LEM

WORKING PAPER SERIES
Estimating the Economy-Wide Rebound Effect Using
Empirically Identified Structural Vector Autoregressions

Stephan B. Bruns a,b
Alessio Moneta c
David I. Stern dCentre for Environmental Sciences, Hasselt University, Belgium.
Department of Economics, University of Göttingen, Germany.
cInstitute of Economics and EMbeDS Department, Scuola Superiore Sant’Anna, Pisa, Italy.
dCrawford School of Public Policy, Australian National University, Canberra, Australia.
ab

2019/27

August 2019
ISSN(ONLINE) 2284-0400

Estimating the Economy-Wide Rebound Effect Using
Empirically Identified Structural Vector Autoregressions
Stephan B. B RUNS1

Alessio M ONETA2David I. S TERN3

August 19, 2019

Abstract
The size of the economy-wide rebound effect is crucial for estimating the contribution thatenergy efficiency improvements can make to reducing greenhouse gas emissions and for understanding the drivers of energy use. Existing estimates, which vary widely, are based oncomputable general equilibrium models or partial equilibrium econometric estimates. The former depend on many a priori assumptions and the parameter values adopted, and the latter do
not include all mechanisms that might increase or reduce the rebound and mostly do not credibly identify the rebound effect. Using a structural vector autoregressive (SVAR) model, we
identify the dynamic causal impact of structural shocks, including an energy efficiency shock,applying identification methods developed in machine learning. In this manner, we are able toestimate the rebound effect with a minimum of a priori assumptions. We apply the SVAR to
U.S. monthly and quarterly data, finding that after four years rebound is around 100%. Thisimplies that policies to encourage cost-reducing energy efficiency innovation are not likely tosignificantly reduce energy use and greenhouse gas emissions in the long run.

Keywords: Energy efficiency, Rebound effect, Structural VAR, Impulse response functions, Independent component analysis.
JEL classification: C32, Q43
1
Centre for Environmental Sciences, Hasselt University, Belgium and Department of Economics, University ofGöttingen, Germany. Email: stephan.bruns@uhasselt.be2
Institute of Economics & EMbeDS, Scuola Superiore Sant’Anna, Pisa, Italy. Email: a.moneta@santannapisa.it
3
Crawford School of Public Policy, Australian National University, Australia. Email: david.stern@anu.edu.au

We thank the Australian Research Council for funding under Discovery Project DP160100756: “Energy Efficiency
Innovation, Diffusion and the Rebound Effect.” We thank Yingying Lu for research assistance in developing the proposal.

1

Introduction

Governments and international organizations are expecting energy efficiency improvements tomake a major contribution to reducing greenhouse gas emissions and improving energy security(Stern, 2017). But energy savings are usually less than the improvement in energy efficiency. The
size of this rebound effect is crucial for estimating the contribution that energy efficiency improvements can make to reducing greenhouse gas emissions as well as for understanding the drivers ofenergy use. The micro-economic direct rebound effect occurs when an energy efficiency innovation reduces the cost of providing an energy service, such as heating, lighting, or transport, and,as a result, users increase the use of the service offsetting some of the energy efficiency improvement. But there are also changes in the use of complementary and substitute goods or inputs andother flow-on effects that affect energy use across the economy known as indirect rebound effects.
Together these constitute the economy-wide rebound effect.
The size of the economy-wide rebound effect is controversial (Gillingham et al., 2016) and insufficiently researched (Turner, 2013). Existing estimates vary widely from “backfire” (also known
as the “Jevons paradox”) where energy use increases following an efficiency improvement to superconservation where energy use falls by more than the efficiency improvement (Stern, 2011b; Saunders, 2013; Turner, 2013). Previous empirical research uses either computable general equilibrium(CGE) models or partial equilibrium econometric models to estimate the economy-wide reboundeffect. The former depend on many a priori assumptions and the parameter values adopted and thelatter do not include all mechanisms that might increase or reduce the rebound and mostly do notcredibly identify the rebound effect. In this paper, we develop a structural vector autoregressive(SVAR) model that is empirically identified using independent component analysis, which imposes
This paper was presented at the 41st IAEE International Conference in Groningen, the 5th Asian Energy Modelling
Workshop in Singapore, the Arndt Corden Department of Economics at the Australian National University, the 4thMonash Environmental Economics Workshop, the 8th Mannheim Energy Conference, the School of Management at
the Polytechnic University of Milan, GREDEG, University of Nice - Sophia Antipolis, and RWI Essen. We thank Don
Fullerton, Mattia Guerini, Rocco Mosconi, and other attendees at our presentations for helpful comments.

2statistical conditions on the shocks. We apply the model to U.S. data, finding that after four years
the economy-wide rebound is around 100%.
Turner (2013) notes that there is a lack of consensus in the rebound literature on what is meantby energy efficiency. Some authors include substitution of capital or materials for energy, suchas the installation of insulation, in their definition of energy efficiency improvements (e.g. Sorrell
et al., 2009) or examine the secondary changes resulting from an initial behavioral energy conserving action (van den Bergh, 2011). We focus on rebound effects due to energy-saving technologicalchange. Therefore, we define energy efficiency improvements as those that save energy due to theadoption of more efficient cost-reducing technology and define the rebound effect as the resulting
behavioral responses of economic agents that cause the actual energy savings to differ from thepotential energy savings.
Most empirical research on the rebound effect focuses on the direct rebound effect (Sorrell
et al., 2009) where households and firms consume more energy services in response to efficiency
improvements that reduce the energy required to provide the same level of service and, therefore,
its cost. Indirect rebound effects include the energy use effects of: the increase in demand forcomplementary energy services (and reduction in demand for substitutes); the increase in the use ofenergy to produce other complementary goods and services; the effect of reduced energy prices dueto the fall in energy demand (Borenstein, 2015); and a long-run increase in total factor productivity,
which increases capital accumulation and economic growth and, as a result, energy use (Saunders,
1992). These direct and indirect effects sum to the economy-wide rebound effect. Others (e.g.
Azevedo, 2014; Gillingham et al., 2016) define changes in prices and growth effects as macro
effects that are distinct from indirect rebound effects.
Estimates of the size of the direct rebound effect tend to be fairly modest positive numbers (Sorrell et al., 2009). It is usually assumed that the indirect rebound is positive and that the economywide rebound will be larger in the long run than in the short run (Saunders, 2008). Turner (2013)
argues, instead, that because the energy used to produce a dollar’s worth of energy is higher than3the embodied energy in most other goods, the effect of consumers shifting spending to goods otherthan energy will mean that the indirect rebound could be negative and the economy-wide reboundmay also be negative in the long run. Borenstein (2015) presents further arguments for negativerebound. Lemoine (2018) conducts a general equilibrium analysis of the rebound effect. Assumingthat all sectors share the same technology, general equilibrium effects amplify the partial equilibrium rebound as investigated by Saunders (1992). With heterogeneous technologies generalequilibrium effects amplify the rebound for low elasticities of substitution and reduce it for highelasticities of substitution between energy and non-energy inputs in production. Backfire is possible for elasticities of substitution less than unity, especially for innovations in those sectors that are
relatively energy inefficient or energy intensive. In general, this analysis shows that the economywide rebound effect is likely to be large and backfire is likely. Lemoine assumes that there is a
fixed endowment of capital and, by using a CES utility function, that all goods are p-substitutes(Stern, 2011a) in consumption. The former assumption reduces the rebound effect, as Saunders(1992) shows. Lemoine comments that allowing for complementarity in consumption could makenegative rebound possible.
As mentioned above, evidence on the size of the economy-wide rebound effect to date depends
on simulation models (e.g. Turner, 2009; Barker et al., 2009; Turner and Hanley, 2011; Broberg
et al., 2015; Adetutu et al., 2016; Koesler et al., 2016; Lu et al., 2017; Wei and Liu, 2017) orpartial equilibrium econometric estimates (e.g. Adetutu et al., 2016; Saunders, 2013; Malpede andVerdolini, 2016; Orea et al., 2015; Shao et al., 2014; Lin and Du, 2015). Turner (2009) finds
that, depending on the assumed values of the parameters in a CGE model, the rebound effect for
the UK can range from negative to more than 100%. Therefore, CGE models do not provide
strong evidence on the size of the economy-wide rebound effect. A simpler simulation method
was proposed by Saunders (1992, 2015). This “Solow approach” uses a CES production function
with factor-augmenting technical change. Assuming that the energy price is constant, the modelcomputes the effect of energy-augmenting technical change on energy use. The method depends
on having good estimates of the production parameters – a notoriously difficult problem (León-4Ledesma et al., 2010) – assumes a very simple structure to the economy, and because the energy
price is held constant is a partial equilibrium approach.1
In between these two extremes, Rausch and Schwerin (2018) build a small general equilibrium
macro-economic model that is calibrated on U.S. annual data from 1960 to 2011. Their model
is a so-called putty-clay model, where once a capital vintage is chosen, no substitutability between energy and capital is possible. The energy efficiency – energy services per unit energy –of energy-using capital vintages is chosen depending on capital and energy prices at the time ofinvestment. Therefore, they do not distinguish between energy-capital substitution and energy augmenting technical change. The calibration compares observed and counterfactual scenarios withno energy efficiency improvements by holding energy and energy-using capital prices constant,finding a rebound of 102%.
Several methods have been proposed to econometrically estimate the rebound effect (e.g. Shaoet al., 2014; Lin and Du, 2015; Galvin, 2014; Saunders, 2013; Orea et al., 2015), but all of theseare partial equilibrium methods and/or do not credibly identify a causal effect of energy efficiencychanges on energy use, which is needed to claim a rebound effect (Gillingham et al., 2016).2 The
best existing approach, in our opinion, is represented by Adetutu et al. (2016), who use a stochastic
frontier model to estimate energy efficiency and then a dynamic panel model to estimate the effectof efficiency on energy use. Again, they control for energy prices and output resulting in a partial
equilibrium estimate. They estimate that in the short run rebound is 90% while in the long run
super-conservation occurs with a negative rebound of 36%.
Historical research hints that the economy-wide rebound effect could be large. Both van Benthem (2015) and Csereklyei et al. (2016) find that energy intensity in developing countries today issimilar to what it was in today’s developed countries when they were at similar income levels. But,
1

If we relax the constant price assumption this would be a single sector CGE model. Brockway et al. (2017) provide
an empirical implementation of Saunders approach that shows the difficulty of getting good empirical estimates of the
production function.
2
For example, some studies (e.g. Lin and Du, 2015) assume that changes in (intra-industry) energy intensity areequivalent to changes in energy efficiency. But energy intensity already incorporates rebound as well as the effects ofmany other variables.

5

van Benthem (2015) argues that the energy efficiency of many products currently sold in developing countries is much better than that of comparable products sold in developed countries when
they were at the same income level. He finds that energy savings from access to more efficienttechnologies have been offset by other trends, including a shift toward more energy-intensive consumption bundles and compositional changes in industry such as outsourcing. Though such studies
cannot identify causal effects, they suggest that the economy-wide rebound effect is close to 100%.
Hart (2018) provides a theoretical model of this shift to more energy intensive goods – this could
be caused by changing patterns of consumption with rising income, and/or improvements in energy
efficiency that reduce the costs of energy services – in other words, the rebound effect.
SVARs have several advantages in the context of estimating the economy-wide rebound effect.
SVAR models are small, multivariate, dynamic, time series econometric models that are estimateddirectly from the data but have restrictions imposed to identify the effects of specific structural
shocks. We use a data-driven approach to identify the model, based on general statistical assumptions, thus avoiding economic-theoretic restrictions. Unlike previous econometric approaches inthe economy-wide rebound literature, impulse response functions derived from SVAR models cancapture general equilibrium effects, as all the variables are endogenous and can evolve in response
to a shock. Moreover, SVAR models can recover the response to true exogenous shocks addressing the credible identification issue. On the other hand, some adaptation may already occur as anenergy efficiency improvement is “installed” that our empirical measure of the energy efficiency
shock will miss. Therefore, our estimate of the rebound effect is a lower bound on the true effect.
SVAR models have more parameters than reduced-form vector-autoregressive (VAR) models.
The reduced form parameters can be estimated directly from the data using standard (e.g. OLS)
regression methods. The structural parameters are then recovered by applying identifying restrictions, which are usually based on economic theory. Instead, we identify SVAR models exclusively
based on statistical theory. There is a quite established econometric tradition of identification
methods based on atheoretical search procedures (e.g. Swanson and Granger, 1997; Bessler andLee, 2002; Demiralp and Hoover, 2003; Moneta, 2008). This literature shows that tests of zero6

partial correlations among the estimated errors of the VAR model allow partial identification of
the SVAR model. This specific approach, although it eschews economic-theoretic assumptions, isbased on graph-theoretic conditions (Pearl, 2009; Spirtes et al., 2000), whose reliability in an economic time-series context is often hard to assess (see Hoover, 2001). Moreover, it typically makes
use of the normality assumption, which can fail to hold in economic data.
Thus, in this paper we use a statistical identification procedure based on a quite different framework. This framework is called Independent Component Analysis, a set of tools that has beenshown to be particularly powerful in the statistical identification of SVAR models (see e.g. Moneta
et al., 2013; Capasso and Moneta, 2016; Gouriéroux et al., 2017; Lanne et al., 2017; Herwartz,2018). Its key assumptions are the statistical independence of the shocks and the non-Gaussianityof the data, which can be easily checked empirically.
The next section of the paper lays out our theoretical approach that guides the development ofan empirical model. The third section presents the econometric approach. The empirical model islaid out in the third section. The fourth section discusses the data and results, and the final section
provides conclusions and discussion.

2

Theoretical Background

Thinking of energy use as the equilibrium outcome of the demand and supply of energy, the major
factors resulting in changes in energy use will be changes in the price of energy and in income –at the macroeconomic level GDP. Building on Blanchard and Quah (1989), King et al. (1991), andKilian (2009), we can represent this vector of three variables as the outcome of cumulative shocks
to GDP, the price of energy, and a residual energy-specific shock:∆xt = µ +∞X

Bj εt−j(1)

j=0where xt = [et , pt , yt ]0 is the vector of the logs of energy use, the price of energy, and GDP,7

respectively, εt = [εet , εpt εyt ]0 is the vector of shocks with var(εt ) = I, and µ is a vector of
constants.3 We interpret εet as an energy efficiency shock, as it represents the exogenous reduction
in energy use that are not due to exogenous shocks to GDP or energy prices.
We measure energy quantity here in BTU rather than a volume index because our focus is on
the standard definition of the rebound effect, which refers to heat units of energy. The price ofenergy is, therefore, the total cost of energy divided by BTUs. Changes in this energy price may
reflect a shift in the energy mix as well as changes in the prices of individual energy carriers. As
energy inputs vary in their productivity or energy quality (Stern, 2010), a shift to higher quality
energy carriers such as primary electricity instead of coal would tend to reduce energy use, ceterisparibus. Shifts in economic activity from more energy intensive sectors to less energy intensive
sectors and vice versa, will also affect energy use (Stern, 2012). Therefore, we consider a second
model that includes measures of the structure of the economy and energy quality. This five variable
and shock framework accounts for the most important other factors:∆x̃t = µ̃ +∞XB̃j ε̃t−j

(2)

j=0

where x̃t = [et , pt , yt , st , qt ]0 and s and q are the logs of structure (in practice the log of industrial
production) and energy quality variables, ε̃t = [ε̃et , ε̃pt , ε̃yt , ε̃st , ε̃qt ] is the vector of shocks, and µ̃
is a vector of constants.
The shocks are identified by estimating a structural vector autoregression as described in thefollowing section. We use the impulse response function of energy with respect to the energy
efficiency shock to measure the rebound effect. Using the subscript i to denote the number of
periods since the energy efficiency improvement, the rebound effect is given by
Ri = 1 −∆ei
Actual=1−∆êPotential

3(3)

As the model assumes it is equally likely that the stochastic component of a technology shock is positive or negative
there should also be a constant negative drift term in log energy use (see King et al., 1991, Equation 2). This may be thecase for log GDP too. Potential cointegrating relations may also need a constant term.

8where ∆ê is the potential change in energy use, and ∆ei is the actual change in energy use.4 We
measure the potential energy savings as the energy efficiency shock.
As an example, if in response to a 1% improvement in energy efficiency actual energy use
declines 0.5%, the rebound effect is 50%. On the other hand, if energy use actually increased by
0.2%, rebound would be 120%. Figure 1 shows the impulse response function of the log of energy
with respect to an energy-specific shock. Initially, energy use is reduced by 1% in response to theshock. Over time these savings decrease and, in this example, eventually energy use increases overits pre-shock level so that there is backfire.
0.6
0.4
0.2

Actual

0

lnE

-0.2
-0.4

Potential
-0.6
-0.8
-1
-1.2

Time

Figure 1: The rebound effect.
Assuming that our model captures the important factors that affect energy use apart from energy
efficiency, there are two important limitations on our ability to identify energy efficiency shocks
and the rebound effect. Not all energy efficiency changes might be captured by our identified
energy efficiency shock and we will not be able to account for instantaneous rebound that takes
place at i = 0.
4
Usually, the rebound effect is presented in terms of the potential and actual energy savings. The signs of these terms
are the opposite of the actual and potential change in energy use, but as a result the rebound formula is identical. As ourmodel tracks changes in energy use, we prefer to define the rebound effect in terms of changes in energy use rather thansavings.

9

Price shocks might affect the rate of energy efficiency improvements too. Note, however, that it
is not changes in prices that directly cause changes in technology in the theory of directed technicalchange. Rather the level of price affects the rate of innovation (Acemoglu, 2002). If the elasticity
of substitution between energy and other inputs is less than unity, then an increase in the priceof energy relative to other inputs will increase the rate of energy-augmenting technical change(Shanker and Stern, 2018).
If energy efficiency improvements are positively correlated with labor-augmenting technicalchange then shocks to GDP due to labor-augmenting innovations will be associated with improvements in energy efficiency. Our energy efficiency shocks can only measure the part of energy
efficiency improvements which are orthogonal to labor augmenting technical change shocks. Ourestimate of the rebound effect will be only that in response to these energy-specific efficiency improvements. If the response of energy use to other innovations is different then we will not capture
the average rebound effect in response to all energy efficiency improvements.
Some of the rebound may happen contemporaneously with the energy efficiency improvement.
For example, a car manufacturer might introduce a new model with a more fuel-efficient engine
that is also larger and heavier than the previous model, so that the fuel economy of the new modelshows less improvement than the engine efficiency improvement.5 New more energy efficienthouses might be larger than existing houses thus requiring more energy services than older houses.
Consumers might also immediately adapt their behavior to the new technology. Our approach tomeasuring the size of the rebound relies on the rebound taking place over a period of time. If all
the rebound occurred instantaneously we would measure 0% rebound.
Figure 2 shows how this will affect the estimated rebound. The observed energy efficiency
shock is a fraction of the true shock. The outcomes show two potential cases where we assume thatthe observed shock is 75% of the true energy efficiency shock. In the energy saving case in Figure
5

In both the U.S. (Knittel, 2011) and Austria (Meyer and Wessely, 2009) the weight and power of cars and lighttrucks increased in recent decades as engine fuel efficiency increased. In the U.S. the actual fuel economy of cars only
increased by 6.5%, while horsepower increased by 80% and weight by 12%. There was also a strong shift towards lighttrucks, which have lower fuel economy and increased in power and weight by even more.

10

0.5
2, the true rebound is 50% but the observed rebound is 1 − 0.75
= 33%. In the backfire case, the0.25
true rebound is 125%, but the observed rebound is 1 + 0.75
= 133%. So, where there are energy

savings our estimated rebound will underestimate the true rebound and where there is backfire ourestimated rebound will exaggerate the rebound. The closer the true rebound is to 100%, the smaller
will this error likely be in percentage points. In the econometric analysis we use both monthly andquarterly data. Monthly data should provide a better estimate of the size of the efficiency shock.
0.4

ShocksOutcomes

0.2

0

lnE

-0.2

-0.4

-0.6

-0.8

-1

-1.2
Real

Observed

Energy Saving

Backfire

Figure 2: Observed vs. real rebound.

3

Empirical Approach

3.1

Structural Vector Autoregressions (SVARs)

We use an SVAR to determine the effect of a permanent exogenous improvement to energy efficiency that we identify as a “technology shock” (Gali, 1999) on macro-level energy use in future
periods. The three-dimensional reduced form VAR model is given by
xt = ξ +pX

Πi xt−i + ut

i=1

11

(4)

where ut is a vector of white noise errors that may be correlated across equations, the Πi arematrices of parameters to be estimated, ξ is a vector of constants, and t indexes time. The structuralmodel is given by:
A0 x t = A0 ξ +p
X

A0 Πi xt−i + εt

(5)

i=1

where the diagonal entries of A0 are unity (normalization), εt = A0 ut , and var(εt ) = I. Now theeffects of shocks on the dependent variables can be independently assessed, as each is associatedwith a particular equation. The matrix A0 is, therefore, the matrix of the contemporaneous effectsof the endogenous variables on each other. This results in a simultaneity and identification problem,
which will be discussed below.
In the structural model (5), we need a total of (n2 − n)/2 restrictions (where n is the number ofvariables) in order to identify the SVAR parameters. Restrictions that involve just the matrix of thecontemporaneous coefficients A0 are called “short-run restrictions,” while restrictions that involvethe long term impact matrix (a combination of A0 with the matrices of lagged coefficients Πi ) arereferred to as “long-run restrictions” (Shapiro and Watson, 1988). Yet this is an area of economics
where theory is contested and uncertain (Bruns et al., 2014) so that imposing zero restrictions onparameters on a priori theoretical grounds is undesirable. Restrictions can also take the form ofequalities or inequalities – known as “sign restrictions” (Kilian and Murphy, 2012; Fry and Pagan,2011). But it is unclear which of the infinite number of impulse response functions admissible
according to a sign restriction is the best (Fry and Pagan, 2011).6
Gali (1999) attempts to identify a technology shock to labor productivity by assuming that only
the “technology shock” can have a long-run impact on labor productivity. However, in the case of
100% rebound, the effect of the energy efficiency shock on energy use will be zero in the long run.
Therefore, it is undesirable, a priori, to impose these types of restriction on the model.7 We achieve
6

Kilian and Lütkepohl (2017) outline various recent approaches to selecting the optimal model from among the set
identified by sign restrictions.
7
Ramey (2016) notes that several papers have questioned Galı́’s (1999) basic identifying assumption that technology
shocks are the only shocks that have a long-run effect on labor productivity including Uhlig (2004) and Mertens andRavn (2011) who look at the effect of changes in capital taxation. Basu et al. (2006) show that accounting for factors

12identification by restricting the space of possible structural models using an alternative framework,in which we maximally exploit the statistical information in the data.

3.2

Independent Component Analysis

Our empirical approach is based on a theorem, first proved by Comon (1994, Th. 11) (see also
Gouriéroux et al., 2017), according to which if we assume that the elements of εt are (mutually)
independent and non-Gaussian (with at maximum one exception), then the invertible matrix A−1
0 ,such that εt = A0 ut , is “almost identifiable.” This means that A−1
0 is identifiable up to a column
permutation and the multiplication of each of its diagonal elements by an arbitrary non-zero scalar.8
In the Independent Component Analysis (ICA) literature, several techniques have been developed
to estimate the matrices A−1
0 and A0 , where they are usually referred to as the mixing and unmixing
matrix, respectively (Hyvärinen et al., 2001). These techniques are usually based on searching forthe linear combinations of the reduced form residuals (ut in our case) that are maximally independent. This is done in the style of unsupervised statistical learning that is typical of the machinelearning research (Hyvärinen et al., 2001).
We apply three ICA techniques to estimate A−1
0 and A0 . Using three different approaches allows us to explore the robustness of our rebound estimates. The first two approaches — distancecovariance (dcov) proposed by Matteson and Tsay (2017) and non-Gaussian Maximum Likelihood(ngml) proposed by Lanne et al. (2017) — have been recently studied in the econometric literature in the context of SVAR models (see Herwartz, 2018) and the third approach is the FastICAalgorithm (Hyvärinen and Oja, 1997), which is the most popular approach to ICA estimation inmachine learning.
The first method is the distance covariance (dcov) approach recently proposed by Matteson and
Tsay (2017). This approach minimizes a nonparametric measure of dependence among n linear
such as capital utilization, fluctuations in true productivity are only about half those in raw TFP.
8
In other words, the matrix is identifiable up to the post multiplication by DP where P is a column permutation
matrix and D a diagonal matrix with non-zero diagonal elements (see Gouriéroux et al., 2017, p. 112).

13

combinations of the observed data (ut ), namely the distance covariance of Székely et al. (2007).
For example, the distance covariance between, say, u1t and u2t is defined as:I(u1t , u2t ) = E|u1t − u∗1t ||u2t − u∗2t | + E|u1t − u∗1t |E|u2t − u∗2t |∗∗∗− E|u1t − u∗1t ||u2t − u∗∗2t | − E|u1t − u1t ||u2t − u2t | (6)∗∗where | · | denotes the Euclidean distance and (u∗1t , u∗2t ) and (u∗∗1t , u2t ) denote two distinct i.i.d.

samples of (u1t , u2t ). On the basis of this measure of dependence, Matteson and Tsay (2017) define an objective function I(θ), whose argument is a vector of rotation angles θ. Each choice of θ
determines a product of rotation matrices G(θ), which in turn determines a mixing matrix A0 (θ)−1and a vector of structural shocks εt (θ) = A0 (θ)ut . Matteson and Tsay (2017) show that the choiceof θ that corresponds to arg minθ I(θ) determines a consistent estimator of A0 (θ)−1 and that thismixing matrix is associated with structural shocks εt (θ) = A0 (θ)ut that are maximally independent (i.e. least dependent). The second ICA estimator we consider in our study is the Maximum
Likelihood estimator proposed by Lanne et al. (2017). In contrast to other ICA estimators, thisapproach is parametric because it assumes that the n structural shocks are distributed according to
specific distributions, besides assuming their mutual independence. The distributions of the shocks
may be different, even belonging to different families of densities with their own parameters, but at
maximum one is allowed to be Gaussian. To construct the likelihood function, one has to choose
the non-Gaussian error distributions. In our application, we employ the t-distribution with different
degrees of freedom. The likelihood function allows us to estimate the unmixing matrix A0 and the
independent components (i.e. the structural shocks εt ).
The third ICA estimator is the fastICA algorithm Hyvärinen and Oja (1997), which is based onminimization of mutual information and maximization of negentropy. These two notions are basedon information theory, and in particular on the notion of differential entropy. Let x be a random
vector and f (x) its density, then the differential entropy H of x is defined as (Papoulis and Pillai,

14

2002)
Z
H(x) = −f (x) ln f (x)dx.

(7)

A fundamental result in information theory is that if x is Gaussian, then it has the largest entropy
among all the random vectors with the same covariance matrix (see again Papoulis and Pillai,2002). Let xG be a Gaussian random vector with the same covariance as x. Negentropy is definedasJ(x) = H(xG ) − H(x)(8)

which is necessarily non-negative and is zero if x is Gaussian. It is then a measure of nonGaussianity (Hyvärinen and Oja, 2000). Let x1 , . . . , xm be a set of (scalar) random variables
and let x = (x1 , . . . , xm )0 . The mutual information I between the m scalar random variables isdefined asI(x1 , . . . , xm ) =mX

H(xi ) − H(x).

(9)

i=1

Mutual information is a measure of (mutual) statistical dependence (Hyvärinen and Oja, 2000).
It turns out that finding linear combinations of the observed variables (e.g. u1t , . . . , unt ) that minimize mutual information (i.e. are maximally independent) is equivalent to finding directions inwhich the negentropy (i.e. non-Gaussianity) is maximized (Hyvärinen, 1997). A potential problem
is that estimating mutual information or negentropy would require estimating the probability density function f (x) (see Equation (3.4)). The FastICA algorithm circumvents this problem using anapproximation of negentropy (see Hyvärinen and Oja, 2000). Given such an approximation, thealgorithm is based on a fixed-point iteration scheme for finding linear combinations of the data thatmaximize non-Gaussianity. Given the tight link between mutual information and negentropy, thisis equivalent to find linear combinations that are maximally independent.
As mentioned above, ICA per se does not deliver full identification of A−1
0 ; one still needs
to find the right order and scale of its columns. The scale indeterminacy is easily solved by post−1−1 such that D is diagonal (withmultiplying the ICA-estimated A−1
0 in ut = A0 εt by a matrix DD

non-zero diagonal elements) and D−1 εt has unit variance. In this manner, the rescaled mixing15

−1matrix becomes A−1
0 D and the rescaled structural shocks become the elements of D εt . Thethcolumn indeterminacy is solved by searching for a column permutation P of A−1
0 such that the ishock maximally (as close as possible) impacts on the ith -variable9 . Notice that if one is interested
in identifying a single shock, say the energy efficiency shock, it is not necessary to recover theentire permutation matrix P : it is sufficient to label one of the n shocks as the shock of interest
if this shock impact maximally on the variable of interest.10 This is, of course, a further a priori
assumption that we impose on the system to achieve identification, jointly with non-Gaussianity(which can be indirectly tested) and independence of the shocks. These assumptions are detachedfrom any specific economic-theoretical model, but still form those a priori conditions needed toachieve SVAR identification.
Lastly, exploiting the sign indeterminacy, each column of A−1
0 DP is multiplied by 1 or -1
in order that the diagonal elements of A−1
0 DP have entries greater than zero, except the entry
corresponding to energy use, the entry (1, 1) in our application, which we set as negative. We
impose this sign-rescaling because we want to study the impact of a reduction of energy use (i.e.
increase in energy efficiency).

3.3

Linear non-Gaussian acyclic model (LiNGAM)

We further probe the robustness of our results by applying an ICA-based identification scheme,
which, besides assuming non-Gaussianity and independence of the structural shocks, makes thefurther assumption of recursiveness. This identification scheme is called Linear Non-Gaussian
Acyclic Model (LiNGAM) (Shimizu et al., 2006; Hoyer et al., 2008; Moneta et al., 2013). Recursiveness here means that there is a particular contemporaneous causal order of the variables
This can be done according to the following procedure: (i) if the maximum entry of A−1
0 D lies in position (row,column) (i, j), then the j th column will permuted to ith column in A−1
DP
;
(ii)
repeatthesameprocedure starting from0
A−1
0 D but computing the maximum entry neglecting all the entries lying in the column and row in which the maximum
was found in the previous step; until the nth column has been permuted.
10
In other words, supposing that energy consumption is the first entry in xt , the j th shock will be labelled as thethenergy shock if the maximum entry of the first row of A−1
column. This would avoid the procedure
0 D lies in the jdescribed in the previous footnote.
916

(which the algorithm is able to identify from the data), such that the unmixing (or, equivalently,
mixing) matrix can be rearranged into a lower-triangular matrix (after a rows/columns permutation). In other words, the contemporaneous causal order of the variables can be represented as adirected acyclic graph Moneta et al. (2013). The standard Choleski identification scheme (Sims,
1980) also makes the assumption that the instantaneous impact matrix (i.e. the mixing matrix) is
lower triangular. In the Choleski scheme, however, the order of the variables that enter in the vector
xt is given a priori and, in many applications, may appear arbitrary. In LiNGAM the ordering is
discovered from the data. Given an arbitrary initial variable order, FastICA is first used to estimate the unmixing matrix A0 and the mixing matrix A−1
0 . Then, in a second step, LiNGAM finds
the right permutation matrix P , which we mentioned above as fundamental to solving the ICA
indeterminacy problem. To obtain P , the algorithm makes use of recursiveness: if the underlying
contemporaneous causal structure among the time-series variables is recursive (acyclic), then there
will be a row-column permutation that makes A0 and A−1
0 lower triangular. Since these matrices
are estimated with errors, the algorithm searches for the row-column permutation which makes
one of these matrices the closest as possible to lower triangular. In comparison with the criterion,
mentioned above, to identify the energy shock simply based on picking the shock that has maximal
contemporaneous impact on the energy time series variable (our baseline rebound effects estimates
will hinge on this criterion), LiNGAM has the clear advantage of providing a complete identification of the mixing and unmixing matrix, with the entire causal graph of the contemporaneous
structure. It has, however, the disadvantage of relying heavily on a lower-triangular scheme, whichis the reason why we use it only for robustness analysis.
We use the R package svars to estimate the dcov and ngml models and own code for the FastICA
and LiNGAM models. More information can be found in the replication package.

3.4

Data

We estimate vector autoregressive models for the United States using monthly and quarterly data.
Identifying restrictions are generally more plausible the more frequent the data is (Kilian, 2009).
17

While estimates using quarterly data may produce a biased estimate of the potential (instantaneous)effect, it is also possible that estimates using monthly data will focus on the short run and underestimate the long-run effects. This is why we estimated models using (separately) both frequencies.
Monthly data run from January 1992 to October 2016 (298 observations), quarterly data from 1973
(first quarter) to 2016 (third quarter) (175 observations). We will also, for the sake of comparison,
estimate the model using quarterly data from 1992 to 2016 (99 observations). All data are taken innatural logarithms and multiplied by 100. Appendix A discusses the sources of the data in detail.

4

Results

4.1

Reduced-form VARs

Based on Kilian and Lütkepohl (2017), we use the Akaike Information Criterion (AIC) to choose
the lag length. Based on the Schwert (1989) criterion, we use a maximum of 5 lags for the quarterly
data and a maximum of 6 lags for the monthly data. We select 3 lags for both frequencies. Identification of the energy efficiency shock requires that at most one of the structural shocks is Gaussian.
Since we do not observe the structural shocks, we cannot directly test their Gaussianity. However,
we can test it indirectly: if the reduced-form residuals, which are linear combinations of independent shocks, are non-Gaussian this implies that the structural shocks are also non-Gaussian, since
a linear combination of independent and normally distributed random variables also has a normal
distribution. Using a Jarque-Bera test with α = 0.05, we find that for all reduced-form VAR models used in the subsequent analysis, at most one of the reduced-form residuals exhibits a Gaussian
distribution11 .
11

We obtain only isolated cases (monthly data, 3 variables model) in which the normality of the reduced-form residual
associated to energy use is not rejected at 0.05 level of significance.

18

4.2

Identification of SVARs

Our focus here is on the energy efficiency shock and the partial identification of the SVAR (i.e.
the labeling of one of the identified shocks as the energy shock), which is sufficient to estimate
the rebound effect. However, we also discuss the GDP and energy price shocks, and we ascertain
whether the estimated SVAR is generally consistent with economic theory. The contemporaneous
effect of an improvement in energy efficiency on GDP should be positive due to the increase in TFP
this represents, while the effect on energy prices should be negative due to the reduction in demandfor energy. However, we expect the contemporaneous effects of energy efficiency improvements
on GDP and energy prices to be small as the transmission of these effects is likely to take sometime. We expect the contemporaneous effects of a positive energy price shock on energy use and
GDP to be, if at all, negative and small, especially in monthly data. We also do not expect strong
contemporaneous effects of a positive GDP shock on energy use and energy prices, but these effects
should be positive.
Table 1 shows the A−1
0 DP matrices obtained by the four identification methods for monthly
data. For the three ICA approaches (dcov, ngml, FastICA) the first column shows what we label
as the contemporaneous impact of the energy efficiency shock. This shock has the largest contemporaneous effect on energy use and comparably small and effects on GDP and energy prices as
expected from economic theory. The bootstrapped confidence intervals of these effects also include
zero. We, therefore, conclude that these energy efficiency shocks conform with economic theory.
Applying LiNGAM, we estimate the causal order as y −→ e −→ p assuming a recursive causal
structure.12 While the effect of energy efficiency improvements on GDP is set to zero, the effecton energy prices is relatively large, but the sign conforms to economic theory.
12
Note that the mixing matrix reported in Table 1 and 2 for LiNGAM results in a lower triangular impact (mixing)
matrix as required by a recursive causal structure. It is important to assess how stable this causal order is when wechange the initial condition of the FastICA algorithm (which constitutes the first step of LiNGAM). We run a simulation
where LiNGAM is iteratively applied to the same data set but resampling the initial conditions each time. LiNGAM
results in this case are 100% stable. A further, and more severe, exercise to check stability is to run a bootstrap in which
we do not only change initial conditions of the algorithm, but also resample the data. In this case, we get the same causal
structure 95.4% of the time. Our conclusion is that the causal order y −→ e −→ p output of LiNGAM is satisfactorily
stable.

19

Regarding the GDP shock, the bootstrapped confidence intervals again suggest that only thecontemporaneous effect on GDP is statistically significant (except for LiNGAM in which there isalso a positive and significant effect of the GDP shock on price). Regarding the energy price shock,
it is again only the contemporaneous effect on energy prices that is statistically significant. Overall,
we conclude that the identified shocks are consistent with economic theory.
Table 2 shows the A−1
0 DP matrices for quarterly data. Results for the energy efficiency shock
are very similar to those obtained for monthly data. The contemporaneous effects on GDP andenergy prices tend to zero and the bootstrapped confidence intervals include zero. For quarterly
data, the energy efficiency shock identified by LiNGAM is also more consistent with economic
theory. LiNGAM suggests the same contemporaneous causal structure as for monthly data (y −→e −→ p).13
Analogously to the findings for monthly data, the GDP shocks only affect GDP and the energy
price shocks only affect energy prices, i.e. the bootstrapped confidence interval does not include
zero. An exception is LiNGAM for which the GDP shock affects all three variables.
Regarding the expected long-run effects, we expect an energy efficiency shock to have a large
negative effect on energy use. Figure 3 shows the impulse response functions for an SVAR identified with the distance covariance method using monthly data. The first column shows the effectof the energy efficiency shock on energy use, GDP, and the energy price. The energy efficiency
shock results in a strong decrease in energy use initially but this effect is eliminated over time resulting in backfire. Therefore, it appears that this shock is transitory, which is consistent with thepresence of a cointegrating relationship (Pagan and Pesaran, 2008).14 We also expect the energy
efficiency shock to have a positive effect on GDP and a negative on energy prices in the longrun. While energy prices decrease first, they eventually return to the initial level, but the effects
13

While resampling initial conditions we also have here complete stability, but bootstrap stability (resampling theobserved data) is a bit lower: 91.8%.
14
We carried out some exploratory analysis of potential cointegrating relations in our model. We found that there
might be either one or two cointegrating vectors. In the latter case, our impulse response functions suggest that the priceshock is also transitory and the GDP shock is permanent.

20

Table 1: Mixing Matrices (A−1
0 DP ) for Monthly Data
εe

εy

εp

-1.68
[-1.73, -1.24]
0.09[-0.18, 0.27]-0.02[-1.89, 1.55]

0.32[-0.59, 0.86]
0.51[0.38, 0.51]
0.57[-1.95, 2.38]

0.29[-0.33, 0.87]
0.03[-0.15, 0.27]
5.04[4.04, 5.06]-1.50[-1.71, -0.82]-0.21[-0.40, 0.16]
0.14[-1.89, 1.37]-0.66[-1.40, 0.52]
0.45[0.25, 0.50]
0.51[-1.87, 2.17]

0.47[-0.32, 0.75]
0.03[-0.15, 0.26]
4.81[4.07, 4.97]-1.61[-1.71, -1.15]-0.13[-0.34, 0.17]
0.07[-1.80, 1.31]-0.39[-1.18, 0.59]
0.49[0.33, 0.50]
0.99[-1.99, 2.67]

0.43[-0.29, 0.73]-0.01[-0.21, 0.26]
4.90[3.84, 4.97]-2.18[-2.33, -2.02]
0.00

0.00
[-0.19, 0.20]
0.69[0.64, 0.73]
1.23[0.39, 2.08]

0.00

Distance covariance (dcov)
etytpt
Non-Gaussian Maximum Likelihood (ngml)
etytpt
FastICAetyt
pt
LiNGAMetyt-1.25[-2.01, -0.36]

pt

0.00
8.93
[8.36, 9.49]

Notes: Bootstrapped 0.90 confidence intervals in brackets. LiNGAM (Causal structure y −→e −→ p): 95.4 % bootstrap stability; 100% initial conditions stability.

21

Table 2: Mixing Matrices (A−1
0 DP ) for Quarterly Data
εe

εy

εp

-1.55
[-1.63, -1.45]
0.16[-0.24, 0.18]
0.05[-2.47, 2.36]

0.51[-0.40, 0.54]
0.71[0.63, 0.72]-0.52[-2.29, 2.58]

0.05[-0.46, 0.45]
0.03[-0.23, 0.18]
8.59[7.51, 8.59]-1.55[-1.62, -1.51]
0.15[-0.21, 0.12]-0.05[-1.20, 1.24]

0.42[-0.32, 0.40]
0.72[0.64, 0.73]-0.74[-1.61, 1.63]

0.14[-0.23, 0.25]
0.07[-0.16, 0.14]
8.85[7.79, 9.30]-1.53[-1.59, -1.52]
0.12[-0.21, 0.11]-0.15[-1.08, 1.13]

0.41[-0.32, 0.40]
0.69[0.67, 0.70]-0.80[-1.31, 1.34]

0.02[-0.21, 0.23]
0.006[-0.12, 0.10]
8.33[8.17, 8.37]-2.05[-2.28, -1.82]
0.00

0.52
[0.23, 0.80]
1.30[1.17, 1.43]
0.731[1.17, 1.43]

0.00

Distance covariance (dcov)
etytpt
Non-Gaussian Maximum Likelihood (ngml)
etytpt
FastICAetyt
pt
LiNGAM
et
yt
pt

-1.90
[-3.94, 0.22]

0.00
15.25
[13.81, 16.75]

Notes: Bootstrapped 0.90 confidence intervals in brackets. LiNGAM (Causal structure y −→e =⇒ p): 91.8 % bootstrap stability; 100% initial conditions stability.

22remain mostly statistically non-significant. The long-run effect on GDP is positive, but statistically
non-significant.
We also see that though the initial effect of the price shock on energy use and GDP is positive(but not statistically significant), in the longer run it has the expected negative and statisticallysignificant effects on both variables. On the other hand, the price shock appears have transitory
effects on the price of energy but what look like permanent effects on at least GDP. The GDP shock
has positive long-run effects on all three variables. The long-run effects, therefore, also conformwith economic theory.

4.3

Rebound effect

Estimates for the rebound effect after 1, 2, 4, and 6 years are presented in Table 3. The estimates of
the rebound effect are very similar for the four methods of identification and approach 100% after
4 years. However, the rebound effect after 6 years tends to be smaller for monthly data (Models
1 to 4) compared to quarterly data (Models 5 to 8). The different estimates of the rebound effectmay result from the different frequencies of the data or from the different time periods covered. We
also estimate the rebound effect using quarterly data for the time span 1992-2016 (Models 9 to 12)
and the differences in estimated long-run rebound effects reduce suggesting that the time period
explains most of the differences. As discussed in Section 2, the quarterly data should estimate alower rebound than monthly data when the rebound is less than unity and a greater rebound thanmonthly data when it is greater than unity.

4.4

Robustness analyses

We extend the VAR with energy use, GDP, and the price of energy by adding two further control
variables – the log of industrial production and the log of energy quality – to reduce potential omitted variable biases in identifying the energy efficiency shock. The A−1
0 DP matrices for monthly
and quarterly data can be found in the Appendix B (Tables 5 and 6). Labeling shocks by the largest
23

Figure 3: Impulse Response Functions for Monthly Data (Distance Covariance). 0.90 confidence
intervals computed using wild bootstrap.

24

Table 3: Rebound EffectModel
1Frequency
Monthly

Period1992-2016

Method
dcov

2

ngml

3

fastICA4

LiNGAM

5

Quarterly

1973-2016

dcov

6

ngml

7

fastICA8

LiNGAM

9

Quarterly

1992-2016

dcov

10

ngml

11

fastICA12

LiNGAM

1 year
0.78[0.61,0.88]
0.76[0.62,0.89]
0.77[0.85, 0.93]
0.90[0.88, 0.92]
0.61[0.34,0.68]
0.61[0.35,0.63]
0.59[0.52, 0.75]
0.63[0.61, 0.64]
0.58[0.35,0.81]
0.45[0.34,0.8]
0.54[0.62, 0.8]
0.71
[0.67, 0.79]

Notes: Bootstrapped 0.90 confidence intervals in brackets.

25

2 years
0.94[0.76,1.04]
0.91[0.76,1.04]
0.92[0.93, 1.06]
0.99[0.98, 1.01]
0.90[0.57,1.03]
0.90[0.6,0.97]
0.88[0.55, 1.14]
0.88[0.84, 0.95]
0.91[0.58,1.2]
0.77[0.58,1.14]
0.88[0.78, 1.16]
0.95[0.87, 1.06]

4 years
1.01[0.91,1.1]
0.99[0.9,1.09]
1.00[0.96, 1.06]
1.01[1, 1.02]
1.16[0.81,1.38]
1.17[0.84,1.32]
1.16[0.80, 1.37]
1.08[1.01, 1.16]
1.09[0.8,1.35]
1.01[0.8,1.31]
1.08[0.89, 1.18]
1.03[0.98, 1.11]

6 years
1.01[0.95,1.08]
0.99[0.94,1.06]
1.00[0.97, 1.04]
1.00[1, 1.01]
1.23[0.94,1.47]
1.24[0.96,1.45]
1.23[0.88, 1.35]
1.12[1.06, 1.18]
1.07[0.87,1.3]
1.03[0.88,1.28]
1.06[0.94, 1.12]
1.02[0.99, 1.08]

Table 4: Rebound Effect (Robustness Analysis, 5 Variable Model)
Model
1

Frequency
Monthly

Period1992-2016

Method
dcov

2

ngml

3

fastICA4

LiNGAM

5

Quarterly

1973-2016

dcov

6

ngml

7

fastICA8

LiNGAM

1 year
0.94[0.65,1.19]
0.98[0.64,1.93]
0.84[0.89, 1.03]
0.96[0.94, 0.98]
0.72[0.52,1.42]
0.63[-0.07,0.63]
0.59[0.55, 1.13]
0.71[0.64, 0.78]

Notes: Bootstrapped 0.90 confidence intervals in brackets.

26

2 years
1.03[0.83,1.32]
1.06[0.83,2]0.94[0.91, 1.07]
0.97[0.95, 1]0.85[0.66,1.92]
0.82[-0.1,0.91]
0.83[0.61, 1.41]
0.84[0.77, 0.93]

4 years
1.09[0.94,1.43]
1.13[0.97,2.22]
0.99[0.91, 1.08]
0.98[0.96, 1.01]
0.93[0.65,1.84]
1.16[0.31,1.46]
1.16[0.78, 1.43]
0.97[0.89, 1.08]

6 years
1.06[0.95,1.33]
1.09[0.97,1.91]
1.00[0.94, 1.07]
0.99[0.98, 1.01]
0.97[0.64,1.64]
1.30[0.54,1.84]
1.28[0.87, 1.36]
1.03[0.96, 1.12]

contemporaneous effect size is not unique for the VAR with five variables as in some cases thesame shock has the largest contemporaneous effect for two variables – GDP and economic structure (industrial production). As our interest is in the robustness of the rebound effect, we focus on
the energy efficiency shock. Table 4 again presents the rebound effect after 1, 2, 4, and 6 years.
The estimated rebound effects are very similar to those for the VARs with three variables.
For LiNGAM, the identified contemporaneous causal structures are much less stable than they
are for the three variable VARs. For monthly data, the most stable structure is y → s → q → e →p. However, this structure reaches only 58% stability under random variation of the algorithm’s initial conditions and 64.5% stability under bootstrap resampling of the data. Therefore, we examined
the robustness of our results under the second most stable causal structure (s → y → q → e → p)
and find that the estimated rebound effect is robust to this second causal structure as well. For
quarterly data, the most stable causal structures is q → y → s → e → p (73% initial conditions
stability, 38.7% bootstrap stability). We also find the rebound effect to be robust if the second moststable structure s → q → y → e → p is used.
In conclusion, LiNGAM does not provide stable and sufficiently reliable results for the VAR
with five variables. It is interesting to note, however, that among the diverse causal structures
suggested by the algorithm (including others we did not present), each of them singularly unstable,
it is always the case that y comes before e and e before p in the contemporaneous causal chain,
which was also the output of the 3-variable model. This probably means that the structure y →e → p is remarkably stable, with the other variables (s, q) playing diverse causal roles that cannot
be described by a recursive scheme. This is why it was important to show results with methods notcommitted to such a scheme (dcov, ngml, FastICA).
The robustness analysis shows that the magnitude of the rebound effect obtained by the VAR
with three variables is robust to controlling for two further determinants of energy use.

27

5Conclusions and discussion

We have produced the first econometric general equilibrium estimate of the economy-wide reboundeffect, which we think is credibly identified, using an empirically identified time-series model.
Estimates of the rebound effect after 4 years are close to 100%, regardless of the method or data
frequency used. As some part of the rebound might occur instantaneously, our estimates may differ
from the true rebound. However, the true rebound is likely to be closer to 100% than the estimatedrebound and our estimates of the long-run rebound are almost exactly 100%. These results arecongruent with the historical research (van Benthem, 2015; Csereklyei et al., 2016; Hart, 2018) thathints that the economy-wide rebound effect could be large. This implies that policies to encourage
costless energy efficiency innovation are not likely to significantly reduce energy use in the longrun, which has important implications for climate mitigation policies. On the other hand, there
are short-run energy savings that will reduce cumulative energy use and greenhouse gas emissions.
15

Our approach is equivalent to applying the same size efficiency improvement to all sectors of

the economy, which could also be seen as a typical or average effect. Policymakers may be moreconcerned with the effects of efficiency improvements in specific sectors. These should be exploredwith more disaggregated models.
Despite this large rebound effect, energy intensity has declined over time in the United States.
How can this fact be compatible with our results? Based on our three variable VAR, there arethree possible mechanisms that can explain this. First, energy efficiency shocks may increase GDP
by more than they increase energy use. In Figure 3 this seems to be the case, if we ignore thevery wide confidence interval around the IRF for the effect of an energy efficiency shock on GDP.
Second, GDP shocks tend to increase GDP by much more than they increase energy use, which isstrongly supported by Figure 3. In a simple aggregate model of the economy, as in Shanker andStern (2018) or Saunders (2008), if the price of energy is constant then labor-augmenting technicalchange cannot change energy intensity because the marginal product of energy is a function of15

Of course, policies that impose costly energy efficiencies on energy consumers are likely to have a reduce energy
use by more than the engineering effect (Fullerton and Ta, 2019).

28

energy intensity alone. However, we see that GDP shocks increase the price of energy as well andthis is what allows energy intensity to decline. Population growth can also increase GDP, but it is
not clear why it would increase energy use by less than GDP. Finally, increasing energy prices can
reduce energy use though they also reduce GDP. However, as shown in Figure 3, they reduce GDP
by more than they reduce energy use in the long run.

29

References
Acemoglu, D. (2002). Directed technical change. Review of Economic Studies 69(4), 781–809.
Adetutu, M., A. Glass, and T. Weyman-Jones (2016). Economy-wide estimates of rebound effects:
Evidence from panel data. Energy Journal 37(3), 251–269.
Azevedo, I. M. (2014). Consumer end-use energy efficiency and rebound effects. Annual Review
of Environment and Resources 39, 393–418.
Barker, T., A. Dagoumas, and J. Rubin (2009). The macroeconomic rebound effect and the world
economy. Energy Efficiency 2(4), 411.
Basu, S., J. G. Fernald, and M. S. Kimball (2006). Are technology improvements contractionary?
American Economic Review 96(5), 1418–1448.
Bessler, D. A. and S. Lee (2002). Money and prices: US data 1869–1914 (a study with directed
graphs). Empirical Economics 27(3), 427–446.
Blanchard, O. J. and D. Quah (1989). The dynamic effects of aggregate demand and aggregate
supply. American Economic Review 79(4), 655–73.
Borenstein, S. (2015). A microeconomic framework for evaluating energy efficiency rebound andsome implications. Energy Journal 36(1), 1–21.
Broberg, T., C. Berg, and E. Samakovlis (2015). The economy-wide rebound effect from improved
energy efficiency in Swedish industries—A general equilibrium analysis. Energy Policy 83,
26–37.
Brockway, P., H. Saunders, M. Heun, T. Foxon, J. Steinberger, J. Barrett, and S. Sorrell (2017).
Energy rebound as a potential threat to a low-carbon future: Findings from a new exergy-based
national-level rebound approach. Energies 10(1), 51.

30

Bruns, S. B., C. Gross, and D. I. Stern (2014). Is there really Granger causality between energy use
and output? Energy Journal 35(4), 101–133.
Capasso, M. and A. Moneta (2016). Macroeconomic responses to an independent monetary policy
shock: A (more) agnostic identification procedure. LEM Working Paper Series 2016/36, Scuola
Superiore Sant’Anna.
Comon, P. (1994). Independent component analysis, a new concept? Signal Processing 36(3),287–314.
Csereklyei, Z., M. Rubio- Varas, and D. Stern (2016). Energy and economic growth: The stylized
facts. Energy Economics 37(2), 223–255.
Csereklyei, Z., M. d. M. Rubio-Varas, and D. I. Stern (2016). Energy and economic growth: The
stylized facts. Energy Journal 37(2), 223–255.
Demiralp, S. and K. D. Hoover (2003). Searching for the causal structure of a vector autoregression.
Oxford Bulletin of Economics and Statistics 65(s1), 745–767.
Fry, R. and A. Pagan (2011). Sign restrictions in structural vector autoregressions: A critical review.
Journal of Economic Literature 49(4), 938–60.
Fullerton, D. and C. L. Ta (2019). Costs of energy efficiency mandates can reverse the sign ofrebound. NBER Working Paper 25696.
Gali, J. (1999). Technology, employment, and the business cycle: Do technology shocks explain
aggregate fluctuations? American Economic Review 89(1), 249.
Galvin, R. (2014). Estimating broad-brush rebound effects for household energy consumption in
the EU 28 countries and Norway: some policy implications of odyssee data. Energy Policy 73,
323–332.
Gillingham, K., D. Rapson, and G. Wagner (2016). The rebound effect and energy efficiency
policy. Review of Environmental Economics and Policy 10(1), 68–88.
31

Gouriéroux, C., A. Monfort, and J.-P. Renne (2017). Statistical inference for independent component analysis: Application to structural VAR models. Journal of Econometrics 196(1), 111–126.
Hart, R. (2018). Rebound, directed technological change, and aggregate demand for energy. Journal of Environmental Economics and Management 89, 218 – 234.
Herwartz, H. (2018). Hodges-Lehmann detection of structural shocks—An analysis of macroeconomic dynamics in the Euro area. Oxford Bulletin of Economics and Statistics 80(4), 736–754.
Hoover, K. D. (2001). Causality in Macroeconomics. Cambridge University Press.
Hoyer, P. O., S. Shimizu, A. J. Kerminen, and M. Palviainen (2008). Estimation of causal effects using linear non-Gaussian causal models with hidden variables. International Journal of
Approximate Reasoning 49(2), 362–378.
Hyvärinen, A. (1997). Independent component analysis by minimization of mutual information.
Helsinki University of Technology.
Hyvärinen, A., J. Karhunen, and E. Oja (2001). Independent Component Analysis. Wiley.
Hyvärinen, A. and E. Oja (1997). A fast fixed-point algorithm for independent component analysis.
Neural Computation 9(7), 1483–1492.
Hyvärinen, A. and E. Oja (2000). Independent component analysis: Algorithms and applications.
Neural Networks 13(4-5), 411–430.
Kilian, L. (2009). Not all oil price shocks are alike: Disentangling demand and supply shocks inthe crude oil market. American Economic Review 99(3), 1053–69.
Kilian, L. and H. Lütkepohl (2017). Structural Vector Autoregressive Analysis. Cambridge University Press.

32

Kilian, L. and D. P. Murphy (2012). Why agnostic sign restrictions are not enough: Understanding
the dynamics of oil market var models. Journal of the European Economic Association 10(5),1166–1188.
King, R., C. Plosser, J. Stock, and M. Watson (1991). Stochastic trends and economic fluctuations.
American Economic Review 81(4), 819–40.
Knittel, C. R. (2011). Automobiles on steroids: Product attribute trade-offs and technologicalprogress in the automobile sector. American Economic Review 101(7), 3368–99.
Koesler, S., K. Swales, and K. Turner (2016). International spillover and rebound effects fromincreased energy efficiency in Germany. Energy Economics 54, 444–452.
Lanne, M., M. Meitz, and P. Saikkonen (2017). Identification and estimation of non-Gaussian
structural vector autoregressions. Journal of Econometrics 196(2), 288–304.
Lemoine, D. (2018, October). General equilibrium rebound from energy efficiency innovation.
NBER Working Paper 25172.
León-Ledesma, M. A., P. McAdam, and A. Willman (2010). Identifying the elasticity of substitution with biased technical change. American Economic Review 100(4), 1330–57.
Lin, B. and K. Du (2015). Measuring energy rebound effect in the chinese economy: An economic
accounting approach. Energy Economics 50, 96–104.
Lu, Y., Y. Liu, and M. Zhou (2017). Rebound effect of improved energy efficiency for different
energy types: A general equilibrium analysis for china. Energy Economics 62, 248–256.
Malpede, M. and E. Verdolini (2016). Rebound effects in (europe). In Proceedings of the Fourth
IAERE Annual Conference, Bologna, Italy, pp. 11–12.
Matteson, D. S. and R. S. Tsay (2017). Independent component analysis via distance covariance.
Journal of the American Statistical Association 112(518), 623–637.
33

Mertens, K. and M. O. Ravn (2011). Technology-hours redux: Tax changes and the measurementof technology shocks. In NBER International Seminar on Macroeconomics, Volume 7, pp. 41–
76.
Meyer, I. and S. Wessely (2009). Fuel efficiency of the Austrian passenger vehicle fleet—analysisof trends in the technological profile and related impacts on CO2 emissions. Energy Policy 37(10), 3779–3789.
Moneta, A. (2008). Graphical causal models and VARs: An empirical assessment of the realbusiness cycles hypothesis. Empirical Economics 35(2), 275–300.
Moneta, A., D. Entner, P. O. Hoyer, and A. Coad (2013). Causal inference by independent component analysis: Theory and applications. Oxford Bulletin of Economics and Statistics 75(5),705–730.
Orea, L., M. Llorca, and M. Filippini (2015). A new approach to measuring the rebound effectassociated to energy efficiency improvements: An application to the US residential energy demand. Energy Economics 49, 599–609.
Pagan, A. R. and M. H. Pesaran (2008). Econometric analysis of structural systems with permanentand transitory shocks. Journal of Economic Dynamics and Control 32(10), 3376–3395.
Papoulis, A. and S. U. Pillai (2002). Probability, random variables, and stochastic processes. Tata
McGraw-Hill Education.
Pearl, J. (2009). Causality: Models, Reasoning, and Inference. Cambridge University Press.
Ramey, V. A. (2016). Macroeconomic shocks and their propagation. In J. B. Taylor and H. Uhlig(Eds.), Handbook of macroeconomics, Volume 2, pp. 71–162. Elsevier.
Rausch, S. and H. Schwerin (2018). Does higher energy efficiency lower economy-wide energy
use? Economics Working Paper 18/299, Center of Economic Research at ETH Zurich.

34

Saunders, H. D. (1992). The Khazzoom-Brookes postulate and neoclassical growth. Energy Journal 13(4), 131–148.
Saunders, H. D. (2008). Fuel conserving (and using) production functions. Energy Economics 30,
2184–2235.
Saunders, H. D. (2013). Historical evidence for energy efficiency rebound in 30 US sectors and atoolkit for rebound analysts. Technological Forecasting and Social Change 80(7), 1317–1330.
Saunders, H. D. (2015). Recent evidence for large rebound: elucidating the drivers and theirimplications for climate change models. Energy Journal 36(1), 23–48.
Schwert, G. W. (1989). Tests for unit roots: A monte carlo investigation. Journal of Business &Economic Statistics 7, 147–160.
Shanker, A. and D. I. Stern (2018). Energy intensity, growth and technical change. CAMA Working
Paper 46/2018.
Shao, S., T. Huang, and L. Yang (2014). Using latent variable approach to estimate China s
economy-wide energy rebound effect over 1954–2010. Energy Policy 72, 235–248.
Shapiro, M. D. and M. W. Watson (1988). Sources of business cycle fluctuations. NBER Macroeconomics Annual 3, 111–148.
Shimizu, S., P. O. Hoyer, A. Hyvärinen, and A. Kerminen (2006). A linear non-Gaussian acyclic
model for causal discovery. Journal of Machine Learning Research 7, 2003–2030.
Sims, C. A. (1980). Macroeconomics and reality. Econometrica 48(1), 1–48.
Sorrell, S., J. Dimitropoulos, and M. Sommerville (2009). Empirical estimates of the direct reboundeffect: A review. Energy Policy 37(4), 1356–1371.
Spirtes, P., C. N. Glymour, and R. Scheines (2000). Causation, Prediction, and Search. MIT press.

35

Stern, D. I. (2010). Energy quality. Ecological Economics 69(7), 1471–1478.
Stern, D. I. (2011a). Elasticities of substitution and complementarity. Journal of Productivity
Analysis 36(1), 79–89.
Stern, D. I. (2011b). The role of energy in economic growth. Annals of the New York Academy of
Sciences 1219(1), 26–51.
Stern, D. I. (2012). Modeling international trends in energy efficiency. Energy Economics 34(6),2200–2208.
Stern, D. I. (2017). How accurate are energy intensity projections? Climatic Change 143(3-4),
537–545.
Swanson, N. R. and C. W. Granger (1997). Impulse response functions based on a causal approachto residual orthogonalization in vector autoregressions. Journal of the American Statistical Association 92(437), 357–367.
Székely, G. J., M. L. Rizzo, N. K. Bakirov, et al. (2007). Measuring and testing dependence bycorrelation of distances. Annals of Statistics 35(6), 2769–2794.
Turner, K. (2009). Negative rebound and disinvestment effects in response to an improvement inenergy efficiency in the UK economy. Energy Economics 31(5), 648–666.
Turner, K. (2013). “Rebound” effects from increased energy efficiency: A time to pause and reflect.
Energy Journal 34(4), 25–42.
Turner, K. and N. Hanley (2011). Energy efficiency, rebound effects and the environmental Kuznets
curve. Energy Economics 33(5), 709–720.
Uhlig, H. (2004). Do technology shocks lead to a fall in total hours worked?
European Economic Association 2(2-3), 361–371.

36

Journal of thevan Benthem, A. A. (2015). Energy leapfrogging. Journal of the Association of Environmental andResource Economists 2(1), 93–132.
van den Bergh, J. C. (2011). Energy conservation more effective with rebound policy. Environmental and Resource Economics 48(1), 43–58.
Wei, T. and Y. Liu (2017). Estimation of global rebound effect caused by energy efficiency improvement. Energy Economics 66, 27–34.

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Appendix A: Data
Monthly Data
As energy intensity is conventionally measured in terms of primary energy we use both primaryenergy quantities and prices that are as close as possible to the price of primary energy. We compile
a data set for the period January 1992 to October 2016, which is restricted by the availability ofmonthly GDP (beginning of sample) and monthly energy use data and prices (end of sample).
Energy Quantities: We use Energy Information Administration (EIA) data on consumption ofprimary energy from various sources measured in quadrillion BTU. This data is reported in theMonthly Energy Review (MER) and available from the EIA website. The primary sources arepetroleum, natural gas, coal, primary electricity (which is reported for several sources), and biomassenergy. We assume that geothermal and solar power is all primary electricity in our computation of
the aggregate energy price index and energy quality. We treat biomass as primary energy whether
it is used to generate electricity or not. We deseasonalize energy quantity and price data using theX11 procedure as implemented in RATS using a multiplicative seasonality model.
Energy Prices and Quality: EIA provide a variety of energy price series. For crude oil we use
the “Refiner Acquisition Cost of Crude Oil, Composite” series from Table 9.1 in the MER. For
electricity prices we use “Average Retail Price of Electricity, Industrial” from Table 9.8 in the MER.
This price averaged $61 per MWh from January 2001 to December 2013. Using data on wholesale
electricity prices provided by the Intercontinental Exchange to EIA (https://www.eia.gov/
electricity/wholesale/#history), over the same period the Northeast Pool wholesale
electricity price also averaged $61. The Mid-Columbia wholesale price averaged $42, Palo Verde$49, and PJM West $54. However, using these wholesale prices would further restrict our sample
to start in January 2001 and not all of the US has liberalized electricity markets. Monthly electricity
prices are not available for 1992-1994 and we used annual prices for this period.
For natural gas prices we use the Henry Hub spot prices available on this page:38

http://www.eia.gov/dnav/ng/hist/rngwhhdA.htm
from January 1997. Prior to that we use EIA’s “Natural Gas Price, Wellhead” from Table 9.10 in
the MER. For the price of coal we use “Cost of Coal Receipts at Electric Generating Plants” from
Table 9.9 in the MER.
Annual biomass prices for 1970 to 2014 are available from this webpage:
http://www.eia.gov/state/seds/data.cfm?incfile=/state/seds/sep_prices/
total/pr_tot_US.html&sid=US
For months after 2014 we applied the growth rate of the price of crude oil as the correlation between
the price of oil and biomass was 0.92 from 1992 to 2014.
All these prices are converted to prices per BTU using standard conversion factors. For primaryelectricity we use the ratio of primary energy to electricity produced to obtain a price for primaryenergy from the price of electricity. Table 7.2 in the MER provides the generation of electricity
from various sources. We use this to get conversion factors for nuclear, hydropower, solar, andwind. For geothermal and solar energy we use the data in this table and the amount of geothermal
power used in electricity generation in MER Table 10.2c. But we apply the derived price to all
geothermal and solar energy as described above.
As monthly energy quantities and prices are often highly seasonal, we deseasonalized eachseries at the fuel level before aggregating using the X11 procedure in RATS and a multiplicative
specification of the seasonal factor. To obtain the price of energy we simply compute the total costof energy in our data and divide by total BTUs of primary energy. As discussed in Section 2, energy
quality is a measure of shifts in the the mix of energy carriers (fuels and primary electricity). If theshare of each energy carrier remains constant then a volume index of energy use, such as a Divisia
Index, will grow at the same rate as the simple sum of BTU. To obtain the energy quality index wecompute a Divisia energy volume index of energy use and divide this by total BTUs. The energy
quality index will increase if there is a shift towards the more expensive energy carriers.

39

GDP: Macroeconomic Advisers have interpolated a monthly GDP series, which appears to beseasonally adjusted, for the U.S. using many of the underlying variables used by the Bureau of
Economic Analysis to update quarterly GDP:
http://www.macroadvisers.com/monthly-gdp/
This data includes nominal and real series, which can be used to compute a monthly GDP deflator,
which we use to deflate energy prices.
Industrial Structure: McCracken and Ng (2015) have compiled a large monthly macroeconomic
data set for the United States (“FRED-MD”), which is available through the Federal Reserve Bankof St. Louis FRED data tool at
https://research.stlouisfed.org/econ/mccracken/fred-databases/
We use their series for industrial production, which is seasonally adjusted. The ratio of industrial
production to GDP is our measure of industry structure.

Quarterly Data
We compiled a quarterly dataset for 1973:1 to 2016:3.
We use quarterly GDP data from the Bureau of Economic Analysis (BEA) National Income andProduct Accounts (NIPA). GDP data is real GDP in chained 2009 dollars. All other data is fromthe same sources as the monthly data. We aggregated the monthly data into quarterly data and deseasonalized the energy series before computing energy quantity and price aggregates as described
for the monthly data.
Monthly oil prices are only available from 1974 and electricity and gas prices from 1976. Monthly
electricity prices are not available for 1984-1994 either. We used annual prices for this missing
data.

40

Appendix B: Mixing matrices for SVARs with five variables
Table 5: Monthly Data.
εe

εy

εp

εs

εq

Distance Covariance
et
yt
pt
st
qt

-1.24
0.22
-0.09
0.14
-0.07

0.40
0.37
-0.01
-0.08
0.03

0.29
0.08
5.02
0.03
0.07

0.52
0.16
-0.23
0.54
0.20

-0.77
-0.05
-0.06
-0.17
0.63

Non-Gaussian Maximum Likelihood
et
yt
pt
st
qt

-1.14
0.02
-0.06
0.26
-0.07

-0.09
0.46
0.73
0.07
0.05

0.43
0.01
4.57
0.10
0.14

0.55
0.07
-1.13
0.42
0.22

-1.01
-0.07
-0.44
-0.21
0.63

FastICA (Negentropy)
et
yt
pt
st
qt

-1.15
-0.12
-0.80
0.13
-0.29

-0.21
0.36
-0.13
-0.45
0.00

-0.12
-0.21
-4.37
0.15
0.05

-0.13
0.24
-2.17
-0.17
-0.04

1.09
0.01
-0.32
-0.02
-0.56

LiNGAM
et
yt
pt
st
qt

-1.57
0.00
-0.88
0.00
0.00

0.19
0.49
1.00
-0.51
0.00

0.00
0.00
4.71
0.00
0.00

0.11
0.00
0.66
0.13
0.02

-0.11
0.00
0.21
0.00
0.63

Notes: LiNGAM (Causal structure: y −→ s −→ q −→ e −→ p) 64.5% bootstrap
stability, 58% initial conditions stability

41

Table 6: Quarterly Data.
εe

εy

εp

εs

εq

Distance Covariance
et
yt
pt
st
qt

-1.28
-0.06
2.20
-0.12
0.45

-0.20
0.63
-2.74
0.48
0.18

0.05
0.22
7.61
0.35
-0.23

0.43
0.19
-0.42
0.96
0.10

-0.79
-0.06
-2.21
-0.01
-0.32

Non-Gaussian Maximum Likelihood
et
yt
pt
st
qt

-1.06
0.15
-1.60
0.11
-0.17

0.44
0.64
-0.68
0.93
0.15

-0.23
0.08
8.11
0.18
-0.06

0.13
-0.21
0.19
0.47
-0.03

-1.04
0.01
-0.26
-0.07
0.64

FastICA (Negentropy)
et
yt
pt
st
qt

-1.07
0.16
-1.39
-0.05
-0.14

0.45
0.64
-1.06
0.21
0.15

0.18
-0.09
-8.10
-0.11
0.06

0.24
-0.09
0.14
0.72
-0.02

-0.91
0.00
-0.19
-0.07
0.57

LiNGAMet-1.36 0.05 0.00 0.34
-0.35
yt
0.00 0.66 0.00 0.00
0.05
pt
0.98 1.35 8.38 2.12
-0.61
st0.00 0.00 0.00 0.76
-0.06
qt
0.00 0.00 0.00 0.00
0.60
Notes: LiNGAM (Causal structure: q −→ y −→ s −→ e −→ p) 38.7% bootstrap
stability, 73% initial conditions stability

42