The final and central question that we seek to answer is that whether incorporating our model with the added information borrowed from Brander et al. can improve the efficiency of BT predictions generated from our meta-equation. In Table 2.3, we set up four hypothetical policy scenarios, which focus on different wetland types with different degree of wetland services, to help examine this concern. Besides, we will also provide a thorough BT prediction, which is free from the trade-off among models via BMA, under these scenarios.
Scenario 1 Scenario 2 Scenario 3 Scenario 4
Intercept 1 1 1 1
Income 43 43 43 43
Acres 10000 10000 10000 10000
Share 0.125 0.125 0.125 0.125
Freshwatermarsh 1 0 1 0
Saltwatermarsh 0 1 0 1
Prairiepothole 0 0 0 0
Watersupply 0.5 0.5 0.1 0.1
Quality 0.5 0.5 0.1 0.1
Flood 0.5 0.5 0.1 0.1
RecFish 0.319 0.319 0.319 0.319
ComFish 0.278 0.278 0.278 0.278
bird 0.38 0.38 0.38 0.38
Amenity 0.5 0.5 0.1 0.1
Habitat 0.5 0.5 0.1 0.1
Publish 0.694 0.694 0.694 0.694
EA 0 0 0 0
PFMPNFI 0 0 0 0
CVM 1 1 1 1
HP 0 0 0 0
TCM 0 0 0 0
R1 0 0 0 0
R2 0 0 0 0
R3 0 0 0 0
Table 2.3: Four Hypothetical policy scenarios for BT exercises
The results in Table 2.2 show that models in group 2 dominate all other group models. As a result, the following BT exercises are carried out by using the estimated coefficients from models in group 2 only. For comparison purposes, we also report the BT predictions generated from models in group 1 to serve as the baseline. In order to avoid the problem that the mean BT wetland estimate tends to be easily affected by extreme values in the dataset, we calculate median value instead of mean value in the following BT exercises.
i) Examination the efficiency of BT estimates
Within a Bayesian framework, a BT prediction based on any specific model will require researchers to construct a model-specific Bayesian predictive distribution (BPD), whose form is p(yf |xf,Mm), where xf is the hypothetical policy value. Since such BPD is not conditional on any model parameters, the construction of BPD can be done through the following steps to margin out model parameters. For a given model Mm, the Gibbs Sampler retains the last r=10,000 draws in posterior simulation. For each of the retained r draws, draw yf from p(yf |xf,θrth,Mm). The resulting 10,000 draws of yf can be viewed as drawing from p(yf |xf,Mm), which nets out all model parameter in model Mm. Table 2.4 presents the median BT wetland values as well as lower and upper bounds of 90% confidence interval for the BPD of each model in groups 1 and 2 under four hypothetical policy scenarios.
As can be seen in Table 2.4, the 90% confidence intervals for all models in group 2 are substantially lower than their respective counterparts in group 1 by up to 54%. This suggests that the added information borrowed from Brander et al. can effectively improve the efficiency of BT predictions through limiting the range of values our model parameters can take and thus lead to more meaningful and reliable policy implication.
ii) Bayesian Model Averaging
One of the appealing features in a Bayesian framework is that we can derive the model probabilities for the models under consideration. With these probabilities, a thorough BT estimate can be generated through BMA by weighting individual BT
estimates over a number of models associated with their probabilities. In this way, researchers are able to avoid from the trade-off among models if they do have hard time making the selection. Here, we would like to take advantage of this feature to provide a thorough BT prediction for policy suggestion purposes.
When carrying out BMA for BT, we need to derive posterior model probability (weight) for each candidate model involved in BMA exercise so that model-specific BPDs can be combined associated with their weights to produce the following model- averaged.
( )
∑ ∫
=
m
m m m m m m f
f y p y M p M d p M y
y
p( | ) ( |θ , ) (θ | ) θ ( | )
The posterior model probability, p(Mm|y), in the above equation can be derived by Bayes’ rule as follows.
=∑
=
j j j
m m
m m
m p y M p M
M p M y p y
p M p M y y p
M
p ( | ) ( )
) ( )
| ( )
(
) ( )
| ) (
| (
Since we neither have any prior information nor preference against any specific model, each candidate model involved in BT exercise is assigned equal prior probability.
As such, p(Mm) and p(Mj) can be cancelled out in the above equation. Besides, since the model-conditioned marginal likelihood, p(y|Mm), has already been derived earlier in Table 2.2, the calculation of posterior model probability for any model would be straightforward.
Table 2.5 lists the posterior probability for each proposed model on three bases:
overall base, within-group base and between-group base. After having the posterior model probability for each model involved in BMA exercise, the model-averaged BPD can be constructed through the following two steps. The first step, which has been outlined above, is again to net out model parameters for each involved model-specific BPD. The second step is to margin over all involved models, which can be done by weight-averaging model-specific draws of yf from model-specific BPDs across all involved models.
51
Group 1 Group 2 Group 3 Group 4
Base 1 Base 2 Base 3 Base 1 Base 2 Base 3 Base 1 Base 2 Base 3 Base 1 Base 2 Base 3
Model 0 0 0 0 0 0 1 0 0 0 0 0 0
Model 1 0 0 0 0 0 1 0 0 0 0 0 0
Model 2 0 0 0 0 0 1 0 0 0 0 0 0
Model 3 0 0 0 0 0 1 0 0 0 0 0 0
Model 4 0 0 0 0 0 1 0 0 0 0 0 0
Model 5 0 0 0 0 0 1 0 0 0 0 0 0
Model 6 0 0 0 0 0 1 0 0 0 0 0 0
Model 7 0 0 0 0 0 1 0 0 0 0 0 0
Model 8 0 0 0 0 0 1 0 0 0 0 0 0
Model 9 0 0 0 0 0 1 0 0 0 0 0 0
Model 10 0 0 0 0 0 1 0 0 0 0 0 0
Model 11 0 0 0 0 0 1 0 0 0 0 0.333 0
Model 12 0 0 0 0.455 0.455 1 0 0 0 0 0 0
Model 13 0 1 0 0.545 0.545 1 0 1 0 0 0.667 0
Base 1: The posterior model probabilities were calculated on the basis of overall 56 models in the model space (overall comparison) Base 2: The posterior model probabilities were calculated on the basis of all 14 models within each group (vertical comparison) Base 3: The posterior model probabilities were calculated on the basis of specific model across all groups (horizontal comparison)
Table 2.5: Posterior Model Probability for all considered models in the model space
Table 2.6 presents the median BMA BT estimates derived from models in group 2 under four hypothetical policy scenarios. Since models 12 and 13 in group 2 account for nearly 100% of the posterior probabilities, the estimates in Table 2.6 could be viewed as the weighted average of the BT estimates generated from these two models.
Median 90% Low 90% High
S1 319.706 60.443 1598.502
S2 77.750 11.879 516.734
S3 134.994 22.758 755.372
S4 33.217 4.668 234.846
Table 2.6: BMA BT prediction based on models in group 2