Book Chapter
Reference
Panel data estimation of the intergenerational correlation of incomes
ABUL NAGA, Ramses, KRISHNAKUMAR, Jaya
ABUL NAGA, Ramses, KRISHNAKUMAR, Jaya. Panel data estimation of the intergenerational correlation of incomes. In: Jaya Krishnakumar, Elvezio Ronchetti. Panel data econometrics : future directions : papers in honour of professor Pietro Balestra . Amsterdam :
North-Holland, 2000. p. 85-106
Available at:
http://archive-ouverte.unige.ch/unige:42045
Disclaimer: layout of this document may differ from the published version.
Intergenerational Correlation of Incomes 1
by
Ramses Abul Naga 2
and Jaya Krishnakumar 3
July 1999
Abstract
Weconsidertheproblemofestimatingtheintergenerationalcorrelationof
incomes inthe context of apanel data frameworkwith measurement errors.
We present single equation estimation methods as well as system methods
under various assumptions regarding the serial correlation of the error term
and taking into account possible correlations among children of the same
family. ApplicationtoasampleofUSparentsandchildrenleadstoestimates
of the orderof 0.42 to0.60 for the coeÆcient of income transmission.
Keywords: Intergenerational mobility,panel data, errors invariables
JEL Classication Codes: I3, J6,C3
1
Wewouldliketo thankPietro Balestra,ChristianSchluterandDirk Vandegaerfor
theircarefulreadingandvaluablecommentsonanearlierversionofthepaper.Suggestions
from the participants of the Economics Seminar at the University of St.Gallen are also
acknowledged.
2
DEEPandIEMS,UniversityofLausanne,Switzerland
3
DepartmentofEconometrics,UniversityofGeneva,Switzerland
Ever since President Johnson's declaration of the "War on Poverty" in the
United Statesinthe year 1964,economists haveincreasinglyturnedtheirat-
tention and eort tounderstanding the mechanismsof incometransmission.
There are several importantequity issues underlyingthe generalquestion of
the intensity of intergenerationalinheritance of income status.
Equityconcerns canbeformulatedalongseveral lines. Onequestionthat
oftenarises concerns the extent towhich childrenraised inpoorfamiliesare
themselveslikelytoendupinthelowerendofthedistributionofincomedur-
ing their adult lives (Atkinson et al.(1983), ch. 1). The continuity problem
in low incomes is often also formulated by contrasting the likelihoods that
childrenraisedindierentsocialenvironmentsface, ofendingup inpoverty.
Economists and social scientists also seek to assess the separate contribu-
tions ofenvironment-relatedfactorsand geneticallytransmittedtraitsinthe
overall transmission of income status ( the famous 'nature versus nurture'
debate).
In immigration countries such as the United States, another question
arises about the rate at which newcomers to the host country are likely
to be assimilatedin terms of economic and educational attainment (Stokey
(1996)). To answer such questions, one needs to make progress both at the
theory level, i.e. in the formulation of models of income transmission and
the dynamicsof wealthdistribution (e.g. Galorand Zeira(1993)),aswellas
in the directionof the empirical analysis of longitudinalincome data.
Theestimationofthe intergenerationalcorrelationofincomesisapartic-
ularly complex task, since the variables of interests, namely the permanent
incomes of parents and children, are typically unobservable quantities. In-
stead, the researcher usually possesses a short time-series (in the order of 3
or 4 observations)on some indicatorof economic attainment such as hourly
earnings, or a family income measure. The resulting measurement error
problem has been shown by previous researchers in the area to render the
OLS estimator biased towards zero (see e.g. Bowles (1972), Atkinson et al.
(1983), Behrman and Taubman(1990)).
Of the various approaches proposed in order to deal with this mea-
surement error problem, two methods have received considerable attention.
planatoryvariable)overtheseveral yearsofavailabledatainordertoreduce
theratiooftransitorytototalincomevariance,and(ii)theuseofoutofequa-
tion instruments(typicallythe father's education)for theincome pertaining
to the parents' generation (cf. for instance Solon(1992)).
The purpose of the present study is to estimate the intergenerational
correlation ofincomeswithinapaneldata framework. Itis rathersurprising
that noexplicit treatment of this problem has been considered in the panel
data perspective 4
,sincemostrecentstudies(atleast theUSones) have been
based on samplesextracted fromlongitudinal earningssurveys 5
.
There are indeed several good reasons for considering this estimation
problem within a panel data framework. Firstly, as shown by Griliches and
Hausman (1986), Hsiao and Taylor (1991) and others, a wide range of er-
rors in variables models may be identied without having to appeal to out
of equation instruments. Secondly, by controlling for unobserved popula-
tion heterogeneity,paneldata estimatorswillusuallyhavemoreexplanatory
power than their cross-section counterparts. Finally, because we have re-
peatedmeasurementsonthe incomesof parentsand children, weare able to
estimate our various models under alternative scenarios regarding the serial
correlation in the transitorycomponents of incomeand test these scenarios.
This latter point extends beyond eÆciency considerations, since the choice
of instruments itselfis dictated by the assumptions regarding the serial cor-
relation inerrors of measurements.
Our study provides a discussion of two further issues: the pooling of
instrumental variable regressions from the various cross-sections into a sys-
tem (of equations) estimator, and the consequences of allowing for multiple
children from the same parent family (entailing cross-sectional correlations
across observations) for eÆcient estimation. When applied to a US sample
of parents and children, the various models and estimators considered here
provide estimates in the rangeof 0.42 to0.60.
The structure of the paper isthe following: section2 discusses the data,
section 3 deals with inference issues, section 4 contains the empiricalappli-
4
ThoughsomeinsightsintothisproblemareprovidedinZimmerman(1992).
5
See for instance Behrman and Taubman (1990), Altonji and Dunn (1991), Solon
(1992),Zimmerman(1992)andBjrklundandJantti(1997)
research.
2 Data
The empirical applications contained in this study draw evidence from the
panel studyof incomedynamics(PSID).The PSIDis aUSlongitudinalsur-
vey of5000 familieswhich hasbeen runningannually since1968. A detailed
description of the history of the panel, and its main features, can be found
inHill (1993). Our sample wasextracted from the mergedfamily-individual
tapeofwaveXXI(1968-1988)ofthe PSID.Asummaryofthecharacteristics
of our sample can be found in Abul Naga (1998). Here we discuss three
sampling aspects of our data which entail direct consequences for the esti-
mation strategies to be pursued in section3 below. These are (i)the extent
to which our data meet the standard denition of a panel, (ii) the problem
of measuring long-run economic status, or permanent income, and (iii) the
presence of multiplechildren from the same parent family. We take up each
of these pointsin turn.
We have a sample of 1426 parent-child pairs. Incomes of parents and
children are observed respectively over the ve year intervals 1967-71 and
1983-87. On the one hand data ondependent and explanatory variables are
not contemporaneously observed, a feature which is not typicalof the stan-
dard framework adopted in panel data analysis (for instance Hsiao (1986),
Chap.3). On the other hand, the incomedata have anatural time ordering,
which is essential for modelling and testing for serial correlation in the dis-
turbances, and alsofor selectingvalidinstruments for explanatoryvariables
(see below). As willbecome apparent in the next section, the estimators we
adopt do not require incomes of parents and children to be contemporane-
ously observed. The estimators will be shown to have obvious panel data
interpretations,but more generally, they may alsobeconsidered within the
framework of multiple(simultaneous) equations systems.
Our main income concept is the total annual family income normalized
by the family's needs, and deated to 1967 dollars (the rst year of the
panel). The needs standard is the oÆcial Orshansky scale which is used
in the US in order to quantify the extent of poverty. The price deator
used is the consumer price index. Such income measures may be treated
(1986), as random draws from a variable with a population mean equal to
the household's permanent income. Thus, in section 3, we propose several
instrumental variables procedures in order to deal with the measurement
error problem inherentin our data.
Astheincomeconcept wehavechosen touseisbased onthe totalfamily
income,we did not nd itnecessary torestrictour sample tofather and son
pairs. Thus, for a daughter froman initialsamplefamily, the income of her
household is matched tothat of the parents' whether or not she happens to
bethe main incomeearner. We have alsoallowed formultiple children from
the same parent family. We note that because of unobserved family back-
ground components, observations on siblings are expected to be correlated.
Under such circumstances, eÆcient estimation will require the adoption of
generalized least squares methods, which are also discussed in some length
in the followingsection.
Table 1 : Summary Statistics
year Parents Children
mean coef.variation mean coef.variation
1 1.73 0.80 2.86 0.70
2 1.88 0.85 3.00 0.71
3 2.00 0.84 3.05 0.76
4 2.15 0.83 3.15 0.77
5 2.31 0.83 3.25 0.83
1. TheresourceconceptisthefamilyincomenormalizedbytheOrshanskyneedsstandard.
2. Therstobservationyear(t=1)pertainsto1967forparents,andto1983forchildren.
Table1 providessome summarystatisticsforthe incomesof parents and
children. Parents appear tobe onthe wholepoorer than the US population
in the late 1960s. This may be explained by noting that approximately a
half of the parents originate from the SEO le (a sample of families whose
havefound thatdisadvantagedgroupsweremore likelytoexitthe paneland
that on the whole, the PSID has evolved towards a sample of middle class
families. This nding may help in accounting for the seemingly lower level
of income dispersion experienced by the sample of children, as well as the
growth in their average income.
3 The Model and the Estimation Methods
We are interested in estimating a relationshipof the form
c;i
=
p;i +
i
(1)
where
c;i
representsthe logarithmofthe permanentincomeofthe child,
p;i
the logarithmof the the permanent income of the parents and
i
arandom
disturbance term specic tothe individual. If correctmeasurements of both
the variablesare available,then a consistent estimationof would begiven
byOLS.Howeverthevariablesinquestionarenotdirectlyobservableandthe
only information one generally has are the actual income of the parent and
the childat dierent pointsintime. Obviouslythese snapshots docontain a
component equaltothe permanent incomebut alsoinclude othertransitory
components. Hence these observations can be thought of as measurements
of thepermanentincomesubjecttoerror. Denotingtheobserved incomesas
y
it and x
it
, we can write:
y
it
=
c;i +u
it
(2)
x
it
=
p;i +v
it
(3)
whereu
it andv
it
areuncorrelatedrandomerrortermswithzeromeans. Thus
the relationshipbetween the observables is
y
it
=x
it +
i +(u
it v
it
) (4)
or
y
it
=x
it +
i +"
it
(5)
with
"
it
=u
it v
it
(6)
and
it i c it p it it it
Let us note that if only the dependent variable is measured with error
(
c;i
in our case), OLS would still consistently estimate , whereas if the
explanatoryvariablehasameasurementerrorthenOLSproducesinconsistent
estimates(cf. Greene(1993)e.g.). Thougharemedytothisproblemisgiven
by the instrumentalvariable(IV) technique, inthe caseof purecrosssection
data, oneneedstoresorttoextraneous instrumentsand this maypose areal
practical diÆculty. However there is an easier way out in the case of panel
data andGrilichesand Hausman(1986)wereamongstthe rstonestopoint
out thatthemodelcanbeidentiedwithouttheuse ofexternalinformation.
SeealsoHsiaoandTaylor(1991),WansbeekandKoning(1991),Biorn(1998)
in this regard. Indeed, the observations onthe explanatory variablerelating
to dierent time periods provide appropriate instruments for the current
period, the gap to be chosen depending on the nature of serial correlation
assumed for the errors.
Ingeneral,twoapproaches havebeen proposed inthe literatureforpanel
datamodelswithmeasurementerrors. Eithertheequationcanberstdier-
enced (toeliminatethe specic eect
i
) andthe instrumentsused inlevels,
or the equation is kept in levels and the instruments dierenced (see Biorn
(1998)).
For our model, special care needs to be taken while choosing the in-
struments as the true unobserved variable is time invariant and hence any
dierencing operationover time willleave us with onlythe dierence of the
corresponding v
it
'sand any \instrument"correlated withthis dierence will
necessarilybecorrelatedwiththesamedierenceappearingintheerrorterm.
For instance, uponrst dierencing we get
y
it y
i;t 1
=(x
it x
i;t 1 )+"
it
"
i;t 1
(8)
But from (3) we have
x
it x
i;t 1
=v
it v
i;t 1
(9)
which also appears in "
it
"
i;t 1
. Thus any variable correlated with (x
it
x
i;t 1
)(condition tobesatised by a validinstrument)willalwaysbecorre-
lated with ("
it
"
i;t 1 ).
case and we will therefore only consider estimating the equation in levels.
For the same reasons, dierencing of instruments is alsoinappropriate.
Theset ofvalidinstrumentsvaries accordingtothetypeofserial correla-
tionassumedfor"
it
. Letusdistinguishtwocases,bothassumingstationarity:
(i) iiderrors and (ii)MA(1) errors. More generalMA processes will alsobe
considered in the application section. The case of AR(1) errors oers no
immediate solution and hencewill not be pursued here.
Case 1: iid "
it 's
Assumption A1:
"
it
iid(0;
2
"
);
i
iid(0;
2
); E(
j
"
it
)=0 8i;j;t
In terms of the individual componentsthe assumptions would be
E(u
it u
js )=Æ
ij Æ
st
2
u
8i;j;s;t E(v
it v
js )=Æ
ij Æ
st
2
v
8i;j;s;t
E(
j u
it
)=0 8i;j;t E(
j v
it
)=0 8i;j;t
where Æ
ij
is the Kronecker delta. However it should be noted that only
the overall variance 2
"
= 2
u +
2
2
v
can beidentied.
Inthiscase,foranyperiodt,alltheobservationsatleastoneperiodapart
are valid instruments. Let y
t
;x
t
; and "
t
denote (N 1) vectors obtained
by piling all the individuals for period t, and dene r
t
= +"
t
. So for the
period1 equationwritten as
y
1
=x
1
++"
1 x
1 +r
1
(10)
z
1
= [x
2 x
3
::: x
T
] is a validinstrument matrix, assuming that we have
observationsfor T time periods. Furthermore,
V(r
1
)=V( )+V("
1 )=
2
I
N +
2
"
I
N
2
r I
N
Giventhis, twomethodsarepossible: applyingtheIV techniqueequationby
equation orcombining all periods into a system and then applying IV with
the whole set of instruments for all the periods together. Letting z
t
denote
the instrument set for the t-th periodequation, the rst methodleads to
^
(t)
=[x 0
t z
t (z
0
t z
t )
1
z 0
t x
t ]
1
x 0
t z
t (z
0
t z
t )
1
z 0
t y
t
(11)
as V(r
t )=
r I
N .
The system estimatorwillbe obtained as follows combining allt's:
y
1
= x
1
++"
1
:
:
:
y
T
= x
T
++"
T
or using appropriatenotations
y=X+(
T I
N
)+"X+r (12)
where
T
is a vector of ones, and y;X and r are (NT 1) vectors. The
resulting system variance covariancematrix isgiven by
V(r)= 2
(
T
0
T I
N )+
2
"
I
NT
Writing the whole set of instrumentsas
Z = 0
B
B
B
@ z
1
0 0
0 z
2 0
: : :
0 0 z
T 1
C
C
C
A
(13)
we willhave
^
(gl s)
=[X 0
Z(Z 0
Z) 1
Z 0
X]
1
X 0
Z(Z 0
Z) 1
Z 0
y (14)
This estimator can be shown to be a weighted average of the \individ-
ual"
^
(t)
'sobtained by (11) for each period,with theweights being inversely
proportional to the corresponding variance matrix. Needless to say that
the same estimatorcan bederived using a Generalised Method of Moments
(GMM)approach(cf. Greene(1993)e.g.) basedonthe orthogonalitycondi-
tions between the instruments and the combined error term 6
. Note that we
could alsowrite a system OLS estimatoras follows:
6
Analternativeestimatoristheso-calledGeneralisedInstrumentalVariablesEstimator
(GIVE)whichisobtainedbyusingtheinstrumentmatrix 1
2
ZandthenapplyingGLSon
themodeltransformedbytheinstrumentmatrix. TherelativeeÆciencyofthisestimator
with respect to the one given in (14) cannot be established in general. We have not
considered the GIVE in our empirical analysis due to the computational complications
involvedinitsimplementation.
^
(ol s)
=[X Z(ZZ) Z X] XZ(ZZ) Z y (15)
Sofar we have assumed independence across dierent individuals in the
sample. Howeverour sampleof parents andchildrencontains siblings andit
willbemore realisticto assumethat the individualeects of children of the
same parents are correlated. This would implynon zero correlationbetween
the specic eects
i
and
j
if i and j belong to the same family. Let us
takethis intoaccount by means of a variable d
ij
such that
d
ij
= (
1 if i,j are siblings
0 if not
(16)
Let usfurther assume that
E(
i
j )=d
ij c
This modiesour variancematrix of as follows:
V( )= 0
B
B
B
B
B
B
B
B
@
2
d
12 c
::: d
1N c
d
21 c
2
::: d
2N c
: : : :
: : : :
: : : :
d
N1 c
:: :::
2
1
C
C
C
C
C
C
C
C
A
(17)
Then
V(r
t )=
+
2
"
I
N
0
(18)
and
V(r)
=(
T
0
T
)+
2
"
I
NT
= 0
B
B
B
B
B
@
0
:::
: : : :
: : : :
: : : :
:::
0 1
C
C
C
C
C
A
(19)
Thus the single equation estimator of accounting for correlations across
siblings would be given by
^
(t)
=[x 0
t z
t (z
0
t
0 z
t )
1
z 0
t x
t ]
1
x 0
t z
t (z
0
t
0 z
t )
1
z 0
t y
t
(20)
and the system estimatorby
^
(gl s)
=[X Z(Z Z) Z X] XZ(Z Z) Z y (21)
Nowlet usproceed to the MA(1)case.
Case 2: MA(1) errors
Assumption A2: "
it
=w
it +
"
w
i;t 1
Here the underlyingassumptions onthe componentsare:
u
it
=
it +
u
i;t 1
(22)
v
it
=
it +
v
i;t 1
(23)
but onceagain only the parameters describing the overall error term can be
identied. Letus denote
1
E("
it
"
i;t+1 )
and we know for anMA(1) process
E("
it
"
is
)=0 for js tj2
Theinstrumentset has tobemodiedaccording tothe newassumptions
asobservationsone periodapart arenolonger validinstruments. Theyhave
to be at least two periodsaway. Thus we would have (taking T=5 as is the
case in our study):
~ z
1
= [x
3 x
4 x
5 ]
~ z
2
= [x
4 x
5 ]
~ z
3
= [x
1 x
5 ]
~ z
4
= [x
1 x
2 ]
~ z
5
= [x
1 x
2 x
3 ]
Havingmade these changes and assumingforthe momentthat there are
no siblings, the expression of the single equation estimator
~
(t)
remains the
same ( equation(11) ) where the new instrument z~
t
is to be used instead of
z
t
. In the system estimation procedure, the varianceof the vector r changes
as we now have non zero correlation between r
t
and r
t 1 or r
t+1
. Again
assuming nosiblings,we have
E(r
t r
0
t )=
2
I
N +
2
"
I
N
0
(24)
E(r
t r
t+1 )=
I
N +
1 I
N
1
(25)
and
E(r
t r
0
s )=
2
I
N
2
for js tj2 (26)
Thus
E(rr 0
)= 0
B
B
B
B
B
B
B
B
@
0
1
2
:::
2
1
0
1
:::
2
: : : : :
: : : : :
: : : : :
2
2
:::
1
0 1
C
C
C
C
C
C
C
C
A
~
(27)
Thus the modied system estimatorwould be written as
~
gl s
=[X 0
~
Z(
~
Z 0
~
~
Z) 1
~
Z 0
X]
1
X 0
~
Z(
~
Z 0
~
~
Z) 1
~
Z 0
y (28)
where
~
Z now contains the new set of instruments.
Let usnow introduce correlations amongsiblings. We get
E(r
t r
0
t
) =
+
2
"
I
N
0
(29)
E(r
t r
0
t+1
) =
+
1 I
N
1
(30)
E(r
t r
0
t+2
) =
2
(31)
Thus
E(rr 0
)= 0
B
B
B
B
B
B
B
B
@
0
1
2
:::
2
1
0
1
:::
2
: : : : :
: : : : :
: : : : :
2
2
:::
1
0 1
C
C
C
C
C
C
C
C
A
~
(32)
Now, the single equation and system estimators taking account of such cor-
relations are given respectively by
~
(t)
=[x 0
t
~ z
t (~z
0
t
0
~ z
t )
1
~ z 0
t x
t ]
1
x 0
t
~ z
t (~z
0
t
0
~ z
t )
1
~ z 0
t y
t
(33)
~
(gl s)
=[X 0
~
Z(
~
Z 0
~
~
Z) 1
~
Z 0
X]
1
X 0
~
Z(
~
Z 0
~
~
Z) 1
~
Z 0
y (34)
Variance - covariance estimators
In order to implement all the above methods in practice, we need prior
estimates ofthevariousvariance-covarianceparametersappearinginthedif-
ferent blocksof V(r). We give below the various sample moments that con-
sistently estimatethe relevant parameters.
As
r
it r
i:
="
it
"
i:
(35)
and
V("
it
"
i:
)=
(T 1)
T
2
"
(36)
where a dot in the place of a subscript indicates the average taken over it,
2
"
can be estimated by
^ 2
"
= 1
N(T 1) X
i X
t (^r
it
^ r
i:
) 2
(37)
wherer^
it
=y
it x
it
^
and
^
canbeeitherthecorrespondingestimatorforthat
time period (given by equation (11)) or a rst stage system OLS estimator
(given by (15)).
Using
V(r
i:
)=V(
i
)+V("
i:
)= 2
+
2
"
=T (38)
we can estimate 2
as follows:
^ 2
= 1
N X
i
^ r 2
i:
1
T
^ 2
"
(39)
Finally, anestimatorof c
is given by
^ c
= 1
N
J X
i;j2J (^r
i:
~ r
::
)(^r
j:
~ r
::
) (40)
J
personsin J and r~
::
isthe mean overthis group.
In the MA(1)case, we need
1
in additiontothe above. Knowing
Cov(r
it
;r
i;t+1 )=
2
+
1
and
Cov(r
it r
i:
;r
i;t+1 r
i:
)=
(T 1)
T
1
we can obtain
^
1
= T
(T 1) 1
(NT) X
i X
t (^r
it
^ r
i:
)(^r
i;t+1
^ r
i:
)
= 1
(T 1) 1
N X
i X
t (^r
it
^ r
i:
)(^r
i;t+1
^ r
i:
)
The above procedures (single equation and system) can be generalised
to MA(2), MA(3),... errors in an exactly analogous way from iid to MA(1)
without much diÆculty. In fact these possibilitieswill alsobe explored em-
pirically.
Specication tests
Before nishingwith the methodology, let us look at the dierent speci-
cation tests that can be carried out inthis context. The rst type of tests
that one can think of are those that concern the nature of serial correlation
assumed for the errors r
t
. For instance, in order to test the assumption of
serial independence, denoted as MA(0), against the alternative MA(1), a
Hausmantest statisticcanbe constructedbasedonthedierencebetween
^
and
~
denotingtheestimatorunderthenull(timeindependence)as
^
andthe
one under the alternative(MA(1)) as
~
. Under the null,both are consistent
with
^
being more eÆcient and under the alternative only
~
is consistent.
Let us consider the single equation case say for t = 1. The instrument will
be z
1
= [x
2 x
3 x
4 x
5
] under H
0
and z~
1
= [x
3 x
4 x
5
] under H
a . The
twoestimators of are given by:
^
=[x 0
1 z
1 (z
0
1
^
0 z
1 )
1
z 0
1 x
1 ]
1
x 0
1 z
1 (z
0
1
^
0 z
1 )
1
z 0
1 y
1
(41)
and
~
=[x
1
~ z
1 (~z
1
^
0
~ z
1 ) z~
1 x
1 ] x
1
~ z
1 (~z
1
^
0
~ z
1 ) z~
1 y
1
(42)
where
^
0
is the estimator of the variance of r
1
under MA(1). It is easily
shown that
V(
^
)=[x 0
1 z
1 (z
0
1
^
0 z
1 )
1
z 0
1 x
1 ]
1
(43)
V(
~
)=[x 0
1
~ z
1 (~z
0
1
^
0
~ z
1 )
1
~ z 0
1 x
1 ]
1
(44)
with V(
^
)<V(
~
)
and
Cov(
^
;
~
)=V(
^
) (45)
Hence
V(
~
^
)=V(
~
) V(
^
) (46)
The test statisticis therefore
(
~
^
) 0
[V(
~
) V(
^
)]
1
(
~
^
)
2
1
with large values rejecting independence.
Similarly, MA(1) can be tested against MA(2) by constructing the test
statistic based on the dierence between the two corresponding estimators.
In this case, the instrument sets for the rst period equation are z
1
=
[x
3 x
4 x
5
] under H
0
and z~
1
= [x
4 x
5
] under H
a
. Other tests such as
MA(2) against MA(3) or MA(1) against MA(3), MA(0) against MA(2) or
MA(0) againstMA(3) can all bedevised similarly.
The same resultshold for the system estimators too,the basicdierence
being the dimension of the variables involved. Let us illustrate the case iid
against MA(1).
We have the system (see (12))
y=X+r (47)
and
Z = B
B
B
@ z
1
0 0
0 z
2 0
: : :
0 0 z
T C
C
C
A
(48)
for the nulland
~
Z = 0
B
B
B
@
~ z
1
0 0
0 z~
2 0
: : :
0 0 z~
T 1
C
C
C
A
(49)
for the alternative. The expressions ofthe estimators willbe
^
=[X 0
Z(Z 0
Z) 1
Z 0
X]
1
X 0
Z(Z 0
Z) 1
Z 0
y (50)
and
~
=[X 0
~
Z(
~
Z 0
~
Z) 1
~
Z 0
X]
1
X 0
~
Z(
~
Z 0
~
Z) 1
~
Z 0
y (51)
Finally the test statisticwillbe given by
(
~
^
) 0
[V(
~
) V(
^
)]
1
(
~
^
)
2
K
Another important category of tests is the so-called overidentication
tests ofSargan(1976). These testsuse covariancesbetween IV residualsand
asetof instrumentsthatneednothavebeen usedinthe estimationtodetect
any model misspecication regarding the validity of the instruments. Under
the nullof nomisspecication,the covariance between the residuals and the
instrumentset in questionwould be closetozero. More explicitly,letus say
we have amodel writtenas
y=X+" (52)
and estimatedby IV usingW asinstruments. Nowsupposewehaveanother
potentialinstrumentset
~
W oforderm(
~
W mayormaynotoverlapwithW).
Then thetest statisticusingthe covariance matrix
^
of "^ 0
~
W is given by (see
Godfrey(1988))
^
"
0
~
W
^
1
~
W"^ 2
m
Now if
~
W is a subset of W (Pagan-Hall (1983) suggestion) then the above
test isequivalent totesting Æ=0in the augmented regression
y=X+WÆ+" (53)
as the above test statistic can be shown to be the same as that of Gallant
and Jorgenson (1979) for testing Æ = 0 in the above model. The latter is
basedonthedierencebetweentheoptimisationcriterionintheunrestricted
model (53) and the restricted model(with Æ=0). It is given by
~
S
^
S
2
m
where
~
S ="~ 0
W(W
^
W) 1
W 0
~
"; "~=y X
~
~
W
~
Æ
^
S ="^ 0
W(W
^
W) 1
W 0
^
"; "^=y X
^
and both models are estimated using W as instruments and
^
calculated
using ".~
Coming to our model say for the single equation t = 1, in the iid case, one
cantestthevalidityof
~
W =[x
2
]asinstrumentsbyestimatingtheaugmented
model
y
1
=x
1 +x
2 Æ+r
1
(54)
and testing Æ = 0. If the test is rejected then x
2
is not a valid instrument
and only [x
3 x
4 x
5
] should be used as the instrument matrix. This can
be repeated for
~
W = [x
2 x
3 ] or
~
W = [x
2 x
3 x
4
]. In the MA(1) case,
W =[x
3 x
4 x
5 ] and
~
W can be either [x
3 ] or[x
3 x
4 ].
Once again the procedure is easily extended for testing other MA struc-
tures and to the case of system estimators.
Letusnowgoontolookattheempiricalresultsonthevariousestimators
and tests discussed above.
This empirical sectionis divided intothree parts. Firstly we examine single
equationestimation of theincometransmissionequation. Then,we consider
a system analysis where all the equations are pooled to arrive at a unique
estimator. Finally, in the third sub-section we carry out specication tests
on the system estimates in aneort to reconcile the various ndingsand to
arriveat generalconclusions pertainingto the serial correlation in the error
term, and hence the appropriate choice of instruments.
All the regressions results reported below include as explanatory vari-
ables ameasurementontheparentfamily'sannualincome,age controls(age
and `age-squared' ofthe parent and childfamilyheads)together with acon-
stant. All estimationstakeaccountofcorrelations acrosssiblings,andwhere
appropriate, are robust toserial correlation up to three lags(orleads).
4.1 Single equation results
For the single equation analysis, we can report estimates for up to 5 years
(year one pertains to observations on the child's 1983 income and her/his
parents' 1967 income, while year 5 pertains to the child's 1987 income and
the parents' 1971 resources). Single equation analysis is very popularin the
empirical literature on intergenerational mobility (see for instance tables 2
and 3of Solon (1992)) and table 2of Bjrklund and Jantti(1997)). We note
that potentially we could estimatea total of 25 equations, not just 5. This
is so, because, in practice, nothingprecludes us fromusing each of x
1 to x
5
toexplainy
1
(i.e. ve separateregressions), andtousethe samevariablesto
explain y
2
etc. Our choicehere toregressy
t onx
t
ispurely expositional,and
has been motivated by our interest in formulating our estimation problem
in the context of the panel data framework. Below we report results for the
rst and last year of the panel only, since these will suÆce to motivate the
need for proceedingin the directionof system levelestimation.
instr: set :
^
x
5 x
4 x
3 x
2
Sargantest
0:550(0:035)
0:544(0:033) 0:511
0:539(0:032) 0:525
0:532(0:031) 1:680
1. The dependent variable is y
1
(the child's 1983 income). The set of explanatory
variablescontainsx
1
(theparents'1967income)togetherwithagecontrolsforthe
parentandchildfamilyheads.
2. Thesingleequationestimatorofisasdenedinequation(20). Amarkindicates
that the corresponding variableis used asan instrument. Theset of instruments
(x
2 to x
5
)containsleadsontheparentfamily'sincomefrom 1968to1971.
3. Standarderrorsappearinsideparentheses.
Intable2wereportresultsforperiodoneofthepanel. Whenx
5
,parental
income in 1971, is used to instrument x
1
, (the corresponding 1967 variable)
the coeÆcientofincomeinheritanceisestimated as0.55. Asabenchmark,
the OLSestimateforthis parameteris0.485withastandarderrorof 0.0245.
As thislatterestimatehoweveris downwardlybiased(and itsstandarderror
is incorrect), it is expected that IV estimates of would be higher; this is
indeed the case for all the guresreported in table 2.
Allfourestimatesoftable 2arewithinthe narrowbracketof 0.53to0.55
(less than one standarderror fromone another). Withasample sizeof 1426
observations, adding extra instruments can generally be recommended on
eÆciencygrounds, sincetheresultingnitesamplebias(whichincreaseswith
the number of instruments included in the regression) is small. The results
of table 2 conrm this pattern: standard errors fall from 0.035 to 0.031 as
the set of instruments is expanded from x
5
to include x
5 to x
2
. Finally, the
Sargantests (lastcolumnofthe table)donot rejectany ofthe specications
at the 5% levelwhen x
4 tox
2
are included inthe set of instruments.
in the disturbances, the story somewhat changes when we turn to period 5
results, reported intable 3.
Table 3: IV results for period ve of the panel
instr: set :
^
x
1 x
2 x
3 x
4
Sargantest
0:599(0:034)
0:577(0:033) 1:796
0:564(0:032) 2:166
0:545(0:031) 14:040
1. The dependent variable is y
5
(the child's 1987 income). The set of explanatory
variablescontainsx
5
(theparents'1971income) togetherwithagecontrols.
2. A *markindicates thatthe Sargantestof over-identifying restrictionsisrejected
atthe5%level.
The estimates of table 3 use lags of family income as instruments, while
thoseoftable2usedleadsofparentalincome. Noterstthattheestimatesof
intable 3 are generally larger(rangingbetween 0.55 and 0.60) than those
of table 2, though these dierences are not statistically signicant. On the
basis of the Sargan tests statistics we may also conclude that the ndings
reject the validity of a one period lagged income as an instrument at the
5% (also atthe 1%) signicance level. Conditional upon x
1
(incomelagged
four periods) being a valid instrument, x
2
, and [x
2
;x
3
] are not rejected as
over-identifying instruments, while the inclusion of x
4
, leads to a rejection
of the orthogonality conditions. Thus, period 5 results suggest an MA(1)
specication for the regression error term, while those of period 1 give no
indication of any potential serial correlation in the disturbances. It is for
this reason that we will also subject our system-level estimators to various
specication tests, given that the set of instruments may be called todoubt
in the light of our ndings.
Weclose thissub-section byreportingsome relatedndingsinthelitera-
ture. Usingeducation of the household head asan instrument, Solon(1992)
mates this same parameter at 0.63 (also using education asan instrumental
variableforasub-sampleofthePSID).SomeUKresultsprovidedbyDearden
et al. (1997) estimate to be in the order of 0.55 to 0.6 when education is
used to instrument parental income. They also report aninteresting nding
that theincomesof daughters tendtobemore correlatedwith theirparents'
economic outcomes than inthe case of sons.
4.2 System estimation
Movingfromthe single equationframework tothe system estimatorisanal-
ogous to the transitionfrom two-stage to three stage least squares methods
in the general simultaneous equations framework. This allows for potential
eÆciency gainsprovidedthe specicationsconsideredarevalidatedusingap-
propriate diagnostics tests.
Table 4 : System estimators
Model System OLS System GLS
MA(0) 0:498(0:026) 0:417(0:022)
MA(1) 0:520(0:027) 0:475(0:024)
MA(2) 0:538(0:027) 0:489(0:025)
MA(3) 0:565(0:030) 0:522(0:027)
1. MA(0) denotesamodel withserially uncorrelatedresiduals, whileMA(q) denotes
amodelwithamovingaveragedisturbancetermoforderq.
2. Thesystem-OLSand system-GLSestimatorsareasdened in equations(14)and
(21)or(34)respectively.
In comparison to the single equation results of tables 2 and 3, we may
note thatthe systemGLSestimateissubstantiallysmallerundertheMA(0)
assumption-noserialcorrelationintheerrors-(avalueof0.42)butthatthis
estimate increases to 0.52 with a standard error of 0.027 under the MA(3)
sitive to the time-series structure of the error term than the system OLS
estimator, thoughwith the OLS estimator we alsond that the estimate of
increases as we move from the MA(0) specication to the MA(3) model,
with valuesranging from0.50 to 0.57.
Asdiscussedinsection3,theexistenceofserialcorrelationintheresiduals
reduces the set of potential instruments for parental income. Thus, while
undertheMA(0)andMA(1)assumptionsallequationscanbeused toarrive
at an estimate of , this is nolonger the case under the MA(2) and MA(3)
specications. Under the MA(2) specication, x
3
cannot be instrumented
usinglagsandleadsofx(whenthepanelspansaperiodof5years). Likewise,
under the MA(3) structure, x
2 , x
3
and x
4
will not possess within-equation
instruments. Accordingly,for the MA(2)specication the period3equation
hasbeendeletedfromthesystemestimator. Likewise,theMA(3)modelonly
poolsperiod one and period 5data toestimate . The resulting patternfor
the standarderrors ofthesystem estimatoris totakeonincreasing valuesas
we movefrom the MA(0) tothe MA(3)models (for both OLS and GLS).
Finally, note that the variance of the individual eect accounts for ap-
proximately 70% of the total variance in the error term:
2
is estimated at
0.354,while 2
"
isestimatedat0.166(resultsnot showninthetable). Viewed
underthisperspective,itisclearthatpaneldatamodelswillprovideabetter
t for microdata with a substantialcomponent of unobserved heterogeneity
than pure cross-section regressions.
4.3 Hausman specication tests
Ournaltaskinthisempiricalsectionofthepaperconsistsintestingvarious
scenariosregardingthetime-seriesstructureofthedisturbancesofthesystem
estimator. Such tests are required in order to provide a sound basis for
discriminating between the various estimates of in table 4, with values
ranging from0.42 to 0.57.
Specication System OLS System GLS
MA(2) vs: MA(3) 11:525 16:539
MA(1) vs: MA(3) 16:942 19:897
MA(0) vs: MA(3) 25:232 52:449
MA(1) vs: MA(2) 8:649 7:199
MA(0) vs: MA(2) 18:661 46:828
MA(0) vs: MA(1) 0:0127 49:246
A *(**)indicates thatthenullhypothesis isrejectedat a5%(1%) probabilityoftype-I
error.
Our benchmark assumption is that of an MA(3) process: the resulting
systemestimatorwillbeconsistent(butineÆcient)undertheMA(2),MA(1),
or MA(0) specications, while, the other way around, the MA(q) estimator
is inconsistent under the MA(r) scenario, for r>q.
Specicationtestsarecarriedoutusingboththesystem-OLSandsystem-
GLSestimators,as,inpractice,asub-optimalweighting schemeforthe GLS
estimatorneednotresultiseÆciencygainsoverOLS(Szroeter(1994)). Com-
paringourteststatisticstoa 2
variablewith6degreesoffreedom,ourresults
(table5)canbesummarizedasfollows: theMA(1)andMA(0)specications
are rejected atthe 1% levelagainst the nullhypothesisof anMA(3) process
using both the system OLS and GLS estimators. The MA(2) hypothesis
(rst lineoftable 5)isnot rejected usingthe system OLS estimator, though
it is rejected at the 5% level (but not at 1%) using GLS. In this sense, the
MA(1) and MA(0) assumptions appear to be inadequate, while, provided
the MA(3) hypothesis is a valid one, the MA(2) model appears to be more
plausible.
Thelast threelines of table 5providefurther resultswhen the nullspec-
ication is modied. A test of the MA(1) specication against the MA(2)
assumption isrejectedagainstMA(2)atthe 1% level. Finally,the GLSesti-
matorrejectstheassumptionofnoserialcorrelationintheerrorterm,against
theMA(1)model,whileOLSfailstorejecttheMA(0)modelagainstMA(1).
The overall pattern, then, is that both estimators reject the MA(q-2) model
againstMA(q),whilesystem-OLSneverrejectstheMA(q-1)scenarioagainst
MA(q). Adopting a more global perspective, the system-GLS estimator is
perhapsto be recommended oversystem-OLS onsuch grounds.
5 Conclusions
In this study, we have carried out the estimation of the intergenerational
correlationofpermanentincomeswithintheframeworkofapaneldatamodel
with measurement errors. This has enabledus toimplementnew estimators
aswellasspecicationtests,astheproblemhasnotbeenexplicitlyconsidered
from this perspective sofar.
The availability of time-series on a cross-section of families has allowed
us to exploit within-sample information to deal with the errors in variables
problemas wellas todevelop system estimators for the parameters of inter-
est. Thepanel data contexthas alsobeen useful intesting various scenarios
regarding the time-series structure of the error term and its implications
for the choice of instruments. Our estimators have also taken into account
possiblecorrelations among childrenbelongingtothe same family.
Forthevariousmethodsusedinthis study, wehaveestimated the coeÆ-
cientofincomeinheritancetobeintherangeof0.42to0.60. Singleequation
estimates varybetween 0.53and 0.60. Results varydepending onthe choice
of instrumentsfor a given time period,but alsofrom one periodto another.
System estimates were found to be between 0.42 and 0.57, with the system
OLS yielding higher values than the GLS counterpart. On the basis of our
specication tests we nd that MA(0)(no serial correlation) isdenitely re-
jected compared to MA(3), MA(2) and MA(1) (at least in the system-GLS
case). TurningtoMA(1),itisrejectedcomparedtoMA(3)butnotcompared
to MA(2). Finally MA(2) is accepted against MA(3). Given that MA(3) is
the least restricted model that our data permit and hence using this as a
reference for comparing the other scenarios, we can say that there is some
supporting evidencefor an MA(2)structure of the errors.
ityfromonegenerationtothenext. Aslopevalue inthe orderof0.5to0.55,
while indicative of a process of regression towards the mean, is nonetheless
far remote froman idealizedequalopportunities situation.
In general, panel data models allow the researcher to relax the indepen-
dence assumption between the specic eect and the explanatory variable.
However, this was not possible inour study because the variable of interest
subject to errors (namely permanent income of the parent family) is time-
invariant. Furthermore,wehavenotconsideredotherdynamicstructuresfor
the error term such AR or ARMA processes, which could provide a useful
directionfor further work.
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