Modern Economy, 2011, 2, 729-734
doi:10.4236/me.2011.25081 Published Online November 2011 (
Copyright © 2011 SciRes. ME
On the Relationship between Water W ithdrawal and
Income: Smooth Transition Regression (STR) Approach
Mohsen Mehrara1, Abdolnaser Hemati1, Ali Sayehmiri1,2
1Department of Economics, Faculty of Economics, University of Tehran, Tehran, Iran
2Department of Economics, Faculty of Humanities, Ilam University, Ilam, Iran
E-mail: ir,,
Received July 2, 2011; revised August 11, 2011; accepted August 22, 2011
The Purpose of this paper is to investigate the relationship between annual water withdrawal (AWW) and
income in some countries. To achieve this end, a smooth transition regression (STR) model on cross section
data of 163 countries in 2006 was used. The findings support the nonlinearity of the link between AWW and
income. The findings, further, reveal the income elasticity of AWW as a bible-shaped curve. Hence, the
policies and management processes in water sector including water allocation between activities and reigns
should take into account the development degree and also focus on income level, water scarcity and the eco-
nomic, social and ecological structure in each country.
Keywords: Water-Income, STR, NRBEKC, Elasticity, Socio-Economic Structure
1. Introduction
Freshwater resources are vital for maintaining human life,
health, agricultural production, economic activities as
well as critical ecosystem functions. As populations and
economies grow, new constraints on freshwater resources
are appearing, raising problems for limits of water avail-
ability. Accordingly, the analysis of the national water
withdrawal intensity measurement becomes an important
policy issue. To serve these purposes, some water with-
drawal efficiency indicators have been developed and
applied to explain differences in performance between
countries and international benchmarking [1]. It should
be noted that the income elasticity of water withdrawal is
one solution used in this paper.
In recent years, the relationship between income elas-
ticity of natural resources use and income has attracted
an increasing attention among academic, non-govern-
mental organizations, and the media. A notable empirical
finding of the recent environmental economics literature
has been the existence of an inverted U-shaped relation-
ship between per capita income and pollution (per capita
emissions) of many local air pollutants [2]. Since this
relationship bears a resemblance to the Kuznets rela-
tionship between income and income inequality, it is
known as the Environmental Kuznets Curve (EKC) and
has spawned a vast number of papers in recent years. In
addition, attempts have been made to estimate EKCs for
a wide range of environmental indicators, including en-
ergy use, deforestation and municipal waste [3-7].
The shape of the EKC, attributed to scale, composition
and technique effects (SCTE) as discussed below, would
also seem to apply to (income elasticity of) water con-
sumption. The main reason to disregard water use in
EKC studies would appear to be a lack of socioeco-
nomic-hydrological data, although some recent investi-
gations and dataset have now resolved somewhat this
problem [8-12].
In this paper, we examine the relationship between
AWW per capita and GDP per capita using Smooth
Transition Regression (STR) model for 163 countries
across the world based on cross section data in 2006. The
following section will provide a brief review of the re-
lated literature. Section 3 introduces the econometric
methodology and empirical results, and the final section
presents the conclusions of the present study.
2. Natural Resources Based on
Environmental Kuznets Curve
The majority of EKC literature examines pollution levels
as a function of income. This has led to the criticism that
such research ignores the natural resource component of
environmental quality [2,13-16]. These studies tend to
treat resource use identical to pollution as an indicator of
environmental quality pointing to natural resources based
on environmental Kuznets curve (NRBEKC). Like pol-
lution, resource use can provide an economic benefit
coupled with an undesired environmental impact. Thus,
many of the theoretical explanations for the existence of
EKCs for natural resources mirror those for pollution.
The inverted U relationship between income and pol-
lution is typically explained in terms of the interaction of
scale, composition and technique effects (SCTEs). The
scale effect (SE) implies that as the scale of the economy
grows (ceteris paribus), AWW will do so. The composi-
tion effect (CE), however, refers to the fact that as
economies develop, there is totally a change in emphasis
from heavy industry to light manufactures and services
sectors, and also from high water intensity to low water
intensity in industrial, agriculture and domestic sectors.
Since the latter are typically less resource intensive than
the former, the composition effect of growth, ceteris
paribus, will reduce water use. Finally, there is the tech-
nique effect (TE). As incomes rise there is likely to be an
increased demand for environmental regulations [5]. The
effect of these regulations must be considered to reduce
water intensity due to improved techniques of production
and consumption.
3. Methodology
3.1. Smooth Transition Regression (STR)
The smooth transition regression (STR) model is a non-
linear regression model that may be viewed as a further
development of the switching regression model intro-
duced by [17]. The STR model originated as a generali-
zation of a particular switching regression model in the
work of [17]. These authors considered two regression
lines and devised a model in which the transition from
one line to the other is smooth. The earliest references in
the econometrics literature are [18,19]. Recent accounts
include [20-25]. The standard STR model is defined as
,,, 1,,,
Gcs utT
 
 
 
 
where is a vector of explanatory
les1, ,
variab, y
are a vector of exogenous variables. Furthermore, φ = (φ0,
φ1, ..., φm) and θ = (θ0, θ1, ..., θm) are ((m + 1) × 1)
parameter vectors and utiid(0, σ2) are given. Transition
function G(γ, c, st) is a bounded function of the continu-
ous transition variable st, continuous everywhere in the
parameter space for any value of st, γ is the slope pa-
rameter and c = (c1, ..., cK)
which is a vector of location
parameters, c1 ... cK. The last expression in Equation
(1) indicates that the model can be interpreted as a linear
model with stochastic time-varying coefficients φ + θG
(γ, c, st). In this paper it is assumed that the transition
function is a general logistic function:
 
,,1 et
Gcs sk
 
where γ > 0 is an identifying restriction. Equations (1)
and (2) jointly define the logistic STR (LSTR) model.
The most common choices for K are K = 1 and K = 2.
For K = 1, the parameters φ + θG (γ, c, st) change mono-
tonically as a function of st from φ to φ + θ. For K = 2,
they change symmetrically around the midpoint (c1 + c2)/
2, where this logistic function attains its minimum value.
The minimum lies between zero and 1/2. It reaches zero
when γ and equals 1/2 when c1 = c2 and γ < .
Slope parameter γ controls the slope and c1 and c2 the
location of the transition function. Transition function (2)
with K = 1 is also the one that [18] proposed, whereas
[18] and Chan & Tong (1986) favored the cumulative
distribution function of a normal random variable. In fact,
these two functions are close substitutes.
The LSTR model with K = 1 (LSTR1 model) is capa-
ble of characterizing asymmetric behavior. As an exam-
ple, it is supposed that st measures the phase of the busi-
ness cycle. Then the LSTR1 model can describe proc-
esses whose dynamic properties are different in expan-
sions from what they are in recessions, and the transition
from one extreme regime to the other is smooth. On the
other hand, the LSTR2 model (K = 2) is appropriate in
situations in which the local dynamic behavior of the
process is similar at both large and small values of st and
different in the middle (For further work on parameter-
izing the transition in the STR framework, see [18].
When γ = 0, the transition function G (γ, c, st) 1/2, and
thus the STR model (1) nests the linear model. At the
other ends, when γ , the LSTR1 model approaches
the switching regression model with two regimes that
have equal variances. When γ in the LSTR2 model,
the result is another switching regression model with
three regimes in which the outer regimes are identical
and the mid regime is different from the other two. It is
noteworthy that an alternative to the LSTR2 model exists,
the so-called exponential STR (ESTR) model. It is Equa-
tion (1) with the follow transition function:
,, exp
Et tt
Gcs sc
 
1 (3)
This function is symmetric around *
c and has, at
low and moderate values of slope parameter γ, approxi-
Copyright © 2011 SciRes. ME
mately the same shape, albeit a different minimum value
(zero), as (2). Because this function contains one pa-
rameter less than the LSTR2 model, it can be regarded as
a useful alternative to the corresponding logistic transi-
tion function. It has a drawback, however. When γ ,
(1) becomes practically linear with (3), for the transition
function equals zero at *
and unity elsewhere. The
ESTR model is therefore not a good approximation to the
STR2 model when γ in the latter is large and c2c1 is at
the same time not close to zero. In practice, the transition
variable st is a stochastic variable and very often an ele-
ment of zt. It can also be a linear combination of several
variables. In some cases, it can be a difference of an
element of zt; see [18] for a univariate example. A spe-
cial case, st = t, yields a linear model with deterministi-
cally changing parameters. When xt is absent from (1)
and st = ytd or st = ytd, d > 0, the STR model becomes
a univariate smooth transition autoregressive model [19]
for more discussion).
3.2. The Modeling Cycle
In this section we consider modeling nonlinear relation-
ships using STR model (1) with transition function (2).
We present a modeling cycle consisting of three stages:
specification, estimation, and evaluation. The specifica-
tion stage entails two phases. First, the linear model
forming the starting point is subjected to linearity tests,
and then the type of STR model (LSTR1 or LSTR2) is
selected. Economic theory may give an idea of which
variables should be included in the linear model but may
not be particularly helpful in specifying the dynamic
structure of the model. Linearity is tested against an STR
model with a predetermined transition variable. If eco-
nomic theory is not explicit about this variable, the test is
repeated for each variable in the predetermined set of
potential transition variables, which is usually a subset of
the elements in zt. Testing linearity against STAR or
STR has been discussed, for example, in [20,21].
The resulting test is more powerful than both the
LSTR1 (K = 1) and LSTR2 (K = 2) models. Assume now
that the transition variable st is an element in zt and let
, where t is an (m × 1) vector. The ap-
proximation yields, after merging terms and parameter-
izing, the following auxiliary regression:
yx s
1, ,tuT, (4)
where 3tttt
 z with the remainder R3(γ,
c, st). The null hypothesis is H0: β1= β2= β3= 0 because
each βj, j = 1, 2, 3, is of the form j
, where, 0j
is a function of θ and c. This is a linear hypothesis in a
linear (in parameters) model. Because under the
null hypothesis, the asymptotic distribution theory is not
affected if an LM-type test is used.
,,1exp/, 0
Gcs sc
4. Empirical Results
The basis of our empirical approach is exactly the same
as that used by many authors in literature. The observa-
tions in this modeling experiment come from AQUSTAT
FAO and WDI database of the [26,27]. The purpose of
the study is to investigate the effect of GDP per capita on
the annual water withdrawal (AWW). The AWW is as-
sumed to be a nonlinear function of the GDP per capita.
Figure 1 demonstrates a clearly nonlinear relationship
between the logarithmic values of GDP per capita (x-axis)
and the AWW (y-axis) as long as kernel fitting curve.
The sample consisted of annual water withdrawal
(AWW) per capita in cubic meter and GDP per capita in
2000 constant dollar for 163 countries of the world. Fit-
ting a linear model to the data yields:
2.356 5.671
()1.644 0.460t
 
, 163n
, ,
1,160 10.27
p (6)
An STR model is fitted to the logarithmic data. The
transition function is defined as a logistic function.
equals the residual standard deviation and
p is the p-value of the RESET test. The test does
indicate serious misspecification of (6). On the other
hand, the residuals arranged according to i
X in as-
cending order and graphed in Figure 2 show that the
linear model is not adequate. It can be seen in
large value in Equation (6).
The results of the linearity tests appearing in Table 1
p-values are remarkably small. Hypothesis H0 is the gen-
Figure 1. Observations of the logarithmic x-axis, and the
logarithm of the y-axis.
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Figure 2. Residuals of (1) (x-axis: the value of ; y-axis:
residual i
Table 1. P-values of the linearity tests of model (6).
Hypothesis p-Value
H0 9.2627e03
H4 2.3688e01
H3 6.2594e01
H2 1.6281e03
eral null hypothesis based on the third-order Taylor ex-
pansion of the transition function. Hypotheses H04, H03,
and H02 are the ones discussed in the Section of method-
ology. Because the p-value of the test of H03 is much
larger than the ones corresponding to testing H04 and H02.
The choice of K = 1 in Equation (7) (the LSTR1 model)
is quite clear. This is also obvious from Figure 1, for
there appears to be a single transition from one regres-
sion line to the other. The next step is to estimate the
LSTR1 model, which yields:
 
 
4.3730.6344.483 0.643
1881.057 0.1
9.560 0.77464.763 0.901
 
 
T = 163, R2 = 2.7975e01, σ = 1.11754, σx = 1.2086,
σ/σL = 0.971 (7)
where, σlx is the sample standard deviation of Lxi, σ is the
residual standard deviation of linear model and σL is that
of non-linear one. It should be noted that there are two
large standard deviations, which suggests that the full
model may be somewhat over parameterized. This is
often the case when the STR model is based on the linear
model without any restrictions. Model (7) is an example
of such a situation. It may appear strange that the need to
reduce the size of the model is obvious in this model
already because it only has a single explanatory variable.
The first reaction of the model would perhaps be to
tighten the specification by removing the nonlinear in-
tercept, Restriction φ = 0 or G(Lxi,
, c) = 0. Another
possibility would be to restrict the intercepts by imposing
the other exclusion restriction φ0 = θ0. In fact, the first
alternative yields a model with a slightly better fit than
the latter one. The model estimated with this restriction
 
(0.032)1.303 0.147
28.223 0.1
0.6274.696 0.490
1 exp19.7657.46 5
ii i
 
T = 163, R2 = 2.3839e-01, σ = 1.205, σLx = 1.209,
The estimated standard deviations of all estimates in
(7) are now appreciably small, and thus further reduction
of the model size is not necessary. The fit of both (7) and
(8) is vastly superior to that of (6), whereas there is little
difference between the two LSTR1 models. The residual
standard deviation of these models is only about one-
tenth of the corresponding figure for (2). Such an im-
provement is unthinkable when economic time series are
being modeled. The graph of the transition function as a
function of the observations in Figure 3 shows that the
transition is indeed smooth.
The test of no additive nonlinearity [H0: β1 = β2 = β3 =
0 in (transition function)] has the p-value of 0.0010. In
testing [H02: β1 = 0 | β2 = β3 = 0, a test based on a first
order Taylor expansion of H (γ2, c
2, x
2i)] and thus one
against another LSTR1component, we find that the p-
value of the test equals 0.017. These results show that
nonlinearity in this data set has been adequately charac-
terized by the LSTR1 model. The tests of no error auto-
correlation and parameter constancy are not meaningful
here in the same way as they are in connection with time
series models, and they have therefore not been applied
to model (8).
We modify this approach by using STR model re-
cently developed by Gonzalez et al. [28].
01 ,,
iii i i
LyLxLxG Lxci
 
  (9)
where εi is i.i.d (0, 2
) and the transition function G is:
GLx cLx c
 
, 0, (10)
5. Analysis
According on empirical result the relationship between
water and income is nonlinear model so we can calculate
the elasticity of water. In STR model, income elasticity
of AWW per capita depends on (log GDP per pita)
level (i). So it allows the parameters to change
smoothly as a function of the threshold or transition
ariable. Indeed, the elasticity of income is explained by
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Figure 3. Transition function of model (3) as a function of the transition variable.
the weighted average of parameters including
0 and
The income elasticity of for country ith is: (E)
 
Lx c
Ly gLx cLx
Lx Lx
 
sible values of the transition variable (GDP per capita).
Elasticity is increased slightly according to the income
level. Moreover, there is strong evidence that the rela-
tionship between per capita income and elasticity of
AWW is bible-shaped.
6. Conclusions
In this specification, a negative value of
1 may also lead
to the increase of elasticity. The estimated parameters in
this part could not be interpreted as elasticity. In Equa-
tion (12), the income elasticity of AWW () has been
presented. All calculations are computed with Matlab
and JMULTI software. The equation used to calculate
elasticity is given as the Equation (12).
This study reports on the use of a smooth transition re-
gression model based on cross section data to estimate
the relationship between AWW and income for 163
countries throughout the world. The findings yielded an
inverse U-shaped curve for world countries in the sample
data. These results justify ideas of (NRBEKC), (STCE)
and (OVW) concepts that are combining ecological and
social benefits as a whole. The findings further show that
income and socioeconomic criteria along with water
scarcity can have an effect on water withdrawal in wa-
ter-scarce countries and water intensity of use.
On the Figure 4, this elasticity is displayed for all pos-
The results, also, have important implications for the
models of water use and economic growth developed by
[29], and the other issues of water-income relationship
by [1,5,8,11]. This model needs to be modified in a
number of ways to account for the water savings accom-
panying a rapid structural transformation of an economy.
This can be accomplished either by estimating the water
savings attributable to the structural transformation of an
economy or revising the way we think about water use
during the transition.
Figure 4. The relationship between GDP per capita and
AWW’s elasticity.
 
19.765 0.489654.696exp19.7657.465
E 0.62680
1 exp19.7657.4651 exp19.7657.465
Lx Lx
Lx Lx Lx
 
 
 
 
7. Acknowledgements
This paper was funded by grant from the University of
Tehran submitted to the author
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