Tilburg University Pseudo panel data Verbeek, M.J.C.M. Publication date: 1993 Link to publication Citation for published version (APA): Verbeek, M. J. C. M. (1993). Pseudo panel data. (Reprint series / CentER for Economic Research; Vol. 122). Unknown Publisher. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. - Users may download and print one copy of any publication from the public portal for the purpose of private study or research - You may not further distribute the material or use it for any profit-making activity or commercial gain - You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 31. May. 2018
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Tilburg University
Pseudo panel data
Verbeek, M.J.C.M.
Publication date:1993
Link to publication
Citation for published version (APA):Verbeek, M. J. C. M. (1993). Pseudo panel data. (Reprint series / CentER for Economic Research; Vol. 122).Unknown Publisher.
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
- Users may download and print one copy of any publication from the public portal for the purpose of private study or research - You may not further distribute the material or use it for any profit-making activity or commercial gain - You may freely distribute the URL identifying the publication in the public portal
Take down policyIf you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.
BoardHarry Barkema1-~elmut BesterEric van Damme, chairmanFrank van der Duyn SchoutenJeffrey James
ManagementEric van Damme (graduate education)Arie Kapteyn (scientific director)Marie-Louise Kemperman (administration)
Sclentific CouncilAnton BartenEduard BomhoffWillem Buiter]acques DrèzeT-heo van de KlundertSimon KuiPerslean-Jacques LaffontMerton MillerStephen NickellPieter RuysJacques Sijben
Université Catholique de LouvainErasmus University RotterdamYale UniversityUniversité Catholique de LouvainTilburg UniversityGroningen UniversityUniversité des Sciences Sociales de ToulouseUniversity of ChicagoUniversity of OzfordTilburg UniversityTilburg University
Research CoordinatorsEric van DammeFrank van der Duyn SchoutenHarry HuizingaArie Kapteyn
CentER, Erasmus University RotterdamUniversity of FrankfurtPrinceton UniversityCentER, LSEUniversity of PittsburghStockholm School of Economics
Address : Warandelaan 2, P.O. Box 90153, 5000 LE Tilburg, The NetherlandsPhone : f31 13 663050Telex : 52426 kub nlTelefaz : f 31 13 G63066E-mail : "center~ htikub5.bitnet"
ISSN 0924-7874
1992
Pseudo Panel Data
byMarno Verbeek
Reprinted from L. Mátyás and P. Sevestre (eds.),The Econometrics of Panel Data,
Dordrecht: Kluwer Academic Publishers, 1992
Reprint Seriesno. 122
:1larno Verbnek
14 PSEUDO PANEL DATA
In the previous chapters much attention was paid to estímation and testingstrategies using panel data in a variety of models.~ In practical situations,however, a true panel data set may not always be available, while repeatedcross sections are. For example, in the United Kingdom, no panel data areavailable on consumer expenditures or laboiir supply. Nevertheless, a randomsample of the population is available each year in the Family ExpenditureSurveys (F.E.S.). Recently, several authors have stressed the fact that paneldata are not indispensible for the identification of many commonly estimatedmodels and that the parameters of interest can often be identified (with orwithout some additional assumptions) from a single cross section or a seriesof cross sections (see, for example, Heckman and Robb [1985], Deaton [1985]and MofTitt [1991]). In this chapter we shall discuss the identification andestimation of panel data models from repeated cross sections. In particular,attention will be paid to linear models with fixed individual effects and tomodels containing lagged endogenous variables.
Models containing individual effects that are correlated with the explana-tory variables ("fixed effects models") often arise natutally from economictheory, for example in life cycle models where the individual effects representmarginal utility of wealth (see, for example, Heckman and MaCurdy [1980],~1laCurdy [1981] and Browning et aL (1985J). Deaton (1985] has shown thatthe slope parameters in such models can usually be identified from a series ofindependent cross sections. In his approach, which will be described below,individuals sharing some common observed characteristics, Gke age or sex,are grouped into cohorts, after which the averages within these cohorts aretreated as observations in a pseudo panel (or synthetic panel). Because theobserved cohort aggregates are error-ridden measurements of the true cohortpopulation values, Deaton [1985] proposes an errors-in-variables approach.In practice this errors-in-variables problem is usually ignored if the numberof observations within each cohort is reasonably large (see, e.g., Browninget al. (1985J, Blundell et al. [1989] and Moffitt [1991]). The latter author,among other things, generalises the identification and estimation results to
' This chapter has benefitted from helpful comments by Bertrand 4felenbetg and TheoNijman.
303
L. ,Nátvás and P. Sevestre (eds.l. The Econnmetrres r,f Pane! Data. 303-315.s~ 1992 K(uwer Academle Publishers. Prrnted in dre Ned~erfands.
304 Pseudo Pane! Dzt:t
models with lagged endogenous variables (excluding fixed effects). Extensionsto linear dynamic models are also providetl by Collado [1991], who, however,does not ignore the errors-in-variables problem.
The structure of this chapter is as follows. In Section 14.1 the estimation
of a linear fixed efiects model is discussed along the lines sketched by Deaton(1985], while in Section 14.2 attention is paid to an instrumental variables
interpretation oí estimators based on grottped data, with a discussion of possi-
ble alternative estimators. The estimation of models with lagged endogenousvariables is briefly discussed in Section 14.3. Finally, Section 14.4 concludes.
14.1 Estimation of a Linear Fixed Efiects Model
To illustrate identification and estimation from pseudo panel data we shallstart with analysing a simple linear model with fixed individual effects, asdiscussed in Chapter 3. Consider the basic covariance model as given in (3-2),
y~: - i,~tA-f a~ -~ ua, t- 1,...;T, (14-1)
where p is the parameter of interest. The index i refers to individuals andthroughout this chapter we shall assume that the available data set is a seriesof independent cross sections, such that observations on N individuals are
available in each period.2
If the individual effects a; are uncorrelated with the explanatory variablesin x,,, model (14-1) can easily be estimated from repeated cross sections bypooling all observations and performing ordinary least squares treating a; ~- u;,
as a composite error term. The drawback of using repeated cross sectionaldata is that the variance of a; cannot be identified. ~loreover, the resultingestimator may be less efficient than when a genuine panel data set would beavailable. This latter point is discussed elaborately in Nijman and ~'erbeek
[1990].
However, in many applications the individual effects a; are likely to becorrelated with the explanatory variables in ~; so that estimation procedurestreating the a; as random drawings from some distribution (as in the ran-dom effects model) lead to inconsistent estimators, unless the correlation isexplicitly taken into account. When panel data are available, this problemcan be solved by treating the a; as fixed unknown parameters. Obviously, thisstrategy no longer applies if no repeated observations on the same individiralsare available.
In a recent paper, Deaton [1985] suggests the use of cohorts to obtainconsistent estimators for ;3 in (14-1) if repeated cross sections are available,even if the individual effects are correlated with the explanatory variables.
2 Because different individuals are observed in each period this implies that i does not runfrom I to N for each t.
l~shmation r~( a l,in~ar !'ixc~l EI(ccts ;iforlcl 30,i
Let us de(ine C cohorts, which are firoups of individuals sharing some cornrnoncharactr.rist.ics. These groups are de(ined such that each individual is a mernb~rof exactly one cohort, which is the same for all periods. For example, a par-ticular cohort may consist of all malr's born in L950-1955, or oi all individualshaving a university degree on .lanuary 1, 1980. It is important to realizcthat the variables on which the cohorts are defined should be observed for altindividuals in thc sample. This rules out variables like "wage earnings in 19~37"or "family size at January lst, 1990" because these variables are typically notobserved for all individuals in the sample.
If we aggregate all observations to cohort level, the resulting model canbe written as
J~~ -?rr.rti~ f á~r f u~r, c- 1, ...,C; t- 1,...,T, (14-2)
where y~r is the average value of all observed y;,'s in cohort c at time t, andanalogously for the other variables in the model. The resulting data set isa pseudo panel or synthetic panel with repeated observations over T periodsand C cohorts. The main problem with estimating A in (14-2) is that á~,depends on t, is unobserved and is likely to be correlated with ~, (viz. if a; isc.orrelated with x„). Therefore, treating á~, as random errors is likely to leadto inconsistent estimators, wtrile treating them as fixed unknown parametersresults in an identification problem unless the variation over t can be ignored(~~t - ir~}. If cohort averages are based on a large number of individualobservations, this assumption seems reasonable and a natural estimator íor,3 to look at is the covariance estimator (or within estimator) on the pseudopanel, given by
pH, - (?~r - ?.~)(y~e - yr:) J , (14-3)r
where i{ - T~T 1~, is the time average of the observed cohort means forcohort c. If the average cohort siae n~ - N~C tends to infinity, consistency ofthis estimator requires that
C T
PLm C7, ~ ~(?~t - ~)ut:a - Qn~--~ ~-t r-r
and
(14-4)
1 C T
Plim C:7, ~ ~(~t - ~)á~~ - ~. (14-5)"~-~ ~-r ~-r
Since á~, -~ a~ (for some a~ ) if the number of observations in cohort c attime t tends to infinity, this argument can be used to prove consistency (asin Moffitt (1991]). However, it is not a priori obvious that ( 14-4) holds whenonly the number observations per cohort (N~C) tends to infinity. This willonly be the case if there are no time effects in u;t, i.e., if plim„~-.o, u„ - 0 forall t. b4oreover, as shown by Verbeek and Nijman [1992a], even if the cohort
t,(~ ~(~r - ~)(~r - ~) ) (~ ~~-r r-~ r-r r-
306 Pseudo Panel Data
sizes are large, the bias in the within estimator on the pseudo panel, AN„ maystill be substantial.
Consequently, using n~ -. oo to approximate the behaviour of i3~„ thoughsimplifying, may not be the best choice in practice, a point to whicl~ we shallreturn below. Procedures have been developed that do not depend on havinglarge numbers oí observations per cohort. A convenient way of analysing theproblem follows from considering the cohort population version of (14-2), assuggested by Deaton [1985]
where the asterisks denote unobservable population cohort means and wherea~ is the cohort fixed effect, which is constant because population cohortscontain the same individuals in all periods. Now i~, and j~, can be consideredas error-ridden measurements of xt, and y~,. In particular, it is assumed thatthe measurement errors are distributed with zero mean, independent of thetrue values. In particular,
where the population cohort means are treated as fixed ( unknown) constants.Although ~, o and o0o are unknown, they can easily be estimated consistently(for tV, T, or both, tending to infinity) using the individual data. Onceestimates for E and o are available, it is easy to adjust the moments in thewithin estimator to eliminate the variance due to measurement error (cf. Fuller(1987]). This leads to the following errors-in-variables estimator
where ~ and ó are estimates of E and o, respectively, and where r- (T -1)~T. As discussed in Verbeek and Nijman ( 1992b], the original estimatorpresented by Deaton [1985] is characterized by r- 1. líowever, eGminatingthe incidental parameters first ( by within transforming the data) and workingout the appropriate moments results in r-(T - 1)~T. Of course, when T islarge, the difference between the two estimators will be negligible.
The asymptotic behaviour of i3o depends crucially on the type of asymp-totics one is willing to apply. If the number of observations in each cohort(n~) tends to infinity, the measurement errors tend to zero and both r and vtend to zero, as well as their estimators. In this case the errors-in-variablesestimator ;3D is asymptotically identical to the within estimator on the pseudopanel Qn,. Consequently, if n~ is reasonably large, most applied studies ignorethe errors-in-variables problem and use standard estimators Like {3~v, see,
Estiniation of a I,~ncar h'ixcd Elïccts M11o~1c1 Zq7
for r'xample, I~rowning rt al. [1985~ (with an average cohort size of 19Q), orBlundell et aL [1989) (with cohort sizes of 3Q0-400).
For finite n„ the r'rrors-in-variables estimator 13D is consistent for ,3 forcithcr C or T going to infinity (or both) provided that the adjusted momr'ntsmatrix of the explanatory variables in (14-8) is nonsingular.
The construction of the cohorts.
In the constr~iction of pseudo panel data, it is clear that there is a trade-offbetween the number of cohorts C and the size of these cohorts. Smaller cohorts(with less observations) imply less precise estimates of the population cohortmeans, so essentially the trade-off is between the number of observations andthe acciiracy of these observations. Using the errors-in-variables estimator!3D it is possible to optimize on this trade-off and to choose an optimal cohortsize. If, instead, one uses the standard within estimator QN„ there is also atrade-off, but in this case it is between bias and variance of the estimator.
In addition to the sizes of the cohorts, the way in which the cohorts areconstructed is also important. Since one would like to have measurement errorvariances that are as small as possible, one would like to construct the cohortssuch that the variation within cohorts is small, while the variation betweendifferent cohorts is large. In other words, the individuals within each cohortshould be as "homogeneous" as possible, while those from different cohortsshould be as "heterogeneous" as possible. Suppose, as an extreme example,that cohorts are defined on the basis of a variable that is independent of thevariables in the model. In that case the true cohort means xit would beidentical for each cohorx c(and equal the overall population mean) and theonly source of variance left in the data that is not attributed to measurementerror would be the variation of ~t over time. If these population means do notchange over time, all variation in the observed cohort means is measurementerror and the errors-in-variables estimator QD does not have a well-definedprobability limit.
In practice, cohorts should be defined on the basis of variables that donot vary over time and that are observed for all individuals in the sample.This is a serious restriction. Possible choices include variables like age (dateof birth), sex, race, residential location, manual or non-manual worker, etc.Identification of the parameters A in the model requires that the expectation ofeach element in ~, conditional on the cohort identifying variables varies witht. This requirement puts a heavy burden on the cohort identifying variables,in particular if only birth cohorts are used (e.g., Blundell et al. [1989]).
The choice of asymptotics.
As said before, the behaviour of the estimators depends crucially on theasymptotics one is willing to apply. In addition to the two dimensions in realpanel data (N and T), there are two additional dimensions, the number ofcohorts C, and the number of observations per cohort n~, which can be used
308 PseuJo Pancl Ur~ta
to define asymptotics. Contrary to N and T, these latter two dimensionsare not cíetermined by the available sample but can be chosen in the analysisstage.
In choosing the desired asymptotics, I would like to start from the viewthat asymptotic theory is not meant as a guideline for how our estimatorswill behave when we get more data. Rather, we appeal to asymptotic theorywhen some dimension of the sample we already have is large enoitgh for thisto be appropriate. This view gives rise to the question which dimension of thesample is "large enough" to use asymptotics on it. An additional importantquestion is which type of asymptotics assures consistency of our estimators.
Table I4-1: Consistent estimators for ,Q
N fixed N -a oo N -~ oo N -~ o0
C fixed C fixed C-- oó C--~ o0
n~ fixed n~ -. oo n~ fixed n~ -~ o0
T fixed
T~ oo Qo
`
~3o,3H,
Qo
Qo
po ~ Qw
QD,i3K,
We shall analyse the second question first. In Table 14-1 we present an
overview of conditions for consistency of our estimators pk, and Qo underthe assumption that the individual effects a; and the explanatory variablesin x„ are not uncorrelated, while u;, and x„ are. When only N and n~ tendto infinity (indicated in the table by ~`) consistency of the estimators dependson the question whether aggregation into coltorts is independent of u,,, i.e.,whether u~, in (14~) can be assumed to be zero. If there are no cohort ortime effects in the error term u;,, this is the case and CT ~ oo is not requiredto attain consistency. If u~, is nonZero, both QK, and po are inconsistentbecause they converge to a random variable instead of a constant. They are,however, unbiased estimators for Q. Overall, we see in the table that the crucial
condition for consistency of the standard within estimator Qu, is whether thenumber of observations per cohort n~ tends to infinity or not. This is obviousbecause p~, and 3o become equivalent when n~ tends to infinity.
There is, unfortunately, no general rule to judge whether n~ is large enoughto use asymptotics in n~. In a recent article, Verbeek and Yijman [1992a]
analyse the bias in 3~v for finite values of n~. Depending on the way in whichthe cohorts are constructed, the bias in the standard withirt estimator maystill be substantial, even if the cohort sizes are fairly large. In general, it
:Lr lnstrumcntal Vari:tblca fntcrprctlticrn ~09
herlds that, for given n~, the bias is srnaller if the cohort are chosen such thatthc rclative maf;nitude of thc mc.rsurement crrors is smaller compared to thewitlrin cohorts variance of ~',. [n practice, however, it may not be that easyto chouse tlle cohorts in such a way.
14.2 An Instrumental Variables Interpretation
In a recent paper, dfo(fitt [1991] proposes an alternative approach to theestimation from repeated cross sections, which generalizes the discussion abovein some directions. His approach starts from the general observation thatgrouping can be viewed as an instrumental variables procedure, a fact whichwas first noted by Durbin (1954]. To illustrate this we first of all decomposeeach individiral effect a; into a cohort effect a~ and individual i's deviationfrom this effec.t. Letting d„ (c - 1, ..., C) equal 1 if individual i is a memberof cohort c and 0 otherwise, we can write
ca; - ~ a~d~; f v;. (14-9)
~-rSubstituting (14-9) into (14-1), one obtains
cy;t - x;t~ f~a~d~~ -~ v; f u;t, (14-10)
~-rwhere-for expository purposes-attention is restricted to a scalar x variable.Because a; and x;, are torrelated, it is not unlikely that v; and x;, are corre-lated as well. Therefore (14-10) can not be estimated consistently using leastsquares procedures.
Now, suppose that instruments for x;, can be found that are (asymptoti-cally) uncorrelated with v; (and u;,). In that case, an instrumental variablesestimator produces a consistent estimator for A and a~. Defining the timedummies D„ - 1 if s- t and 0 otherwise, we can use these dummies,interacted with the cohort dummies, as instruments for x;,, i.e., we derivea linear predictor from the reduced form' "
C T Cxta - ~~7r,eed~,D,a f ~?'s,rdr; -t-w;,. (14-11)
~-r,-r c-r
The linear predictor for x;, from this is given by i;, - i~,, which is the averagevalue of x;, in cohort c at time t. The resulting IV estimator for Q is thengiven by
Indeed, this equation suffers from ezact multicollinearity. Arbitrary restrictions can beused to solve this problem without affecting the resulting linear predictor.
~
310 Pseudo Pancl Data
which is identical to the standard within estimator ~3H. on the pseudo panel ofcohort data, as presr.nted in the previous section.
Of course, conditions for consistency of (3~vt are identical to those for Ata..These conditions imply that the instruinents for x;, vary with t (at least one
element in y~ is nonzero) and are (asymptotically) uncorrelated with v; and
u;t. Given the presence of d~; in (14-17) these conditions reduce to'
N T
phm 1 ~ ~(x~~ - i~)u;~ - ~„~,~ ~YT ;-t ~-~
and
(14-13)
1 N T
plim ~ ~(i~, - i~)v; - 0. (14-14)„~,~ NT ~-t ~-t
It is easy to verify that these two conditions are equivalent to (14-4) and(14-5), respectively. Under these conditions the instruments for x;, are asymp-totically valid ( for n~ -r oo).
Recently, Moffitt [1991] presented a more general class of instrumentalvariables estimators. Instead of using cohort dummies, time dummies and afull set of interaction dummies, IVfoffitt's approach allows a more parsimoni-uous parametrization. Because the errors-in-variables problem is neglected,n~ y oo is assumed thoughout.
Moffitt starts by introducing the linear projection of ~; upon a vector oftime invariant variables, ~, say,
o; - ~ B -~ v; (14-15 )
and the linear projection of x;, upon time invariant and time varying variables
x.~ - Í~,71 ~- ~7Z f w~~ . (14-16)
Now (1~I-15) is substituted into (14-1) and ttiis new equation is estimatedusing the linear projector of x;, from (1-I-16), as an instrument for x,,. Con-sistertcy of this IV estimator requires that at least one element of 7t is differentfrom zero, and that
1 N T
plim N7,~~i„u„-~n~-ro ~-( tcl
and
(14-17)
1 .N Tplim -~~2~tv~ - 0. (14-13)„~-~ NT ~-i a-t
Since x;, and o; are correlated this latter condition is nontrivial.
Since tiloffitt's instrumental variables procedure generalizes the standardwithin estimator on the pseudo panel, for which conditions for consistency
~ Ideally, notation should indicate that i reflects diHerent inàividuals in each period, Cor
example by letting t run from 1 to T, while i runs írom 1~ tY(t - 1) to Nt tor a given
value ot t.
Estirnation of Lincar Uvnrunic ~1lorlrls ~11
have been analysed in mom depth above, it is worthwhile to analyse thr~yuc~stion whether rnorr~ attractivic estimators can be construeted from t.hisprocednre. `fhis is a tough question, bccause it is very ríilricult to verifywhether conditions (14-17) and, particularly, (14-18) hold, Iet alone to testthem. Essentially, conrlition (14-18) says that, eonditional upon ia., f and v,should be (isymptotically) uncorrelated. Suppose, withont loss of generality,that f r is orthogonal (or orthogonalized) with respect to w,. Then, (14-18)says that f r and v; are asymptotically uncorrelated. The only case where thisis a priori obvious is the case where f r does not vary with i and asymptoticsare taken for N~ oo.
Even when a genuine panel cíata set is available, consistency of the IVestimator is not guaranteed, whatever asymptotics one wants to use. Only inthe extreme case, where tv; contains individual dummies (such that v; - 0),consistency follows cíirectly (provided (14-17) holds), independent of the choiceof instruments for z;,. ~Vhen z;, is used as its own instrument, the standardwithin estimator (or covariance estimator) for A is obtained. Ilowever, whenw; does not contain a dummy for each individual, some source of unobservedheterogeneity is left in v;, which may be correlated with the instruments forx;, (conditional on w,). If these instruments vary with t only, consistency isobtained for N y oo, as in the case of pseudo panel data.
In summary, we have seen in this section that it is possible to give aninsttumental variables interpretation to the (standard) estimators based on thepseudo panel. This interpretation suggests a whole new range of estimators.However, while consistency of the standard within estimator is achieved whencohort sizes tend to infi~ity, there is no such guarantee for alternative estima-tors in this class, unless the instruments for x;, do not vary with i conditionalupon the instruments for n;. Having instruments for individual time seriesdata that do not exhibit individual variation, may not be a fortunate situationand may lead to inaccurate estimators.
14.3 Estimation of Linear Dynamic Models
In this section we shall brieRy discuss the estimation of models containinglagged endogenous variables using repeated cross sections. Our starting pointis the autoregressive model without exogenous variables, as discussed in, e.g.,~ndetson and Hsiao [1981] and Nickell [1981],
yec - óy;.e-, f ar f utc, t- 1. .... T. (14-19)
In this section, we shall analyse the question to what extent the results ofSection 14.1 can be generalized to capture the autoregressive model. Note thatit is an inherent feature of model (14-19) that a; and y;,t-1 are correlated.
g12 Pscudo Nanel Data
Defining cohorts in exactly tlie same fashion as b~fore, we obtain
where it is important to realize that aggregates with a different time index arebased on difíerent individuals. The cohort population version of this model is
where it is assumed that an initial observation for 1- 0 is available (such
that y~,-1 -(1~T)~; ly~,,-1). This estimator is consistent for ó providedT-. oo (see Collado [1991] for details). The inconsistency for finite T is dueto exacily the same cause as the inconsistency of the covariance estimator inthe autoregressive model based on a genuine panel data set (see Chapter 6),viz. that the (within) transformed regressor is correlated with the transformederror term. As this correlation is of the order 1~T it disappears when T getslarge.
Just as in the case of true panel data, it is possible to construct a consistentestimator for ó for finite T, by using instrumental variables techniques (cf.Collado [1991]). Let us, instead of taking within transformed data, take firstdifferences, denoted by r.~. Then we have
The least squares estimator based on (14-24) corrected for measurement error,given by
1 C T2
óo - C(T - 1) ~~(Dy~.e-i) - 2000~-i ~-z
1 C T
~~ Dy~,~-iJy~~ ,C(T - 1) ~-i ~-z
(14-25)
is inconsistent, even for T--. oo. Now suppose we use y~,,-2 as an instrumentfor .,y~,,-1. Of course, j~,,-z is not a valid instrument because it is correlatedwith ,,~~,, but it is possible to correct for this. Since, from (1.1-22),
Thus, we can correct for the invalidity of the instrument by adding (an estimatefor) o0o to the covariance matrix. Because data on an individual level areavailable, this variance can be identified from the micro data.
A discussed before, if cohort sizes tend to infinity one can ignore the
errors-in-variables problem and set áoo to zero. In this case, bó is still
inconsistent for finite T, while the instrumental variables estimator b presentedin (14-27) is consistent.
Several generalizations of the ideas outlined in this section are possible.First, alternative (adjusted) IV estimators for model ( 14-19) can be devel-oped, for example using 0j~,,-Z as instruments along the lines of Chapter 6.Second, exogenous variables can be added to the model as well. This is a
straightforward extension of the methods presented in the preserrt section andSection 14.1. (See Collado [1991] for details.)
14.4 Concluding Remarks
In this cliapter we have briefly discussed the problem of estimating paneldata models from a time series of independent cross sections. In particular,attention was paid to the estimation of a static "fixed effects" regression modeland to an autoregressive model with individual effects.
The approach suggested by Deaton [1985] is to divide the populationinto a number of cohorts, being groups of individuals sharing some commoncharacteristics, and to treat the observed cohort means as error-ridden mea-surements of the population cohort means. The resulting estimator for thestatic linear model with fixed effects is a corrected within estimator basedon the cohort agóregates. This estimator is consistent when the number ofcohorts or the number of sample periods (or both) tend to infinity. If thenumber of observations in each cohort is large, it may be valid to ignore theerrors-in-variables problems and to treat the pseudo panel set as a genuinepanel data set.
The latter assumption is made by ,tifotFtt [1991], who considers a generalframework of instrumental variables estimators (of which explicit grouping isa special kind}. The conditions for consistency of estimators from this classare nontrivial, except for some very special cases.
314 Pseudo f'ancl Dat.z
The estirnation of dynamic panel data models from repeated cross sectionsis considered hy Cc~I1:Lrln (l~~l], who ~xt~nds the results of D~~aton (1~4.5] tomoclcls with :~ lag;ed endogenous variable appParing on t.he right-hand sidc.A straightforwarcl ~xtension is shown to yield consistent estimators for thc~nuntbc~r of tirne periods tencling to infinity only. I3y using an instnimentalvariables estimator based on the grouped data, consistency can be achievedfor finite T.
Finally, it should be mentioned that the fact that pseudo panel data arecreated from repeated cross sections also has some advantages. Many paneldata sets suffer from (possibly non-random) attrition, a problem which isclearly not present in cross sectional data sets. Moreover, pseudo panel datacan provide a relatively long time series of observations compared to genuinepanel data.
Rc(crcnccs 315
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Moffitt, R., [1991]: Identification and Estimation of Dynamic Models with a Time Seriesoí Repeated Cross Sections; Drown Uniueraity, Providence RI, mimeo; forthcoming inJournal of Econometrics.
Nickell, S. [1rJ81]: Biases in Dynamic Models with Fixed Effects: Econametrrca, Vol. 49,pp. 1417-1426.
Nijman, Th.E. - Vetbeek, M. [1990]: Estimation of Time-Dependent Parameters inLinear ?vfodels Using Cross-Sections, Panels, or Both; Journal of Econometrics, Vol.46, pp. 333-346.
Vcrbeek, M. - Nijman, Th.E. [1rJrJ2a]: Can Cohort Data Be Treated As Genuine Yanel
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Reprint Series, CentER, Tilburg University, The Netherlands:
No. l G. Marini and F. van der Ploeg, Monetary and fiscal policy in an optimisingmodel with capital accumulation and finite lives, The Economic Joumal, vol. 98.no. 392, 1'JRB, pp. 772 - 7RG.
No. 2 F. van der Ploeg, International policy coordination in interdependent monetaryeconomies, Joumal of lntemational Economics, vol. 25, 1988, pp. 1- 23.
No. 3 A.P. Barten, The history of Dutch macroeconomic modelling (1936-1986), in W.Driehuis, M.M.G. Fase and H. den Hartog (eds.), Cha[lengesfor MacroeconomicMode!ling, Contributions to Eeonomic Analysis 17R, Amsterdam: North-Holland,1988, pp. 39 - 88.
No. 4 F. van der Ploeg, Disposable income, unemployment, inflation and state spendingin a dynamic political-economic model, Public Choice, vol. 60, 1989, pp. 211 - 239.
No. 5 Th. ten Raa and F. van der Ploeg, A statistical approach to the problem ofnegatives in input-output analysis, Economic Modelling, vol. 6, no. 1, 1989, pp. 2- 19.
No. 6 E. van Damme, Renegotiation-proof equilibria in repeated prisoners' dilemma,Joutnalof Economic Theory, vol. 47, no. 1, 1989, pp. 206 - 217.
No. 7 C. Mulder and F. van der Plceg, Trade unions, investment and employment ina small open economy: a Dutch perspective, in J. Muysken and C. de Neubourg(eds.), Unemployment in Europe, London: The Macmillan Press Ltd, 1989, pp. 200- 229.
No. 8 Th. van de Klundert and F. van der Ploeg, Wage rigidity and capital mobility inan optimizing model of a small open economy, De Economist, vol. 137, nr. 1,1989, pp. 47 - 75.
No. 9 G. Dhaene and A.P. Barten, When it all began: the 1936 Tinbergen modelrevisited, Economic Modelling, vol. 6, no. 2, 1989, pp. 203 - 219.
No. 10 F. van der Ploeg and A.J. de Zeeuw, Conflict over arms accumulation in marketand command economies, in F. van der Ploeg and A.J. de Zeeuw (eds.), DynamicPolicy Cames in Economics, Contributions to Ewnomic Analysis 181, Amster-dam: Elsevier Science Publishers B.V. (North-Holland), 1989, pp. 91 - 119.
No. 11 J. Driffill, Macroeconomic policy games with incomplete information: someextensions, in F. van der Ploeg and A.J. de Zeeuw (eds.), Dynamic Policy Camesin Economics, Contributions to Economic Analysis 181, Amsterdam: ElsevierScience Publishers B.V. (North-Holland), 1989, pp. 289 - 322.
No. 12 F. van der Ploeg, Towards monetary integration in Europe, in P. De Grauwe etal., De Europese Monetaire Integmtie: vier visies, Wetenschappelijke Raad voor hetRegeringsbeleid V 66, 's-Gravenhage: SDU uitgeverij, 1989, pp. 81 - I06.
No. 13 R.J.M. Alessie and A. Kapteyn, Consumption, savings and demography, in A.Wenig, K.F. Zimmermann (eds.), Demographic Change and EconomicDevelopment, Berlin~Heidelberg: Springer-Verlag, 1989, pp. 272 - 305.
No. 14 A. Hoque, J.R. Magnus and B. Pesaran, The exact multi-period mean-squareforecast error for the first-order autoregressive model, Joumal of Econometrics,vol. 39, no. 3, 1988, pp. 327 - 346.
No. 15 R. Alessie, A. Kapteyn and B. Melenberg, The effects of liquidiry constraints onconsumption: estimation from household panel data, European Economic Review,vol. 33, no. 2~3, 1989, pp. 547 - 555.
No. IG A. Holly and J.R. Magnus, A note on instrumental variables and maximum likeli-hood estimation procedures, Annales d'Économir et de Statistique, no. 10,April-June, 1988, pp. 121 - 138.
No. 17 P. ten Hacken, A. Kapteyn and I. Woittiez, Unemployment benefits and thelabor market, a micro~macro approach, in B.A. Gustafsson and N. AndersKlevmarken (eds.), The Politica! Economy oj Social Securiry, Contributions toEconomic Analysis 179, Amsterdam: Elsevier Science Publishers B.V.(North-Holland), 1989, pp. 143 - 164.
No. 18 T. Wansbeek and A. Kapteyn, Estimation uf the error-components model withincomplete panels, Joumal of Econometricr, vol. 4l, no. 3, 1989, pp. 341 - 361.
No. 19 A. Kapteyn, P. Kooreman and R. Willemse, Some methodological issues in theimplementation of subjective poverry definitions, The Journal of HutnanResources, vol. 23, no. 2, 1988, pp. 2?2 - 242.
No.20 Th. van de Klundert and F. van der Plceg, Fiscal poliry and finite lives ininterdependent economies with real and nominal wage rigidity, Oxford EconomicPapers, vol. 41, no. 3, 1989, pp. 459 - 489.
No. 21 J.R. Magnus and B. Pesaran, The exact multi-period mean-square forecast errorfor the first-order autoregressive model with an intercept, Jouma! ofEconometrics, vol. 42, no. 2, 1989, pp. 157 - 179.
No. 22 F. van der Ploeg, Two essays on political economy: (i) The political economy ofovervaluation, The Economiclournal, vol. 99, no. 397, 1989, pp. 850 - 855; (ii)Election outcomes and the stockmarket, European Journal of Political Economy,vol. 5, no. 1, 1989, pp. 21 - 30.
No. 23 J.R. Magnus and A.D. Woodland, On the maximum likelihood estimation ofmultivariate regression models containing serially correlated error components,Internariona! Economic Review, vol. 29, no. 4, 1988, pp. 707 - 725.
No. 24 A.J.J. Talman and Y. Yamamoto, A simplicial algorithm for statiunary pointproblems on polytopes, Mathetnatics of Operations Research, vol. l4, no. 3, 1989,pp. 383 - 399.
No. 25 E. van Damme, Stable equilibria and fonvard induction, Joumu! of EconomicTheory, vol. 48, no. 2, 1989, pp. 476 - 496.
No. 26 A.P. Barten and L.1. Bettendorf, Price formation of fish: An application of aninverse demand system, European Economic Review, vol. 33, no. 8, 1989, pp. 1509- 1525.
No. 27 G. Noldeke and E. van Damme, Signalling in a dynamic labour market, Rcviewoj Econnmic Studirs, vol. 57 (1), no. 189, 1990, pp. 1- 23.
No. 28 P. Kop Jansen and Th. ten Raa, The choice of model in the construction ofinput-output coefficients matrices, International Economic Review, vol. 31, no. 1,1990, PP. 213 - 227.
No. 29 F. van der Pfoeg and A.1. de Zeeuw, Perfect equilibrium in a model ofcompetitive arms accumulation, International Economic Ret-~iew, vol. 31, no. 1,1990, pp. 131 - 146.
No. 30 J.R. Magtus and A.D. Woodland, Separability and aggregation, Economica, vol.57, no. 226, 1990, pp. 239 - 247.
No. 3l F. van der Ploeg, International interdependence and policy coordination ineconomies with real and nominal wage rigidity, Creek Economic Review, vol. 10,no. 1, June 1988, pp. I- 48.
No. 32 E. van Damme, Signaling and forward induction in a market entry context,Openztions Research Pmceedings 1989, Berlin-Heidelberg: Springer-Verlag, 1990,pp. 45 - 59.
No. 33 A.P. Barten, Toward a levels version of the Rotterdam and related demandsystems, Contributions to Operations Research and Economics, Cambridge: MITPress, 1989, pp. 441 - 465.
No.34 F. van der Plceg, International coordination of monetary policies underalternative exchange-rate regimes, in F. van der Ploeg (ed.), Advanced Lecturesin Quantitative Economics, London-Orlando: Academic Press Ltd., 1990, pp. 91- 121.
No. 35 Th. van de Klundert, On socioeconomic causes of 'wait unemployment', EuropeanEconomic Review, vol. 34, no. 5, 1990, pp. 1011 - I022.
No. 36 RJ.M. Alessie, A. Kapteyn, J.B. van Lochem and TJ. Wansbeek, individualeffects in utiliry consistent models of demand, in J. Hartog, G. Ridder and J.Theeuwes ( eds.), Panel Data and Labor Market Seudies, Amsterdam: ElsevierScience Publishers B.V. (North-Holland), 1990, pp. 253 - 278.
No. 37 F. van der Plceg, Capital aocumulation, in~ation and long-run contlict ininternational objectives, Orforá Economic Papers, vol. 42, no. 3, 1990, pp. 501 -525.
No. 38 Th. Nijman and F. Palm, Parameter identification in ARMA Processes in thepresence of regular but inrnmplete sampling, loumal of Time Series Analysis, vol.11, no. 3, 1990, pp. 239 - 248.
No. 39 Th. van de Klundert, Wage differentials and employment ín a two-sector modelwith a dual labour market, Metrtxconomica, vol. 40, no. 3, 1989, pp. 235 - 256.
No. 40 Th. Nijman and M.F.J. Steel, Exclusion restrictions in instrumental variableseyuations, Econometric Reviews, vol. 9, no. l, 1990, pp. 37 - 55.
No. 4l A. van Soest, I. Woittiez and A. Kap[eyn, Labor supply, income taxes, and hoursrestrictions in the Netherlands, Jouma[ of Human Resources, vol. 25, no. 3, I990,pp. 517 - 558.
No. 42 Th.C.M.J. van de Klundert and A.B.T.M. van Schaik, Unemployment persistenceand loss of productive capacity: a Keynesian approach, Jouma! of Macro-economicr, vol. 12, no. 3, 1990, pp. 363 - 380.
No. 43 T-h. Nijman and M. Verbeek, Estimation of time-dependent parameters in linearmodels using cross-sections, panels, or both, Jouma! of Economerrics, vol. 46, no.3, 1990, pp. 333 - 346.
Nu. 44 E. van Damme, R. Selten and E. Winter, Alternating bid bargaining with asmallest money unit, Games and Economic Behavior, vol. 2, no. 2, 1990, pp. 188- 201.
No. 45 C. Dang, The D,-triangulation of R' for simplicial algorithms for computingsolutions of nonlinear equations, Mathematicr of Operations Resrarch, vol. 16, no.1, 1991, pp. 148 - 161.
No. 46 Th. Nijman and F. Palm, Predictive accuracy gain from disaggregate sampling inARIMA models, Jouma! of Business dc Economic Statistícs, vol. 8, no. 4, 1990, pp.405 - 415.
No. 47 J.R. Magnus, On certain moments relating to ratios ofquadratic forms in normalvariables: further results, Sankhya: The Indianlournal of Statistícs, voL 52, seriesB, part. l, 1990, pp. I- 13.
No. 48 M.F.J. Steel, A Bayesian analysis of simultaneous equation models by combiningrecursive analytical and numerical approaches, Jouma! of Econometrics, vol. J8,no. 1~2, 1991, pp. 83 - 117.
No. 49 F. van der Ploeg and C. Withagen, Pollution control and the ramsey problem,Environmenta! and Resource Economics, vol. 1, no. 2, 1991, pp. 215 - 236.
Nu. ~0 F. van der Ploeg, Money and capital in interdependen[ economies withoverlapping generations, Economica, vol. 58, no. 230, 1991, pp. 233 - 256.
No. 51 A. Kapteyn and A. de Zeeuw, Changing incentives for economic research in theNetherlands, European Ecanomic Review, vol. 35, no. 2~3, 1991, pp. 603 - 611.
No. 52 C.G. de Vries, On the relation between GARCH and stable processes, Journa!of Econometrics, vol. 48, no. 3, 1991, pp. 3l3 - 324.
No. 53 R. Alessie and A. Kapteyn, Habit formation, interdependent preferences anddemographic effects in the almost ideal demand system, The Economic Journal,vol. 101, no. 406, 1991, pp. 404 - 419.
No. 54 W. van Groenendaal and A. de Zeeuw, Control, coordination and conflict uninternational commodity markets, Economic ,NortcUing, vol. 8, no. l, 1991, pp. 90- 101.
No. 55 F. van der Ploeg and AJ. Markink, Dynamic policy in linear models with rationalexpectations of future events: A computer package, Computer Science inEconomics and Management, vol. 4, no. 3, 1991, pp. 175 - 199.
No.56 H.A. Keuzenkamp and F. van der Ploeg, Savings, investment, governmentfinance, and the current account The Dutch experience, in G. Atoeoskoufis, L.Papademos and R. Portes ( eds.), Ezternal Cvnstrainu on Macroeconnmèc Policy:The European Experience, Cambridge: Cambridge University Press, 1991, pp. 219- 263.
No. 57 Th. Nijman, M. Verbeek and A. van Soest, The efficienc.y of rotating-paneldesigns in an analysis-of-variance model, lournal ojEconomerrics, vol. 99, no. 3,1991, pp. 3ï3 - 399.
No. 58 M.F.J. Steel and J.-F. Richard, Bayesian multivariate exogeneity analysis - anapplication to a UK money demand equation, Journal oj Econometrics, vol. 49,no. 1~2, 1991, pp. 239 - 274.
No. 59 Th. Nijman and F. Palm, Generalized least squares estimation of linear modelscontaining rational future expectations, International Economic Review, vol. 32,no. 2, 1991, pp. 383 - 389.
No. 60 E. van Damme, Equilibrium selection in 2 x 2 games, Revista Espanola deEconomia, vol. 8, no. 1, 1991, pp. 37 - 52.
No. 61 E. Bennett and E. van Damme, Demand commitment bargaining: the case ofapex games, in R. Selten ( ed.), Came Equilibrium Models 111 - StrategicBatgaining, Berlin: Springer-Verlag, 1991, pp. 118 - 140.
No. 62 W. Giith and E. van Damme, Gorby games - a game theoretic analysis ofdisarmament campaigns and the defense efficiency - hypothesis -, in R.Avenhaus, H. Karkar and M. Rudnianski ( eds.), Defense Decision Maldng -Analytical Suppont and Crisis Managemenr, Berlin: Springer-Verlag, 1991, pp. 215- 240.
No. 63 A. Rcell, Dual-capacity trading and the quality of the market, Journal ojFinancialIntermediation, vol. 1, no. 2, 1990, pp. 105 - 124.
No. 64 Y. Dai, G. van der Laan, A.J.J. Talman and Y. Yamamoto, A simplicialalgorithm for the nonlinear stationary point problem on an unboundedpolyhedron, Siam loumal oj Optimimtion, vol. 1, no. 2, 1991, pp. 151 - 165.
No. 65 M. McAleer and C.R. McKenzie, Keynesian and new classical models ofunemployment revisited, The Economic loumal, vol. 101, no. 406, 1991, pp. 359- 381.
No. 66 A.J.J. Talman, General equilibrium programming, Nieuw Archiejvoor lfrctkunde,vol. 8, no. 3, 1990, pp. 387 - 397.
No.67 J.R. Magnus and B. Pesaran, The bias of forecasts from a first-order
No. 68 F. van der Ploeg, Macroeconomic poliry coordination issues during the various
phases of economic and monetary integration in Europe, European Economy -
The Economicr of EMU, Commission of the European Communities, special
edition no. l, 1991, pp. 136 - 164.
No. G9 H. Keuzenkamp, A precursor to Muth: Tinbergen's 1932 model of rationalexpectations, The Economic Journal, vol. IOI, no. 408, 1991, pp. 1245 - 1253.
No. 70 L. Zou, The target-incentive system vs. the price-incentive system under adverse
selection and the ratchet effect, Journa! ojPublic Economics, vol. 46, no. l, 1991,
PP. 51 - 89.
No.71 E. Bomhoff, Between price refotm and privatization: Eastern Europe in
No. 77 A. de Zeeuw, Note on 'Nash and Stackelbecg solutions in a differential game
model of capitalism', Joumal oj Economic Dynamics arul Control, vol. 16, no. 1,
1992, PP. 139 - 145.
No. 78 J.R. Magnus, On the fundamental bordered matrix of linear estimation, in F. van
der Ploeg (ed.), Advanced Lecrures in Quantitative Economics, London-Orlando:
Academic Press Ltd., 1990, pp. 583 - 604.
No. 79 F. van der Ploeg and A. de Zeeuw, A differential game of international pollution
control, Systems and Control Leners, voL 17, no. 6, 1991, pp. 409 - 414.
No. 80 Th. Nijman and M. Verbeek, The optimal choice of controls and pre-experimen-
tal observations, Journal ojEconometncs, voL 51, no. 1~2, 1992, pp. 183 - 189.
No. 81 M. Verbeek and Th. Nijman, Can cohort data be treated as genuine panel data",Empirical Economics, vol. 17, no. 1, 1992, pp. 9- 23.
No. 82 E. van Damme and W. Guth, Equilibrium selection in the Spence signaling game.in R. Selten (ed.), Gume Equilibrium Modelt II - Methodr, MoraLr, and ,Narkets.Berlin: Springer-Verlag, 1991, pp. 263 - 288.
No. 83 R.P. Gilles and P.H.M. Ruys, Characterization of economic agents in arbitrarycommunication structures, Nieuw Archief voor Wiskunde, vol. 8, no. 3, 1990, pp.325 - 345.
No. 84 A. de Zeeuw and F. van der Ploeg, Difference games and policy evaluation: aconceptual framework, Oxford Economic Papers, vol. 43, no. 4, 1991, pp. 612 -636.
No. 85 E. van Damme, Fair division under asymmetric information, in R. Selten (ed.),RaáonaL Intemction - Essays in Honor ojJohn C. Harsanyi, Berlin~Heidelberg:Springer-Verlag, 1992, pp. 121 - 144.
No. 86 F. de Jong, A. Kemna and T. Kloek, A contribution to event study methodologywith an appGcation to the Dutch stock market, Joumal oj Banking and Finance,vol. 16, no. 1, 1992, pp. 11 - 36.
No. 87 A.P. Barten, The estimation of mixed demand systems- in R. Bewley and T. VanHoa (eds.), Contnibutions to Consumer Demand and Econometrics, Essays inHonour of Henri Theil, Basingstoke: The Macmillan Press Ltd., 1992, pp. 31 - 57.
No. 88 T. Wansbeek and A. Kapteyn, Simple estimators for dynamic panel data modelswith errors in variablcs, in R. Bewley and T. Van Hoa ( eds.), Contriburionr toConsumer Demand and Econometrics, Essays in Honour of Henri Theil,Basingstoke: The Macmillan Press Ltd., 1992, pp. 238 - 251.
No. 89 S. Chib, J. Osiewalski and M. Steel, Posterior inference on the degrees offreedom parameter in multivariate-t regression models, Economics Leners, vol.37, no. 4, 1991, pp. 391 - 397.
No. 90 H. Peters and P. Wakker, Independence of irrelevant alternatives and revealedgroup preferences, Economerrica, vol. 59, no. 6, 1991, pp. 1787 - 1801.
No. 91 G. Alogoskoufis and F. van der Ploeg, On budgetary policies, growth, andexternal deficits in an interdependent world, JoumaL oj the lapanese andInremationaL Economies, vol. 5, no. 4, 1991, pp. 305 - 324.
No. 92 R.P. Gilles, G. Owen and R. van den Brink, Games with permission structures:The conjunctive approach, Inremational JournaL of Game Theory, vol. 20, no. 3,1992, pp. 277 - 293.
No. 95 A. van den Nouweland, P. Borm and S. Tijs, Allocation rules for hypergraphcommunication situations, lnrenaationalloumal of Game Theory, vol. 20, no. 3,1992, pp. 255 - 268.
No. 96 E.J. Bomhoff, Monetary reform in Eastern Europe, European Economic Review,vol. 36, no. 2~3, 1992, pp. 454 - 458.
No. 97 F. van der Ploeg and A. de Zeeuw, International aspects of pollution control,Environmental and Resource Economies, vol. 2, no. 2, 1992, pp. 117 - 139.
No. 98 P.E.M. Borm and S.H. Tijs, Strategic claim games corresponding to an NTU-game, Games and Economic Behavior, vol. 4, no. 1, 1992, pp. 58 - 71.
No. 99 A. van Soest and P. Kooreman, Coherenc,y of the indirect translog demandsystem with binding nonnegativity constraints, Jourrwl oj Ecatometrics, vol. 44,no. 3, 1990, pp. 391 - 400.
No. 100 Th. ten Raa and E.N. Wolff, Secondary products and the measurement ofproductiviry growth, Regional Science and Urban Economics, voL 21, no. 4, 1991,pp. S81 - 615.
No. 101 P. Kooreman and A. Kapteyn, On the empirical implementation of some gametheoretic models of household labor supply, TFteJournalojNuman Resources, vol.25, no. 4, 1990, pp. S84 - 598.
No. 102 H. Bester, Ber[rand eyuilibrium in a differentiated duopoly, InternarionalEconomie Review, vol. 33, no. 2, 1992, pp. 433 - 448.
No. 103 J.A.M. Potters and S.H. Tijs, The nucleolus of a matrix game and other nucleoli,Mathematicr of Operations Reseanch, vol. 17, no. 1, 1992, pp. 164 - 174.
No. 104 A. Kapteyn, P. Kooreman and A. van Soest, Quanti[y rationing and concaviry ina flexible household labor supply model, Review ojEconomics anct Stanstics, vol.72, no. 1, 19t)D, PP. SS - 62.
No. 105 A. Kapteyn and P. Kooreman, Household labor supply: What kind of data can
tell us how many decision makers there are'?, European Economic Review, vol. 36,no. 2~3, 1992, pp. 365 - 371.
Nu. I0G Th. van de Klundert and S. Smulders, Reconstructing growth theory: A survey,Dr Economist, vol. 140, no. 2, 1992, pp. 177 - 203.
No. 107 N. Rankin, Imperfect competition, expectations and the multiple effects ofmonetary growth, Thr Economic Journal, vol. 102, no. 413, 1992, pp. 743 - 753.
No. 108 J. Greenberg, On the sensitivity of von Neumann and Morgenstern abstractstable sets: The stable and the individual stable bargaining set, InternationalJournal of Game Theory, voL 21, no. 1, 1992, pp. 41 -~~.
No. 109 S. van Wijnbergen, Trade reform, policy uncertainty, and the current account: Anon-expected-utiliry approach, American Economic Review, vol. 82, no. 3, 1992,
PP. 626 - 633.
No. I 10 M. Verbeek and Th. Nijman, Testing for selectivity bias in panel data models,International Economic Review, vol. 33, no. 3. 1992, pp. 681 - 703.
No. 111 Th. Nijman and M. Verbeek, Nonresponse in panel data: The impact onestimates of a life cycle consumption function, Jnurnal nf Applied Econometrics,vol. 7, no. 3, 1992, pp. 243 - 257.
No. 112 I. Bomze and E. van Damme, A dynamical characterization of evolutionarilystahle states, AnnaLs of Operadorct Research, vol. 37, 1992, pp. 229 - 24J.
No. 113 P.J. Deschamps, E~tpectations and intertemporal separability in an empiricalmodel of consumption and investment under uncertainty, Empirical Econornics,vol. 17, no. 3, 1992, pp. 419 - 450.
No. 114 K. Kamiya and D. Talman, Simplicial algorithm for computing a core elementin a balanced game, Joumal ojrhe Opemtions Research, vol. 34, no. 2, 1991, pp.222 - 228.
No. l IS G.W. imbens, An efficient method of moments estimator for discrete choicemodels with choice-based sampling, Econometrica, vol. 60, no. 5, 1992, pp. 1187 -1214.
No. 11G P. Borm, On perfectness concepts for bimatrix games, OR Spekrrum, vol. 14, no.1, 1992, PP. 33 - 42.
No. 117 A.P. Jurg, [. Garcia Jurado and P.E.M. Borm, On modifications of the conceptsof perfect and proper equilibria, OR Spektrum, voL 14, no. 2, 1992, pp. 85 - 90.
No. 118 P. Borm, H. Keiding, R.P. McL.ean, S. Oortwijn and S. Tijs, The compromisevalue for NTU-games, lntemarional Joumal oj Game Theory, vol. 21, no. 2, 1992,pp. 17S - 189.
No. ] l9 M. Maschler, J.A.M. Potters and S.H. Tijs, The general nucleolus and thereduced game property, Internanonal Joumal of Game Theory, vol. 21, no. 1,1992, pP. 85 - 106.
No. 120 K. W~rneryd, Communication, rnrrelation and symmetry in bargaining,Economict Lettets, vol. 39, no. 3, 1992, pp. 295 - 300.
No. 121 M.R. Baye, D. Kovenock and C.G. de Vries, It takes two to tango: equilibria ina model of sales, Games and Economit Behavior, vol. 4, no. 4, 1992, pp. 493 -510.
No. 122 M. Verbeek, Pseudu panel data, in L. M5ty~4s and P. Sevestre (eds.), TheEconometrics of Panel Data, Dordrecht: Kluwer Academic Publishers, 1992, pp.303 - 315.
p~1 RnY oni~~ ~nnn i~ Tii Ri iQr~ TLJC AICTLJCCI ANDSBibliotheek K. U. Brabanti~i u u u uuuu i ~~ i I f II~I~~
