Within a semiparametric framework we propose a test of shape invariance of Engel curves, which is a necessary condition for base independence. Using Canadian family expenditure data for 1996 we reject shape invariance for the fuel and clothing share equations.

Dianqin Wang

Thanasis Stengos

English

Research Papers in Economics

Testing for Shape Invariance of Semiparametric Equivalence
Scales 
Thanasis Stengos Dianqin Wang
University of Guelph University og Guelph
Abstract
Within a semiparametric framework we propose a test of shape invariance of Engel curves,
which is a necessary condition for base independence. Using Canadian family expenditure
data for 1996 we reject shape invariance for the fuel and clothing share equations.
The first author would like to acknowledge financial support from the SSHRC of Canada.
Citation: Stengos, Thanasis and Dianqin Wang, (2007) "Testing for Shape Invariance of Semiparametric Equivalence Scales."
Economics Bulletin, Vol. 4, No. 21 pp. 1-11
Submitted: March 12, 2007.  Accepted: June 12, 2007.
URL: http://economicsbulletin.vanderbilt.edu/2007/volume4/EB-07D10005A.pdf1 Introduction
Equivalence scales constitute a useful conceptual tool for designing economic
policy such as an equitable tax and beneﬁts y s t e mt h a tw o u l dp u to ne q u a l
footing (bring into equivalence) the living standards of families of diﬀerent
size and composition. Preferences that obey the property that equivalence
scales will not vary across household income levels are said to follow base
independence. This is an important restriction on preferences that allows
for a meaningful interpretation of equivalence scales for policy purposes, (see
Blackorby and Donaldson (1993) and Blundell and Lewbel (1991)).
In parametric demand systems base independence is many times rejected.
However, it is not clear whether this rejection reﬂects a genuine dependence of
the equivalence scale function on total expenditures, or is the result of restric-
tive parametric assumptions about the functional form of the share equations.
Recently, to deal with this issue Gozalo (1997) and Pendakur (1999), Wilke
(2003) and Stengos, Sun and Wang (2006) have introduced nonparametric
and semiparametric models of equivalence scales. Gozalo (1997) treats both
the share equations and the equivalence scale function as purely nonpara-
metric in the context of a system of Engel curves, whereas Pendakur (1999),
Wilke (2004) and Stengos, Sun and Wang (2006) allow for general unspec-
iﬁed share equations that can be estimated nonparametrically, whereas the
equivalence scale is estimated as a parameter in each equation and as a re-
sult it is directly comparable with parametric estimates. In this paper we
will propose a simple test for shape invariance which is a necessary condition
for base-independence. This condition simply states that under base inde-pendence the (nonparametric) share equation curves for diﬀerent household
types will have the same curvature. As in Gozalo (1997) we use bootstrap-
ping to construct conﬁdence intervals. We ﬁnd that shape invariance holds
for food shares, but not for fuel and clothing shares.
The paper is organized as follows. In the next section we present our test
of shape invariance between nonparametric Engel curves. We then proceed
to present the results of the estimation of a system of Engel Curves using the
1996 Canadian Family Expenditure Survey.
2 Testing for Shape-Invariant equivalence scales
Base independence implies shape invariance for a system of the equations
that belong to a demand system. The same holds true if one looks at the
system without temporal variation as a system of Engel curves. The lat-
ter are linked through shape invariance for each household type and they
have the same curvature. This restriction constitutes the testable part of the
base-independence hypothesis. Shape invariance is only a necessary but not
suﬃcient condition for base independence. Hence, rejection of shape invari-
ance is useful since it would directly invalidate base independence, whereas
acceptance of shape invariance would not necessarily imply the validity of
base independence.
In the analysis below we restrict our attention to a single good in a system
of Engel expenditure share equations, where price remains constant. As in
Pendakur (1999) for exposition purposes we consider two household types,
type a and type b.L e t x be the log of total expenditure and let y = f(x)
b et h es h a r eo ft h es i n g l eg o o dp u r c h a s e db yt h eh o u s e h o l d . U n d e rb a s e -
2independence, the expenditure shares of type a and type b are related by
fb(xb) − η = fa(xb − δ) (1)
where ft(xt) is the share of a single good by household type t = a,b, δ is
the log of base-independent equivalence scales and η is the elasticity of the
equivalence scale function with respect to price. If δ and η are independent
of x, then they are said to be base independent. We consider a system of
Engle curves where prices are given and hence under base independence both
δ and η are constants.
S u p p o s ew eh a v ent observations for household type t, t = a,b on x and
y, xt,j and yt,j for j =1 ,2,···,n t. A nonparametric estimate of ft(x) based
on (xt,j,y t,j) for j =1 ,···,n t is deﬁned by
ˆ ft(x)=
rt(x)
dt(x)
(2)
where r(x) and d(x) are given by
rt(x)=
1
ntht
nt X
i=1
yt,iK
µ
xt − xt,i
ht
¶
(3)
dt(x)=
1
ntht
nt X
i=1
K
µ
xt − xt,i
ht
¶
(4)
K (.) is the kernel function and ht is the bandwidth. We use the Gaussian
kernel, K
³
xt−xt,i
ht
´
= 1 √
2πe
−1
2(
xt−xt,i
ht )
2
. We will determine the optimal band-
w i d t hb yc r o s s - v a l i d a t i o nm e t h o do v e rt h er a n g e(0,2sn
−1
5
t ), where s is the
standard deviation of x.
We will estimate the parameter δ, the log of base-independent equivalence
3scale, using the loss function introduced by Stengos, Sun and Wang (2006)
that is an improved variant of the one suggested by Pendakur (1999). If the
two households have similar preferences under base independence, we would
expect that the ranges of the two sample sets (xa,y a) and (xb −δ,yb −η) to
overlap and the nonparametric regressions based on (xa,y a) and (xb−δ,yb−
η) should be the same. We will deﬁne ˆ gb(x) as nonparametric estimates
based on points (xb,j − δ,yb,j) for j =1 ,···,n b, compared to ˆ fb(x) based on
points (xb,j,y b,j) for j =1 ,···,n b. The points (xa, ˆ fa(xa)) and (xb−δ,ˆ gb(xb))
determine the shape of expenditure share curves of household type a and type
b. Since x =
⎛
⎝ xa
xb − δ
⎞
⎠ and n = na + nb, the sets of points (x, ˆ fa(x)) and
(x, ˆ gb(x)) can be used to construct a measure of the closeness of the two
curves. We run the following artiﬁcial regression
ˆ gb(x)=η + β ˆ fa(x)+u (5)
The hypothesis of shape invariance is given as β =1 . We let a and b
denote the OLS estimates of η and β respectively from equation (5) above.
To compute the standard error of b, we use a wild bootstrap procedure sim-
ilar to the one presented in Gozalo (1997)). Using the notation introduced
above, under the null hypothesis of shape invariance of the share equations,
(x, ˆ gb(x)−a) should be similar to (x, ˆ fa(x)). Let us deﬁne the vector of resid-
uals u0 =( ˆ gb(x) − a) − bb fa(x), and let y∗
0 denote ˆ gb(x) − a and let y∗
0(i)
be the i − th element of y∗
0. Similarly we deﬁne u0(i) as the i − th element
of u0 and we also deﬁne ¯ u =
Pn
i=1 u0(i). The jth bootstrap observation is
produced by the following:
4For each i from 1 to n,w e
1. randomly generate the bootstrap residual u(i) from a distribution Fi
such that for Z ∼ Fi,
EFiZ =0
EFiZ
2 =( u0(i) − ¯ u)
2
EFiZ
3 =( u0(i) − ¯ u)
3
2. set y∗
j(i) to be y∗
0(i)+( u(i)+¯ u)
We use a distribution Fi deﬁned in Gozalo (1997), which has a density func-
tion piφai,κ +(1−pi)φbi,κ,w h e r eφc,κ is the density function of N(cκ,c2(1−
κ2)), and pi,a i, and bi are deﬁned as following:
pi =
³
5+
√
5
´
/10
ai = u0(i)
³
1 −
√
5
´
/2
bi = u0(i)
³
1+
√
5
´
/2
κ =(
√
3 − 1)/2
T h eb o o t s t r a pp r o c e d u r ep r o d u c e sas e r i e so fy∗
j, for j =1 ,···,B,where
B is the number of bootstrap replications. Using the series y∗
j,w eo b t a i nt h e
empirical distriution of b, using equation (7) above, which gives the bootstrap
standard error of b. We proceed to test the hypothesis H0 : β =1 , using the
bootstrap standard error in the construction of the t−ratio for H0.
3 Empirical Results
5We now apply the test for shape invariance to data taken from the 1996
Canadian Family Expenditure Survey to estimate expenditure share curves
for cloth and fuel. The data set we examine covers 8741 households. Af-
ter controlling for regional characteristics we deﬁne the household type HT
by the number of seniors (ns), the number of adults(na),t h en u m b e ro f
youths(ny), and the number of children(nc) as
HT =1 0 0 0 ns +1 0 0 na +1 0 ny + nc.
For example HT =2 1 1r e f e r st oah o u s e h o l dw i t ht w oa d u l t s ,o n ey o u t h
and one child, while HT =1 2 0 0refers to a household with one senior and
two adults. We will focus on those types of household with more than 50
observations in the survey. There are 23 such household types, the most
popular being the one with two adults, which will serve as the reference
household. For a description of the data, see Stengos, Sun and Wang (2006).
The distribution of the household types is summarized in Table1.
It is worth noting that using the above classiﬁcation scheme to select
data into diﬀerent cells allows for controls for age aﬀects, household size and
the number of children which are among the most important characteristics
in consumer demand. In eﬀect our approach is similar in spirit to the par-
tial linear semiparametric model of Blundell, Duncan and Pendakur (1998),
Lyssiotou, Pashardes and Stengos (1999) and Yatchew, Sun and Deri (2003),
where characteristics enter the linear part of partial linear speciﬁcation.1
1Note that since we are only using one-dimensional kernels our estimates do not suﬀer
from the curse of dimensionality problem that typically plagues multidimensional kernel
estimates. That is true even in the case of the smallest cell of 50 observations.
6Table 1: Distribution of Household Types
HT Members Obs HT Members Obs
10 1y o u t h 52 210 2 adults, 1 youth 442
20 2y o u t h s 91 211 2a d u l t s ,1y o u t h ,1
child
249
100 1 Adult 1064 212 2a d u l t s ,1y o u t h ,2
children
100
101 1 adult, 1child 182 220 2 adults, 2 youths 352
102 1 adult, 2 children 128 221 2a d u l t s ,2y o u t h s ,1
child
122
110 1a d u l t ,1y o u t h 254 300 3a d u l t s 142
111 1 adult, 1 youth, 1
child
99 1000 1s e n i o r 907
120 1 adult, 2 youths 66 1100 1s e n i o r ,1a d u l t 350
200 2a d u l t s 1542 1200 1s e n i o r ,2a d u l t s 84
201 2 adults, 1 child 566 2000 2s e n i o r s 678
202 2 adults, 2 children 861 2100 2s e n i o r s ,1a d u l t 67
203 2 adults, 3 children 343
We use the loss function introduced by Stengos, Sun and Wang (2006) to
estimate δ of the reference household of two adults (HT=200) and the other
household types. We then proceed to test for shape invariance for the system
of fuel and clothing share equations. Table 2 presents the results of these
tests. It is clear that the hypothesis of shape invariance and consequently
of base independence is rejected in most cases, except for a few categories
for fuel.2 This suggests that shape invariance and hence base independence
does not hold for these share equations and as such equivalence scales will
be dependent on expenditures.
2To compute standrad error of the coeﬃcient estimate b we use 999 bootstrap
replications.
7Table 2: The bootstrap testing results contain the mean of b and p-values
for the null hypothesis of shape invariance
HT bcloth p − value for bcloth bfuel p − value for bfuel
10 -0.0326 0.0000 -0.0053 0.0000
20 0.0661 0.0000 0.1978 0.0000
100 0.0088 0.0000 1.0130 0.74345
101 0.1164 0.0000 0.1637 0.0000
102 0.0426 0.0000 0.1956 0.0000
110 0.2413 0.0000 0.5219 0.0000
111 0.0269 0.0000 -0.3654 0.0000
120 0.1043 0.0000 0.1336 0.0000
201 0.2251 0.0000 0.8950 0.0000
202 0.0904 0.0000 1.0885 0.2378
203 -0.2112 0.0000 1.0820 0.2846
210 0.0195 0.0000 0.5607 0.0000
211 0.1294 0.0000 0.7432 0.0000
212 0.1646 0.0000 0.4066 0.0000
220 0.0098 0.0000 0.8348 0.1123
221 0.1387 0.0000 0.5108 0.0000
300 0.0958 0.0000 0.8896 0.2223
1000 -0.0726 0.0000 0.8516 0.1423
1100 -0.0437 0.0000 0.9893 0.7135
1200 0.1352 0.0000 0.5794 0.0000
2000 0.1513 0.0000 0.7365 0.0000
2100 0.0395 0.0000 0.2192 0.0000
8References
[1] Blackorby C. and D. Donaldson, 1993, Adult-equivalence scales and the
economic implementation of interpersonal comparisons of well-being, So-
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[2] Blundell R. and A. Lewbel, 1991, The information content of equivalence
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[3] Blundell R., Duncan, A. and K. Pendakur, 1998, Semiparametric estima-
tion and consumer demand, Journal of Applied Econometrics 13, 453-461.
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[5] Lyssiotou, P., Pashardes, P. and T. Stengos, 1999, Preference Heterogene-
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[6] Pendakur, K., 1999, Estimates and tests of base-independent equivalent
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[7] Stengos, T., Sun, Y. and D. Wang, 2006, Estimates of Semiparametric
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[8] Wilke, R., 2003, Semiparametric Estimation of Regression Functions un-
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9[9] Yatchew, A., Sun, Y. and C., Deri, 2003, Eﬃcient estimation of semi-
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10

Testing for Shape Invariance of Semiparametric Equivalence Scales

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Testing for Shape Invariance of Semiparametric Equivalence Scales

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