Base Paper For 4th Methodology
Base Paper For 4th Methodology
Base Paper For 4th Methodology
Energy Economics
On income and price elasticities for energy demand: A panel data study
Jiti Gao, Bin Peng ⁎, Russell Smyth
Monash University, Australia
a r t i c l e i n f o a b s t r a c t
Article history: Obtaining reliable cross-country estimates of the income and price elasticity of energy demand requires a panel
Received 18 August 2020 data model that can simultaneously account for endogeneity, heterogeneity over time, cross-sectional heteroge-
Received in revised form 24 January 2021 neity, nonstationarity and cross-sectional dependence. We propose such an integrated framework and apply it to
Accepted 5 February 2021
a very large dataset of 65 countries over the period 1960–2016 recently assembled by Liddle and Huntington
Available online 19 February 2021
(2020). We find that while the elasticities of income and price are non-linear, the income elasticity is generally
Keywords:
in the range 0.6 to 0.8 and the price elasticity in the range −0.1 to −0.3. We also find that the income elasticity
Elasticity has been declining since the 1990s, which broadly corresponds to increasing awareness of the negative external-
Energy policy ities associated with burning fossil fuels associated with the Kyoto Protocol. From a policy perspective, that the
Panel data analysis income energy elasticity is less than one, and has been declining since the 1990s, bodes well for climate change
mitigation because it suggests that energy intensity will fall with economic growth.
JEL classification: © 2021 Elsevier B.V. All rights reserved.
C23
O13
Q11
https://doi.org/10.1016/j.eneco.2021.105168
0140-9883/© 2021 Elsevier B.V. All rights reserved.
J. Gao, B. Peng and R. Smyth Energy Economics 96 (2021) 105168
Table 1
Selected studies of income and price elasticities of energy demand. IE, PE and NP stand for income elasticity, price elasticity, and nonparametric, respectively.
Galli (1998) Point estimates for 10 developing Asian countries over 1973–1990 IE 1.18; PE −0.32
Medlock and Soligo (2001) Point estimates for 28 countries (including 7 non-OECD) by sector over IE 0.5–1 depending on sector; PE −0.1
1978–2003
Gately and Huntington (2002) Point estimates for 96 countries over 1971–1997 IE 1 for non-OECD and 0.6 for OECD countries
van Benthem and Romani Point estimates for 17 developing countries over 1978–2003 IE not comparable; PE −0.1
(2009)
Karimu and Brännlund (2013) NP IE and PE for 17 OECD countries over 1960–2006 IE is nonlinear and fluctuates with income level; PE −0.2
van Benthem (2015) Point estimates for 58 countries (28 non-OECD) over 1978–2006 IE near unity; PE insignificant for non-OECD
Labandeira et al. (2017) A meta-analysis of PE of energy demand based on studies since 1990 Long-run PE −0.6; Short-run PE −0.2
Huntington et al. (2019) Review of IE and PE in major industrializing countries since 2000 IE & PE vary across countries and sectors, but generally
inelastic
Liddle and Huntington (2020) Point estimates for an unbalanced panel of 41 non-OECD countries over IE around 0.7; PE insignificant
1973–2016
Liddle et al. (2020) NP IE and PE for 26 middle income countries over 1996–2014 IE in the range 0.6–0.8; PE insignificant or close to 0
shocks. One way to address this issue is to model energy demand using included are nonstationary over the relevant time period. This finding
a non/semi-parametric approach. Semi-parametric modelling allows is not surprising. For instance, Phillips and Moon (1999) point out that
one to examine how the elasticities change over time. This allows for almost every macroeconomic variable in the Penn World Table is non-
more precise estimates than the point estimates which are averages. stationary; of which, GDP per capita is one.
Comparison of the time-varying estimates with the point estimates To address the above issues, our main contribution is to propose a
can also be informative. For instance, if the time-varying estimates do (time-varying) structural equation panel data model, which simulta-
not change much over time, it might be appropriate to use accepted neously allows for endogeneity, heterogeneity across countries and
point estimates as a rule of thumb for forecasting (Liddle and over time, nonstationarity and cross-sectional dependence, and to illus-
Huntington, 2020). In this line of research, there are, however, very trate its application to estimating income and price elasticities of energy
few non/semi-parametric panel data estimates of income and/or price demand using a large cross-country dataset. We incorporate an interac-
elasticity of energy demand. The few examples include Karimu and tive fixed effects structure into the analysis in order to capture depen-
Brännlund (2013), Nguyen-Van (2010), Park and Zhao (2010), Chang dence among countries and time-varying individual heterogeneity. To
et al. (2016), and Liddle et al. (2020). address nonstationarity, we incorporate the approach of Casas et al.
In addition to heterogeneity over time, it is also important to account (2021), which allows for nonstationary regressors, provides a data
for heterogeneity across countries. Traditionally, fixed effects have been driven method to select the number of unobservable factors and does
employed to model individual heterogeneity. Since the seminal studies not limit these factors to be stationary.
by Pesaran (2006) and Bai (2009), interactive fixed effects have been in- It is important to note that while we use the energy elasticity litera-
corporated into panel data models to not only capture time-varying in- ture to illustrate the application of our modelling framework, our pro-
dividual heterogeneity, but also measure cross-sectional dependence. posed framework could be applied to several other areas of energy
In energy demand studies, it is well-recognised that country level and environmental economics that routinely employ cross-country
data exhibit cross-sectional dependence, so it is important to account panel data, such as the Environmental Kuznets Curve and the relation-
for cross-sectional dependence and time-varying individual heteroge- ship between income inequality and environmental degradation.
neity using interactive fixed effects, when using a panel dataset, such While these studies typically address one or more of cross-sectional de-
as the one to be considered in this paper. pendence, endogeneity, heterogeneity or nonstationarity, none of these
We next consider endogeneity. The price of energy is likely deter- studies simultaneously address all of them.
mined endogenously by the same components that determine energy The study that is perhaps closest to ours is Liddle et al. (2020). That
demand (see, e.g., Huntington et al., 2019). GDP per capita, as a measure study also employs a subset of the Liddle and Huntington (2020)
of income, is also almost certainly endogenous. Micro-studies of energy dataset for 26 countries spanning 1996–2014 to estimate income and
demand often address price endogeneity (e.g., Miller and Alberini, price elasticity of energy demand using a semiparametric local linear
2016). While some macro energy studies that have produced point esti- dummy variable estimation method. We differ from that study in the
mates have explicitly addressed price endogeneity (see e.g., Burke and following ways. First, Liddle et al. (2020) do not account for endogeneity
Abayasekara, 2018), most macro studies using panels data models just of income or price. We explicitly address endogeneity using a traditional
employ fixed/time effects to partially account for endogeneity TSLS type estimator for both parametric and semiparametric ap-
(e.g., van Benthem and Romani, 2009; Nguyen-Van, 2010; Chang proaches. Second, the modelling framework in Liddle et al. (2020) re-
et al., 2016). quires the right-hand side variables (including GDP per capita and
Finally, we consider nonstationarity. Many studies on energy de- price) to be stationary. Third, to capture heterogeneity, they use fixed
mand assume that the regressors are stationary over time. A common effects, while we add a factor structure to capture individual and time-
source of misspecification for time series is the violation of the assump- varying heterogeneity. Our results differ from Liddle et al. (2020) in im-
tion of stationarity, which will lead to spurious results (Engle and portant ways. While Liddle et al. (2020) find that the price elasticity is
Granger, 1987). In this regard, a few studies note the importance of ad- largely insignificant and occasionally positive, we find that both income
dressing nonstationarity (e.g., Park and Zhao, 2010; Liddle et al., 2020) and price elasticities are significant and vary over time. Specifically, we
and some, that have produced point estimates, have used error- find that income and price elasticity of energy demand are positive and
correction models (see, e.g., Galli, 1998). negative respectively, consistent with exconomic theory, and that in-
We examine the impact of income (measured by GDP per capita) come has had a stronger impact on energy demand than price over
and price on energy demand for 65 countries from 1960 to 2016 using the time period 1960–2016.
a dataset assembled by Liddle and Huntington (2020). Unit root tests The rest of the paper is organised as follows. Section 2 describes the
for the variables in our study suggest that most of the time series details of the dataset. Section 3 presents a benchmark parametric model
2
J. Gao, B. Peng and R. Smyth Energy Economics 96 (2021) 105168
and the corresponding results. We present the details of our proposed Table 2
semiparametric model and results in Section 4. Section 5 concludes. Ad- Percentage of countries exhibiting nonstationarity for
each variable.
ditional details regarding the estimation method, which is supplemen-
tal to that presented in the main text, is summarized in Appendix A. Variable Percentage
TFC 64.62%
GDP 72.31%
2. Data
Price 92.31%
3
J. Gao, B. Peng and R. Smyth Energy Economics 96 (2021) 105168
Table 3
Results of the constant parameter models from all countries (i.e., Dataset 1). The numbers in parentheses are the 95% confidence interval. The value of the number of factors is calculated as
in Step 2 of Appendix A.2, but for the parametric models.
Two-way fixed effects model Eq. (4) The structural equation model Eqs. (1)–(3)
Table 4
Results of the constant parameter models from the high income countries (i.e., Dataset 2). The numbers in parentheses are the 95% confidence interval. The value of the number of factors is
calculated as in Step 2 of Appendix A.2, but for the parametric models.
Two-way fixed effects model Eq. (4) The structural equation model Eqs. (1)–(3)
for which the long-run elasticities, EG and EP, are the summations of the
3.2. Benchmark results coefficients associated with GDP and Price respectively - see Liddle et al.
(2020) for example.
We present results based on Eqs. (1)–(3) for three cases: The results are presented in Tables 3–6. In the FE model, Tables 4–6
Case 1: y2, it = log GDPit and xit = log GDPi, t−1; show that the coefficients on GDP, Price, EG, and EP show signs which are
Case 2: y2, it = log Priceit and xit = log Pricei, t−1; inconsistent with economic theory from case to case. However, once
Case 3: y2, it = (logGDPit, log Priceit)′ and xit = (logGDPi,t−1, one accounts for endogeneity, the estimates for each of the coefficients
log Pricei,t−1)′. and elasticities in the structural equation model Eqs. (1)–(2) have the
We follow Liddle et al. (2020) in including the one period lag of GDP expected sign and are significant in all of Tables 3–6, although the mag-
and Price, but we treat them as IVs. In addition to the coefficients of the nitudes vary from dataset to dataset. Moreover, in the structural equa-
model, we also examine the long-run GDP and Price elasticities (de- tion model, the estimates not only have the expected sign, but are
noted by EG and EP hereafter). For the purpose of illustration, consider significant in pretty much all cases with all datasets.
Case 3 only. Inserting Eq. (2) into Eq. (1) yields the values of EG and Below, we focus on the results of the structural equation model only.
EP, which are the first and second elements of B0′β0 respectively.4 In First, across Tables 3–6, the coefficient on GDP is always positive, while
the following, we report the estimated values of β0, EG and EP. Note the coefficient on price is always negative, consistent with theory. With
that β0 includes the coefficients presented in Eq. (1), and so measures the full sample (Dataset 1), the magnitude of the elasticity for GDP is
the impact of GDP and Price in the current period on current TFC, i.e. close to 1 in Table 3. While Liddle and Huntington (2020) find that the
short-run elasticities. income elasticity is generally less than unity (e.g., 0.7) for their full sam-
ple, a number of previous studies find the energy income elasticity of
4
The long-run elasticities are calculated in the same way as in Liddle et al. (2020, eq. 5)
demand to be close to 1 (see, e.g., Galli, 1998; Medlock and Soligo,
and Liddle and Huntington (2020, eq. 2) after inserting Eq. (2) into Eq. (1). The coefficients 2001; van Benthem, 2015). When Liddle and Huntington (2020) split
in Eqs. (1) and (2) can be interpreted as the short-run coefficients. the sample into high- and middle-income panels, they find similar
4
J. Gao, B. Peng and R. Smyth Energy Economics 96 (2021) 105168
Table 5
Results of the constant parameter models from the middle income countries (i.e., Dataset 3). The numbers in parentheses are the 95% confidence interval. The value of the number of fac-
tors is calculated as in Step 2 of Appendix A.2, but for the parametric models.
Two-way fixed effects model Eq. (4) The structural equation model Eqs. (1)–(3)
Table 6
Results of the constant parameter models from the dataset of Liddle et al. (2020) (i.e., Dataset 4). The numbers in parentheses are the 95% confidence interval. The value of the number of
factors is calculated as in Step 2 of Appendix A.2, but for the parametric models.
Two-way fixed effects model Eq. (4) The structural equation model Eqs. (1)–(3)
results for each panel. However, when we divide the dataset into high y2,it ¼ B0 ðτt Þ xit þ λ2,it þ ε2,it , ð6Þ
d1
and middle income groups (i.e., Dataset 2 and Dataset 3), the elasticity
for GDP is higher than 1 for high income countries, but lower than 1 where τt = t/T for short. Note that both λ1, it and λ2, it correspond to the
for the middle income countries. The results for the elasticity of GDP factor structure of the constant parameter model. The detailed form of
in Dataset 4, reported in 6, is also close to 1. these terms are clear from the reduced form.
The point estimates on price vary from −0.0376 in Table 6 to We can write Eqs. (5) and (6) in a reduced form as follows.
−0.2566 in Table 4. These results are similar to those in Liddle and
Huntington (2020) for their high income panel, although for their mid- yit ¼ π0 ðτ t Þxit þ Λi f t þ εit , ð7Þ
dle income panel, price was generally insignificant. For all Tables 3–6,
looking at the absolute values of the coefficients, EG and EP, the results where yit = (y1, it, y2, it′)′, and the other variables are defined accord-
of the structural equation model suggest that GDP has a stronger impact ingly. Again, further comments on the model are provided in
on TFC than Price. This result is also consistent with the point estimates Appendix A.1, and the detailed estimation procedure is summarized in
of income and price elasticities reported in studies such as Galli (1998) Appendix A.2.
and Liddle and Huntington (2020).
4.2. Constancy test
4. A semiparametric approach
Before presenting the results from the semiparametric model, we
In this section, we estimate a time-varying structural equation panel first conduct a constancy test to demonstrate that a semiparametric ap-
data model. proach is needed. In particular, we focus on Eq. (7), and test whether the
coefficient of π0 is a constant.
4.1. The time-varying structural equation model Formally, we state the null and alternative hypotheses below:
y1,it ¼ y02,it β0 ðτt Þ þ λ1,it þ ε1,it , ð5Þ ℍ1 : Prfπ0 ðÞ ≡ π 0 g < 1 for all π0 ∈ℝðpþ1Þd :
p1
5
J. Gao, B. Peng and R. Smyth Energy Economics 96 (2021) 105168
0.1
Table 7
Constancy test.
p-value
0
Case 1 0.0000
Case 2 0.0000
Case 3 0.0000
-0.1
Price
needed. We summarize the technical details in Appendix A.3.
We conduct the constancy test for Case 1 to Case 3, and present the -0.3
results in Table 7. The p-values equal 0.000 in all tests, which strongly
indicate that the constant parameter model is not flexible enough for
studying energy demand. -0.4
4.3. Results
-0.5
1.4 1.6
1.4
1.2
1.2 1
GDP
0.8
0.6
0.4
1
0.2
0
1960 1968 1976 1984 1992 2000 2008 2016
GDP
0.8
0.1
0
0.6
-0.1
-0.2
Price
-0.3
0.4
-0.4
-0.5
0.2 -0.6
1960 1968 1976 1984 1992 2000 2008 2016 1960 1968 1976 1984 1992 2000 2008 2016
Fig. 1. Estimates of β0(⋅) for Case 1 using the Data of All Countries (i.e., Dataset 1). Black, Fig. 3. Estimates of β0(⋅) for Case 3 using the Data of All Countries (i.e., Dataset 1). Black,
blue and red solid lines stand for the estimated β0(⋅) using the bandwidths h, hL and hR, re- blue and red solid lines stand for the estimated β0(⋅) using the bandwidths h, hL and hR, re-
spectively. The black dotted lines stand for the 95% confidence interval associated with the spectively. The black dotted lines stand for the 95% confidence interval associated with the
black solid line. black solid line.
6
J. Gao, B. Peng and R. Smyth Energy Economics 96 (2021) 105168
1.8 1.1
1.6
1.4 1
1.2
GDP
1 0.9
GDP
0.8
0.6 0.8
0.4
0.2 0.7
1968 1976 1984 1992 2000 2008 1996 1998 2001 2004 2006 2009 2012 2014
0.1
0.1
0
0
-0.1
Price
Price
-0.1
-0.2 -0.2
-0.3 -0.3
-0.4 -0.4
1960 1968 1976 1984 1992 2000 2008 2016 1996 1998 2001 2004 2006 2009 2012 2014
Fig. 4. Estimates of β0(⋅) for Case 3 using the High Income Countries Data (i.e., Dataset 2). Fig. 6. Estimates of β0(⋅) for Case 3 using the Data of Liddle et al. (2020) (i.e., Dataset 4).
Black line stands for the estimated β0(⋅). The black dotted lines stand for the 95% confi- Black line stands for the estimated β0(⋅). The black dotted lines stand for the 95% confi-
dence interval. dence interval.
capita and are a determinant of energy demand. As shown in Fig. 2, the and elasticities in Figs. 3 and 9 is consistent with the dematerialization
coefficient on energy prices in absolute values increased markedly over process emphasised by authors such as Brookes (1972) in which energy
a similar time-frame to which real GDP was increasing in Fig. 1. Again, intensity initially increases as low-income countries increase their in-
the time-varying pattern in the corresponding elasticities for Case 2, dustrial bases and then declines over time with a sectoral shift from
depicted in Fig. 8, are very similar. energy-intensive heavy industry to light industry and then finally to
When we control for energy prices in Case 3, the time varying pat- the less energy-intensive commercial sector. Studies, such as Medlock
tern in the income coefficient and elasticity looks very different. The and Soligo (2001), document this phenomenon using the quadratic of
mildly non-linear inverted U-shaped pattern in the income coefficients
1.4
1.6
1.4
1.2 1.2
1
GDP
0.8
0.6 1
0.4
0.2
GDP
0 0.8
1960 1968 1976 1984 1992 2000 2008 2016
0.5
0.6
0.4
Price
0.2
1960 1968 1976 1984 1992 2000 2008 2016
-0.5
1960 1968 1976 1984 1992 2000 2008 2016 Fig. 7. Elasticity for Case 1 using the Data of All Countries (i.e., Dataset 1). Black, blue and
red solid lines stand for the estimated elasticity using the bandwidths h, hL and hR, respec-
Fig. 5. Estimates of β0(⋅) for Case 3 using the Middle Income Countries Data (i.e., Dataset tively. The black dotted lines stand for the 95% confidence interval associated with the
3). Black line stands for the estimated β0(⋅). The black dotted lines stand for the 95% con- black solid line.
fidence interval.
7
J. Gao, B. Peng and R. Smyth Energy Economics 96 (2021) 105168
0.05
1.8
1.6
0
1.4
1.2
GDP
-0.05 1
0.8
0.6
-0.1
0.4
0.2
1960 1968 1976 1984 1992 2000 2008 2016
-0.15
Price
-0.2
0.05
-0.25 0
-0.05
Price
-0.3 -0.1
-0.15
-0.35 -0.2
-0.25
-0.4 -0.3
1960 1968 1976 1984 1992 2000 2008 2016 1960 1968 1976 1984 1992 2000 2008 2016
Fig. 8. Elasticity for Case 2 using the Data of All Countries (i.e., Dataset 1). Black, blue and Fig. 10. Elasticity for Case 3 using the High Income Countries Data (i.e., Dataset 2). Black
red solid lines stand for the estimated elasticity using the bandwidths h, hL and hR, respec- line stands for the estimated elasticity. The black dotted lines stand for the 95% confidence
tively. The black dotted lines stand for the 95% confidence interval associated with the interval.
black solid line.
varying estimates for GDP in Chang et al. (2016) and Liddle et al.
(2020). Most countries have experienced a decrease in energy intensity,
GDP in a parametric specification, as do Chang et al. (2016) in a defined as the ratio of energy consumption to real GDP, over the last two
non-parametric framework for specific countries, such as China and decades. Since 1990, global energy intensity has declined at an average
South Korea. rate of 1.2% per year, while for lower-middle income countries this fig-
The finding that the income coefficient and elasticities in Figs. 3 and ure has been higher at 1.8% per year (see e.g. Chen et al., 2019). Hence,
9 have been declining since the 1990s is consistent with the time- the decline in the income coefficient and elasticities in Fig. 3 and Fig. 9
since the 1990s is consistent with the increase in autonomous energy
efficiency over the same period.
1.4
1.2
1.2
1
1
GDP
0.8
0.8
GDP
0.6
0.6
0.4
0.4
0.2
1960 1968 1976 1984 1992 2000 2008 2016 0.2
0
1960 1968 1976 1984 1992 2000 2008 2016
0.05
0
0.5
-0.05
-0.1
Price
-0.15
-0.2
Price
0
-0.25
-0.3
-0.35
1960 1968 1976 1984 1992 2000 2008 2016
Fig. 9. Elasticity for Case 3 using the Data of All Countries (i.e., Dataset 1). Black, blue and -0.5
1960 1968 1976 1984 1992 2000 2008 2016
red solid lines stand for the estimated elasticity using the bandwidths h, hL and hR, respec-
tively. The black dotted lines stand for the 95% confidence interval associated with the
Fig. 11. Elasticity for Case 3 using the Middle Income Countries Data (i.e., Dataset 3). Black
black solid line.
line stands for the estimated elasticity. The black dotted lines stand for the 95% confidence
interval.
8
J. Gao, B. Peng and R. Smyth Energy Economics 96 (2021) 105168
1.1 It is worth noting that using the semiparametric approach, two un-
observable factors are identified (i.e., r = 2) for all Cases 1–3, which is
1 estimated in Step 2 of the estimation approach provided in Appendix
A.2. However, the parametric approach identifies only one unobserv-
GDP
0.9 able factor (see Table 3). This results further reinforces the need to ex-
amine the elasticity of energy demand in a semiparametric setting,
0.8
given that more information can be retrieved. Finally, we emphasize
that as shown in Figs. 1–3 and 7–9, our approach is not sensitive to
0.7
the choices of bandwidth.
1996 1998 2001 2004 2006 2009 2012 2014
4.3.2. Results for Datasets 2–4
In Figs. 4 and 10 we present the Case 3 coefficient and elasticity re-
0.05 sults for the high-income panel (Dataset 2). The coefficient on GDP
per capita is positive and significant and generally in the range 0.8 to
0 1, although it peaked in the mid-1970s at 1.2. The coefficient on GDP
per capita is highly non-linear. It first exhibits a U-shaped pattern
-0.05
followed by an inverted U-shaped pattern. The latter, consistent with
Price
9
J. Gao, B. Peng and R. Smyth Energy Economics 96 (2021) 105168
was generally in the range 0.6 to 0.8 between 1996 and 2014. However, the time-varying structural equation panel data model Eqs. (5)–(7).
comparing the results for price from Datasets 3 and 4 with those in The estimation procedure for the constant parameter model Eqs. (1)–
Fig. 3.b in Liddle et al. (2020), we find more evidence that the coefficient (3) can be considered as a much simplified version, so is omitted.
on price is negative and significant for extended periods. Possible expla- Section A.3 provides the details of the constancy test.
nations for the different results is that in Liddle et al. (2020) the data
have been detrended nonparametrically and that Liddle et al. (2020) A.1. Some facts on the models
do not account for endogeneity of price.
For the parametric model Eqs. (1)–(3), we partition π0 as:
5. Conclusion 0 1
π0,1
B 1d C
Estimating energy elasticities with panel data models present five π0 ¼ @ A:
π0,2
main modelling challenges: ie. addressing endogeneity, heterogeneity pd
over time, heterogeneity across units, nonstationarity and
cross-sectional dependence. While there is a large literature estimating Simple algebra shows that multiplying Eq. (3) by (1, −β0′) yields:
energy elasticities with panel data, these five issues have not been fully
addressed in an integrated framework. Our contribution has been to y1,it −y02,it β0 ¼ π0,1 −β00 π0,2 xit þ 1, −β00 Λi f t þ 1, −β00 ε it , ðA:1Þ
propose such a framework and apply it to a very large panel recently as-
sembled by Liddle and Huntington (2020) that includes data on energy From Eq. (1) it follows:
prices as well as GDP and spans 56 years. Analysing such a large dataset
π0,1 −β00 π0,2 ¼ 0,
which extends back to 1960 means that not only do we employ a model ðA:2Þ
1, −β00 Λi ¼ λ01,i ,
that simultaneously addresses a range of issues that has plagued this lit-
erature, but we present time-varying estimates for energy income and
price elasticity that are, by far, for the longest time period and widest Once π0 is recovered, we can estimate β0(⋅) using π0, 1 − β0′π0, 2 = 0
cross-section of countries available. from Eq. (A.2) provided that π0, 2 has full row rank (and, thus, d ≥ p).
We find that the elasticities of income and price are time-varying. Similarly, for the model Eqs. (5)–(7), we partition π0(τ) as:
For Dataset 1, in Case 3 in which we include income and price in the 0 1
π0,1 ðτ Þ
one specification, the income elasticity is generally in the range 0.6 to B 1d C
π 0 ðτÞ ¼ @ A:
0.8, while the price elasticity in the range − 0.1 to −0.3, although the π0,2 ðτ Þ
income elasticity has been decreasing since the 1990s. The results for pd
the income elasticity are generally consistent with the notion of dema-
terialization in the energy literature. The decline in the income elasticity Then, similar to Eq. (A.2), the following equalities hold:
since the 1990s is consistent with the effect of the Kyoto Protocol reduc-
π0,1 ðτt Þ−β00 ðτt Þπ0,2 ðτt Þ ¼ 0,
ing energy intensity. ðA:3Þ
From a policy perspective, that the income energy elasticity is less 1, −β00 ðτ t Þ Λi ¼ λ1,it :
than one, and has been declining since the 1990s, bodes well for climate
change mitigation because it suggests that energy intensity will fall with Once π0(⋅) is recovered, we can estimate β0(⋅) using Eq. (A.3) pro-
economic growth. More generally, consistent with the conclusion in vided that π0, 2(⋅) has full row rank uniformly.
Chang et al. (2016), our findings point to transnational institutions,
such as the Kyoto Protocol and the Paris Agreement, having an impor- A.2. The estimation procedure for Eqs. (5)–(7)
tant role in reducing energy intensity, both as commitment devices for
the signatories and as barometers of changing public attitudes toward We focus on the estimation of π0(⋅) of Eq. (7). Write Eq. (7) in matrix
fossil fuels and climate change. form as follows.
Appendix A where
The Appendix A presents materials supplemental to the main text. 1. π is (p + 1) × d, Xi = (xi1, …, xiT)′, tr{⋅} stands for the trace operation;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
First, Appendix A.1 elaborates on the two models considered in the 2. K h,τ ¼ diag K h ðτ1 −τ Þ, . . . , K h ðτT −τ Þ , Kh(⋅) = K(⋅/h)/h, and K
main text. In Appendix A.2, we describe the estimation procedure for (⋅) and h are the kernel function and bandwidth respectively;
10
J. Gao, B. Peng and R. Smyth Energy Economics 96 (2021) 105168
−1
0
column rank.
estimate r by
The detailed steps are as follows.
Step 1: For ∀τ ∈ [0, 1], we estimate π0(τ) and F = (f1, …, fT)′ by ( ⁎ ⁎
! ⁎
!)
b
η‘þ1 b
η‘ b
η‘
br ¼ arg min ⋅I ⁎ ≥ε N þI ⁎ <ε N ,
1 1≤‘≤J b⁎
η‘ b0
η b0
η
πbðτÞ, b
F ¼ argmin Q ðπ, F Þ subject to F 0 F ¼ IJ , J≥r, ðA:6Þ
T n n o
o−1
⁎
where ε N ¼ ln max b η0 , N .
where J is a user chosen large constant and r is defined as in Section 4.1. Step 3: Update the estimates of π0 and F by
Step 2: Let b
ηj be the jth largest eigenvalue of the estimated sample co-
1
variance matrix e, e
π F ¼ arg min Q ðπ, F Þ subject to F 0 F ¼ IJ , J≥br :
T
N
0
b ðτ Þ ¼ 1 ∑ K
Σ b0 Y i −X i πb0 ðτÞ K h,τ :
h,τ Y i −X i π 0ðτ Þ
0
N i¼1 e ie
exit −Λ
eit ¼ yit −π e i ¼ 1 Y i −X i πe0 e
Let further u f t and Λ T F, where
0
Define a mock eigenvalue e e e
F ¼ f 1 , . . . , f T . Calculate the statistic:
0
0 0
N
b
η0 ≔∥ N1 ∑i¼1 b 2 b
Y i −X i π ðτÞ K h,τ Y i −X i π ðτ Þ ∥2 , where ∥ ⋅ ∥2 stands for
1 τ −τ
the spectral norm. We estimate r by LNT ¼ pffiffiffiffiffi ∑∑1031 u ejs 131 K t ⁎ s ,
eit u
⁎ h
N T h i, j t≠s
b
η‘þ1 b
η b
η
br ¼ sup arg min ⋅I ‘ ≥εN þ I ‘ <ε N , where h ∗ is a bandwidth, and ∑i, j and ∑t≠s read ∑N N
1≤‘≤J b
η‘ b
η0 b
η0 i=1∑j=1 and
τ∈½0, 1
T T
∑t=1∑t=1, t≠s, respectively. By Corollary 2.1 of Casas et al. (2021),
−1 LNT→DN(0, σ2ℓ), where σ2ℓ = 2δ4u ∫ K2(w)dw, in which δ2u can be estimated
where εN ¼ ln max η b0 , N . The values of br in Tables 3–6 are cal-
2
∑t¼1 ∑i¼1 1031 u
T N
culated in the same fashion, but for the parametric models of Section 3. by BNT ¼ NT
1 eit consistently.
Step 3: Update the estimates of π0(τ) and F by
1 Appendix B. Supplementary data
πeðτÞ, e
F ¼ arg min Q ðπ, F Þ subject to F 0 F ¼ I J , J≥br : ðA:7Þ
T
Supplementary data to this article can be found online at https://doi.
eðτÞ and e
π F are the final estimates of π0(τ) and F. org/10.1016/j.eneco.2021.105168.
Finally, we comment on the choice of kernel function and the band-
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