Regression systems for unbalanced panel data: a stepwise maximum likelihood procedure

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1 Journal of Econometrics 1 (004) Regression systems for unbalanced panel data: a stepwise maximum likelihood procedure Erik Birn Department of Economics, University of Oslo, P.O. Box 1095 Blindern, Oslo 0317, Norway Accepted 3 October 003 Abstract The estimation of systems of regression equations with random individual eects from unbalanced panel data, where the unbalance is due to random attrition or accretion, bygeneralized least squares (GLS) and maximum likelihood (ML) is considered. In order to utilize the previous results for the balanced case, it is convenient to arrange the individuals in groups according to the number of times theyare observed. It is shown that the GLS estimator can be interpreted as a matrix weighted average of the group specic GLS estimators with weights equal to the inverse of their respective covariance matrices. A stepwise algorithm for solving the ML problem, which can be interpreted as a compromise between the solution to the group specic ML problems, is presented. c 003 Elsevier B.V. All rights reserved. JEL classication: C13; C3; C33 Keywords: Panel data; Unbalanced panels; Regression equation systems; Heterogeneity; Covariance estimation 1. Introduction Systems of regression equations have a long history in econometrics. Notable examples are systems of demand equations for inputs or consumer goods derived from producers or consumers optimizing behaviour. The reduced form of a structural model also has this format. This historyis substantiallyshorter in panel data econometrics. Starting from the textbook model for single equations with balanced panel data and random eects (see e.g., Greene (003, Chapter 13)), Avery(1977), Tel.: ; fax: address: erik.biorn@econ.uio.no (E. Birn) /$ - see front matter c 003 Elsevier B.V. All rights reserved. doi: /j.jeconom

2 8 E. Birn / Journal of Econometrics 1 (004) Baltagi (1980), and Magnus (198) generalized this framework and the estimation procedures to seeminglyunrelated regression systems, Magnus (198) byconsidering maximum likelihood (ML). 1 These generalizations also assumed balanced panel data, which is a veryrestrictive assumption from a practical point of view, since data sets where the time series of the dierent units have unequal length, often is the rule, rather than the exception. Single equation extensions of the textbook model to models with unbalanced panel data and random individual eects were considered in BiHrn (1981) and Baltagi (1985); see also Verbeek and Nijman (1996). Substantial eciencymay be lost bydropping observations from an unbalanced data set to make it balanced; see Matyas and Lovrics (1991) and Baltagi and Chang (1994). The purpose of this paper is to integrate, for random eects situations, the regression system ML approach to balanced panel data with the single equation approach to unbalanced panel data, when the attrition or accretion is random. As a preliminary to the ML problem, the generalized least-squares (GLS) problem is considered. Time specic random eects, which often are of secondaryinterest for micro data from households, rms, etc., are ignored.. Model and notation Consider a system of G regression equations, indexed by g = 1;:::;G, with observations from an unbalanced panel with N individuals, indexed by i = 1;:::;N. The individuals are observed in at least one and at most P periods. Let N p denote the number of individuals observed in p periods (not necessarilythe same and not necessarilyconsecutive), p = 1;:::;P, and let n be the total number of observations, i.e., N = P N p and n= P N pp. Assume that the individuals are ordered in P groups such that the N 1 individuals observed once come rst, the N individuals observed twice come second, etc. Let M p = p k=1 N k be the cumulated number of individuals observed up to p times, so that the index sets of the individuals observed p times can be written as I p =(M p 1 +1;:::;M p )(p =1;:::;P; M 0 = 0). We may, formally, consider I 1 as a cross section and I p (p =;:::;P) as a balanced panel with p observations of each individual. Let, for equation g, x git, of dimension (1 H g ), be the regressor vector, u git the genuine disturbance specic to individual i, observation t, g the (H g 1) coecient vector (including the intercept term), and gi a latent eect specic to individual i. Hence, t is a sequence index, not a time index. Stacking the G equations for 1 For further extensions to interdependent simultaneous equation systems, see Krishnakumar (1996). The joint occurrence of unbalanced panel data and random two-wayeects raises special problems and will not be considered here. Quadratic unbiased and ML estimation of a single equation combining unbalanced panel data and random two-wayeects is considered in Wansbeek and Kapteyn (1989). Fixed period specic eects can be included without notable diculties.

3 observation (i; t), we have 3 E. Birn / Journal of Econometrics 1 (004) y it = X it + i + u it = X it + it ; (1) where y it =(y 1it ;:::;y Git ), X it =diag(x 1it ;:::;x Git ), u it =(u 1it ;:::;u Git ), =( 1 ;:::; G ), i =( 1i ;:::; Gi ), and it = i +u it. The dimensions of X it and are (G H) and (H 1), respectively, where H = G g=1 H g. We assume that i and u it have zero expectations and covariance matrices and u, respectively, and are mutually uncorrelated and uncorrelated with X it. It follows that E( it js) = ij ( + ts u ), where ij = 1 for i = j, 0 for i j, and ts = 1 for t = s, 0 for t s. 4 Let now y i(p) =(y i1 ;:::;y ip), X i(p) =(X i1 ;:::;X ip), and i(p) =( i1 ;:::; ip), i I p ; ;:::;P, and write the model as y i(p) = X i(p) +(e p i )+u i(p) = X i(p) + i(p) ; () E( i(p) i(p))=i p u + E p = K p u + J p (p) = (p) ; (3) where (p) = u + p ; ;:::;P (4) and I p is the p dimensional identitymatrix, e p is the (p 1) vector of ones, E p =e p e p, J p =(1=p)E p, and K p =I p J p. The latter two matrices are symmetric and idempotent and have orthogonal columns, which facilitates inversion of (p). The (G G) matrices of overall within individual and between individual (co)variation in the s of the dierent equations can be expressed as p W = ( it i )( it i ) ; B = p( i )( i ) ; (5) i I p i I p t=1 respectively, where i =(1=p) p t=1 it; i I p and =(1=n) P within individual, and between individual (co)variation in the s of the individuals i I p p i. The observed p times are expressed as 5 W (p) = p ( it i )( it i ) ; i I p t=1 B (p) = p ( i (p) )( i (p) ) ; (6) i I p 3 This equation can be given a more general interpretation, allowing for cross-equational coecient constraints. If the coecient vectors are not disjoint across equations because at least one coecient occurs in at least two equations, we can redene as the complete coecient vector (without duplication) and the regressor matrix as X it =[x 1it ;:::;x Git ], where the kth element of x git contains the observations on the variable in the g th equation which corresponds to the kth coecient in. If the latter coecient does not occur in the gth equation, the kth element of x git is zero. 4 Even if E( it js )=0 for all i j in the data generating process, it is not unlikelythat non-random attrition, or accretion, mayresult in E( it js i Ip;j Iq) 0 for some or all p q. Such cases are ruled out bythis formulation. 5 Note that W = P W (p), but B P B (p) because the unweighted group means occur in the latter and the global means (weighted group means) in the former.

4 84 E. Birn / Journal of Econometrics 1 (004) respectively, where (p) =(1=N p ) i I p p i. The within and between matrices of the y s and x s are dened similarly. We show in Appendix A.1 that ( P E(W )=(n N) u ; E(B )=(N 1) u + n ) pp : (7) n Similarly, E(W (p) )=N p (p 1) u ; E(B (p) )=(N p 1) u + p(n p 1) : (8) Hence, ˆ u = W n N ; ˆ = B ((N 1)=(n N))W n ( P N pp )=n (9) would be unbiased estimators of u and if the it s were known, 6 while unbiased estimators based on the disturbances from the individuals observed p times would be ˆ u(p) = W (p) N p (p 1) ; ˆ (p) = 1 [ B(p) p N p 1 W ] (p) : (10) N p (p 1) 3. GLS estimation Before addressing the ML problem, we consider the GLS problem for when u and are known, i.e., the problem of minimizing Q = P i I p i(p) 1 (p) i(p) with respect to. Since 1 (p) = K p 1 u + J p ( u + p ) 1, we can rewrite Q as Q = i I p i(p)[k p 1 u ] i(p) + i I p i(p)[j p 1 (p) ] i(p): GLS estimation for the individuals observed p times: We may, as a preliminary to full GLS estimation, applygls on the observations for the individuals observed p times, denoted as group p, separatelyfor p =1;:::;P. We then minimize Q (p) = i I p i(p) [K p 1 u ] i(p) + i I p i(p) [J p 1 (p) ] i(p), and obtain ˆ GLS(p) = X i(p)[k p 1 u ]X i(p) + 1 X i(p)[j p 1 i Ip i I p (p) ]X i(p) X i(p)[k p 1 u ]y i(p) + X i(p)[j p 1 i Ip i I p (p) ]y i(p) ; (11) 6 Consistent residuals can replace the s in (9) to obtain consistent estimates of u and in practice.

5 E. Birn / Journal of Econometrics 1 (004) with covariance matrix V( ˆ GLS(p) )= X i(p)[k p 1 u ]X i(p) + X i(p)[j p 1 i Ip i I p (p) ]X i(p) 1 : (1) Since J 1 =1, K 1 =0, we have 1 (1) = 1 (1) =( u+ ) 1, Q (1) = i I 1 i(1) ( u+ ) 1 i(1). The expressions for ˆ GLS(1) and V( ˆ GLS(1) ) are simplied accordingly. GLS estimation for all observations: The GLS estimator of for known ( u ; ), obtained from 9Q=9 = 0, is 1 ˆ GLS = i(p)[k p 1 u ]X i(p) + i(p)[j p 1 i I p i I p i(p)[k p 1 u ]y i(p) + i I p i I p (p) ]X i(p) i(p)[j p 1 (p) ]y i(p) (13) with V( ˆ GLS )= i I p i(p)[k p 1 u ]X i(p) + i I p i(p)[j p 1 (p) ]X i(p) 1 (14) : It follows from (11) (14) that the overall estimator ˆ GLS can be interpreted as a matrix weighted average of the group specic estimators, with weights equal to the inverse of their respective covariance matrices: 1 ˆ GLS = V( ˆ GLS(p) ) 1 V( ˆ GLS(p) ) 1 ˆ GLS(p) : 4. ML estimation We next consider ML estimation of,, and u when assuming normalityof the individual eects and the disturbances, i.e., i IIN(0; ), u it IIN(0; u ). Then the i(p) X i(p) s are independent across i(p) and distributed as N(0 Gp;1 ; (p) ). The log-likelihood function of all y s conditional on all X s for the individuals in group p and for all individuals then become, respectively, L (p) = GN pp L = ln() N p ln (p) 1 Q (p)(; ; u ); (15) L (p) = Gn ln() 1 N p ln (p) 1 Q(; u; ); (16)

6 86 E. Birn / Journal of Econometrics 1 (004) where Q (p) = Q (p) (; u ; )= i I p [y i(p) X i(p) ] 1 (p) [y i(p) X i(p) ]; Q = Q(; u ; )= Q (p) (; u ; ) and (p) = (p) u p 1 (cf. Appendix A.). Two ML problems, corresponding to the two GLS problems, can be dened, one based on the balanced subsample from group p onlyand solved bymaximizing L (p), and one based on the complete unbalanced data set and solved bymaximizing L. The rst problem (when p 1) is similar to the ML problem for a system of regression equations for balanced panel data, considered in Magnus (198). The second problem, which is, of course, the most interesting one in practice, is more complicated. Since the individuals are observed a varying number of times, dierent gross disturbance covariance matrices, (p), all of which are functions of and u, are involved. The ML problem for group p: We split the problem into: (A) Maximization of L (p) with respect to for given ( u ; ) and (B) Maximization of L (p) with respect to ( u ; ) for given. It is convenient to arrange the disturbances from individual i; i I p, in the (G p) matrix Ẽ i(p) =[ i1 ;:::; ip ], write i(p) = vec(ẽ i(p) ), where vec is the vectorization operator, and dene W (p) = i Ip Ẽ i(p) K p Ẽ i(p); B (p) = i Ip Ẽ i(p) J p Ẽ i(p): (17) Comparing these expressions with (6), we see that W (p) = W (p), while in general B (p) B (p), but theycoincide when (p) (whose plim is zero) is omitted from the latter expression. Subproblem (A) is identical with the GLS problem for group p, since maximization of L (p) with respect to for given u and is equivalent to minimization of Q (p). This gives (11). To solve subproblem (B), we need expressions for the derivatives of L (p) with respect to u and. In Appendix A., we show that 9L (p) 9 u 9L (p) 9 = 1 [N p 1 (p) + N p(p 1) 1 u 1 B (p) (p) 1 (p) 1 u = 1 [N pp 1 (p) p 1 W (p) 1 u ]; B (p) (p) 1 (p)]: (18) From the rst-order conditions 9L (p) =9 u = 9L (p) =9 = 0 we get N p 1 (p) + N p(p 1) 1 u = 1 B (p) (p) 1 (p) + 1 u W (p) 1 u ; N p p 1 (p) = p 1 (p) B (p) 1 (p) : (19)

7 E. Birn / Journal of Econometrics 1 (004) For p =1,weget ˆ (1) = ˆ u(1) + ˆ (1) = B (1) =N 1, while for p 1, ˆ (p) = ˆ (p) ()= B (p) W (p) ; ˆ u(p) = ˆ u(p) ()= N p N p (p 1) and hence, ˆ (p) = ˆ (p) ()= 1 N p p [ B (p) W (p) p 1 (0) ] : (1) These estimators of u and are approximatelyequal to those in (10) when N p is not too small. The complete stepwise algorithm for solving jointlysubproblems (A) and (B) for group p then consists in switching between (11) and (0) (1) and iterating until convergence. Oberhofer and Kmenta (1974) and Breusch (1987) have, for the single equation case, demonstrated monotonicityproperties of such a sequential procedure which ensure that its solution converges to the ML estimator even if the likelihood function is not globallyconcave. Baltagi and Li (199) have generalized this result to the two-wayrandom eects model. The ML problem for the complete data set: We split the problem of maximizing L into: (A) Maximization of L with respect to for given ( u ; ) and (B) Maximization of L with respect to ( u ; ) for given. Subproblem (A) is identical with the full GLS problem, since maximization of L with respect to for given u and is equivalent to minimization of Q, which gives (13). To solve subproblem (B), we need expressions for the derivatives of L with respect to u and. Since (16) and (18) imply 9L 9 u = 1 9L 9 = 1 [N p 1 (p) + N p(p 1) 1 u 1 B (p) (p) 1 (p) 1 u W (p) 1 u ]; [N p p 1 (p) p 1 B (p) (p) 1 (p)]; () the rst-order conditions 9L=9 u = 9L=9 = 0 reduce to [N p 1 (p) + N p(p 1) 1 u ]= [ 1 B (p) (p) 1 (p) + 1 u W (p) 1 u ]; N p p 1 (p) = p 1 B (p) (p) 1 (p) ; (3) which must be solved jointlyfor u and. Unlike the situation with group-specic estimation, no closed-form solution to subproblem (B) exists. Inserting for W (p) and B (p) from (0), we can express (3) in terms of the group-specic estimators of (p) and u, denoted as ˆ (p) and ˆ u(p). We then get N p 1 (p) [I G ˆ (p) 1 (p) ]+ N p (p 1) 1 u [I G ˆ u(p) 1 u ]=0;

8 88 E. Birn / Journal of Econometrics 1 (004) N p p 1 (p) [I G ˆ (p) 1 (p)]=0: (4) This wayof writing the rst-order conditions illuminates the compromise nature of the overall estimators of u and relative to the group specic ones: u,, and (p) = u + p should be chosen so that I G ˆ (p) 1 (p) and I G ˆ u(p) 1 u are zero in a matrix weighted average sense taken across the dierent groups. This property maybe used to choose initial values for u and, e.g., averages, so as to speed up the solution to the full problem for the unbalanced data set after having solved the group-specic problems. The complete stepwise algorithm for solving jointlysubproblems (A) and (B) consists in switching between (13) and (3) and iterating until convergence. To prevent the solution to converge towards a local maximum some additional crossing-type conditions which mayrequire some further structure on to be imposed on u and may be needed. The specic properties of these conditions are not explored here. Solving (3) mayrequire separate iteration loops. This is a motivation to consider the single equation case, G = 1, in which u and are scalars, denoted as u and, respectively. Then (3) reduces to [ N p u + p + N ] [ p(p 1) B (p) u = (u + p) + W ] (p) (u) ; [ N p p u + p ] = [ ] p B (p) (u + p) ; where B (p) and W (p) are scalars. This gives two equations in u and = = u: [ ] [ ] u Np 1+p + N B (p) p(p 1) = (1 + p) + W (p) ; u [ ] Np p = 1+p [ ] p B (p) (1 + p) ; whose numerical solution maybe found rather easily, even for large P. These G single equation solution values can be used to provide starting values for the diagonals of u and when solving (3) for the G equation system. Some of the procedures described above have been programmed in the Gauss software code, Version 3., and applied, in ML estimation, for the case G =; P =; N 1 = 18433; N = 3887, H 1 = H 6 38 and data covering 0 calendar years, with fast convergence and short computation time, in Wangen and BiHrn (001). Quite often, the one-step feasible GLS, combining (13) and (14) with (10), gave results deviating considerablyfrom those obtained bythe stepwise ML procedure.

9 E. Birn / Journal of Econometrics 1 (004) Acknowledgements I thank an editor, four referees, Terje Skjerpen, and Knut R. Wangen for helpful comments. Appendix A. Proofs A.1. Proof of (7) Let T i denote the number of observations of individual i, i.e., T i = p for i I p. Then (5) can be written as T i W = ( it i )( it i ) ; B = T i ( i )( i ) ; t=1 where i =(1=T i ) T i t=1 it and =( N T i i )=( N T i). Obviously, where E(W )=E(W uu ); E(B )=E(B )+E(B uu ); (A.1) W uu = B uu = B = T i (u it u i )(u it u i ) = t=1 T i (u i u)( u i u) = T i ( i )( i ) = T i u it u it t=1 T i u i u i ; ( N ) T i u i u i T i u u ; ( N ) T i i i T i ; and u i =(1=T i ) T i t=1 u it, u =( N T i u N i )=( T i), and =( N T i i )=( N T i). Since E(u i u i )= u=t i, E(u u )= u = T i, and E( )=( Ti ) =( T i ), we have ( ) T E(W uu )=(n N) u ; E(B uu )=(N 1) u ; E(B )= n i : n Combining these expressions with (A.1), we obtain E(W )=(n N ) u ; E(B )=(N 1) u + ( n ) T i ; n (A.) which generalize single equation results in Searle et al. (199, p. 71). Inserting N T i = P N pp = n and N T i = P N pp, we get (7).

10 90 E. Birn / Journal of Econometrics 1 (004) A.. Proof of (18) We utilize the following matrix results: (a) tr(abcd)=tr(cdab) = vec(a ) (D B)vec(C) = vec(c ) (B D)vec(A); (b) 9 ln A 9A =(A ) 1 ; (c) 9tr(CB 1 ) = (B 1 CB 1 ) ; 9B (d) J p C + K p D = C D p 1 ; as J p and K p have ranks 1 and p 1, respectively, see Lutkepohl (1996, pp. 41 4, ), Magnus (198, Lemma.1), and Magnus and Neudecker (1988, pp ). Let the part of Q (p) which relates to individual i be Q i(p) =ˆ i(p) 1 (p) ˆ i(p) =ˆ i(p)[k p 1 u ]ˆ i(p) +ˆ i(p)[j p 1 (p) ]ˆ i(p) = vec(ẽ i(p) ) [K p 1 u ]vec(ẽ i(p) ) + vec(ẽ i(p) ) [J p 1 (p) ]vec(ẽ i(p) ): From (a) it follows that Q (p) = i I p tr(q i(p) )= i I p [tr[ẽ i(p) 1 (p)ẽi(p)j p ]+tr[ẽ i(p) 1 u Ẽ i(p) K p ]] = i I p [tr[ẽ i(p) J p Ẽ i(p) 1 (p) ]+tr[ẽ i(p) K p Ẽ i(p) 1 u ]]; which, when using (17), can be rewritten as Q (p) =tr[ B (p) 1 (p) ]+tr[ W (p) 1 u ]: From (d) we nd (p) = J p (p) + K p u = (p) u p 1, while (b) and (c) give and 9Q (p) 9 u 9Q (p) 9 9 ln (p) 9 u 9 ln (p) 9 = 1 B (p) (p) 1 (p) 1 u = p 1 (p) B (p) 1 (p) W (p) 1 u ; = 9 ln (p) +(p 1) 9 ln u = 1 (p) +(p 1) 1 u ; 9 u 9 u = 9 ln (p) 9 = p 1 (p) : Combining these results, we obtain (18).

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