ESSAYS ON IMPROVING THE ECONOMETRIC ESTIMATION OF WETLANDS VALUES VIA META ANALYSIS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduat[.]
Introduction
Wetlands, an environment at the interface between terrestrial ecosystems and aquatic systems, are found under a wide range of hydrological conditions and only a limited set of plant species can survive in such areas (Brinson, 1995) As a result, wetlands have historically been considered wastelands and only could be improved by drainage (Mitsch & Gosselink, 1986) In the last few decades, this situation has changed since the public became aware of the contributions of wetlands through the provision of functions such as habitats for species, protection against floods, water purification, amenities and recreational opportunities Drainage is now recognized as destructive of these dimensions of wetlands
Because wetlands services often have no market price, it is difficult for policy makers to balance the benefits of wetlands restoration and/or conservation against the cost and to determine an efficient allocation of scarce public resources to wetlands projects, especially in difficult financial times (Lambert, 2003) Partly in response to this situation, there is now substantial empirical literature on wetland valuation Given the costs of original site-specific benefit studies, public agencies have expressed strong interest in generalizing the findings already in the literature, in the hope of cheaper and quicker first-cut benefit approximations for policy-relevant sites (Boyle and Bergstrom,
1992) This idea has stimulated the use of benefit transfer (BT) technique, in particular meta-analysis.
Meta-analysis is a statistical procedure designed to synthesize experimental results across independent studies that address related research questions In the case of wetland valuation, meta-analysis uses the summary statistics reported in several primary studies conducted at various sites as the data points for analysis This implicitly allows each study to carry some weight in obtaining the parameter estimates, rather than put the entire weight in any single study (Moeltner, 2007) In such a framework, the meta- regression equation can thoroughly consider the effects of user characteristics, site attributes and methodological differences across the collected primary studies As a result, one can gain a more accurate representation of the population relationship than is provided by the individual study estimators
Most of the existing wetland meta-analytical studies focus on deriving the marginal effects of explanatory variables on wetland value and looking for systematic trends for future inference Several studies within this framework have been conducted by Brouwer et al (1999), Woodward and Wui (2001), Borisova-Kidder (2006), and Brander et al (2006) Although these studies do provide some useful information on what factors determine a wetland’s value as well as their marginal effects, there still remain some debates associated with these existing studies
The first potential issue is that a pooling effect might lead to unreliable results Since wetlands provide various types of values such as direct use, indirect use, future use and existence values, the elicitation of wetland values from respondents will require specific methodologies based on which of the above values are to be measured In empirical studies, wetland valuations are usually performed by the following techniques: travel cost (TC), replacement cost (RC), net income factor (NFI), contingent valuation (CV), hedonic price (HP), energy analysis (EA), production function (PF) and market price (MP) However, these techniques differ considerably in terms of the welfare measures they estimate (Freeman, 1993; Carson et al., 1996; Brander et al., 2006) For example, the welfare measure of is estimated by compensating variation or by equivalent variation HP and TC capture the welfare change through differences in Marshallian consumer surplus NFI measures the change through differences in producer surplus PF uses the differences in both consumer and producer surplus However, there is no sound basis in welfare theory for MP and RC
The way that existing wetland meta-analysis studies deal with such diverse valuation methods is by pooling all the observations with corresponding dummy variables to denote the respective valuation methods However, since each valuation method is based on a different welfare theory (or none at all), the value estimate derived from a particular primary study could be quite different (conceptually as well as empirically) from those of other studies As a result, although pooling all data points can provide rich information on wetland types, locations and demographics in the derivation of meta-regression function, methodology-specific effects, unless controlled, it might make it difficult to uncover the true relationships between explanatory variables and wetland values
The second potential issue is that a meta-analytical regression function with higher R-squared value does not necessarily lead to the best benefit transfer results R- squared value is the proportion of variation in a data set that can be explained by a statistical model In the existing wetland meta-analysis valuation literature, most studies involve a large number of study-specific variables, such as study characteristics and wetland characteristics, in order to have good explanatory power for the variations in their data set However, not all of the included variables are expected to have significant effects on wetland values This inclusion of insignificant variables, although it can boost R-squared value, might result in over- or under-estimates and decrease the efficiency of
Research Objective and Hypothesis
The existing wetland meta-analyses aim to provide useful information about the determinants of wetland values and also some suggestions for future benefit transfer However, pooling all data points, which may be based on dramatically different welfare theories, in benefit transfer estimation might fail to uncover the real relationship between explanatory variables and wetland value In addition, since benefit transfer is very sensitive to any right hand side variables in the value function, the inclusion of too many insignificant variables might lead to unsatisfactory benefit transfer estimates Based on these concerns, the objectives of this study are as follows
First, we base our wetland meta-analysis on four proposed models, which are designed to exhibit different levels of pooling effects, to assess the determinants of wetland values as well as to provide the comparisons between our proposed models Besides, we further modify our proposed models by dropping insignificant explanatory variables in the above models to avoid BT exercises being too sensitive to those variables The detail specification of each model will be introduced in a later section
Second, we perform two hypothesis tests to exam two aforementioned concerned issues: pooling observations and eliminating variables Since it is believed that pooling more observations can enrich the information contained in the wetland data set, the first test will be performed based on the hypothesis that the model exhibiting the larger pooling effect has more efficient and consistent parameter estimates In addition, some may claim that insignificant variables might still have effects on wetland values and thus the second test will be performed for four proposed models based on the hypothesis that the full model (with a full set of variables) is better than its corresponding reduced model (with a subset of variables)
This essay is organized as follows In the next section, we give an overview of the empirical wetland valuation literature and show the resulting descriptive statistics based on wetland type, wetland function, valuation methodology and geographical location In the subsequent section, we describe the setup for a meta-regression and full details of model description for the proposed four models This section also gives the regression output and interpretation of the results In addition, we also perform two hypothesis tests and the implications of the results are also provided here Moreover, in this section we explore the performance of each hypothesized model in this study in a value transfer exercise.
Data and Descriptive Statistics
The data chosen to illustrate the concerns addressed in this study are extracted from Borisova-Kidder (2006) Details of the selection criteria for data points can be found in her paper This dataset has been selected because previous efforts only focus on deriving marginal effects of wetland functions and attributes on economic benefits However, we believe there might be some empirical relationships that can be further investigated and thus provide some prospects for future inference Overall, the dataset contains 72 observations on wetland values, stemming from 33 studies, which consist of
22 journal articles, 7 research reports or academic papers, 2 chapters in books, 1 PhD dissertation and 1 Master’s thesis Those studies are listed in Table 1.1 with short descriptions
Although the sample is not as large as would be ideal, it still has several desirable properties As shown in Table 1.1, the selected studies have good coverage of the geographic area in the U.S where the main wetlands are located Among the collected studies, 1 study was conducted in the 1960s, 5 studies in the 1970s, 15 studies in the 1980s and 12 studies in the 1990s As can be seen, most of them were conducted within the last 20 years, since 1988, and thus apply modern survey and estimation methodologies In addition, the data quality is acceptable since 24 out of 33 studies were published in refereed journals While checking Borisova-Kidder’s dataset against original sources, we corrected 2 errors in data entry prior to any following analyses 1,2
In order to have a general perspective on frequency distribution as well as the simple statistics of values regarding our dataset, these 72 observations are summarized in Table 1.2 according to the following four categories: wetland type, wetland service, valuation methodology, and geographical locations In order to synthesize wetland values from all different services and different survey years, the wetland values have been standardized into 2003 dollars per acre by using the consumer price index (CPI) of each corresponding year
1 The first correction is the entry of wetland value per acre in the study “Economic Value of Wetlands Systems” by Farber and Costanza (1987) The wetland value per acre entered by Kidder in the above study is $678 However, the abstract of original study indicates estimates of the value should range from $6,400 to $10,000 per acre by using EA As a result, the entry for wetland value per acre of the above study should be corrected from $678 to the half way point between $6,400 and $10,000, i.e $8,200
2 The second correction is the entry of “wetland type” in study "The influence of wetland type and wetland proximity on residential property values" by Doss and Taff (1996) The entry of the wetland type in Kidder’s data set was classified as “saltwater marsh” However, Doss and Taff’s study was conducted in Minnesota where the only wetland type in that region is swamp Hence, the classification for wetland type of the above study should be “swamp”
Categories Frequency Minimum Median Mean Maximum
Table 1.2: Statistics Summary for Wetland Value Based on Various Categories
As shown in Table 1.2, four major wetland types in the U.S have been considered in this dataset and the valuation literature was largely focused on freshwater wetlands, followed by saltwater wetlands Of the 72 observations, 39 are for freshwater wetlands,
19 for saltwater wetlands, 7 for swamps and 7 for prairie potholes Although there is only a limited set of wetland functions included in this dataset, the most common ones emphasized in literature are considered in this study In addition, a broad range of valuation methods, such as energy analysis, production function, market prices, net factor income, replacement cost, travel cost, hedonic pricing and contingent valuation method, were applied in the collected studies Among these valuation methods, the one most commonly used in the collected studies is contingent valuation method, followed by replacement cost method and net income factor method Furthermore, eight different geographical locations 3 of wetlands are considered in this study, and the Northern Crescent, Fruitful Rim and Mississippi Portal areas account for almost half of the observations in this dataset
In terms of the wetland values summarized in these four categories, it is noted that the mean, median and maximum values of wetlands vary considerably by wetland type, wetland service, valuation method and geographical location of wetland The average wetland value in this dataset is $15,841 per acre The median value, however, is only
$366 per acre This indicates that the distribution of wetland values is highly skewed with a long tail of high values In fact, this might be a common issue faced by researchers who apply meta-analysis to non-market value data However, no study was found to analyze the possible problems that might be caused by such type of data
Since the distribution of wetland values in our meta-dataset is highly skewed to the right, it indicates that the dataset likely contains more noise than usual Besides applying a log-normal distribution to correct the skewed distribution, it remains difficult for researchers to give a thorough consideration to the sources of variation and its associated problems However, if those concerns can be partly or fully controlled, more reliable results and useful guidance can be expected As a result, the empirical work carried out in the following section can help us understand the aforementioned concerns, and provide some results and implications that have not yet been found in previous literature.
Empirical Implementation
In the previous section, simple descriptive statistics provide a general perspective on how wetland values vary across wetland types, wetland services, valuation
3 These eight geographical locations of wetlands share the same definition with the ERS Farm Resource Regions defined by USDA methodologies, and geographical locations In this section, we will follow the existing wetland meta-analysis studies to estimate model parameters within the classical framework to assess the relative importance of all potential factors in our proposed models Since most of the existing studies focus on chasing high R squared value (or Adjusted R squared value) to get better explanatory power for their value functions, we will depart from this view by placing less weight on high R squared value and more on other measures of goodness-of-fit for BT
For the model specification, we use log-linear model, which is commonly used in non-market value data There are several reasons for using the natural log form for the dependent variable instead of a linear one First, it was used by most of the existing wetland meta-analyses Following this setting, it would be convenient to compare the results across studies Second, it can considerably improve the model’s fit as well as mitigate heteroskedasticity if any exists (Brander, 2006) A natural log transformation often reduces heteroskedasticity since it compresses the scale in which the variables are measured Third, this functional form has the ability to capture the curvature in the valuation function because it allows explanatory variables to influence the dependent variable in a multiplicative rather than an additive manner (Borisova-Kidder, 2006) Finally, the issue of the highly skewed wetland values can be substantially improved by taking natural log
The dependent variable in the meta-regression function is the natural log of the wetland value per acre The explanatory variables in this study can be grouped in three major different matrices: socio-economic characteristics inX s , study characteristics in
X c , and wetland physical and geographical characteristics inX g The model can be expressed in the matrix from as follows ε β β β + + +
Ln( ) where c is the intercept term, ε is a vector of residuals and the β vectors contain the estimated coefficients on the respective explanatory variables In order to minimize the chance of estimating study-specific effects, we followed a rule that each of the explanatory variables should have a minimum of 3 observations
The only one variable in socio-economic characteristics is the annual U.S household income (INCOME), which has been converted into 2003 U.S dollars Variables indicating the method of valuation used in each study are included in study characteristics Although there are a total of 8 valuation methods found in our dataset, the production function, market price and net factor income methods have been combined into a single variable (PFMPNFI) for regression estimation 4 Finally, energy analysis (EA), contingent valuation (CV), hedonic price (HP), travel cost (TC), replacement cost (RC) and the above combined method (PFMPNFI) mentioned above will be used in this study
Wetland physical and geographical characteristics include any variable related to the attributes of wetlands, such as wetland size, type, function, and geographical location The wetland size variable (ACRE) characterizes the amount of wetland acres valued in the study The wetland share variable (SHARE) is continuous with a range between 0 and
1 The purpose of this variable is to serve as an index of the scarcity of wetland As for wetland types, there are 4 mutually exclusive variables (FRESHWATERMARSH, SALTWATERMARSH, SWAMP and PRAIRIEPOTHOLES) indicating wetland types that will be used in the regression estimation
In terms of wetland services, 9 wetland functions (WATERSUPPLY, QUALITY, FLOOD, RECFISH, COMFISH, BIRDHUNT, BIRDWATCH, AMENITY, and HABITAT, which are not mutually exclusive) are considered in our meta-regression Because the variance-covariance analysis indicates that BIRDHUNT and BIRDWATCH are highly correlated to each other, these two services are combined into a single variable BIRD in the regression estimation 5 What is worth paying attention to is that most of the services are provided to some extent by most of the wetlands; however, fewer are mentioned in the source studies Consequently, the services mentioned above can be thought of as the ones that were highlighted by the source studies
4 In our dataset, PF has four observations and MP has only one observation In order to avoid the estimation for a variable with only one or few observations, PF, MP and NFI, which rely on similar basis in welfare theory, are combined into single variable
5 A tempt had been tried to include BIRDHUNT only, BIRDWATCH only and BIRD in meta-regression function However, there are not much difference between the above three alternatives As a consequence, variable BIRD was selected to be included in the underlying analysis
Finally, the regional variables, defined by the nine ERS Farm Resource Regions, characterize the geographical location of wetlands Some regions have few observations, such as Prairie Gateway, which has only one observation To avoid the problem that might arise from creating a variable with fewer than three observations, regions with similar geographical characteristics have been combined, which results in the following 4 newly defined variables Region 1 (R1) includes Northern Crescent and Northern Great Plains Region 2 (R2) includes Fruitful Rim and Southern Seaboard Region 3 (R3) includes Heartland and Mississippi Portal Region 4 (R4) includes Prairie Gateway and Eastern Uplands The detailed definitions and descriptive statistics for dependent and explanatory variables included in the model estimation are listed in Table 1.3
Since our dataset consists of observations stemming from 8 valuation methods including EA, HP, MP, NFI, RC, CV, TC and PF, four models are proposed according to the following strategies to address the aforementioned two concerns as well as to perform their associated hypothesis tests
Model I 72 observations stemming from 8 valuation methods are included This model contains all observations in the dataset and can serve as the baseline for comparison with other models
70 observations stemming from 7 valuation methods are included In Table 1.2, it is noted that the value estimates generated from EA method dominate those from other methods As a result, the 2 EA observations were dropped from this model
63 observations stemming from 5 valuation methods are included 7 observations generated by TC and HP are subsequently dropped from this model TC and HP are theoretically different from the remaining 5 valuation methods since they use revealed preference techniques that is designed to look at actual human behavior and deduce the real world cost of people’s willingness to place on real properties
28 observations stemming from 1 valuation method (CV) are included There are many reasons to separate CV For example, it is a stated preference method, so CV is based on what people say they would do, as opposed to what people are observed to do In addition, it is the only method that can be used to measure all types of wetland values, such as use value, nonuse value, existence value, option value, and bequest value
As a result, it is worth having a further look at the stand-alone effect for
Conclusion
This study attempts to assess if any systematic trends can be distilled from the selected U.S wetland valuation studies as well as to identify what factors determine the value of U.S wetlands Moreover, this study is also aimed to provide alternative thinking to researchers who might be interested in applying meta-analysis to wetland valuation by comparing four models proposed with different level of pooling effects to examine if observations produced by different valuation methods, which might not be directly comparable, can be pooled in a wetland meta-analysis study
The major findings of our analysis can be summarized as follows First, income was found to be an important determinant of wetland values An increase in income will lead to higher wetland values This finding is consistent with those in Borisova-Kidder
(2006) and Brander et al (2006) Second, among several wetland types saltwater marsh and prairie pothole, compared to freshwater marsh, were shown to have a negative relationship with wetland value In particular, prairie pothole has the lowest value among all types of wetland Third, most of the wetland services except water quality and amenity were shown to have no significant influence on wetland value Fourth, among valuation methods only energy analysis was shown to have a significant effect on wetland value Other geographical factors, e.g distance from population center, may have impact Last, we cannot find any evidence for wetland value being significantly affected by its regional location
In addition to the above findings, the two tests performed in this study also provide some guidance for future similar studies First, test results obtained by the bootstrapping method suggest that value estimates produced by different valuation methods, although they might not be directly comparable, can be pooled in the estimation of meta-equation The test result based on asymptotic theory contrasts with the above finding; however, it might be inappropriate especially when the sample size is small Besides, the test result also suggests that models with most of the insignificant variables removed have better goodness of fit and forecast performance We believe that the above test results can benefit future studies in the similar context
Finally, we would also like to stress some possible improvements that can be made by this study to obtain more robust results First, our dataset might be smaller than ideal for estimating efficient estimates for site attributes, valuation methods and wetland services If more primary studies can be included, a more robust result can be expected Second, the choice of model specification for meta-analysis might be somewhat arbitrary, which points out another issue known as model uncertainty If such model uncertainty can be reduced, the efficiency of BT estimates could be further advanced
Number of obs Published Journal
Southern Journal of Agricultural Economics
Ecology and management of tidal marshes
5 Bergstrom et al 1990 1986 LA 1 Y Ecological
8 Cangelosi et al 2001 1996 MI,OH 2 N -
Economic and Management of Water and Drainage in Agriculture
Loomis 1992 1989 CA 3 Y Water resources research
Journal of Environmental Economics and Management
Journal of Agricultural and Resource Economics
Journal of Environmental Economics and Management
Table 1.1: Source Studies Used in Meta-Regression (Continued)
Table 1.1: Source Studies Used in Meta-Regression (Continued)
American Journal of Agricultural Economics
21 Johnston et al 2002 1995 NY 2 Y Coastal management
Tobin 1989 1987 IL,IA 2 Y Professional geographer
23 Loomis et al 1991 1990 CA,WI 2 Y
Economic and Management of Water and Drainage in Agriculture
Journal of Environmental Economics and Management
Journal of Agricultural and Resource Economics
Batie 1987 1984 LA,MA 1 Y Marine Resource
33 Whitehead and Blomquist 1991 1989 KY 1 Y Water Resources
LNVALUE Logarithm of value per acre of wetland, U.S year 2003 dollars 72 5.608
ACRES Amount of wetland acres used in the study valuation 72 356640.190
Share of wetland acres in the area by FIPS codes as reported by the NRI 1997 data
FRESHWATER MARSH Coded as 1 if a wetland is a freshwater marsh, 0 if not 39 0.542
SALTWATER MARSH Coded as 1 if a wetland is a saltwater marsh, 0 if not 19 0.264
SWAMP Coded as 1 if a wetland is a swamp,
PRAIRIE POTHOLE Coded as 1 if a wetland is a prairie pothole, 0 if not 7 0.097
Reduced damage due to flooding and/or stabilization of the sediment for erosion reduction, coded as 1 if a wetland function is noted in the study, 0 if not
Reduced costs of water purification, coded as 1 if a wetland function is noted in the study, 0 if not
Increased water quantity, coded as
1 if a wetland function is noted in the study, 0 if not
Improvements in recreational fisheries either on or off site, coded as 1 if a wetland function is noted in the study, 0 if not
Improvement in commercial fisheries either on or off site, coded as 1 if a wetland function is noted in the study, 0 if not
BIRDHUNT Hunting of wildlife, coded as 1 if a wetland function is noted in the study, 0 if not
Table 1.3: Description of wetland meta-analytical variables (Continued)
Table 1.3: Description of wetland meta-analytical variables (Continued)
Recreational observation of wildlife, coded as 1 if a wetland function is noted in the study, 0 if not
Amenity values provided by proximity to the environment, coded as 1 if a wetland function is noted in the study, 0 if not
HABITAT Nonuse appreciation of a species, coded as 1 if a wetland function is noted in the study, 0 if not
CVM 1 if study used Contingent
HP 1 if study used Hedonic Pricing
TCM 1 if study used Travel Cost Method,
RC 1 if study used Replacement Cost
1 if study used Production Function or Market Prices or Net Factor Income Method, 0 if not
EA 1 if study used Energy Analysis
PUBLISH 1 is study is a journal article, 0 if not 50 0.694
1 if study conducted in Northern crescent or Northern great plains, 0 if not
R2 1 if study conducted in Fruitful rim or Southern seaboard, 0 if not 22 0.306
R3 1 if study conducted in Heartland or
1 if study conducted in Prairie gateway=1 or Eastern uplands, 0 if not
Amacher, G S., R.J Brazee, J.W Bulkley, R.A Moll, 1989 “Application of wetland valuation techniques: examples from Great Lakes Coastal wetlands.” School of Natural Resources, University of Michigan, Ann Arbor
Arhonditsis, G B., C A Stow, L J Steinberg, M A Kenney, R C Lathrop, S J Mcbride, K H Reckhow, 2006 “Exploring ecological patterns with structural equation modeling and Bayesian analysis,” Ecological modeling, 192: 385-409
Batie, S.S and J.R Wilson, 1978 "Economic values attributable to Virginia's coastal wetlands as inputs in oyster production." Southern journal of agricultural economics, July, pp 111-118
Bell, F.W., 1989 “Application of wetland valuation theory to Florida fisheries.” Department of Economics, Florida State University, Tallahassee, Florida
Bell, L., 1997 "Marginal value of coastal wetlands to Florida's commercial fisheries." In Coultas and Hsieth eds Ecology and management of tidal marshes, p 231-257
Bergstrom, J C., J R Stoll, J P Titre and V L Wright, 1990 "Economic value of wetlands-based recreation." Ecological economics 2: 129-147
Blomquist, G C and J C Whitehead, 1998 "Resource quality information and validity of WTP in contingent valuation." Resource and Energy Economics 20: 179-196
Borisova-Kidder A., 2006 “Meta-analytical estimates of values of environmental services enhanced by government agricultural conservation programs,” Dissertation, Department of Agricultural, Environmental, and Development Economics, Ohio State University
Boxall, P.C., W.L Adamowicz, J Swait, M Williams, J Louviere, 1996 “A comparison of stated preference methods for environmental valuation,” Ecological Economics, 18: 243-253
Boyle, K.J & J.C Bergstrom, 1992 “Benefit transfer studies: myths, pragmatism, and idealism,” Water Resour Res 28: 657-663
Brander L.M., R.J.G Florax, J.E Vermaat, 2006 “The empirics of wetland valuation: a comprehensive summary and a meta-analysis of the literature,” Environ Resour Econ, 33: 223-250
Breaux, A., S Farber, J Day, 1995 "Using natural coastal wetlands systems for wastewater treatment and economic benefit analysis." Journal of environmental management 44: 285-291
Brinson, M M., F R Hauer, L C Lee, W L Nutter, R D Rheinhardt, R D Smith, and
D Whigham, 1995 "A guidebook for application of hydrogeomorphic assessments to riverine wetlands," Technical Report WRP-DE-11, U.S Army Engineer Waterways Experiment Station, Vicksburg, MS NTIS No AD A308 365
Brouwer, R., I.H Langford, I.J Bateman, T.C Crowards, R.K Turner, 1999 “A meta- analysis of wetland contingent valuation studies,” Regional Environmental Change, 1: 47-57
Cangelosi, A., R Wither, J Taverna, and P Cicero, 2001 "Wetlands restoration in Saginaw Bay," in the Revealing the economic value of protecting the Great Lakes National Oceanic and Atmospheric Administration and Northeast-Midwest Institute
Carson, R.T., N.E Flores, K.M Martin, and J.L Wright, 1996 “Contingent valuation and revealed preference methodologies: comparing the estimates for quasi-public goods,” Land Econ 72(1): 80-99
Carson, R.T., N.E Foores, K M Martin, J.L Wright, 1996 “Contingent valuation and revealed preference methodologies: comparing the estimates for quasi-public goods,” Land Economics, 72:80-99
Chabreck, R., 1991 "Wildlife harvest in wetlands of the United States," in P.E Greeson, J.R Cooper J and J Loomis "Economic value of wildlife resources in the San Joaquin Valley: hunting and viewing values”, in the Economic and management of water and drainage in agriculture, eds Diner & Zilberman, Kluwer Academic Publishers
Chalmers, T.C., 1991 “Problems induced by meta-analyses.” Stat Med, 10:971-80
Creel, Michael & J Loomis, 1992 “Recreation value of Water to Wetlands in the San Joaquin Valley: Linked Multinominal Logit and Countdata Trip Frequency Models” Water resources research Vol 28 No10, p2597~2606
Danielson, L E and J A Leitch, 1986 “Private vs public economics of prairie wetland allocation.” Journal of Environmental Economics and Management vol 13: 81-92
Dillman, B., J Lawrence, and D Hook, 1993 "Non-market valuation of freshwater wetlands: the Francis Biedler Forest Report 135, South Carolina water resources research institute, Clemson Univ
Doss, C.R and S J Taff, 1996 "The influence of wetland type and wetland proximity on residential property values." Journal of agricultural and resource economics 21 (1): 120-129
Duncan M J., J C Spence, W K Mummery, 2005 “Perceived environment and physical activity: a meta-analysis of selected environ-mental characteristics,” Int J Behav Nutr Phys Act, 2:
Farber, S 1987 “The Value of Coastal Wetlands for Protection of Property against Hurricane Wind Damage.” Journal of Environmental Economics and Management 14:143-151
Farber, S and R Constanza, 1987 "The economic value of wetlands systems." Journal of Environmental Management 24: 41-51
Farber, S., 1988 "The value of coastal wetlands for recreation: an application of travel cost and contingent valuation methodologies." Journal of Environmental Management 26: 299-312
Freeman, A.M., 1993 “The Measurement of Environmental and Resource Values,” Resources for the Future Washington, DC
Glass, G.V., 1976 “Primary, secondary and meta-analysis of research,” Educ Res 5: 3-8
Gupta, T.R and J.H Foster, 1975 "Economic criteria for freshwater wetland policy in Massachusetts." American Journal of Agricultural Economics 57: 40-5
Hastings, W K., 1970 “Monte Carlo Sampling Methods Using Markov Chains and Their applications,” Biometrika, 57: 97-109
Johnston et al., 2002 "Valuing estuarine resource services using economic and ecological models." Coastal management, 30 (2002), 47-66
Koop, G., 2004 “Bayesian Econometrics,” John Wiley & Sons, Ltd
Lambert A., 2003 “Economic Valuation of Wetlands: an Important Component of Wetland Management Strategies at the River Basin Scale,” Ramsar Convention Bureau publication
Lant, C.L and G.A Tobin, 1989 "The economic value of riparian corridors in Corn belt Flood plains: a research framework." Professional geographer 41 (3): 337-349
Loomis, Hanemann, Kanninen and Wegge, 1991 "WTP to protect wetlands and reduce wildlife contamination from agricultural drainage", in the economic and management of water and drainage in agriculture, eds Diner & Zilberman, Kluwer Academic Publishers
Lynne, G.D., P Conroy and F.J Prochaska, 1981 "Economic valuation of marsh areas for marine production processes." Journal of environmental economics and management 8: 175-186
Mitsch, W.J & J.G Gosselink, 1986 Wetlands, Van Nostrand Reinhold, New York
Moeltner, K., J B Kevin and W P Robert, 2007 “Meta-analysis and benefit transfer for resource valuation-addressing classical challenges with Bayesian modeling,” Journal of Environmental Economics and Management, 53(2): 250-269
Montgomery, J and B Nyhan, 2008 “Bayesian Model Averaging: Theoretical developments and practical applications,” A poster based on an earlier version of this paperwas presented at the Society for Political Methodology Summer Conference, State College, PA, July 18-21
Mullarkey, D., 1997 "Contingent valuation of wetlands: testing sensitivity to scope." PhD Dissertation, Department of agricultural and applied economics, Univ of Wisconsin-Madison
Pate, J and J Loomis , 1997 "The effect of distance on WTP values: a case study of wetlands and salmon in California." Ecological economics 20, p 199-207
Poor, J., 1999 "The value of additional central flyway wetlands in Nebraska's rainwater basin wetland region." Journal of agricultural and resource economics 24 (1): 253-
Ragkos, A., A Psychoudakis, A Christofi, & A Theodoridis, 2006 “Using a functional approach to wetland valuation: the case of Zazari–Cheimaditida,” Regional Environmental Change, 6:193-200
Roberts, L., and J Leitch, 1997 "Economic valuation of some wetland outputs of Mud Lake, Minnesota-South Dakota" Agricultural economics report no 381, Department of agricultural economics, North Dakota Agricultural experiment station, North Dakota State University
Rosenberger, R S and J B Loomis, 2001 “Benefit transfer of outdoor recreation use values: A technical document supporting the Forest Service Strategic Plan,” Gen Tech Rep RMRS-GTR-72 Fort Collins, CO: U.S Department of Agriculture, Forest Service, Rocky Mountain Research Station
Rosenberger, R S and T D Stanley, 2006 “Measurement, generalization, and publication: Sources of error in benefit transfers and their management,” Ecological Economics, 60(2): 372-378
Schulte-Hostedde B., D Walters, C Powell and D Shrubsol, 2007 “Wetland management: An analysis of past practice and recent policy changes in Ontario,” Journal of Environmental Management, 82(1): 83-94
Shabman, L.A and S.S Batie, 1987 "Mitigating damages from coastal wetland." Marine Resource Economics 4: 227-248
Thibodeau, F.R and B.D Ostro, 1981 “An economic analysis of wetland protection,” Journal of Environmental Management, 12: 19-30
Thomas D.C., M Jerrett, N Kuenzli, T.A Louis, F Dominici, S Zeger, J Schwarz, R.T Burnett, D Krewski, D Bates, 2007 “Bayesian model averaging in time-series studies of air pollution and mortality,” J Toxicol Environ Health A, 70: 311-315
Verralland, R.J and P.D England, 2005 “Incorporating expert opinion into a stochastic model for the chain-ladder technique,” Insurance: Mathematics and Economics, 37(2): 355-370
Whitehead, J., 1990 "Measuring willingness to pay for wetlands preservation with the contingent valuation method" Wetlands 10 (2): 187-201
Whitehead, J., and G Blomquist, 1991 "Measuring contingent values for wetlands: effects of information about related environmental goods" Water resources research
Wong, A C M & J Wu, 2002 “Small sample asymptotic inference for the coefficient of variation: normal and nonnormal models,” Journal of Statistical planning and Inference, 104:78-82
Woodward, R.T & Y.S Wui, 2001 “The economic value of wetland services: a meta- analysis,” Ecological Economics, 37: 257-70
Zucker D.M., O Liberman, O Manor, 2000 “Improved small sample inference in the mixed linear model : Bartlett correction and adjusted likelihood,” Journal of the Royal Statistical Society, 62:827-838.
Introduction
Meta-analysis, a quantitative analysis of statistical summary, has received wide attention in wetland valuation since it can synthesize statistical results from previous studies and provide thorough estimates by allowing each study to carry some weight in obtaining parameter estimates rather than put the whole weight on any single study (Chalmers, 1991; Glass, 1976) In the literature, the ultimate goal of most wetland meta- analyses is aimed to provide benefit transfer (BT) predictions to non-valued wetland sites through their estimated meta-equations especially when primary studies are impeded by time or budget constraints However, due to the nature of observation screening process in a wetland meta-analysis study, most of the meta-equations were found to be estimated with a limited set of data Examples of studies that fall into this category are Brouwer et al (1999), Woodward and Wui (2001), Borisova-Kidder (2006), and Brander et al
The analysis in essay 1 does not depart much from the limitations that previous studies faced and is only based on a 72-observation U.S wetland dataset, which was assembled by Borisova-Kidder (2006), to support BT in the U.S In fact, this thin 72- observation dataset in the analysis of essay 1 confronts us with two difficulties First, the number of observations may be smaller than idea for estimating effects on valuation methodology, wetland type, services, and etc Second, this dataset is relatively noisy In practice, the combination of these two effects might result in inefficient parameter estimation and BT predictions 10 , which in turn leads us to unreliable policy suggestions
As a result, the purpose of this study is to improve the efficiencies of both the model parameter estimation and the BT predictions in our wetland meta-equation
In order to tackle the difficulties, we propose to augment our study with information from a second dataset in a published wetland meta-analysis study conducted by Brander et al (2006) for introducing additional information to our thin dataset The reason that we choose Brander et al.’s study is because their data consists of 215 observations extracted from 80 studies in 25 countries across 5 continents As a result, we hypothesize this large-scale multi-country (mostly OECD country) dataset has richer information than ours on the effects of valuation methodology, wetland type, services, and etc that would effectively inform our study of U.S wetland values
An intuitive way to carry out the above idea is simply combining Brander et al.’s 215-observation multi-country dataset with our 72-observation U.S dataset to estimate the meta-equation However, this method is impeded by technical limitations in the following two aspects 11 First, each dataset does not share the same set of variables Unless some compromise is made on both datasets, it is impossible to combine these two sets of data Second, if these two datasets can be combined in some way, Brander et al.’s dataset will bring not only additional information into our independent variables, which is a desire to our study, but also non U.S wetland values into our dependent variables, which certainly will undermine the goal of studying U.S values Under this circumstance, the analytical method in essay 1, while a common practice of wetland valuation in the literature, suffers from inability to meet our objectives
In this study, we turn to a Bayesian modeling framework to take advantage of its flexible structure in the following two features First, the selected information contained in Brander et al.’s 215-observation global dataset can enter our meta-analysis through the refinement of our Bayesian model priors In this way, the added information will benefit the independent variables in our meta-equation through refining the range of values our model parameters can take (Verralland & England, 2005) without introducing undesired
10 In this study, efficiency has the same meaning as precision when we refer to BT predictions
11 In this essay, we do not have Brander et al.’s row data The two limitations addressed here are based on the assumption that their row data is available to the public non U.S wetland values to our model dependent variables Second, model uncertainty in the derivation of BT prediction can be addressed by Bayesian model averaging (BMA) With this feature, this study can employ a set of competing models to generate a thorough
BT prediction without trading off among them
The hypothesis of this study is that incorporating our analysis with the added information extracted from Brander et al.’s global wetland study can enhance our model performance and thus lead to more efficient BT predictions Centering on this hypothesis, this study is organized as follows Section II describes the settings of our Bayesian model Section III provides the details on posterior distribution, model space selection, and prior refinement In this section, we show the work on how we perform a series of monotonic adjustments to unify the inconsistencies between two studies before any information from Brander et al can be used as our model priors Section IV is the empirical implementation that provides results of model estimation Besides, the performance of models with and without added information is also compared in this section In section V, the efficiency of BT predictions will be examined by comparing the variation of BT predictions for models with and without the added information under several hypothetical
BT exercises A thorough BT prediction, which is averaged over competing models through BMA, is provided in this section Section VI gives the conclusion of this study.
Model Description
The model constructed in this study takes a standard linear form with the assumption that the measured logged wetland value per acre is a function of a number of variables in the following three groups: socio-economic characteristics, study characteristics, and wetland physical and geographical characteristics Since the observations in our meta-dataset vary considerably in terms of the valuation methodology used, site located, service or function provided, and etc., the heteroskedasticity issue might be present in this dataset (Boxall et al., 1996; Carson et al., 1996) In order to reflect this possibility, we allow the error terms in our model to carry this heteroskedasticity at the observation-specific level Based on this setting, our model can thus be written as ε β +
In the above equation, y denotes the logged wetland value per acre reported in each study X denotes both study and wetland characteristics associated with observation y β is a corresponding vector of regression coefficients ε i is the regression error, which is assumed to be normally distributed with mean zero and variance-covariance matrix h −1 Ω, where h=(σ 2 ) − 1 and ω i is the observation-specific variance weight N is the total number of observations in our dataset Conditional on the observation-specific variance weight, the likelihood function of our meta-regression model can be expressed as:
After having the likelihood function, the next step is to specify the prior distribution for all parameters in our meta-equation In this study, we use an independent Normal-Gamma prior for β and h The vector of regression coefficients β is assumed to follow a multivariate normal distribution with mean β and variance-covariance V, both of which are common assumptions in the literature The shared component h is assumed to follow a gamma distribution with shape v and scale s − 2 The above prior specifications can be written in the following functional forms:
Since it is convenient to work with error precision rather than variance, therefore, we define λ ≡(λ 1 ,λ 2 ,Lλ N )'≡(ω 1 − 1 ,ω 2 − 1 ,L,ω N )' Here, the variance weights λ are assumed to follow a gamma distribution and depend upon a hyper-parameter v λ The prior specifications of above two parameters can be written in the following functional forms:
Posterior Distribution, Model Space Selection, and Prior Refinement
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:
= β β The full conditional posterior distribution of h is to follow a Gamma distribution:
2 =( − β)'Ω − ( − β)+ The full conditional posterior distribution of each λ i is to follow a Gamma distribution:
The full conditional posterior distribution of v λ is to follow
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
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 technique 12 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.
Empirical Implementation
i) Prior Assignment and model settings
In order to examine if the model performance is improved by incorporating the added information borrowed from Brander et al.’s study as well as by accounting for heteroskedasticity issue in our dataset, the 14 models in the model space are set to receive two different treatments One is either accounting for hereroskedasticity or not and the other is either incorporating informed priors or not By crossing these two treatments, it results in 4 model groups as follows:
In practice, if informed priors are used, the borrowed information contained in Brander et al.’s 215-observation global dataset will carry some weight in determining our model parameters However, if diffuse priors are used, our model parameters will be fully determined by the information contained in our 72-observation U.S dataset According this fact, models in group 1 can serve as the baseline for model comparisons since they are structured without informed priors and heteroskedasticity Models in group 4, compared with those in group 1, are set to work with both Models in group 2, compared with those in group 4, are set to work with the added information only Models in group 3, compared with those in group 4, are set to work with the heteroskesdasticity issue only ii) Bayesian Model Estimation
Since the posteriors of all model parameters are in the form of full conditional distributions, a posterior simulation is required for netting out all other model parameters to obtain the marginal distribution of interest In this study, we apply Gibbs Sampler (GS) with a built-in Metropolis-Hastings (Hastings, 1970) algorithm to simulate those posterior distributions specified in the previous section
The 72-observation U.S dataset is used in the estimation of our meta-equation and a run of 25,000 replications is taken In order to eliminate the effects from the starting values arbitrarily chosen by us, 15,000 burn-in replications are discarded and only 10,000 draws are retained in the posterior simulation The posterior means and numerical standard errors of the coefficients on independent variables in each model are reported in Appendix C (Table C.1 to Table C.14) iii) Model Performance
Our study hypothesis as well as the assumption we put on the error term to reflect heteroskedasticity can be examined by comparing the performance of each model in all groups via the following two indexes: mean absolute percentage error (MAPE) and model-conditioned marginal likelihood a.) Mean Absolute Percentage Error (MAPE)
MAPE is a measure of accuracy in a fitted value of quantitative forecasting It usually expresses accuracy as a percentage, which can be defined by the following formula:
1 ˆ where y i is the observed wetland value in our dataset, yˆ is the fitted value produced i from our meta-equation, and N is the total number of fitted points Because this index is measured in percentage error, it allows us to compare the error of fitted value in each model that differs in level A smaller MAPE suggests a relatively good fit for quantitative forecasting Since we retain 10,000 replications in the posterior simulation, the MAPE reported for each model in Table 2.2 is the value averaged over 10,000 MAPEs generated from those retained replications b.) Model-Conditioned Marginal Likelihood
In the Bayesian framework, the model-conditioned marginal likelihood, defined as p(y|M m ), is a measurement indicating how likely the observed data generated by model M m and thus can serve as an index for model comparison The derivation of model-conditioned marginal likelihood begins with the simple Bayes’ rule
M p y pθ = θ θ , where θ is a vector of parameters and
(y M m p θ , p(θ|M m ) and p(θ|y,M m ) are the likelihood, prior and posterior respectively This equation can be rearranged to give an expression for the model- conditioned marginal likelihood,
= θ Since p(y|M m ) does not depend upon θ , its value can be calculated at any arbitrary point θ * In our case, the model-conditioned marginal likelihood reported for each model in Table 2.2 is the value calculated at the mean of all model parameters in vector θ
Group 1 Group 2 Group 3 Group 4 log(P(y|M)) MAPE log(P(y|M)) MAPE log(P(y|M)) MAPE log(P(y|M)) MAPE
Table 2.2: Average MAPEs and Logged Marginal likelihood values iv.) Estimation results
Table 2.2 pair wisely lists the values of MAPE and logged model-conditioned marginal likelihood for each model in all groups Since models in group 1 neither incorporate informed priors nor account for heteroskedasticity issues, both values derived from this group are the baseline for model comparison to see if there is any improvement that can be observed from our study hypothesis as well as the assumption we made on the regression error
In terms of MAPEs, the results show that the models in groups 2 and 4 with informed priors do perform better in the accuracy of model prediction by approximate 5% over their counterparts in groups 1 and 3, which is a weak evidence to support our study hypothesis Comparing models between groups 1 and 3 as well as comparing between 2 and 4, the results suggest that models with accounting for heteroskedasticity have slightly better predictive performance However, since the differences in MAPEs are small, we might suspect but not conclude that the heteroskedasticity is present in our dataset
In terms of logged marginal likelihood values, the results show that models in groups 2 and 4 with informed priors score better values (smaller in absolute value) than their respective counterparts structured with diffuse priors in groups 1 and 3 According to Kass and Raftery (1995), if the difference in marginal likelihood value is greater than 5, a model with smaller absolute value is considered to be decisively superior As a result, we can conclude that the added information borrowed from Brander et al does effectively improve our model performance by increasing the probability of observing the samples points in our dataset Besides, it is also noted that models in groups 1 and 2 without accounting for heteroskedasticity score higher marginal likelihood values than those in groups 3 and 4 respectively Since the difference in value is greater than 5, we can conclude that the heteroskedasticity issue is not present in our dataset
By comparing models within each individual group, the results show that model performance measured by both MAPE and model-conditioned marginal likelihood is gradually improved from FULL model to REDUCED model This suggests that sequentially eliminating insignificant variables from FULL model does not ignore information contained in those dropped variables Instead, it can eliminate the unexpected noise in our dataset and further enhance the model performance This implicitly suggests that if a single model should be selected for BT exercises, the one with less insignificant variables might be a good choice for this task.
Hypothetical Policy Scenarios for Benefit Transfer Predictions
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
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(y f |x f ,M m ), where x f 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 M m , the Gibbs Sampler retains the last r,000 draws in posterior simulation For each of the retained r draws, draw y f from p(y f |x f ,θ r th ,M m ) The resulting 10,000 draws of y f can be viewed as drawing from p(y f |x f ,M m ), which nets out all model parameter in model M m 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
The posterior model probability, p(M m |y), in the above equation can be derived by Bayes’ rule as follows
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(M m ) and p(M j ) can be cancelled out in the above equation Besides, since the model-conditioned marginal likelihood, p(y|M m ), 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 y f from model-specific BPDs across all involved models
Base 1 Base 2 Base 3 Base 1 Base 2 Base 3 Base 1 Base 2 Base 3 Base 1 Base 2 Base 3
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
Table 2.6: BMA BT prediction based on models in group 2
Conclusion
Meta-analysis for benefits transfer often confronts researchers with issues such as small and noisy datasets, which results in inefficient BT predictions and less informed policy suggestions In this study, we propose to overcome the above issues in essay 1 by borrowing information from a second dataset in a published wetland meta-analysis conducted by Brander et al (2006) for introducing additional information to our thin dataset In this study, we turn to a Bayesian model to have such information entered our model through the refinement of priors to improve the efficiency of our model BT prediction
In addition to the above task, this study also seeks to provide some suggestions regarding whether or not to drop insignificant variables in our meta-equation This task has been done through the comparison of 14 models, which were generated from the 13 steps in the procedure of the backward elimination model selection These 14 models also play the role that allows us to examine our study hypothesis under different model specifications
Our results showed that models with the added information borrowed from Brander et al score higher logged marginal likelihood values than their respective counterparts no matter when heteroskedasticity is taken into account or not in our model
This suggests that such added information can effectively improve our model performance by increasing the probability to observe the sample points in our dataset In terms of MAPEs, our results also showed that the accuracy of model forecasting is improved by 5% when the added information has been introduced into our model
The other central question that we seek to answer is that whether the added information borrowed from Brander et al can improve the efficiency of the BT predictions in our meta-equation In this study, we carry out this examination through performing several BT prediction exercises based on four hypothetical policy scenarios set up by us The results showed that the 90% confidence intervals of models with the added information from Brander et al have been narrowed down by 54% This again supports the previous finding
By comparing the 14 models, the results showed that the model performance measured by both MAPE and model-conditioned marginal likelihood is gradually improved while the insignificant variables were sequentially removed from our meta- equation As a result, we might conclude that no information will be ignored when insignificant variables are dropped from our meta-equation This finding is consistent with that in essay 1
Finally, we also want to address that the model performance can be further improved if the variance-covariance matrix of coefficient estimates in Brander et al can be obtained Just like most published studies, Brander et al only reported coefficient estimates and variances in their meta-regression equation Since some prior information in this study came from the combination of two or more variables, the derivation of their variances requires the summation of variances and co-variances over several variables
We believe that the model performance can be further improved if this information can be included in the derivation of our model priors
Median 90% Low 90% High Difference Median 90% Low 90% High Difference Model 0
S1 277.255 30.748 2459.133 2428.385 179.381 22.116 1398.191 1376.074 S2 88.806 8.606 928.616 920.010 60.239 6.591 541.202 534.610 S3 144.442 16.777 1204.718 1187.941 85.591 10.549 666.572 656.023 S4 46.266 4.585 458.149 453.563 28.743 3.230 253.771 250.541 Table 2.4: BT predictions for models in group 1 and 2 under four hypothetical policy scenarios (Continued)
Table 2.4: BT predictions for models in group 1 and 2 under four hypothetical policy scenarios (Continued)
Amacher, G S., R.J Brazee, J.W Bulkley, R.A Moll “Application of wetland valuation techniques: examples from Great Lakes Coastal wetlands.” School of Natural Resources, University of Michigan, Ann Arbor, June, 1989
Bell, L "Marginal value of coastal wetlands to Florida's commercial fisheries." In Coultas and Hsieth eds Ecology and management of tidal marshes, p 231-257,
Bergstrom, J C., J R Stoll, J P Titre and V L Wright, 1990 "Economic value of wetlands-based recreation." Ecological economics 2: 129-147
Blomquist, G C and J C Whitehead, 1998 "Resource quality information and validity of WTP in contingent valuation." Resource and Energy Economics 20: 179-196
Borisova-Kidder A., 2006 “Meta-analytical estimates of values of environmental services enhanced by government agricultural conservation programs,” Dissertation, Department of Agricultural, Environmental, and Development Economics, Ohio State University
Boxall, P.C., W.L Adamowicz, J Swait, M Williams, J Louviere, 1996 “A comparison of stated preference methods for environmental valuation,” Ecological Economics, 18: 243-253
Brander L.M., R.J.G Florax, J.E Vermaat, 2006 “The empirics of wetland valuation: a comprehensive summary and a meta-analysis of the literature,” Environ Resour Econ, 33: 223-250
Brinson, M M., F R Hauer, L C Lee, W L Nutter, R D Rheinhardt, R D Smith, and
D Whigham, 1995 "A guidebook for application of hydrogeomorphic assessments to riverine wetlands," Technical Report WRP-DE-11, U.S Army Engineer Waterways Experiment Station, Vicksburg, MS NTIS No AD A308 365
Cangelosi, A., R Wither, J Taverna, and P Cicero "Wetlands restoration in Saginaw Bay," in the Revealing the economic value of protecting the Great Lakes National Oceanic and Atmospheric Administration and Northeast-Midwest Institute, 2001
Carson, R.T., N.E Foores, K M Martin, J.L Wright, 1996 “Contingent valuation and revealed preference methodologies: comparing the estimates for quasi-public goods,” Land Economics, 72:80-99
Chabreck, R "Wildlife harvest in wetlands of the United States," in P.E Greeson, J.R Cooper J and J Loomis "Economic value of wildlife resources in the San Joaquin Valley: hunting and viewing values”, in the Economic and management of water and drainage in agriculture, eds Diner & Zilberman, Kluwer Academic Publishers,
Chalmers, T.C., 1991 “Problems induced by meta-analyses.” Stat Med, 10:971-80
Duncan M J., J C Spence, W K Mummery, 2005 “Perceived environment and physical activity: a meta-analysis of selected environ-mental characteristics,” Int J Behav Nutr Phys Act, 2:
Glass, G.V., 1976 “Primary, secondary and meta-analysis of research,” Educ Res 5: 3-8
Hastings, W K., 1970 “Monte Carlo Sampling Methods Using Markov Chains and Their applications,” Biometrika, 57: 97-109
Koop, G., 2004 “Bayesian Econometrics,” John Wiley & Sons, Ltd
Lambert A., 2003 “Economic Valuation of Wetlands: an Important Component of Wetland Management Strategies at the River Basin Scale,” Ramsar Convention Bureau publication
Moeltner, K & R Woodward, 2007 “Meta-functional benefit transfer for wetland valuation: making the most of small samples,” paper presented at the W1133 annual meetings in Richmond, VA, March 28-30, 2007
Rosenberger, R S and T D Stanley, 2006 “Measurement, generalization, and publication: Sources of error in benefit transfers and their management,” Ecological Economics, 60(2): 372-378
Thibodeau, F.R and B.D Ostro, 1981 “An economic analysis of wetland protection,” Journal of Environmental Management, 12: 19-30
Verralland, R.J and P.D England, 2005 “Incorporating expert opinion into a stochastic model for the chain-ladder technique,” Insurance: Mathematics and Economics, 37(2): 355-370
Woodward, R.T & Y.S Wui, 2001 “The economic value of wetland services: a meta- analysis,” Ecological Economics, 37: 257-70.