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Cameron's censored logistic regression model: Reply
JOURNAL
OF ENVIRONMENTAL
ECONOMICS
AND
MANAGEMENT
20, 303-304 (1991)
REPLY Cameron’s
Censored
Logistic
Regression
Model:
Reply
TRUDY ANN CAMERON Department of Economics, University of California, Los Angeles, 405 Hilgard Allenue, Los Angeles, California 90024-1477
Received May 14, 1990; revised July 12, 1990
I a m certainly pleased that David Patterson and John Duffield have been able to tie up some loose ends left by my paper on censored logistic regression m o d e ls for contingent valuation referendum data [l]. In particular, the covariance matrix result from Lehmann that they identify and demonstrate will be especially helpful to researchers who will be applying these m o d e ls in e m p irical settings. Theirs is a valuable contribution to the working knowledge in this area. I rebut only one aspect of their comments. Patterson and Duffield note my suggestion that “. . . a referendum type contingent valuation survey be treated like a conventional regression study with W T P as the dependent variable.” However, they seem unconvinced that this constitutes an important contribution. W h e n I first began working with referendum data, it seemed to m e that the regression interpretation was a very useful insight-rarely and only m inimally and implicitly exploited by previous e m p irical researchers. Like Patterson and Duffield, one of the original reviewers of my Journal of Environmental Economics and Management submission also discounted this interpretation. After some further reflection on the problem, and another careful tour through the existing e m p irical literature, I decided (not without some trepidation) to change the title to “A New Paradigm. . . ” in the hope that this aggressive advertising stance m ight wake a few people up. Clearly, this has had the intended effect! I disagree strongly with Patterson and Duffield concerning the practical value of the regression interpretation. They state that “. . . it is our experience that in many contingent valuation surveys estimating the overall value of the resource is the primary goal of the study, at least for policymakers, and the relationship of W T P to covariates is of secondary interest.” True, if all you want is a static reading of the current marginal m e a n valuation of a resource, covariates are less interesting. But I submit that information of real value for forecasting and simulation purposes can be extracted from a fitted censored logistic (or normal) regression m o d e l when covariates are present. You cannot use a calibrated m o d e l to simulate the effects of a policy unless your m o d e l has a “crank” that can be turned by that policy. In some cases, the attributes of a resource can be expected to change gradually over time. For example, global climate change or acid rain can produce systematic changes in the qualities of different environmental resources. Suppose a researcher wished to predict the effect of 10 more years of continued acidic deposition upon the social value of recreational fishing in the Adirondacks. If you had a censored regression m o d e l that included as a covariate the cross-sectional 303 009s0696/91
$3.00
Copyright 0 1991 by Academic Press, Inc All rights of reproduction in any form resewed
304
TRUDY
ANN
CAMERON
variation in current acidification levels in lakes, you could plausibly simulate the anticipated social damages of continued acid rain. In other cases, isolated major events such as oil spills can cause the attributes of a resource like recreational fishing to change suddenly. If objectively measured levels of pollutants have been included in valuation models, then the anticipated effect of these catastrophic events upon social values could potentially be simulated using the calibrated models. A researcher might be interested in the effects on values as the level of a pollutant first peaks and then falls off over time. Economic damage estimates like these could be extremely valuable in litigation. Similar arguments apply when survey information on the individual sociodemographic characteristics of respondents is available. As in the formal modeling of demands for market goods, the nonmarket value of a resource can be expected to depend upon individual incomes and upon the prices of substitutes and complements. However, “shift factors” that implicitly influence the preference function should also be entertained. These may include age, sex, race, family size, geographic area, and a host of other variables. With demographic phenomena such as the “baby boom bulge” moving through the age distribution, migration, and immigration, the aggregate social value of a regional resource can be expected to change. Shifts in these demographics cannot be simulated unless the model contains such variables explicitly. Perhaps referendum contingent valuation studies have in the past been used primarily for estimating just the overall value of a resource because few people have recognized that covariates can readily be incorporated whenever they are available. Indeed, with the help of Patterson and Duffield’s additional insights, regression type models are even more accessible to researchers. Now, researchers and agencies responsible for the design of contingent valuation surveys can be encouraged to collect a full range of useful covariates. Richer models are almost always more desirable, since they help us to better understand how individual characteristics and resource attributes affect resource values. It should be noted that earlier correspondence with Patterson and Duffield uncovered two errors in my paper. One, which they mention in the current comment, is that for Tables II and IV, the last column of numbers in each table should have been identical to the second-last column. The formulas in my Eq. (14) are correct but their computer implementation contained a one-character typographical error. The second error is in my Eq. (131, line 2, p. 361, which should read a2 log
L/a&
dK
=
- ( l/K)*cXir(
Ii
-
1 + Ri
+
sit&},
r=l
, . . . , p.
None of the empirical results in the paper are affected by this change because numerical derivatives were used in the optimization. In addition, I have detected that footnote 34 in my paper should refer to a*, rather than a*. REFERENCE T. A. Cameron, A new paradigm for valuing non-market goods using referandum likelihood estimation by censored logistic regression, J. Enuiron. Econom. 355-379 (1988).
data: Maximum Management 15,
OF ENVIRONMENTAL
ECONOMICS
AND
MANAGEMENT
20, 303-304 (1991)
REPLY Cameron’s
Censored
Logistic
Regression
Model:
Reply
TRUDY ANN CAMERON Department of Economics, University of California, Los Angeles, 405 Hilgard Allenue, Los Angeles, California 90024-1477
Received May 14, 1990; revised July 12, 1990
I a m certainly pleased that David Patterson and John Duffield have been able to tie up some loose ends left by my paper on censored logistic regression m o d e ls for contingent valuation referendum data [l]. In particular, the covariance matrix result from Lehmann that they identify and demonstrate will be especially helpful to researchers who will be applying these m o d e ls in e m p irical settings. Theirs is a valuable contribution to the working knowledge in this area. I rebut only one aspect of their comments. Patterson and Duffield note my suggestion that “. . . a referendum type contingent valuation survey be treated like a conventional regression study with W T P as the dependent variable.” However, they seem unconvinced that this constitutes an important contribution. W h e n I first began working with referendum data, it seemed to m e that the regression interpretation was a very useful insight-rarely and only m inimally and implicitly exploited by previous e m p irical researchers. Like Patterson and Duffield, one of the original reviewers of my Journal of Environmental Economics and Management submission also discounted this interpretation. After some further reflection on the problem, and another careful tour through the existing e m p irical literature, I decided (not without some trepidation) to change the title to “A New Paradigm. . . ” in the hope that this aggressive advertising stance m ight wake a few people up. Clearly, this has had the intended effect! I disagree strongly with Patterson and Duffield concerning the practical value of the regression interpretation. They state that “. . . it is our experience that in many contingent valuation surveys estimating the overall value of the resource is the primary goal of the study, at least for policymakers, and the relationship of W T P to covariates is of secondary interest.” True, if all you want is a static reading of the current marginal m e a n valuation of a resource, covariates are less interesting. But I submit that information of real value for forecasting and simulation purposes can be extracted from a fitted censored logistic (or normal) regression m o d e l when covariates are present. You cannot use a calibrated m o d e l to simulate the effects of a policy unless your m o d e l has a “crank” that can be turned by that policy. In some cases, the attributes of a resource can be expected to change gradually over time. For example, global climate change or acid rain can produce systematic changes in the qualities of different environmental resources. Suppose a researcher wished to predict the effect of 10 more years of continued acidic deposition upon the social value of recreational fishing in the Adirondacks. If you had a censored regression m o d e l that included as a covariate the cross-sectional 303 009s0696/91
$3.00
Copyright 0 1991 by Academic Press, Inc All rights of reproduction in any form resewed
304
TRUDY
ANN
CAMERON
variation in current acidification levels in lakes, you could plausibly simulate the anticipated social damages of continued acid rain. In other cases, isolated major events such as oil spills can cause the attributes of a resource like recreational fishing to change suddenly. If objectively measured levels of pollutants have been included in valuation models, then the anticipated effect of these catastrophic events upon social values could potentially be simulated using the calibrated models. A researcher might be interested in the effects on values as the level of a pollutant first peaks and then falls off over time. Economic damage estimates like these could be extremely valuable in litigation. Similar arguments apply when survey information on the individual sociodemographic characteristics of respondents is available. As in the formal modeling of demands for market goods, the nonmarket value of a resource can be expected to depend upon individual incomes and upon the prices of substitutes and complements. However, “shift factors” that implicitly influence the preference function should also be entertained. These may include age, sex, race, family size, geographic area, and a host of other variables. With demographic phenomena such as the “baby boom bulge” moving through the age distribution, migration, and immigration, the aggregate social value of a regional resource can be expected to change. Shifts in these demographics cannot be simulated unless the model contains such variables explicitly. Perhaps referendum contingent valuation studies have in the past been used primarily for estimating just the overall value of a resource because few people have recognized that covariates can readily be incorporated whenever they are available. Indeed, with the help of Patterson and Duffield’s additional insights, regression type models are even more accessible to researchers. Now, researchers and agencies responsible for the design of contingent valuation surveys can be encouraged to collect a full range of useful covariates. Richer models are almost always more desirable, since they help us to better understand how individual characteristics and resource attributes affect resource values. It should be noted that earlier correspondence with Patterson and Duffield uncovered two errors in my paper. One, which they mention in the current comment, is that for Tables II and IV, the last column of numbers in each table should have been identical to the second-last column. The formulas in my Eq. (14) are correct but their computer implementation contained a one-character typographical error. The second error is in my Eq. (131, line 2, p. 361, which should read a2 log
L/a&
dK
=
- ( l/K)*cXir(
Ii
-
1 + Ri
+
sit&},
r=l
, . . . , p.
None of the empirical results in the paper are affected by this change because numerical derivatives were used in the optimization. In addition, I have detected that footnote 34 in my paper should refer to a*, rather than a*. REFERENCE T. A. Cameron, A new paradigm for valuing non-market goods using referandum likelihood estimation by censored logistic regression, J. Enuiron. Econom. 355-379 (1988).
data: Maximum Management 15,