-observation global wetland meta-dataset, as the added information to refine the prior means and standard errors on the coefficients of independent variables in our wetland meta-equation. Since Brander et al.’s dataset has a number of differences from ours in several aspects, such as the unit of measurement used for wetland size, the wetland value reported in different years, the default category used for dummy variables, and etc., a series of adjustments need to be made before any of their coefficients can be used as our model priors.
In the practical application, we start with the estimated meta-equation in Brander et al. and make a series of the following adjustments in a monotonic manner. First, the wetland values and GDP per capita in their meta-equation are converted from 1999 dollar
12 Backward elimination, a variable selection technique, begins by running regression with all independent variables included in the model. The procedure deletes the variables whose coefficient has the largest p- value from that model. The resulting equation is examined for the variable now contributing the least, which is then deleted, and so on. The procedure stops when all coefficients remaining in the model are statistically significant at a level specified by the user.
value into 2003 dollar value. Second, GDP per capita in their meta-equation is converted into household income as we have. Third, the measurement unit HECTARE used in their meta-equation is converted into ACRE as ours. Forth, the variables REPLACEMENT COST and WOODLAND are set to be the default dummy variables in their meta- equation in order to have the consistent settings as ours. Fifth, variable BIODIVERSITY is merged into variable HABITAT in their meta-equation due to these two variables are similar in their definition. Besides, variables PRODUCTION FUNCTION, MARKET PRICES, NET FACTOR INCOME, and OPPORTUNITY COST are combined into a single variable PFMPNFI as we have.
To avoid the dramatic difference from Brander et al.’s estimated coefficients bringing unreasonable effects into our model priors due to the inherent differences between two studies, the last step of this series of adjustments is to rescale the estimated coefficients in Brander et al.’s meta-equation by setting the elasticities of the selected variables equal to those of the corresponding variables in our study at the variable means.
Since the 14 models in our model space contain different set of variables, the refinement of priors for each model is somewhat different in its calculation. As a result, we show the work for MODEL 0 only and its detailed calculation can be found in appendix A. Prior means and standard errors for the other 13 models can be found in appendix B.