5. Empirical Results
5.4. Results of bootstrap simulations
For complicated models like the present one, it is useful t o apply L i b o ~ t ~ t r a p "
techniques t o indicate the statistical reliability of the results. Bootstrap pro- cedures estimate reliability by replicating the estimates over a subsample of the original sample. Under ideal conditions (such as normality), a bootstrap estimate can provide an unbiased estimate of the distribution of parameters of interest.
In the case a t hand, we are interested in estimating the robustness of our estimates of the hedonic impact of global warming. The amenity impact, call it H , is a function of the set of included variables
(0
as well as the sanlple d a t a (x) for N observations; that is, H = h ( i , s , N ) . We calculate a "da,ta bootstrap" distribution of H by taking repeated subsamples of Mc
N . We also consider a "specification bootstrap," which considers a,lternative specifications of the hedonic equation.Bootstrap Replications for the Data
We first present a data bootstrap estimate of the hedonic impact of climate change, H = h ( i , x , N ) , where we examine only the potential sampling error.
For this estimate, we hold the specification
(0
fixed and subsample from the 3105 counties. For this purpose, we use the basic weighted equations in Table 3 along with the uniform national climate change scenario. T h e exact procedure is to replicate the estimate with repeated subsamples of theSeries: SUMNTSLS Sample 1 88 Observations 88
Mean 0.041 070
Median 0.034666
Maximum 0.2241 18 Minimum -0.090342 Std. Dev. 0.056233 Skewness 0.502075 Kurtosis 3.509577 Jarque-Bera 4.649285 Probability 0.097818
Hedonic porometer (H)
Figure 9. Distribution of impacts of C 0 2 doubling using d a t a bootstrap estimation. (The parameters are the estimated impact of a C 0 2 doubling as a fraction of aggregate wages. These estimates correspond t o the two-stage lea,st squares specification in Table 3. The bootstrap sample is determined by drawing half of the counties a t random with replacement.)
county data. For these, we conduct our subsampling by choosing half of the sample randomly and without replacement.6
Figure 9 shows the distribution of estimates of the hedonic impact of climate change. These replications give a mean positive 4.2% amenity im- pact, with a standard deviation of 5.6%. This distribution indicates t h a t with t h e present d a t a set we cannot judge whether the amenity impact is positive or negative. Indeed, about 80% of bootstrap samples have a positive hedonic value of climate change. Similar results are given by the bootstrap replications for the OLS estimates.
T h e unsettling result here is that the central tendency of the bootstrap estimate is markedly larger than that of the basic result shown in Table 5 (the mean of the bootstrap sample in Figure 9 is well determined). This shows the fragility of the relationship. In principle, the estimated central tendency of the bootstrap sample should be the same as t h e underlying regression, but non-normal errors can lead t o differences in mean estimates.
'This procedure was suggested by John Hartigan, who also made a number of helpful comments on the bootstrap procedure. The procedure employed is a "delete-half'' jack- knife, in which each observation has a probability of one-half of inclusion. For a discussion of bootstrap approaches, see Efron and Tibshirani (1993).
Bootstrap Replications for the Specifications
We have in addition developed a novel procedure that is called a specification bootstrap t o test for the sensitivity of the results t o different specifications.
The idea here is as follows. As described in the last section, we are inter- ested in estimating the hedonic impact of climate change, H = h ( ( , x, N).
In addition to sampling issues, there are clearly uncertainties about the specification. Say that the hedonic function is estimated with variable set ( = ( c l , . . .
,
(r,-), where the vector ( represents the Ii possible included variables.The standard approach t o specification is t o use maximum likelihood a,s a method of inclusion. This is useful but fails to give a measure of the sensitivity of the out come t o alternative specifications. Instead, we propose using a specification bootstrap. The simplest approacli is to assume that we are unsure about which variables t o include. We represent inclusion of the it11 variable as (; = 1 while exclusion is
ci
= 0. Assuming that we are uninformed about which variables belong in the relationship and which should be excluded, we take samples of M of theIi'
possible variables. (There are obvious extensions for nonindepe~ldence and not-equally-likely cases, but those are not pursued here.) We then exanline the distribution of H for a sample of the potential specifications.For this purpose, we set M = 6; that is, we assume that 6 of the non- central 56 included variables are the appropriate ones. The actual specifi- cation chosen is a two-stage least squares estimate with a constant, a linear temperature term, the two density variables and their instruments for the first stage, a,nd 6 randomly chosen included variables from the other 56 vari- ables included in the preferred equation shown in Table 3. We randomly sample 100 from the 32 million possible specificatio~ls. Figure 10 shows the estimates from the specification bootstrap simulation. The central tendency for the TSLS specification yields a negative hedonic relatioilship with a,n amenity impact of -3.5% and a standard deviation of 3.6%. (The corre- sponding preferred estimate for the TSLS is -0.17.) The central tendency of the specification bootstrap is the mirror image of the d a t a bootstrap, being less than the preferred estimate. The uncertainty in the specification boot- strap is somewhat less than that in the d a t a bootstrap in part because three variables were included in all specifications.
Series: SPENTSGA Sample 1 100 Observations 100 Mean -0.030640 Median -0.023424 Maximum 0.041513 Minimum -0.133293 Std. Dev. 0.032370 Skewness -0.933334 Kurtosis 4.423724 Jarque-Bera 22.96431 Probability 0.000010
Hedonic parameter, H
Figure 10. Distribution of impacts of COz doubling using specification bootstrap estimation. (The parameter is the estimated impact of a C 0 2 doubling as a fraction of aggregate wages, H. These estimates correspond to the two-stage least squares specification in Table 3 and the central estimate of the hedonic parameter shown in Table 5. The specification bootstrap is generated by drawing G of 56 variables.)
Conclusion on Uncertainty of Estimate
Weighing the alternative specifications along with the bootstrap results, we conclude that it is not possible with the available evidence t o determine tlie amenity impact of climate change for the USA. The preferred estimated impact of an equilibrium C 0 2 doubling, using tlie TSLS regression and uni- form climate change, is -0.37% of total wages. For the simplest specifications shown above, t l ~ e impact ranges between -1.9 (f O.lG)% and $3.1 ( f 0.24)%
of wages. The range of estimates in the specifications in Table 4 is from -3.0% to +8.0%. The range of estimates for different wage series shown in Table G is from -6.0% t o +4.6%. The estimates for the different climate models range from -15.9% t o +26.7% of wages. For the d a t a bootstrap the estimated impact is $4.2 (f 5.6)% while for the specification bootstrap the estimated impact is -3.5 (f 3.6)% of total wages.
Taking all these estimates, the best judgment would seem to be that we are a t this time unable t o reliably determine the impact of global warming on
climate amenities. The most liltely value of the amenity impact is -0.35% of total wages (or -0.17% of total output). I regard the estiinated varia.tion from the bootstrap replications as most reliable and estimate that the uncerta,inty of the estimate is about 5% of wages (or 2.5% of output).'
5.5. Results of alternative studies
It is useful t o consider how this research compares with earlier research on the subject. The only comparable study is that of Hoch and Drake (1975).
They used a technique quite simi1a.r t o that shown in Table 2, estimating OLS regressions of wages on climate and other variables for three samples of workers, using wages for specific occupations, in 43 t o 86 metropolitan areas. T h e climate data were relatively comprehensive, altllough in the end only precipitation, temperature, and their interactions were used.
Their statistical results are difficult t o interpret because of the iluinerous uilpooled samples and inconsistent inclusioil of variables. In addition, they did not consider the potential for simultaneous-equation bias! nor did they allow for the possibility of a systema.tic North-South wage differential. Tlle only robust result is the strong negative sign on summer temperature (in 33 of the 34 subsamples reported in the Appendix), with a snlaller coefficient but illconsistent sign on winter temperatures. This result led Hoch and Drake t o conclude that a. 1 ° F warming would lead to a gain in living standards of a,pproximately USs1.6 billion in 1974. This represents a semi-elasticity of 0.11% for 1974 GDP. Scaling this to a. ba.seline warming of $OF (or 4.5OC) for an equilibrium COz doubling used in this study yields a ga.in of 0.88% of US GDP. In the CIAP study, the effect on amenities tllrough wages was the single largest impact. All other impacts totaled USS0.67 billion, or about 40% of the effect on amenities.
The results of the present study are quite different from the earlier Hoch a.nd Drake study. I interpret the inconsistent results a,s an indicatioil of the fra.gility of the estimates t o both data and specifica.tion differences. The larger sample size in the current study allowed us t,o control for other fac- tors, such as the North-South wage differential, which explains much of OLS temperature-wage correlation. Other differences are that the present study has a much larger and more comprehensive sample - 3105 counties and com- prehensive wage d a t a - and that it has a correction for regional cost of living differentials.
'These figures are derived by combining the basic estimate in Table 5 with the bootstrap replicatiolls in Figures 9 and 10 along with the results of alternative specifications. Each of these four sets of estimat,es is equally weighted and assumed t o be llormally distributed.
6. Conclusions
The present study estimates the impact of greenllouse warilling on cliinate amenities. The amenities associated with climate change include the effects on the value of directly "consumed" climate as well as those on leisure and other nonmarket activities t h a t are complementary with climate. In the first case, climate may affect preferences directly, as in the cases of direct enjoyment of beautiful blue skies or cold, crisp nights in the mountains.
T h e second case - which is more complex and probably more important - comes as climate interacts with other goods and services in the production of amenities; this would include the combillation of warm weather, high surf, and surfboards in the production of surfing amenities or other consunlptioil activities such as hiking, sledding, sunbathing, gardening, or powcler-snow skiing. Additional cases of indirect effects would arise througll the impact of climate on pollution and health.
Analytically, measuring the value of clinlate is difficult because climate is a public good rather than a private good bought and sold on markets. Be- cause there are no market transactions for climate, we must infer its value indirectly from individual choices in other areas. T h e area studied here is in- dividual locational choices as they interact with the labor market, using the technique of wage hedonics t o estimate the impact of climate on econoinic well-being. T h e first step of the estimation is t o determine the correlation between climate (as an exogenous variable) and wages (as an enclogenous variable) for t h e USA. This estimation addresses the issue of simultaneous- equatioil bias, which has generally been overlookecl up to now. Using d a t a on 3105 counties, our preferred relationsllip indicates a small positive relation- ship of nlean temperature and real wages. The interpretation of this result is t h a t warming would be associated with an decrease in ecollonlic welfare.
T h e second step of the process is t o combine estimates of the amenity value of cliinate with climate change projections. This is acconlplished by using the results of simulations of a number of CiCMs t h a t calculate the impact of greenhouse warming on climate. The GCMs project a n average increase of 8OF (4.5OC) along with an increase in precipitation of 4% from a n equilibrium C 0 2 doubling. Using the uniform climate change scenario, we e s t i ~ n a t e t h a t an equilibrium C 0 2 doubling would be associated with 0.35%
higher wages averaged across US counties; this is the equivalent of about 0.17% of GDP.
Under hedonic theory, wage differentials associated with different cli- inates represent the amounts necessary t o compensate people for the asso- ciated amenities. Our preferred equation projects t h a t an equilibrium CO;!
doubling would produce disamenities amounting t o 0.35% of wages or about 0.17% of GDP. However, alternative specificatioils give marliedly different estimates. Weighing all the different specifications and bootstraps, the most likely impact is a disamenity of 0.35% of tota,l wages (or 0.17% of total out- put) with an uncertainty or standard error on this estimate of 5% of wages (or 2.5% of output).
How do amenity impacts compare with other estimated economic im- pacts of climate change? Up t o now, the only sectors with rigorous esti- mates of the impact of climate change a.re a,griculture, energy, and sea-level rise. Although different studies have slightly different results, it is fair t o say that the sum of the reliable estimates of the impact of global wa.rming is very close t o zero for the USA. This number consists of small losses from sea.-level rise (less than 0.1% of GDP), a small gain in heating and cooling (less than 0.1% of GDP), and no net impact on agriculture. T h e present re- sults suggest that inclusion of amenities does not change the overall picture dramatically, but the uncertainties surrounding amenity impacts swamp the impacts identified t o date.
We must emphasize that the estimated relatioilship between clima.te change and amenity values is extremely fragile. The bootstrap estimates for d a t a and for specification uncertainty indicate that it is difficult t o deternliile whether the amenity value of climate change will be positive or negative.
Physicists have grown accustomed t o the Heisenberg uncertainty prin- ciple, which concerns the limits t o observability of physical systems. There ma,y be a behavioral uncertainty principle operating in the social sciences.
This principle holds that because of the complexity of human systems and the difficulty of establishing cause-effect relationships, it is sometimes impos- sible t o accurately forecast the impact of exogenous or policy cha,nges. In the case a t hand, we are blessed with reliable and comprehensive d a t a covering a n enormous ra.nge of experience for the variables of interest of tempera,ture, wages, and demographics. Yet the underlying complexity of labor markets is so great, and the wage-temperature relationship so noisy, tha.t it a.ppears that we cannot a.ccurately project the impact of global warming on climate amenities over the coming decades.
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