Proof. Assume that there is an alternative partition. Now there must be a substructure in the alternative partition that, first, contains a equation that is in a substructure of original partition, and, second, contains another equation that is not in substructure of original partition. Therefore these substructures are not disjoint, which contradicts Lemma 1.
Appendix B. Calculating the causal effect
Consider a causal model with continuous variables and differentiable model functionsgwith additive interventions that define deep model function
f: Rn→Rn. Let h(y, x, a, x0, a0)≡ f(y, x, a) x−x0 a−a0 = g(y, x+a) x−x0 a = 0
be a structure of the corresponding causal model.
LetJh denote the Jacobian determinant of hderivated over v = (y, x, a),
that is Jh = ∂h1 ∂v1 ∂h1 ∂v2 · · · ∂h1 ∂vn ... ... ... ... ∂hn ∂v1 ∂hn ∂v2 · · · ∂hn ∂vn
and let Jh(i, j) denote a corresponding determinant where ith row has been
replaced by the derivate ∂h ∂a0 j, that is Jh(i, j) = ∂h1 ∂v1 · · · ∂h1 ∂vi−1 ∂h1 ∂a0 j ∂h1 ∂vi+1 · · · ∂h1 ∂vn ... ... ... ... ... . .. ... ∂hn ∂v1 · · · ∂hn ∂vi−1 ∂hn ∂a0 j ∂hn ∂vi+1 · · · ∂hn ∂vn .
Now the causal effect is defined by equation dvi
da0
j =−
Jh(i,j)
Jh .
With some elementary but heavy algebra, one can show that dvi
da0
j =
dvi
dxj,
where the latter derivate is calculated from set of equationsg(y, x) = 0. This means, that in the additive interventions case causal effects can be calculated simply from the model equations.
Appendix C. Definition of carbon emissions
In this appendix we derive a identity from the definitions of the dataset. This identity is the source the second problem presented in Section 4.
First, the dataset in use (WDI) defines carbon dioxide emissions as a linear function of fossil fuel combustion and cement manufacturing. The amount of carbon dioxide emissions caused by combustion is determined by the chemical composition of the fuel. The emitted amount of carbon dioxide is calculated by multiplying the amount of fuel usage by a constant factor prescribed by the chemical properties of the fuel. Thus, the total carbon dioxide emissionsCtis a linear combination of the usage of oilEtoil, solid fuels
Esolid
t , natural gasE gas
t , and gas flaringE f lare
t , in addition to emissions from
cement manufacturing St, all measured in per capita term. More formally,
that is,
Ct≡αoilEtoil +αsolidEtsolid+αgasE gas
t +αf lareE f lare
t +St, (C.1)
where αoil, αsolid, αgas, αf lare > 0 are the related ratios of emissions to fuel
quantity. (See Boden et al., 2009)
Second, total energy use Et can be defined as the sum of oil Etoil, solid
fuelsEtsolid, natural gasEtgas, and other energy sourcesEtother, such as nuclear
energy and renewable fuels, which do not cause emissions in the aforemen- tioned sense. Gas flaring does not result in energy production. Therefore
Et≡Etoil+Etsolid+E gas
t +Etother.
To clarify the notation we define two sets of variable: the set of energy commodities affecting carbon dioxide emissions, C ={oil, solid, gas, f lare}, and the set of energy commodities that amount to total energy use, E = {oil, solid, gas, other}.
Next let’s define the proportions of fuel commodities in terms of total energy use, qti ≡ E i t Et ,
where qi
t ≥ 0 for all i ∈ E and
P
i∈Eqti = 1 for any t. By rearranging and
plugging this into identity (C.1) to eliminate Ei
t for each i, we get
Ct≡Et
X
i∈C∩E
qtiαi+αf lareEtf lare+St.
By interpreting the sum term as the average emissions rate of energy con- sumption, we can identify it ascarbon intensity and denote it by At, so that
Ct ≡EtAt+αf lareE f lare
t +St. (C.2)
This is simply an accounting identity derived from the definitions of the dataset, so it must be satisfied by the observed values.
To derive an algebraically more convenient form, we note that gas flaring and cement manufacturing amount only to a percent of total carbon emissions in the data, thus they can be omitted, i.e. set to zero. Therefore taking a natural logarithm of equation (C.2) gives
ct=et+at,
where the variables are the corresponding logarithms of the capital letter variables.
Appendix D. Short run models
Appendix D.1. Causal ordering of VECMs
ModelB in Section 2.6 consists of equations (5)–(8) and forms a structure which can be solved to get graph
y10 z01 ∆y11 y11 z11 ∆y21 y21 · · · y20 z02 ∆y21 y12 z12 ∆y22 y22 · · · y3 0 ∆y31 y13 ∆y23 y23 · · · ε11 ε2 1 ε31 ε12 ε2 2 ε3 2 ... (D.1)
when the values yi
0 are given.
Model A replaces equation (8) with z02 = 0. The resulting graph equals (D.1) when the dotted arrows are removed and variables z2
t are put in the
first row. Model A lacks the causal relationships represented by the dotted arrows in graph (D.1) and therefore has a different causal effects.
Appendix D.2. Problem 2 in the short-run model
Consider the VAR model equation (13) for timet = 0,1, . . . , T. To avoid the problem described in Section 4.1 we omit the equation describing yt2. That is, xt=a0+ p X i=1 Aixt−i+εt, t= 0,1, . . . , T,
where xt = (ct, et, yt) is a vector of logarithms of observed variables, a0 = (ac 0, ae0, a y 0) is a vector of constants,Ai = [Aci Aei A y i] is a matrix of parameters
for lag i, and εt = (εct, εet, ε y
We supplement this with equation (12) for each t. Our exogeneity equations are c0 = c0
0, e0 = e00, y0 = y00, and εt = ε0t for
all t= 0,1, . . . , T.
Now we have 4(T + 1)) model equations andT + 4 exogeneity equations and a equal amount of variables, 5T + 5, forming a structure. Now we get graph a0 a1 c0 e0 c1 e1 y1 c2 e2 y2 y0 ε0 ε1 ε2 ,
which shows the same problematic causal ordering as in the long-run model.
Appendix E. Explaining the fit
In this appendix we consider Ang’s model’s relation to the data. In the section 4.2 we have assumed that Ang’s model is correct (only the interpre- tation is wrong) and disregarded the problem shown in Section 4.1. So how can we account for the good fit of the model? As we have shown, in Ang’s model changes in carbon emissions result from changes in carbon intensity. Therefore we look into this variable.
Carbon intensityathas decreased in France (Figure E.4) because of a de-
cline in the share of heavily polluting fuels like coal. They have been replaced or outgrown by the use of oil, natural gas and nuclear energy (Kaufmann, 1992). Especially in the case of France, is seems that nuclear energy has had a significant impact (see Iwata et al., 2010).
This is in line with the traditional CKC-hypothesis: the decline in carbon intensity is a result of economic growth, however this is contrasted by a growth in energy use, resulting in an “inverted U”-shape between emissions and output.
Because model equation (9) is equivalent with (19), we should see an “inverted U”-shape also in carbon intensity. Figure E.4 does not suggest
such a relationship for France in the period 1960-2006. But why then does Ang get significant estimates?
9.0 9.2 9.4 9.6 9.8 10.0
−6.4
−6.2
−6.0
−5.8
Logarithm of GDP per capita
Logar
ithm of carbon intensity
Figure E.4: The logarithm of Carbon intensity of energy use, measured in kilograms of carbon dioxide emissions per kilogram of oil equivalent energy, and the logarithm of per capita GDP.
The delusive fit of the parabola can be explained by inspecting the fitted model in Ang (2007). Ang reports the parameter estimates β0 = −161.38,
β1 = 2.25, β2 = 31.11, and β3 = −1.67. By using these in model (9), rearranging, and taking the conditional expected value we get equation
ct−2.25et+ 161.38 = 31.11yt−1.67yt2. (E.1)
The left hand side of equation (E.1) is the part that is alloted for output to explain. Becauseβ3 is negative, the value of the right hand side as a function of yt is an upside-down parabola. Suppose this is truly so. Then, to satisfy
The values of the left side terms are plotted against yt in Figure E.5
along with the regression curve (dashed curve), i.e the right hand side of equation (E.1).30 The misleading goodness of the fit of the “inverted U”- shaped curve found by Ang (2007) can be explained by observing, that most of the observations lie on the right hand side of the parabola.
8.8 9.0 9.2 9.4 9.6 9.8 10.0 10.2 143.5 144.0 144.5 145.0 145.5 146.0 146.5
Logarithm of GDP per capita
The par
t not e
xplained b
y energy use
Figure E.5: The solid line is the relationship between the logarithm of per capita CO2
emissions unexplained by energy use and the logarithm of per capita GDP. The dashed line is the corresponding regression curve, i.e. the right hand side of equation (E.1), and the vertical line indicates the tipping point.
However, this does not give a reason to suspect the existence of a tipping point in the carbon intensity or an "inverted U"-shape. As in Ang’s model,
30
The seemingly biased fit in Figure E.5 might be due strong sensitivity to the rounding of the parameter values. Alternative parameter that have the same rounded value can shift the regression curve to give an opposite bias. On the other hand, this might be due to problem described in Section 4.1.
there exists a CKC if and only if the plot in Figure E.5 is polynomial, we can not arrive to our result simply by assuming it. That is, if we can not justify Figure E.5 (or Figure E.4) representing a polynomial shape (with a tipping point in the sample), we can not justify the existence of a carbon Kuznets curve.
Why the do we see a parabola shape in Figure 1? Here energy use is not fixed, thus the parabola is formed by the rising energy use and the declining carbon intensity. But in Ang’s model the term 31.11yt−1.67y2t plots a curve
for a fixed energy level. Therefore this term cannot be plotted in Figure 1, without letting energy use vary. This connection between output and energy use is the source of the problem in Section 4.3.
Appendix F. Mathematical derivations for section 5.3.
The magnitude of the causal effect of output yt on carbon emissions ct
in structure (24) with additive interventions can be assessed by applying the implicit function theorem to get the total derivative
dct dyt =− 0 −1 −1 −(β2+ 2β3yt) 1 1−β1 −e′ 0 1 1 −1 −1 0 1 1−β1 0 0 1 , where we denotee′ = ∂et
∂yt. By calculating the determinants, we get expression
−(β2+ 2β3yt)−e ′
(−1 + (1−β1))
−1 ,
which can be simplified to determine the (total) causal effect dct
dyt
= (β2+ 2β3yt) +e
′
β1. (F.1)
Now causal effect (F.1) can be compared to the biased interpretation in expression (17). We see that the model specification of Ang (2007) is biased by the term −e′
Next we show that the tipping point of CKC is at a higher level of output when bias exists. In the unbiased case the tipping point y∗
t is such that the
causal effect (25) equals zero. This is equivalent to
y∗ t = −β2−e′ β1 2β3 .
Similarly, in the biased case the tipping point y∗∗
t satisfies y∗∗ t = −β2 2β3 . Now, whene′
β1 >0, addingβ2 to both sides givesβ2+e′
β1 > β2. Because
β2 is positive andβ3is negative according to the CKC-hypothesis, we see that
−β2−e′ β1 2β3 > −β2 2β3 .
By noting the tipping points, we get
y∗ t = −β2−e′ β1 2β3 > −β2 2β3 =y∗∗ t .
That is, the true tipping point occurs at a higher level of output.
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