Tilburg University

Estimation of the error-components model with incomplete panels

Wansbeek, T.J.; Kapteyn, A.J.

Publication date:1990

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Citation for published version (APA):Wansbeek, T. J., & Kapteyn, A. J. (1990). Estimation of the error-components model with incomplete panels.(Reprint series / CentER for Economic Research; Vol. 18). Unknown Publisher.

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18

im i iim i iu i i i ii ii ui ii u i Nui im i i ui uiEstimation of the Error-Components Model with

Incomplete Panels

byTom Wansbeek and

Arie Kapteyn

I5`y

,

J~,~Q,.0`,Q~ ,

t"~ ,

Reprinted from Journal of Econometrics,Vol. 41, No. 3, 1989

Reprint Seriesno. 18

CENTER FOR ECONOMIC RESEARCH

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Anton P. Barten, directorEric van DammeJohn DriffillArie KapteijnFrederick van der Ploeg

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Estimation of the Error-Components Model with

Incomplete Panels

byTom Wansbeek and

Arie Kapteyn

Reprinted from Journal of Econometrics,Vol. 41, No. 3, 1989

Reprint Seriesno. 18

Journal oC Econometrics 41 (1989) 341-361. North-Holland

ESTIMATION OF THE ERROR-COMPONENTS MODELWITH INCOMPLETE PANELS~`

Tom WANSBEEKGronutgen Uniuersiry, Groningen, The Netherlunds

Arie KAPTEYNTrlburg Uniuersity, TilAurg, The Netherlunds

Received March 19R6, final version received September 19R8

The crror-components model (ECM) is probably the most frcquently used approach to analyzepanel data in econometrics. When the panel is incomplete, which is the rule rather than theexception when the data come [rom large-scale surveys, standard estimation methods cannot beapplied. We first discuss estimation in the fixed-efTects analogue of the ECM, and then present twoestimators (quadratic unbiased and maximum likelihood) for the ECM. Some simulation resultsare given to assess finite-sample properties and computational burden of the various methods.

1. Introduction

Analysis of panel data (i.e., time series of cross-sections) by means of theerror-components model (ECM) has attracted a lot of attention in economet-rics; see, e.g., Balestra and Nerlove (1966), Wallace and Hussain (1969),Nerlove (1971a, b), Mazodier (1972), Fuller and Battese (1974), Taylor (1977),Mazodier (1978), Baltagi (1981), Wansbeek and Kapteyn (1982), and Dielman(1983). The recent monograph by Hsiao (1986) presents an excellent overview.A problem that often occurs in practice, but which is by and large ignored inthe literature, is the phenomenon of missing observations, i.e., not all cross-sectiona] units are observed during all time periods. If that occurs, we havewhat we will call an incomplete panel or rncomplete data. Note that we use theword `incomplete' to denote the absence of all information for a certaincross-section unit for a certain time period, and not to be the absence ofinformation on some variables only.

'Detailed and helpful comments [rom two anonymous referees are grate(ully acknowledged. Wethank Anton Markink and Paul Flapper for their expert programming support, Brent Moultonand Theo Nijman for their comments on an earlier version, and Jaap Verhees tor checking themany algebraic manipulations.

0304-4076~89,S3.SO~s 1989, Elsevier Science Publishers B.V. (North-Holland)

342 T. {t~"unsbeek m~d A. Kaptet~n. Estimution ojlhe errorcomponenrs model

In their pioneering study, Balestra and Nerlove (1966) analyzed panel datawhere the cross-section units were states of the USA, the time periods beingyears. The data set was constructed by the authors from existing sources andin such a context incompleteness in the above sense will not be a majorconcern. However, with the growing attention for micro-data in econometrics,data are increasingly obtained from large-scale surveys and there incomplete-ness will be the rule rather than the exception. For example, individuals maydisappear from a panel after a few waves because they refuse to cooperate anylonger, because they leave the household that is participating in the panel, orby death. Moreover, incompleteness may in fact be part of a sample design inwhich part of a panel is replaced by new units in each wave (rotating panels).The advantage of such a design is that it eases the task of respondents, therebyreducing attrition from the panel. Since attrition generally causes selectionbias [see, e.g., Hausman and Wise (1977)], which can only be remedied bymeans of fairly elaborate and computationally costly models, a sample designwhich minimizes attrition is of practical interest.

The absence of some observations makes most of the results obtained in theerror-components literature inapplicable. A solution was suggested by Fullerand Battese (1974), in which a dummy exogenous variable is added to the setof reQressors for each missin~ observation. After this amendment of the dataset, the usual methods can be applied. In many practical cases, however, thismeans that hundreds of regressors have to be added. This is clearly computa-tionally impractical and an alternative is called for.

Biern (1981) seems to be the 6rst to discuss error-components models withmissing observations. He discusses maximum-likelihood (ML) estimation inthe case where a fixed proportion of the sample is replaced with each newwave, and does not allow for observations that may be missing randomly. Hewrites down the likelihood of this model, but then observes that it involvesboth the inverse and the determinant of the error-covariance matrix. Given thesize of this matrix in practice, ML estimation without further provisions doesnot seem to be of practical value. Hence, he concentrates on the special casewhere at each wave exactly half of the sample is replaced and where there is notime-specific efTect. Then, the inverse and the determinant can be worked ou[rather easily. Baltagi (1985) studies some aspects of the error-cóvariancematrix for a general `missing pattern' in a model without time-specific effects.

In this paper we will first consider the relatively simple case where theeftects (pertaining to the cross-section units and to the time periods) are fixedrather than random, as in the ECM. This case is dealt with in section 2. Theseresults are of interest in themselves and are used, after section 3 where somebasic aspects of the random-efíects model are discussed, in section 4 as astarting point for deriving quadratic unbiased estimators in the (random-eftects) ECM. Section 5 discusses ML estimation of the ECM. Section 6contains some simulation results, and section 7 concludes. Some of the more

r. Wuiisbeek mrd A. Kap(eyn, Estimution of the error-eompone~rfs model 343

technical points are relegated to appendices; the subject of incomplete panelsis closely related to the topic of `unbalanced data' in the ANOVA literature,from which it is well-known that the results are invariably rather messy.

For clarity, it must be emphasized that some interesting problems lie outsidethe scope of the present paper. We only consider the single-equation model,and do not look at SUR-type or simultaneous equations. The only role that`[ime' plays in the ECM is via the (supposedly independent) time-specificefTects and we do not look at lagged endogenous variables or a more sophisti-cated modelling of the time component in the error structure; nor do we gointo the problems caused by selective nonresponse. For all these extensions,we first need the basic apparatus given in the present paper.

2. The fixed-eflects model

Consider the following regression model:

Yn~ - ah } Yr } xh~a t uh„ (2.1)

where h ( h- 1, ..., H) denotes, say, households, and t( t- 1, ..., T), say,years; xh, is a k-vector of explanatory variables; and ~i a k-vector ofparameters. The error term uh, has the usual `classical' properties: its varianceis az; ah and y, are fixed constants. Therefore, we will sometimes refer to thismodel as the `fixed-efïects' (FE) model. When we have a complete panel, i.e.,all households h are observed for all years t, it is well-known that OLS in (2.1)is equivalent to OLS in the transformed model

Yn~ -Yn.-Y.~ }Y..- (xhr - xh.- x.r } X.. )~Q f eti,, (2.2)

where a dot in the place of an index denotes the average over that index. Away to prove this result is to first write (2.1) in vector format,

Y-(rr~lrt~af(Ir~t~~)YtX~3tu. (2.3)

Note that we have ordered the observations such that the data on the Hhouseholds are ordered in Tconsecutive sets; the index t`runs slowly' and theindex h`runs fast'. In (2.3), it is simple to show that the projector perpendicu-lar to the regressors corresponding to the household and time efTects, viz.(rr~ ~H. Ir~ iH). is given by Er~ EN, where Er- Ir- ~rrrrT and EN isdefined analogously. Moreover, it is rather straightforward to see that applica-tion of this projector to both sides of ( 2.1) efTectuates the transformationshown in (2.2).

When we have incomplete data, these simple projection and transformationresults no longer hold. We will now derive the corresponding more generalresults for the incomplete case.

344 T. Wansbeek unJ A. Kupte~m, Estimaliai oJ the error-components model

Let N, ( N, 5 H) be the number of observed households in year t. LetN-~~N,. Let D~ be the (N, x H) matrix obtained from the ( H x H) identitymatrix from which rows corresponding to households not observed in year thave been omitted, and consider

Di DtiH

Z-( Zi ~ Zz )- :h'xll NxT D

r

The matrix Z gives the dummy-variable structure for the incomplete-datamodeL (For complete data, Zt - tr~ I~r, Zz - Ir~ tlr.)

Next, let

~ll - ZíZ„ ~T- ZZZz, A - ZZZI, (2.5)

where all is the diagonal (H x H) matrix with hth element indicating thenumber of years for which the hth household has been observed; aT is thediagonal (T x T) matrix with t th element the number of observations in yearr; and A is the (T x H) matrix of zeros and ones indicating the absence orpresence of a household in a certain year. (For complete data, d Ir - T Irr,ÓT- H ÍT, and Á- cTCy.) Further deline

Z-Zz-Zi4itIA~ ~-~IN-Zi(ZíZi) 1Zí)Zz~, (2.6)

Q-vT-Aa~r~A' ~-ZZZ~. (2.7)

In the complete data case, Q- H ET. In the incomplete data case, Q has nospecific structure. If each household is observed at least twice (and H~ T),rank(Q) - T- 1. In order to avoid unnecessary complications, we assume inthis section that this condition holds. (If a certain household is observed onlyonce, it conveys no useful information as we [hen have the case of a singleobservation with its own dummy variable.)

Now consider the following matrix:

P - Pi - Pz - ~ IN - Zld~l'Zi ) - ZQ-Z'. (2.8)

Lemnia. P is the projecrion matrix onto the null-space oj Z.

Proof. The proof is in three steps:

(i) P is idempotent: ZíZ- 0 [clear from (2.6)], Z'Z - Z2Z - Q. AsZQ-Q - Z, PZ - ZQ-QQ-Z' - ZQ-Z' - Pz. Also, Pi - Pl and P1Pz-PZP1-Pz. So Pz-Pi fP2 -P~Pz-PZPt-Pt-Pz-P.

T. Wunsbeek mrd A. Kupre~~n, Eslinmtion oJ lhe error-compa~ertls mrlde! 345

(ii) PZ - 0: PZ - Pl(Z~, Z2) - PZ(Z~, ZZ) -(0, Z) -(0, ZQ-Q) - 0.

(iii) rank(P) f rank(Z) - N: rank(P) - tr(P) - tr(P~) - tr(PZ) -(N - H)- tr(Q-Z'Z) - (N - H) - tr(Q-Q) - (N - H) - (T - 1); rank(Z) -H~T-1.

Together, ( i), (ii), and (iii) prove the lemma ( e.g., Balestra (1973, lemma 9)].Q.E.D.

So, P generalizes the expression ET ~ E~~ to the incomplete-data model.Note that there is an asymmetry in the way that we deal with both dimensions(households and years): P contains the generalized inverse of the (T x T)matriz Q for which no closed-form expression is available in general. Alterna-tively, we could have derived an expression [or P that contains tlie inverse ofan analogous (H x H) matrix, but as H~ T in most practical situations, ourchoice is the most favorable one [rom the point of view of compu[ation. Theasymmetry and its inherent lack of elegance is the kind of uncomeliness thatone has to face when dealing with incomplete or unbalanced data.

How to use P? We will give the generalization of the transformation givenin (2.2). Let v(N x 1) denote a vector of variables occurring in the regressionequation; in (2.3), v- y or v is a column of X. We are interested in the formof Pv. Let

~, - Z,u ( H x 1) (2.9)

and~z - ZZU ( T x 1) (2.10)

denote the sum of elements of v over years and households, respectively, andlet

~ - Q-Zv - Q-(~z - Ad~~l~l). (2.11)

(The choice of generalized inverse is arbitrary.) Now

Pv - v- Z1dH'Ziv - ZQ-Z'u - v- Z1dH~~i - Z~. (2.12)

In scalar format this reads as

1 1( PU ) rh - ~rh - -~l h } -ph~ - ~r ~S,, S,,

(2.13)

with o,, the hth column of A and S,, the hth diagonal element of dN. So, OLSon data from an incomplete panel with fixed year and household eli'ects

346 T. t4''uiisheek u~id A. Kuprer~~, Es~imu~ion of !he erro~-compaaents model

amounts to OLS in a model without these efi'ects when the variables have beentransformed according to (2.13). Then using a standard regression package forthe transformed data one should not forget to adjust the standard errors andthe R'- printed by the program for the loss of degrees of freedom. When thereare k`true' regressors ( apar[ from the dummies), the printed standard errorsshould be multiplied by {( N- k)~( N- H- T f 1 - k)} l. Analogously, theOLS residual variance estimate printed by the program should be multipliedby ( N- k),( N- ll - T f 1 - k) to obtain an unbiased estimate of a'-.

3. The random-effects model

ln the usual ECM formulation the ~h and Y, in (2.1) are i.i.d. randomvariables with mean zero, mutually independent and independent of the xh,and uh,. Let aZ be the variance of uh„ then the covariance matrix of thecomposite error term eni - utir } ~h t Yr ~S

Sl - aZIN-~ aiZiZí f aZZ2Zz,

with

ai - var(ati), a; - var(Yr)-

(3.1)

For any sort of efficient estimation of ihe parameter vector ~3 we need anexpression for the inverse of 12. This is given by the following:

Lemma.

a'-.2-' - V - VZZQ-1ZZV,

x.here

V-IN-Z,d~,~Z~ (NxN),

Q-dr-A~~~'A' (TxT),

,a'~~~-4i~} áZlfr

i(HxH),

(3.2)

az .~r-4r} Zlr (TxT). (3.6)

aZ

T. lYmishi~ek mid A. Kuptet'n. Esuw~orion oJ rhe errnr-componenb' ~iuxlel

Proof.

a'-ZíVZz-d~--Ac9~~~f1'-Q- zlT.az

so

a z

347

Q - Zí VZ, - ; IT.aZ

From (3.3),

V-~-h,~-Zi~dii-ZiZi) ~Zi

i-l,v}Zi~d~~-~ii~ Zi

az-1N~} zZ1Zí,a

so inverting the expression in (3.2) yields

( V - VZzQ - ~Zí V ~ ~ - V ~ } Zz ~ Q - Zí VZz ) ~ Zí

z z-1Nf aZZ~Zíf aZZZZí

a a(3.10)

- a-zfl. Q.E.D.

The expression for Tl-1 is somewhat messy and is asymmetric in householdsand years. In contrast with the complete-data case, no closed-form expressionfor fl-' ( or for the eigenvalues and eigenvectors) can be given in general. Thisis only possible for some very specific and `neat' patterns of missing data.However, for practicál purposes the expression for l2-1 is quite useful ascompared to the situation where fl is inverted numerically. The aspect of maininterest is the computariona! complexiry, i.e., the number of computations, andhence the computing time, as a function of the number of observations. In ourcase, where T typically is a small number and N and H are large, we areinterested in the computíng time of the GLS estimator of the regressioncoefiïcients in terms of N and H, and take T to be a constant.

We can make the following observations. The computation of Q involves aconstant number of evaluations of the type j'!l-~g, with j and g being

34R T. Wansheek und A. Kupte~~n, Eslimulion of ~he error-compmienls n~odel

N-vectors ( f and g may be one of the various regressors or the regressand).This, in its turn, involves a constant number of evaluations of the type f'Vg[cf. (3.2)], where f and g now also comprise the columns of Zz. Now,f'Vg - J'g - f'Z~~;~iZíg; j'g can be computed in O(N) time and f'Zid~~1Zígin O(H) time [at least, when the elements of f and g are displayed in a(T x H) matrix, then f'Z~ and Zíg are simply the H-vector of column totals,computable in O(H) time]. If the structure of Sl would not have beenexploited, Sl would have to be inverted numerically, which requires in princi-ple O(H3) time. (The latter statement neglects recent developments in com-plexity theory, which allow for a reduction of the exponent 3 to a somewhatlower figure. Up until now, this development has theoretical significance only.)

The implication of the above is that the expressions used in inverting J2 maylook a little deterring, but that they enable efficient computation (viz. comput-ing time linear in H) of the GLS estimator of the regression coefficients.

4. Quadratic estimators of the variance components

Statistically efficient estimation of the ECM can take place along two lines.One is to use ML (see section 5). The other is to estimate the variancecomponents (az, ai, and a2) by a quadratic unbiased estimation method(t1Uïr; beiow the E in t1UE can aiso stanà ïor estimator) anà [o estima[e theregression coefficients by GLS with these estimates inserted in 12. In thissection we derive QUE's for ai and aZ. The estimator for aZ from the FEmodel is unbiased under RE assumptions as well, so we concentrate on aiand az.

An intuitively appealing approach to derive QUE's for a~ and aZ is toestimate the (FE) model and to use the FE residuals, averaged over householdsor averaged over years, as the basic ingredients. By residuals we mean in thepresent context the residuals with respect to the X-part of the FE model, notthose with respect to all regressors (X and Z); the latter are not informativeabout ai and a2 as the variation of interest is projected out.

We first assume that X does not contain a vector of ones. Let e be theN-vector of FE residuals, i.e., e- y- Xb with b the FE estimate of Q, and let

9i~ - e'Zzdr'Zíe,

9r- e'Zid H~Zíe,

k~~ - tr( X'PX )-1 X'ZZdT ~ZZ X,

kr - tr( X'PX )-~X'Zld ~~1Zi X.

(41)

(4.3 )

(4.4)

T. Wansbeek and A. Kapteyn, Estimarion of the error-components mode! 349

Lemma.

E(9rr) -(T ~- ky)az f Tai f Na2, (4.5)

E( 9r) -( H t kT) az f Nai f Ha2. (4.6)

Proof. We first prove (4.5). Let

M - IN - X ( X'PX ) -1X'P, (4.7)

then by definition e- My - Me. As PZl - 0 and PZz - 0, PSl - azP, so

MT2 - Sl - a zX ( X'PX )-1X'P,

MTlM'-dl-azX(X'PX)-1X'P-azPX(X'PX)-1X'

fazX(X'PX)-1X',

ZZMSlM'Zz - ZZSlZz f azZZX(X'PX)-1X'Zz(4.10)

- a zdT f aiAA' f aZdT f a zZZ X( X'PX )-~ X'Zz.

Using (4.10) and tr(4T1AA')- T,

E(4H) - E~e~Zz4r1Zie)

- tr E( ~ T 1Z2Mee'M'Zz )

- tr(~T1ZZMTlM'Zz) (4.11)

- tr(azlTt ai~T1AA't aZdTf az4T1Z2X(X'PX)-1X'Zz)

-(Tfky)azfTai fNaZ.

The proof of (4.6) is analogous. Q.E.D.

We obtain QUE's for ai and aZ when solving (4.5) and ( 4.6) for ai and a?and using an unbiased estimator for az. The latter may be based on

E(e'Pe)-az(N-T-Hfl-k), (4.12)

i.e., the estimator of a z in the FE model.

350 T. Wunsbeek and A. Kaptey~n, Esrimation oJthe e~ror-componenrs mode!

So far, we have assumed that X does not contain a vector of ones. When, inaddition to X, there is an intercept S in the regression, we have

e-My-MtNSfMe-tNSfe (4.13)

[cf. the line below (4.7)], so ín that case we will use the centered residualsf- ENe rather than e. Computing the expectation of the redefined qH and qT(with f substituted for e) is essentially the same but is more complicated. Seeappendix A for results.

When the effects a~, and yr are normally distributed, explicit expressions forthe variance of the QUE's of ai and aZ can be derived. This is indicated inappendix B.

5. ML estimation and the intormation matrix

For complete data, ML estimation in error-components models has beenstudied by Amemiya (1971). Applying the general results obtained by Magnus(1978), the órst-order conditions for ML estimation of Q and the parametersin dl are

!3- (X'Sl-'X)-'X'fl-lv,

tr( flé isl )- e'SlB le, B- a z, ai , or aZ ,

(5.1)(5.2)

with e- y- Xf3 and Slé 1 being the derivative of 12-r with respect to B.appendix C, it is shown that

Slá:l - a-zt -S1-1 f ai zRdHZR' ~- a2 zVZzQ-zZ2V },

~ozl - -ai'Rd~(zR',

f1á21 - -aZ'VZZQ-zZZV,

withR-( IN - VZzQ 1Zí ) Zl,

and thattr(Sl;:rJl)-a-z{-NfPi}PzÍ~

tr(f2é:1J2) - -a; zp;, i - 1,2,where

azPr-H- 2tr(~Hli-QylA'Q-IAQ~tl),

al

az-T--tr 1Pz - az Q- .z

In

(5.3)

(5.4)

(5.5)

(5.7)(5.8)

(5.9)

(5.10)

T. Wansbeek and A. Kapreyn, Esrimation oJ rhe error-components mode! 351

A few comments on these results are in order. When complete data areavailable, there do not exist closed-form expressions for the MLE's of thevariance components, and this holds true a fortiori for the incomplete-datacase.

Again, the formulae do not look particularly attractive. Yet it is simple tosee that all expressions of interest can be computed in O(H) time. [Since thenumber of iterations is unknown, this dces of course not guarantee that thecomputation of the MLE's until convergence also can be done in O(H) time.]

Both for the purpose of computing asymptotic standard errors of the MLestimators and to test hypotheses it is useful to compute the informationmatrix. The information matrix is derived in appendix C to be

with

and

f X'12-'X 0 l

L o 1~J

~oo ~oi ~oz~- ~io ~ii ~iz ,

~zo ~zi ~zz

a~~-a- {Nt9ii}9zz-2(PitPz-9iz) .

-z z~o;-tÍ'~o-a a; ( P;-4u-9iz), i-1,2,

~,~ - ar za~ z9;~, i, j-1,2,

z zr a9ii - trl ly - 2(áHl f ~H1A'Q-IAd yl) ,

1 al

z za I9zz-tr IT- ázQ- ,

z

a44zr - 9iz - iz tr(dylA'Q-zA~tr1).

aiaz

[(k f 3) x(k f 3)],z

(5.11)

(5.12)

(5.16)

(5.17)

(5.18)

All these expressions can be computed in O(H) time.The problem of estimation with incomplete panels can be considered as a

`missing-observations' problem. By `missing' we mean that for some (h, t)pairs both yh, and xti, are not observed. In the context of missing observa-

352 T. Wansbeek anJ A- Kapteyn, Eslimarion oJ the erroi-camponents niodel

tions, the EM algorithm for obtaining ML estimates has attracted a lot ofattention, recently also in econometrics. Fair's treatment of the Tobit model ispossibly the best-known example (Fair (1977)j. In our context the EM ap-proach suggests the following. Choose starting values for the model parame-ters, and compute the distribution of the `missing' yh, (with x~„ - 0 withoutloss of generality) conditioned on the observed data and the starting values.Next write down the likelihood function for the complete panel (with its nicelystructured Sl), take its expectation with respect to the missing `yti,', andmaximize it to obtain new parameter values, etc. As far as we know, the EMalgorithm has not yet been elaborated for the ECM with incomplete observa-tions. It could ofier a useful alternative to our approach, but a preliminaryinspection suggests that working out the EM approach produces a lot of inessyalgebra, much of the same type as with our approach. This apparently isinherent to the problem at hand.

6. Some simulation results

We consider the following simple version of model (2.1): ai - var(a,,) - 400,aZ - var(Y~) - 25, aZ - var(uh~) - 25, xnr - ( 1, Xti~)~, R -(Qo~ Qi)~ -(25, 2)',N- iuu, T- 5. The scalars x,,, were generated according to [he schemeintroduced by Nerlove (1971a, p. 367) and subsequently used by, e.g., Arora(1973), Baltagi (1981), and Heckman (1981):

xti, - O.lt t O.Sxti.r-1 f wh„ (6.1)

with the w,,, uniform [- ~, Z] and x,,o- 5 f lOw~,o. Three cases are consid-ered:

1. No observations are missing: `complete datá .2. Each period 20`,~ of the households left in the panel is removed randomly:

'random attrition'.3. In period 1 we start with 40 households. In period 2, 20 new households are

added. In period 3, 20 households remaining from period 1 are removedand 20 new households are added. In period 4, the 20 households stillremaining from period 1 are removed and 20 new households are added,etc.: 'rotating panel'.

The x values have been drawn once and are used in all experiments. Giventhe x values we generated values of yti, according to model ( 2.1) in each newsimulation run. For each case 50 runs are made, all using the same pattern of

T. Wa~ubeek andA. Kapreyn, Estimation oJrhe error-components mode! 353

Table 1

Computational burden of diftecent estimation methods.'

Set up OLS GLS ML

Complete data 100 414 2026Random attrition 68 333 3296Rotating panel 53 297 3663

'In CPU scconds relative to OLS on the complete data. The computing time for ML includesthe time required to generate starting values by means of the GLS procedure.

`missings' in the `random attrition' case. For the ML estimation we always usethe estimates of the two-step GLS procedure as starting values.

Table 1 gives an indication of the average computing time required by thetwo estimation methods for each case (GLS and ML), plus OLS. It is quiteobvious that GLS is a lot cheaper than ML. Furthermore, computation timefor GLS decreases if more observations are missing, whereas computationtime for ML increases. For complete data (100 households each period) MLrequires about 5 times more CPU seconds than GLS. In the rotating panelcase (40 or 60 households each period), ML takes about 12 times more CPUseconds than GLS. Of course, the precise magnitude of these figures dependon the particular design matrix chosen.

Table 2 presents means and variances of various estimates obtained in the50 simulation runs. After each figure, its variance over the 50 runs is given. Asto the regression coefficients, it appears that OLS, though unbiased conditionalon the x~,,, performs badly, and that GLS and ML give nearly identical results,both in terms of point estimates and of sampling variances. The latter aresomewhat smaller than those obtained with FE. The sampling variances of alincrease over the three cases, which are ordered by a decreasing number ofobservations, and go markedly up when moving to the `rotating panel' casewith its short time series.

Regarding the estimation of the variance components, it is striking that onaverage the unbiased QUE's are in all cases at least as close to the true valueas the MLE's, and in a majority of cases much better. If one is interested inthe values of the variance components, the results suggest a preference for theQUE's - iteration does not seem to pay off.

The column ` variance of GLS estimate' gives the means of the estimates ofthe variance of the estimates of ai , aZ , and a Z, respectively. Although thevariance formulae indicated in appendix B are exact, the estimates actuallyused are not unbiased, because we have to plug in estimates of aZ, ai, and aZ.The ` variance of the ML estimate' is based on the information matrix. It turnsout that the variance estimates for GLS and ML show some agreement, bothbetween the two and with the corresponding sampling variances, but there aresome striking exceptions - variances of variances can be volatile.

354 T. Wansbeek and A. Kapteyn, Estintatian oJlhe enor-components model

uÓ iaLU .,,G -'m y

eCO J7 ~

V

~ EY

uó ~~ Ec -h u~~ ~~a~

u{~ ~~ E

u

v~ iv.~ E

O u

2'Í-- j

`u`uENÁG

ó ó óó ó óX X X

~.;~ ~D N~ f~ ~ ~ N 1~ ~ tT

v V O " 0~0 v V~ N

8 P~ ~[~ N ~ 1~ vl~ ~ . ~ r ~~O m ~O N f~ ~p N

~ O S O~ 1~ .~i ~ O 1~ vl ~o O OO ~O Ó~

ó~O.~o vv o:oo.P..C ovov~""m

O~n ,,,,, m oo O V,-., ~O m N oo N t~ 1~.ó ? ... vi v ~ ? ... .ti oc .ó ~ ~. r ..N.-. m N N N.r m N N N~--i m ti m

~ v O

ó titi m e~i

ti

V

~

ÉU

~O ~ .~. t~~,~óo~vo ~., o, ~ o0.ó ~: oo v, ~N .-n m N N

N~~

ó~aO .-~

OX p~ ~N X ó.v ~ ~~ N~ Om ~n. y m~D vi

~ vo ~

m ~ o`.. vi r

p ~ ~ .-. ~nvo ~W~O,O .- ~; V oo P ~

á ó,-, ~ à á oo, ó o v`.O m ~ vi O~ ~.. oo p m ao~, C o0 p.`C~l m N N N.r Nl N N

~ rN V~1Ov o

óá ó~ó.-: o ~-:

~N ~~-

P et vl N I~ t~.-mi ~N... .m-~ N .~i v.... m~lt~ O~ N ~D vlc~ a` ~ó -~ r- ~N O N~--~ N O

vt N Q Vl V1 vl N O Vl v't Vl N p Vt vlN RQ N N N 00 N N N O N NR

~Cl O Ó b ~C Q3 b Ó O ~R bl b Ó b

[O

T. tVunsbeek und A. Kuptevn, Estiniation oJ rhe errot-con~ponen(s mode! 355

7. Concluding remarks

Altogether, missing observations lead to less elegant expressions for estima-tors, and computer programs are accordingly more complicated. Yet, in termsof computational complexity missing observations do not constitute majorproblems. In view of the large difference in computational cost, GLS is to bepreferred over ML in the ECM. Statistically, ML and GLS do not seem tobehave very difïerently in finite samples. In the ECM, the FE estimator is alsoa viable alternative. lt has a somewhat larger sampling variance, but one hasnot to make the assumption that the random efïects are independent from theregressors, which may be troublesome in many applications.

Appendix A: QUE's in a model with an intercept

When an intercept term is present, we develop the QUE's starting fromquadratic functions of the ceqtered residuals:

q;, -I'Z~aT'Zít, (A.i)

qT-t'Zta~,`Z~I- (A.z)

In this appendix we evaluate the expectation of qy, which makes clear howunbiased estimators of ai and az can be constructed. As f- ENe - ENMe,

9á - t'M'ENZZ~TtZ2ENMe (A.3)

and, by elaborating EN,

E(9i!) - E(91! ) } I~,ZZ0T1Z2CNl~,M~M 'iN~Nz

- 2 t N MllM'Zzd r1Zí ~ NrN. (A.4)

As ZZ4T'Zít,v - tN, this carries over into

E(qí!) - E(9t!) - tNMSlM'tN~N

- E(qu) - {'NTltNfaZ1NX(X'PX)-tX'tN},N (A.5)

- E(9H) - a2(1 f ko) -~ai~i } az~z)IN,

where the second equality is based on (4.9) and Zt4TtZitN - tN and where in

356 T. Wunsbeek and A. Kuptey~n, Estimation oj the error-components model

the last step the following definitions have been used:

ko - tNX( X~PX) 1X~t,vrN,N

2~t - tNZtZítN- mn,h-I

T2í,Z - INZZZZCN- Il~.

t-1

(A.6)

(A.7)

(A.8)

The interpretation of m~, is that it is the number of times (2 ~ m~, 5 T) thathousehold h was in the panel, whereas n, (2 5 n~ 5 H) denotes the number ofhouseholds in the panel at time l.

So the presence of an intercept term introduces an adjustment of E(qy); seethe ]ast line of (A.5). The same adjustment has to be made to E(qT).

Appendix B: On the variances of the QUE's

Let again B denote any one of the variance components a z, ai , or a2. TheirQUE's can be written as

B - e'MENWENM'E - j'Wf, (B.1)wtth

YV - aP f bZidH1Zi t cZzdTtZZ. (B.2)

Here a, b, and c are constants that can be chosen such that ( B.1)-(B.2)generate the QUE's. According to multivariate normal theory,

var(9) - 2 tr( MENWENM'Sl)Z. (B.3)

Elaborating ( B.3) is a tedious affair but is in principle straightforward. Wehave elaborated and programmed (B.3) to compute the variances of the QUE'sin the simulations reported in section 6. Since the formulae for the variancesare ugly and do not yield any insights, they are not given here. The formulaeare available from the authors on request.

Appendix C: Derivation of results on ML

Let B denote any one of the parameters az, ai, or a2. Then

,flé 1-a-z{-aBl2-1 f (V- VZZQ-1ZZV)e~

- a-z{ -aéSl-t ~- VB- VBZZQ-1ZZV

t VZzQ-1QeQ-1ZZV - VZZQ-1Z2VB } .

(C.1)

T. It~mubezk und A. Kaptel'n, Estinvution oJ J~e erroi-compone~us nialrl 357

There holds

Qe - d TB f Aa Jr ~d J~ed H 'A~ - d~ f Zí VB ZZ,

so

(C.2)

VZZQ-~QeQ-~ZíV- VZzQ-tdreQ- tZíVi- VZZQ -1Zí VBZ,Q-~ZíV.

(C.3)

Substitution of (C.3) into (C.1) yields

Slé~-a-Z{-oéll-~t(IN-VZZQ tZí~Ve~IN-ZZQ-tZíV~

f VZzQ-~4reQ-1ZíV }.

In view of the definition of V[see (3.3)], there holds

VB- Z~~ii~~neá~~1Z~.

(C.4)

(C.5)

This expression and the definition of R[see (5.6)J allow for writing (C.4) as

!2B 1- o- Z{- aé !l -1 -~ R á~i 14 ire4 ir1R' f VZiQ -~~ rQ - iZí V}.

(C.6)

It remains to substitute for aé, ~~, and ~J~8 in order to establish (5.3)-(5.5).There holds

~a,.o~ - az r,., áTa; - o,aZ

áT,; - - á, r, .z

(c.~)

aZz~ Jro' - ai JJi ~ ~ J~a; - - ó01 J~ ~

~4 i~a, - 0, (C.8)

and this leads to (5.3)-(5.5) directly.For further results, we need an expression for Slé tfl. In view of (C.6),

this means that we first have to consider R'dl and ZíV12. Now, since

J.Econ D

35R T. ~Vunsheek arrd A. Kuprein, Esrimaiion oJd~e enor-compo~~e~us madel

V~- I,v f (ai~a)'-ZiZ(, !1 - azV-~ f aZZ2Z2. Hence

ZZV12- Z;V~ozV-~ t a?ZzZz)

- a zZZ -~ az ZZ VZZ Zz

a'-' a zZz } az Q- Z Ir Zí (C.9)

az

- a zZZ ~ a2 QZZ - a zZ2

- aiQZ-

and

R'S2 - Z, ( IN - ZzQ-'ZZV )S2

- Z~ S2 - ZíZZQ-'ZZ VSl

- Z~ S2 - aZ Z~ ZZQ -~QZZ

- azZi -~ aZZ~ZZZZ f aiZiZ1Zí - azZjZZZZ

z-azZi f ai dr~- ázJ~i Zi

i

z'- ai~~iZí.

(C.10)

These expressions, with (5.3)-(5.5), yield

(a~-~~aa2~,fl-a-z{-INfRd~~1Z~-~VZZQ-~ZZ}, (C.11)

(a~l-`~aai)S2- -a~ zRd~~1Zí, (C.12)

(a~-1~aa2~~l- -oZVZzQ-1Z2. (C.13)

To verify (5.7) and (5.8) we have to take traces of these expressions. We use

T. Wa,aheek und A. KupreVn. Esrrmulion of the errar-compaients moJe! 359

the following facts:

tr(Rá~~'Z{) - [r(Z~RáH~~

- trZ~(IN- VZzQ-~Zí)Z~d~~~

- tr~ZiZ~ - ZiVZzQ-~A)d~~~

az- tr ZiZI - ázdiilA'Q lA d~~l

i

az az- tr ds- ?IH- ?dn~A'Q-~A áNi

al al

az- H- Qz [r(d~~~ f d~~lA'Q-lAd~~~~

i

-Pi [cf. (5.9)].

tr( VZzQ 1Zí) - tr(Z2VZZQ-1)

azQ- i- tr azfr Q-z

az- T- Z trQ-~az

- pz [cf. (5.10)] .

(C.14)

(C.15)

This establishes ( 5.7) and (5.8).We finally derive the elements of the information matrix ( 5.12). The element

of ~(cf. ( 5.12)] corresponding with parameters B and B', say, is tr(Slé 1Sl,flé.1J2),and is hence obtained by taking the trace of the product of any two right-hand

360 T. Wmisbeek miJ A. Kuptey~n, Eslimurro~~ o~ die errnr-con~por~enrs moJef

sides of (C.11)-(C.13). The following facts are used

z z0tr(R~~~~ZíR~y1Zí ~ - tr ly- áz (du~ f d~z1A'Q-~A.~~rl~

i

- 91, [cf. (5.16)],

tr(VZ,Q-~ZíVZzQ-1Zí~ - tr(ZíVZZQ-i)z

z za ~-tr IT- ázQ-

z

- qzz [cf. (5.17)],

(C.16)

(C.17)

t.( vZ-n- ~7íR~ ~,1Z; )- tr( Z~ VZZQ-~ZZR~~~1)'`~ 1L ,. .

- Zz Ir(~y'A'Q-zA0 yl ) (C.18)a~az

a4

- 9rz [cf. (5.18)],

where the second equality sign in (C.18) is based on

ZíR- Zí(~.v - VZ,Q- 1Zí I Z~

- A - ZíVZzQ- ~A

- (IT- (Q- oz IT)Q ')A

az- áZ Q-lA.

(C.19)

With the aid of (C.16)-(C.18), the elements of the information matrix followdirectly.

T. l4'unsheek und A. Kaptevn, Fstinwtion o~ rhe error-conrponerus model 361

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Reprint Series, CentER, Tilburg University, The Netherlands:

No. 1 G. Marini and F. van der Ploeg, Monetary and fiscal policy in anoptimising model with capital accumulation and finite lives,The Economic Journal, Vol. 98, No. 392, 1988, pp. 772 - 786.

No. 2 F. van der Ploeg, International policy coordination in interdependentmonetary economies, Journal of International Economics, Vol 25, 1988,PP. 1 - z3.

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.),Challenges for Macroeconomic Modelling, Contributions to EconomicAnalysis 178, Amsterdam: North-Holland, 1988. PP. 39 - 88.

No. 4 F. van der Ploeg, Disposable income, unemployment, inflation andstate spending in a dynamic political-economic model, Public Choice,vol. 60, 1989, pp. ul - z39.

No. 5 Th. ten Raa and F. van der Ploeg, A statistical approach to theproblem of negatives 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, Journal of Economic Theory, vol. 47, No. 1, 1989,pp. 206 - 217.

No. 7 C. Mulder and F. van der Ploeg, Trade unions, investment andemployment in a 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 capitalmobility in an optimizing model of a small open economy, De Economist137, nr. 1, 1989, pp. 47 - 75.

No. 9 G. Dhaene and A.P. Barten, When it all began: the 1936 Tinbergenmodel revisited, Economic Modelling, Vol. 6, No. 2, 1989.pp. 203 - 219.

No. 10 F. van der Ploeg and A.J. de Zeeuw, Conflict over arms accumulationin market and command economies, in F. van der Ploeg and A.J. deZeeuw (eds.), Dynamic Policy Games in Economics, Amsterdam: ElsevierScience Publishers B.V. (North-Holland), 1989. Pp. 91 - 119.

No. 11 J. Driffill, Macroeconomic policy games with incomplete information:some extensions, in F. van der Ploeg and A.J. de Zeeuw (eds.),Dynamic Policy Games in Economics, Amsterdam: Elsevier SciencePublishers B.V. (North-Holland), 1989. PP. 289 - 322.

No. 12 F. van der Ploeg, Towards monetary integration in Europe, in P.De Grauwe e.a., De Europese Monetaire Integratie: vier visies,Wetenschappelijke Raad voor het Regeringsbeleid V 66, 's-Gravenhage:SDU uitgeverij, 1989, pp. 81 - 106.

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-square forecast error for the first-order autoregressive model,.Journal of Econometrics, Vol. 39. No. 3. 1988. PP- 327 - 346.

No. 15 R. Alessie, A. Kapteyn and B. Melenberg, The effects of liquidityconstraints on consumption: estimation from household panel data,European Economic Review 33. No. 2~3. 1989. PP. 547 - 555.

No. 16 A. Holly, J.R. Magnus, A note on instrumental variables and maximumlikelihood estimation procedures, Annales d'Économie et deStatistique, No. 10, April-June, 1988, pp. 121 - 138.

No. 1~ P. ten Hacken, A. Kapteyn and I. Woíttiez, Unemployment benefits andthe labor market, a micro~macro approach, i n B.A. Gustafsson and N.Anders Klevmarken (eds.), The Political Economy of Social Security,Contributions to Economic Analysis 1~9, Amsterdam: Elsevier SciencePublishers B.V. (North-Holland), 1989, pp. 143 - 164.

No. 18 T. Wansbeek and A. Kapteyn, Estimation of the error-components modelwith incomplete panels, Journal of Econometrics Vol. 41, No. 3. 1989.PP. 341 - 361.

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