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MASSIMILIANO MAZZANTI
UNIVERSITY OF FERRARACERIS CNR MILAN
PALERMO, SEPTEMBER 12th 2012
Environmental-Economic Performances in a
Dynamic Setting: Heterogeneous Slopes and Structural Breaks
Issues and Concepts
Usefulness of Slope heterogeneity analysis in environmental economics & policy
I will deal with examples under the umbrella of EKC and IPAT conceptual framework (structural change, decomposition analysis..)
Rationales Econometric rationale
(efficiency, correlation between units)
Better food for thought for policy and management (specific firm, sector, country effects)
More effective communication to non economist’s (the average coefficient problem..)
Econometric matters even (more) in policy and non economics fields…
A lawyier ‘expert’ for US Republicans on climate change recently affirmed in a congressional hearing on climate science:
“EPA cant declare GHG are a health problem, since emissions have been rising for a century, but public health has improved over the same period… “
Papers of Reference
Nicolli F. Mazzanti M. Iafolla V. 2012, Waste Dynamics, Country Heterogeneity and European Environmental Policy Effectiveness, J of Environmental Policy and Planning, i-first
Marin G. Mazzanti M., 2012, The relationship between environmental and labour productivities, J of Evolutionary Economics, i-first
Mazzanti M. Musolesi A., The heterogeneity of Carbon Kuznets Curves for advanced countries. Comparing homogeneous, heterogeneous and shrinkage/Bayesian estimators 2012, Applied Economics forth. and FEEM nota di lavoro 2010
Slope heterogeneity in the environmental economics applied literature
Recent advancements in EKC have focused on sub country and specific country heterogeneity in income-emission relationships
Seminal paper by List and Gallett (1999), Ecological Economics, on CO2 – income relationships at state level in the US
Recent working paper by Martinez Espineira on Bird abundance and GDP growth in a panel of Canadian regions
We already tried to focus on specific homogeneous areas rather than OECD or full sample
G7
0
0,5
1
1,5
2
8 8,5 9 9,5 10 10,5
log(y)
log
(co
2)
CANADA FRANCE GERMANY ITALYJAPAN UK USA FITTED VALUES (BAYES)FITTED VALUES (FE) FITTED VALUES (SWAMY)
Source: Mazzanti, Musolesi and Zoboli, 2010, Applied Economics
2040
6080
100
Em
issi
ons
(199
0=10
0)
1990 1995 2000 2005 2010Year
CO2 NOxSOxItaly Industrial emissions,
NAMEA (ISTAT)
01.
0e+0
72.
0e+0
73.
0e+0
74.
0e+0
75.
0e+0
7CO
2
DA DB DC DD DE DF DG DH DI DJ DK DL DM DN
1990 2007
CO2 emissions of manufacturing sectors: different dynamics
1.2
1.4
1.6
1.8
log(
CO
2 pe
r ca
pita
)
9 9.2 9.4 9.6 9.8 10log(GDP per capita)
AUSTRALIA
1.3
1.4
1.5
1.6
1.7
log(
CO
2 pe
r ca
pita
)
9 9.2 9.4 9.6 9.8 10log(GDP per capita)
CANADA
.4.6
.81
1.2
log(
CO
2 pe
r ca
pita
)
8 8.5 9 9.5 10log(GDP per capita)
JAPAN
.8.9
11.
11.
2lo
g(C
O2
per
capi
ta)
9.2 9.3 9.4 9.5 9.6 9.7log(GDP per capita)
NEW ZELAND
.6.8
11.
21.
4lo
g(C
O2
per
capi
ta)
9 9.5 10log(GDP per capita)
NORWAY
1.6
1.7
1.8
1.9
log(
CO
2 pe
r ca
pita
)
9.4 9.6 9.8 10 10.2log(GDP per capita)
USA
EKC, CO2 diversity in long run trends, while most studies focus on average coefficient estimations (e.g. OECD)
.8.9
11.
11.
2lo
g(C
O2
per
capi
ta)
8.5 9 9.5 10log(GDP per capita)
AUSTRIA
.2.4
.6.8
11.
2lo
g(C
O2
per
capi
ta)
8 8.5 9 9.5log(GDP per capita)
GREECE
.81
1.2
1.4
log(
CO
2 pe
r ca
pita
)
8.5 9 9.5 10log(GDP per capita)
IRELAND
.4.6
.81
1.2
log(
CO
2 pe
r ca
pita
)
8.5 9 9.5 10log(GDP per capita)
ITALY
.2.4
.6.8
1lo
g(C
O2
per
capi
ta)
8 8.5 9 9.5log(GDP per capita)
PORTUGAL
.4.6
.81
1.2
log(
CO
2 pe
r ca
pita
)
8 8.5 9 9.5 10log(GDP per capita)
SPAIN
EU South
1.2
1.3
1.4
1.5
1.6
log(
CO
2 pe
r ca
pita
)
8.5 9 9.5 10log(GDP per capita)
BELGIUM
11.
11.
21.
31.
41.
5lo
g(C
O2
per
capi
ta)
9 9.2 9.4 9.6 9.8 10log(GDP per capita)
DENMARK
.6.8
11.
21.
4lo
g(C
O2
per
capi
ta)
8.5 9 9.5 10log(GDP per capita)
FINLAND.9
11.
11.
21.
3lo
g(C
O2
per
capi
ta)
9 9.2 9.4 9.6 9.8 10log(GDP per capita)
FRANCE
1.3
1.35
1.4
1.45
1.5
log(
CO
2 pe
r ca
pita
)
9 9.2 9.4 9.6 9.8log(GDP per capita)
GERMANY
11.
11.
21.
31.
4lo
g(C
O2
per
capi
ta)
9 9.2 9.4 9.6 9.8 10log(GDP per capita)
NETHERLANDS.9
11.
11.
21.
31.
4lo
g(C
O2
per
capi
ta)
9 9.2 9.4 9.6 9.8 10log(GDP per capita)
SWEDEN
1.25
1.3
1.35
1.4
1.45
log(
CO
2 pe
r ca
pita
)
9 9.2 9.4 9.6 9.8 10log(GDP per capita)
UK
EU North
The way to deal with heterogeneity are many, some pragmatic some more
technically refined
We here hold attention on SUR – Seemingly Unrelated Regression -
contexts
Some notes on SURE models (Zellner’s 1962)
Applied both in cross section and panel contexts Need to test Systems of equations (by OLS and GLS)
e.g. Household demand function (food, housing, clothing)…typical cross section example
Seminal Grunfeld and Zellner papers on firm data 10 firms observed over 20 years
SUR ‘deliveries’ Higher efficiency wrt Fixed effects (constrained SUR which
accounts for correlation between units) Slope heterogeneity based output
In a nut-shell
1. Fixed effect model (LSDV or better within if N high in the panel) Two ways or without T dummies (testparm) E.g. the latter likely to be more efficient…
2. Then what if you are unsatisfied with homogeneous slopes?
3. First, we may try to look at LSDV dummies sign stability and significance reg i f c mu1-mu10, nocons (STATA)
Then, we may try to investigate whether the non observable heterogeneity affects slopes as well
SUR
In case we face a pretty long time series and a limited number of covariates, we can try to go further
The issue is the contemporaneous correlation between cross sectional units
E.g. systems of 10 equations with T=20y1t = a1 + b1 x1t + 1t
y2t = a2 + b2 x2t + 2t
....y10t = a10 + b10 x10t + 10t
Unconstrained and Constrained SUR
SURE…. ‘Model of apparently not related individuals’ Common effects captured by error terms related to unobservable information
We need to reshape reshape wide y x1 x2, i(year) j(cod) (note: j = 1 2 3 4 5 6 7 8 9 10) We now have created new variables
STATA estimates ‘i equations’, 1 to 10. Say we have 2 for learning
global i3(i3 f3 c3), f and c covariates, y dep var
global i8(i8 f8 c8)
. sureg (i3 f3 c3) (i8 f8 c8), corr (estimate only two equations here, 3 and 8)
This is an unconstrained SURE command in STATA
Unconstrained and Constrained SUR
TESTS
1. Through (Chi2) Breusch Pagan I check the correlation between errors SUR consistent and more efficient than OLS systems (often similar estimates but
lower s.e) 2. Towards Het slopes..
test [i3]x13=[i8]x18 (accum) We test all slope’s equality, 3 and 8 are here 2 equations Null Hp is equality (poolability)
If Not rejected, Constr SUR, that accounts for correlation but still witness slope homogeneity
Define constraints – might be burdensome but just in terms of do file construction, then apply SUREG
Very similar to LSDV FE, but we have individual variance in the errors and we account for correlations
* the test on equality may give different results when picking up different ‘couples’
If we apply a constrained SURE when the null is eventually rejected, s.e. rise due to the imposition of a non valid constraint
Applications
Nicolli et al., JEPP 2012
Waste Kuznets curves in the EUEUROSTAT data for EU15 over 1995-2008
Slope heterogeneity highlights various performances on decoupling
Waste generated. SURE Model, constrained slopes.
Constrained slope SUREConstrained slope SURE –
all variables
CONS 0.95*** 1.19***
CONS2 -0.03*** -0.038***
DENSPOP … -0.29***
POLIND … -0.002
TP [CONS per capita, millions of €]
7.521 6.311
Breusch-Pagan test of independence (p-value)
0.000 0.000
Note:. (…) means not included; significance at 10%, 5% and 1% denoted by *, ** and ***, respectively.
Note:. (…) means not included; significance at 10%, 5% and 1% denoted by *, ** and ***, respectively.
SUR: landfilled waste
Constrained SUR
Constrained SUR – all covariates
CONS 1.49*** 4.27***
CONS2 -0.10*** -0.19***
DENSPOP … -3.68***
POLIND … -0.82***
TP [€] 1,659.39 47,328.06
Breusch-Pagan test of independence (p-value)
0.000 0.000
Waste generated. SURE Model, unconstrained model
Countries CONS CONS2 TP [€] Delinking evidence
Austria 84.31*** -4.33*** 16,646.52 Absolute
Belgium -3.73 0.210 7,075.36 No delinking
Denmark -11.26 0.62 8,051.13 No delinking
France 3.57 -0.17 33,767.68 No delinking
Germany 1.89*** -0.12*** 1,633.113 Absolute
Greece 17.36*** -0.91*** 13,548.99 Absolute
Italy -8.73*** 0.487*** 7,842.28 No delinking
The Netherlands 9.28*** -0.47*** 16,578.1 Relative
Portugal 8.89*** -0.48*** 9,983.131 Absolute
Spain 24.03*** -1.29*** 10,885.79 Absolute
Sweden -17.5*** 0.96*** 8,700.899 No delinking
United Kingdom 5.09*** -0.25*** 21,529.84 Relative
Breusch-Pagan test of independence (p-value) = 0.000 F test of slope homogeneity (p-value) = 0.000
Marin and Mazzanti, JEE, 2012
Environmental and labour productivity dynamics in Italy
Sector based lensNAMEA (ISTAT) dataset on economic
and environmental accounts: sector branches (e.g. Food, coke & refinery) over 1992-2009
IPAT/EKC framework
SUR constrained estimates (manufacturing) - All emissions
SUR[manuf]
SUR[manuf]
SUR[manuf]
ln(CO2/L) ln(NOx/L) ln(SOx/L)
ln(VA/L) 2.8517***[0.03]
-3.4261***[0.17]
-11.6507***[0.41]
ln(VA/L)2 -0.2745***[0.003]
0.3455***[0.02]
1.1463***[0.04]
Stagnation 0.0189***[0.001]101.91%
-0.2257***[0.02]79.79%
-0.8337***[0.05]43.45%
Breusch-Pagan test of independence (Chi2)
448.746*** 376.77*** 632.504***
Test of aggregation bias (Chi2)
16589.74*** 19992.81*** 3418.68***
N*T 238 238 238
Period 1990-2006 1990-2006 1990-2006
Turning point(s) 180.3537***[1.94]
142.3858***[7.83]
161.0529***[3.52]
Shape (VA/L) Inverted U shape U shape U shape
What hides behind aggregate shapes?
s.e < FE case
SUR unconstrained estimates for CO2 (dependent variable: ln(CO2/L) )
Branch ln(VA/L) ln(VA/L)2 Shape(VA/L)
TP
VA/L
Stagn. Stagn. (%) ConstantMin Year Max Year
DA2.4189***
[0.23]- Linear - 37.99 1990 47.95 2000
0.1836***[0.05]
120.15%0.4785[0.87]
DB16.2782***
[1.46]-2.2945***
[0.21]Inv. U shape
34.7145***[0.55]
23.34 1990 34.78 2000-0.0533[0.03]
94.81%-
19.1502***[2.5]
DC45.1774***
[1.53]-6.5425***
[0.22]Inv. U shape
31.5834***[0.11]
25.11 1991 32.58 20010.0189[0.03]
101.91%-
69.4115***[2.6]
DD15.94***
[2.41]-2.2944***
[0.36]Inv. U shape
32.2564***[0.68]
22.94 1990 32.99 2001-0.0056[0.03]
99.44%-
18.8895***[4.04]
DE-25.5248*
[15.32]3.6168*
[2]U shape
34.0792***[5.65]
40.95 1990 51.46 20010.1121***
[0.03]111.86%
54.5399*[29.3]
DF0.1429***
[0.02]- Linear - 96.92 2006 266.04 1995
0.0237[0.03]
102.40%12.9344***
[0.09]
DG22.4233***
[5.2]-2.6966***
[0.61]Inv. U shape
63.9241***[1.46]
57 1990 82.71 2004-0.1081***
[0.03]89.75%
-35.1641***
[11.01]
DH38.0536***
[7.09]-4.9565***
[0.94]Inv. U shape
46.4664***[0.55]
40.12 1990 49.18 20060.027[0.02]
102.74%-
63.5661***[13.41]
DI-
38.0085***[2.6]
5.2095***[0.35]
U shape38.3977***
[0.32]37.13 1991 50.17 2006
-0.0098[0.02]
99.02%81.1714***
[4.86]
DJ42.916***
[6.9]-6.0225***
[0.94]Inv. U shape
35.2677***[0.6]
32.65 1990 43.03 2002-0.1531***
[0.04]85.80%
-65.9376***
[12.6]
DK110.5257**
*[14]
-14.156***[1.81]
Inv. U shape
49.5927***[0.29]
42.19 1993 50.09 2000-0.0151[0.05]
98.50%
-206.9535**
*[27.11]
DL30.8633***
[2.75]-3.8026***
[0.36]Inv. U shape
57.8702***[1.42]
37.38 1990 49.21 20010.0064[0.02]
100.64%-
54.1085***[5.24]
DM-
85.8531***[7.86]
11.4303***[1.04]
U shape42.7552***
[0.19]38.02 1993 47.11 2000
0.0817**[0.04]
108.51%170.497***
[14.86]
DN44.0742***
[8.46]-6.1415***
[1.22]Inv. U shape
36.1696***[0.87]
28.91 1991 36.11 20000.07***[0.02]
107.25%-
70.8135***[14.68]
Breusch-Pagan test of independence (Chi 2): 186.514***
constrained SUR estimates for manufacturing confirm the result of FE estimates. It is worth noting that as expected SUR estimates are more efficient than FE, with lower standard error and ‘Stagnation’ structural break that becomes significant.
This gain in efficiency depends on the high correlation among the disturbances of the different sectors
Unconstrained SUR estimates highlight an high degree of heterogeneity of the slopes across sectors, as confirmed by the test of the aggregation bias.
sectors that are robustly associated to absolute delinking are DG and DJ, both included in the EU ETS, and quite critical manufacturing sectors as far as pollution effects are concerned.
All other sectors show either linear (as DF, highly critical sector for GHG related environmental effects, with regional hot spots) or U shaped
false inference
The use of heterogeneous estimators can be motivated by the possible heterogeneity bias associated with the use of pooled estimators. As pointed out by Hsiao (2003), if the true model is characterised by heterogeneous intercepts and slopes, estimating a model with individual intercepts but common slopes could produce the false inference that the estimated relation is curvilinear.
Homogeneous slope models tend to capture EKC shapes even in presence of some outliers, they generally provide better
fits….
….but may hide the average structural relationship characterising the countries
Emerging Methodological issue
WE defined 39 constraints, I am not showing the do file…
Mazzanti and Musolesi, 2010, 2012
EnvKuznetsCurves Focus on advanced countries Looking at country / regions heterogeneity, income-time effects, structural breaks due to time related events
Structural breaks in a panel
Environmental Policy shocksOil shocks
Those can be captured by the time related component of the income-environmental relationship..
Disentangle income and time effects… Further look at separated effects by country Back to the heterogeneity issue
EU south
-.5
0.5
1lo
g(C
O2 p
er
capita)
1960 1970 1980 1990 2000year
lco2pc fitted_step93fitted_ramp93 fitted_step97
fitted_ramp97
North America and Oceania
.6.8
11.2
1.4
log(C
O2 p
er
capita)
1960 1970 1980 1990 2000year
lco2pc fitted_step93fitted_ramp93 fitted_step97
fitted_ramp97
EU North.7
.8.9
11.1
1.2
log(C
O2 p
er
capita)
1960 1970 1980 1990 2000year
lco2pc fitted_step93fitted_ramp93 fitted_step97
fitted_ramp97
Basic Model
(1) it ity f x
(2) 20 1 2it i it it ity x x
1,... , 1,...,i N t T yit is the logarithm of CO2 emissions per capita, xit is the logarithm of per capita GDP, i is individual effects and εit is the error term
Similar to many other studies (Azhomau et al 2006, JPE) we do not control for other possible determinants There are reasons for this specification. The first is data availability over long time series Second, this specification allows for a greater comparability with existing studies.
Homogeneous panel estimations (SURE, DOLS) We implement several tests of cross section
independence and in all cases they strongly reject the null hypothesis that the errors are independent across countries..
Heterogeneous panel estimations (MG, PMG, Bayes) This situation corresponds to our empirical framework
where: (i) per capita GDP presents high variation across countries, (ii) the different groups of countries cannot be characterised by a common slope and, consequently, there is a high risk of estimating a false curvilinear relation
Semi parametric Income and time effects
Homogeneous Panel estimators
Least Square Dummy (LSD) estimator allowing for individual fixed effects
The Dynamic ordinary least squares (DOLS) estimator The PMG estimator proposed by Pesaran et al. (1999)
which can be considered as an ‘intermediate’ estimator since it allows intercepts, short-run coefficients and error variances to differ freely across cross-sections while holding long-run coefficients the same The first three estimators (FEM, DOLS, PMG) assume
that all cross-section units are independent.
The Driscoll-Kraay (DK) (1998) non-parametric
estimator, which corrects the variance-covariance matrix for the presence of spatial as well as serial correlation
Seemingly Unrelated Regressions (SUR) specification proposed by Zellner (1962) allowing cross section correlation
Heterogeneous Panel estimators
the Swamy (1970) random coefficient, which is a weighted average where the weights are inversely proportional to their variance-covariance matrices
Mean Group (MG) estimator proposed by Pesaran and Smith (1995) for dynamic random coefficient models.
Bayesian approaches the hierarchical Bayes approach (Iterative) Empirical Bayes
benchmark (homogeneous)
Table 3 –Estimators allowing for cross sectional dependence: DK, SUR, DSUR
ModelDC SUR DSUR
coef. t-stat
.
coef.
t-stat.
coef.
t-stat.
coef.
t-stat
.
coef. t-stat
.
coef.
t-stat
.
coef.
t-stat
.
coef.
t-stat
.
coef.
t-stat
.
Group of countries
Umbrella EU north EU south Umbrella EU north EU south Umbrella EU north EU south
GDPpc (linear) 3.716 5.97
16.888
9.96
2.862
4.87
3.072
15.133
15.202
26.165
2.498
13.287
3.253
5.667
10.996
6.062
3.337
4.654
GDPpc (quadratic)
-0.173 -5.23
-0.890
-9.89
-0.132
-4.14
-0.138
-12.54
-0.796
-25.67
-0.113
-11.30
-0.031
-4.613
-0.096
-5.979
-0.038
-4.211
EKC shape inverted U inverted U inverted U inverted U inverted U Inverted U inverted U inverted U inverted U
Turning point ($1995)
46,160.715 13,195.623 51,067.782 68,216.025 14,030.586 63,139.216 87,040.245 14,449.242 33,796.922
Turning point range
out in out out in out out in Out
Table 4 – Heterogeneous estimators: Swamy, MG, Hierarchical Bayes Model Swamy MG Hierarchical Bayes
Group of countries Umbrella EU north EU south Umbrella EU north EU south Umbrella EU north EU south
coef. t-stat.
coef. t-stat. coef. t-stat.
coef. t-stat.
coef. t-stat. coef. t-stat.
coef. t-stat. coef. t-stat. coef. t-stat.
GDPpc (linear) 0.473 4.778 17.492 4.135 0.464 6.705 0.475 3.006 12.262 4.966 0.436 4.955 3.600 36.327 17.494 201.080 2.178 25.326
GDPpc (quadratic) … … -0.922 -4.229 … … … … -0.654 -5.070 … … -0.163
-3.630 -0.922 -36.888 -0.088
-2.667
EKC shape monotonic inverted U monotonic monotonic inverted U monotonic inverted U inverted U inverted U
Turning point ($1995) 13,172.68 11,785.41 62,501.4 13,159.87 236,806.82
Turning point range in in out in out
(…) means not included given not significance
Benchmark (heterogeneous)
for both the Umbrella group and southern European countries, most heterogeneous estimators provide evidence of a linear CO2-GDP
relationship. The estimated elasticity is always slightly lower than 0.5, which is a sign of relative de-linking.
Modelling income and time
A more general and, at the same time, an identifiable EKC specification is given by assuming that the income effect, the effect of (time invariant) unobserved heterogeneity, the effect of time and the idiosyncratic effect are separable:
y_{it}= c{i} + f(x_{it})+ g(t,i)+ ε{it}
where the effect of the time invariant unobserved variables is captured by introducing individual- fixed effects,
HOW THE ISSUES OF SLOPE HETEROGENEITY, NON CONSTRAINED
FUNCTIONAL FORM AND TIME RELATED UNOBSERVED FACTORS AFFECT THE
ESTIMATION OF THE EKC
f(x_{it}) and g(t,i).
Moving on…two steps…
Non constrained functional form and common time effect
we estimate the model with common nonparametric trend in order to avoid the omitted time related factors bias
A nonparametric random growth model
we include individual time trends by adopting a nonparametric extension of the random growth model
''
' ( )it i it iti
y i i s x f t 1
''
' ,it i it iti
y i i s x t 1
Semi parametric models
(relative fit tested by F tests)
Joint factor
GAM Model types
One can cope with fixed effects by applying differences and then use GAM (Azhomau et al. Wp2009)
Analogy: within vs LSDV model
We estimate ai+ f(xit)+uit considering ai as dummies to estimate in the non parametric part (e.g. Basile and Girardi, JEG10; Criado, Valente and Stengos, wp2009)
With N low the estimation is efficient and properly designed
You can also implement your model treating the subject intercepts as random effects (sample population). computationally inefficient if you have large numbers of random effects
GAM individual fixed effects (eq 8 with f(t)=0)
Eu south
Eu NorthUmbrella
These results, thus, are on the one hand quite similar to those commented on above for parametric panel models, in terms of economic significance, but on the other hand at the same time highlight the limits of parametric formulations.
Unobserved common time trends
introducing a common (non parametric) trend of the kind:
y_{it}= c{i} + f(x{it})+ g(t)+ ε{it}
GAM with individual fixed effects and nonparametric common trend (eq 8)
EU southEU North
Umbrella
overall time evolution of per capita emissions is driven more by the unobserved common factors related to various time effects
We believe that the issue is not what penalizes northern EU with regard to income related dynamics, but what has advantaged northern EU regarding the time related effects (over the all period, from the energy shock in the 70's 80's to the environmental policy era in the 90's).
Random growth model
We finally propose a nonparametric variant of the random growth model:
y_{it}= c{i} + f(x{it})+ g{i}(t) + ε{it}
which consists at generalising GAM by making interacting the country's indicator variable with the nonparametric trend.
One main reason is that even countries belonging to similar geographical/economic groups tend to `specialize' with respect to innovation, energy and also policy.
1. Including individual time effects
It is interesting to note that, beyond the economic policy's insights, including individual time effects is also important from a statistical point of view.
Indeed, both the Akaike/Bayesian Information Criterion - AIC and BIC strongly support such specification against the common time effects specification
(non linear) CO2-time shapes, inverted U North America, monotonic Oceania and South EU, Negative for EU NORTH
2. Individual time and income effects
A more general specification can be obtained by considering both individual time effects and individual income effects
it does improve very marginally upon the random growth -- homogeneous income effect specification.
Nevertheless, on the side of economic significance, we highlight that the only two countries showing an inverted U EKC / negative shape for both the income-carbon and CO2-time relationships are Sweden and Finland.
Main evidence
Overall, the countries differ more on their carbon-time relation than on the carbon-income relation which is in almost all cases monotonic positive.
Just a few Nordic countries show a bell curve in both income and time related factors.