The unbalanced nested error component regression model

Transcription

1 Journal of Econometrics 101 (2001) 357}381 The unbalanced nested error component regression model Badi H. Baltagi*, Seuck Heun Song, Byoung Cheol Jung Department of Economics, Texas A&M University, College Station, TX , USA Department of Statistics, Korea University, Sungbuk-Ku, Seoul , South Korea Received 1 December 1998; received in revised form 31 August 2000; accepted 2 October 2000 Abstract This paper considers a nested error component model with unbalanced data and proposes simple analysis of variance (ANOVA), maximum likelihood (MLE) and minimum norm quadratic unbiased estimators (MINQUE)-type estimators of the variance components. These are natural extensions from the biometrics, statistics and econometrics literature. The performance of these estimators is investigated by means of Monte Carlo experiments. While the MLE and MINQUE methods perform the best in estimating the variance components and the standard errors of the regression coe$cients, the simple ANOVA methods perform just as well in estimating the regression coe$cients. These estimation methods are also used to investigate the productivity of public capital in private production Published by Elsevier Science S.A. JEL: C23 Keywords: Panel data; Nested error component; Unbalanced ANOVA; MINQUE; MLE; Variance components 1. Introduction The analysis of panel data in econometrics have relied on the error component regression model which has its origin in the statistics and biometrics * Corresponding author. Tel.: # ; fax: # address: badi@econ.tamu.edu (B.H. Baltagi) /01/$ - see front matter 2001 Published by Elsevier Science S.A. PII: S ( 0 0 )

2 358 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381 literature, see Hsiao (1986), Baltagi (1995) and MaH tyah s and Sevestre (1996). A huge bulk of this econometrics literature focuses on the complete or balanced panels, yet the empirical applications face missing observations or incomplete panels. Exceptions are Baltagi (1985), Wansbeek and Kapteyn (1989) and Baltagi and Chang (1994). This paper considers the incomplete panel data regression model in which the economic data has a natural nested groupings. For example, data on "rms may be grouped by industry, data on states by region and data on individuals by profession. In this case, one can control for unobserved industry and within industry "rm e!ects using a nested error component model. See Montmarquette and Mahseredjian (1989) for an empirical application of the nested error component model to study whether schooling matters in educational achievements in Montreal's Francophone public elementary schools. More recently, see Antweiler (1999) for an application of the determinants of pollution concentration as measured by observation stations in various countries over time. This paper proposes natural extensions of the analysis of variance (ANOVA), maximum likelihood (MLE) and minimum norm quadratic unbiased estimators (MINQUE) and compares their performance by means of Monte Carlo experiments. Statisticians and biometricians are more interested in the estimates of the variance components per se, see Harville (1969, 1977), Hocking (1985), LaMotte (1973a, b), Rao (1971a, b), Searle (1971, 1987) and Swallow and Monahan (1984) to mention a few. Econometricians, on the other hand, are more interested in the regression coe$cients, see Hsiao (1986) and Baltagi (1995). Monte Carlo results on the balanced error component regression model include Nerlove (1971), Maddala and Mount (1973) and Baltagi (1981). For the unbalanced error component regression model, see Wansbeek and Kapteyn (1989) and Baltagi and Chang (1994). None of these studies deal with the nested and unbalanced error component model. The only exception is Fuller and Battese (1973). This paper generalizes several estimators in the literature to the nested unbalanced setting and reports the results of Monte Carlo experiments comparing the performance of these proposed estimators. The type of unbalancedness considered in this paper allows for unequal number of "rms in each industry as well as di!erent number of time periods across industries. Section 2 describes the model and the estimation methods to be compared. Section 3 gives the design of the Monte Carlo experiment and summarizes the results, while Section 4 gives an empirical illustration applying these estimation methods to the study of productivity of public capital in private production. Section 5 gives our conclusion. 2. The model We consider the following unbalanced panel data regression model: y "x β#u, i"1,2, M, j"1,2, N and t"1,2, ¹, (1)

3 where y could denote the output of the jth "rm in the ith industry for the tth time period. x denotes a vector of k nonstochastic inputs. The disturbance of (1) is given by u "μ #ν #ε, i"1,2, M, j"1,2, N and t"1,2, ¹, (2) where μ denotes the ith unobservable industry speci"c e!ect which is assumed to be i.i.d. (0, σ ),ν denotes the nested e!ect of the jth "rm within the ith industry which is assumed to be i.i.d. (0, σ ) and ε denotes the remainder disturbance which is also assumed to be i.i.d. (0, σ ). The μ 's, ν 's and ε 's are independent of each other and among themselves. This is a nested classi"cation in that each successive component of the error term is imbedded or &nested' within the preceding component, see Graybill (1961, p. 350). This model allows for unequal number of "rms in each industry as well as di!erent number of observed time periods across industries. Model (1) can be rewritten in matrix notation as y"xβ#u, (3) where y is a N ¹ 1, X is a N ¹ k, β is a k1 parameter vector, and u is a N ¹ 1 disturbance vector. Eq. (2) in vector form yields u"z μ#z ν#ε, (4) where μ"(μ,2, μ ), ν"(ν,2,ν,2,ν ), ε"(ε,2, ε,2, ε ), Z "diag(ι ι ), Z "diag(i ι ), ι and ι are vectors of ones of dimension N and ¹, respectively. By diag(ι ι ) we mean diag(ι ι,2,ι ι ). I is an identity matrix of dimension N, and denotes the Kronecker product. Note that the observations are stacked such that the slowest running index is the industry index i, the next slowest running index is the "rm index j and the fastest running index is time. Under these assumptions, the disturbance covariance matrix E(uu) can be written as Ω"σ Z Z #σ Z Z #σ diag(i I ) "diag[σ (J J )#σ (I J )#σ (I I )], (5) where J "ι ι and J "ι ι are matrices of ones of dimension N and ¹. It is clear from Eq. (5) that Ω is a block diagonal matrix with the ith block given by J B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} Λ "σ (J J )#σ (I J )#σ (I I ), i"1,2, M. (6) Replacing J by its idempotent counterpart ¹JM where JM "J /¹ and by N JM where JM "J /N, we get Λ "N ¹ σ(jm JM )#¹ σ (I JM )#σ (I I ). (7)

4 360 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381 Replacing I by E #JM and I by E #JM, where E "I!JM and E "I!JM and collecting terms with the same matrices, see Wansbeek and Kapteyn (1982, 1983), one gets the spectral decomposition of Λ : Λ "λ Q #λ Q #λ Q, (8) where λ "σ, λ "¹ σ#σ and λ "N ¹ σ#¹ σ#σ. Correspondingly, Q "I E, Q "E JM and Q "JM JM. The λ, p"1, 2, 3, are the distinct characteristic roots of Λ of multiplicity N (¹!1), N!1 and 1, respectively. Note that each Q, for p"1, 2, 3 is symmetric, idempotent with its rank equal to its trace. Moreover, the Q 's are pairwise orthogonal and sum to the identity matrix. The advantages of this spectral decomposition are that Λ "λ Q #λ Q #λ Q, (9) where p is an arbitrary scalar, see Baltagi (1993). Therefore, we can easily obtain Ω as Ω"diag[Λ]"diag[λQ #λ Q #λ Q ] (10) and σ Ω"diag σ λ Q # σ λ Q # σ λ Q "diag[i I ]!diag[θ (I JM )]!diag[θ (JM JM )], (11) where θ "1!σ /λ and θ "σ /λ!σ /λ. This allows us to obtain GLS on (3) as an ordinary least squares (OLS) of yh"σ Ωy on XH"σ ΩX. The typical element of yh is given by (y!θ y!θ y ) where y " y /¹ and y " y /N ¹. This is known in the econometrics literature as the Fuller and Battese (1973) transformation. Note that the OLS estimator is given by βk "(XX)Xy. (12) This is the best linear unbiased estimator when the variance components σ and σ are both equal to 0. Even when these variance components are positive, the OLS estimator is still unbiased and consistent, but its standard errors are biased, see Moulton (1986). The OLS residuals are denoted by u( "y!xβk. The within estimator in this case can be obtained by transforming the model in (3) by Q "diag(i E ) and then applying OLS. Note that Q Z "Q Z "0 because E ι "0. Therefore, Q sweeps away the μ 's and ν 's whether they are "xed or random e!ects. This yields βi "(X Q X )X Q y, (13)

5 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} where X denotes the exogenous regressors excluding the intercept and β denotes the corresponding (k!1) vector of slope coe$cients. β"(α, β ) and the estimate of the intercept can be retrieved as follows: α "(y!xm βi ), where the dots indicate summation and the bar indicates averaging. Following Amemiya (1971), the within residuals u for the unbalanced nested e!ect model are given by u "y!α ι!x βi where m" N ¹. Next, we consider methods of estimating the variance components Analysis of variance methods These are methods of moments-type estimators that equate quadratic sums of squares to their expectations and solve the resulting equations for the unknown variance components. These ANOVA estimators are best quadratic unbiased (BQU) estimators of the variance components in the balanced error component model case, see Graybill (1961). Under normality of the disturbances they are even minimum variance unbiased. However, for the unbalanced model, BQU estimators of the variance components are a function of the variance components themselves, see Searle (1987). Unbalanced ANOVA methods are available but optimal properties beyond unbiasedness are lost. We consider four ANOVA-type methods which are natural extensions of those proposed in the balanced error component literature: (1) A modi"ed Wallace and Hussain (WH) estimator: Consider the three quadratic forms of the disturbances using the Q, Q and Q matrices obtained from the spectral decomposition of Ω in (8): q "uq u, q "uq u, q "uq u, (15) where Q "diag(q ), Q "diag(q ) and Q "diag(q ). Substituting OLS residuals u( for u in (15) we get q(, q( and q(, see Wallace and Hussain (1969) and Baltagi and Chang (1994). Taking expected values, we obtain E(q( )"E(u( Q u( )"δ σ #δ σ #δ σ, E(q( )"E(u( Q u( )"δ σ #δ σ #δ σ, E(q( )"E(u( Q u( )"δ σ #δ σ #δ σ, (16) where the δ 's are given by δ "m!n!tr(xq X(XX)), δ "tr[(xz Z X)(XX)(XQ X)(XX)], δ "tr[(xz Z X)(XX)(XQ X)(XX)], (14)

6 362 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381 δ "n!m!tr(xq X(XX)), δ "tr[(xz Z X)(XX)(XQ X)(XX)], δ "m!t!2tr[(xz Z Q X)(XX)] # tr[(xz Z X)(XX)(XQ X)(XX)], δ "M!tr(XQ X(XX)), δ "m!2tr[(xz Z X)(XX)] # tr[(xz Z X)(XX)(XQ X)(XX)], δ "t!2tr[(xz Z Q X)(XX)] # tr[(xz Z X)(XX)(XQ X)(XX)], (17) with m" N ¹, n" N and t" ¹. Equating the q( 's to their expected values E(q( ) in (16) and solving the system of equations, one gets the Wallace and Hussain (1969)-type estimators of the variance components. These are denoted by WH. (2) A modi"ed Wansbeek and Kapteyn (WK) estimator: Alternatively, one can substitute within residuals in the quadratic forms given by (15) to get q, q and q, see Amemiya (1971) and Wansbeek and Kapteyn (1989). Taking expected values of q, q and q we get E(q )"E(u Q u )"(m!n!k#1)σ, E(q )"E(u Q u ) "[n!m#tr(x Q X )X Q X ]σ#(m!t)σ, E(q )"E(u Q u ) "[M!1#tr(X Q X )X Q X!tr(X Q X )X JM X ]σ # [t!n ¹/m]σ#[m!N¹/m]σ. (18) Equating q to its expected value E(q ) in (18) and solving the system of equations, we get the following Wansbeek and Kapteyn-type estimator of the variance components which we denote by WK: σ "u Q u /(m!n!k#1), σ " u Q u![n!m#tr(x Q X )(X Q X )σ ], m!t Most of the algebra involved is simple but tedious and all proofs are available upon request from the authors.

7 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} σ "(u Q u![m!1#tr(x Q X )X Q X! tr(x Q X )X JM X ]σ![t!n ¹/m]σ )/[m!n¹/m]. (19) (3) A Modi"ed Swamy and Arora (SA) estimator: Following Swamy and Arora (1972), we transform the regression model in (3) by premultiplying it by Q, Q and Q and we obtain the transformed residuals u, u and u, respectively. Let q "u Q u, q "u Q u and q "u Q u. Since q is exactly the same as q the resulting expected value of q is the same as that given in (18). The expected values of q and q are E(q )"E(u Q u ) "(n!m!k#1)σ#[m!t!tr(x Z Z Q X ) (X Q X )]σ, E(q )"E(u Q u ) "(M!k)σ#[t!tr(XZ Z Q X)(XQ X)]σ # [m!tr(xz Z X)(XQ X)]σ. (20) Equating q to its expected value E(q ) and solving the system of equations, we get the following Swamy and Arora-type estimators of the variance components which we denote by SA: σ "u Q u /(m!n!k#1), σ " u Q u!(n!m!k#1)σ m!t!tr(x Z Z Q X )(X Q X ), σ " u Q u!(m!k)σ![t!tr(xz Z Q X)(XQ X)]σ. m!tr(xz Z X)(XQ X) (4) Henderson Method III: Fuller and Battese (1973) suggest an estimation of the variance components using the "tting constants methods. This method uses the within residual sums of squares given by q H"u u. Also, the residual sum of squares obtained by transforming the regression in (3) by (Q #Q ) (i.e., the regression of y!y on x l!x l, for l"1,2, k). This is denoted by q H"u H u H where u H is the residual vector of the (Q #Q ) transformed regression. Finally, this method uses the conventional OLS residual sum of squares denoted by q H"u( u(. If the x variables do not have constant values for measurement of group and nested subgroups, q H is exactly the same as that for (21)

8 364 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381 the WK method, the resulting expected value of q H is the same as that given in (18). Also, the expected value of q H and q H are given by E(q H)"σ[m!M!k#1] # σ[m!t!tr(x Z Z Q X )(X (Q #Q )X )], E(q H)"σ[m!k]#σ[m!tr(XZ Z X)(XX)] # σ[m!tr(xz Z X)(XX)]. (22) Equating q H, for i"1, 2, 3 to its expected value E(q H) in (22), we obtain the Henderson Method III estimator of the variance components, see Fuller and Battese (1973). These are denoted by HFB: σ "u Q u /(m!n!k#1), u H σ " u H!(m!M!k#1)σ m!t!tr(x Z Z Q X )(X (Q #Q )X ), σ " u( u(!(m!k)σ![m!tr(xdiag(i J )X)(XX)σ ]. m!tr(xz Z X)(XX) (23) 2.2. Maximum likelihood estimator Since λ, for p"1, 2, 3 are the distinct characteristic roots of Λ then Λ "(λ )(λ)(λ). Let ρ "σ/σ, ρ "σ/σ and Ω"σΣ, then the log-likelihood function can be written as log "C! m 2 log σ!1 2! 1 2 log(n ¹ ρ #¹ ρ #1) (N!1) log(¹ ρ #1)! 1 2 uσu/2σ. (24) The "rst-order conditions give closed form solutions for β and σ conditional on ρ and ρ : βk "(XΣK X)XΣK y, (25) σ( "(y!xβ)σk (y!xβ)/m. (26) However, the "rst-order conditions based on ρ( and ρ( are nonlinear in ρ and ρ even for known values of β and σ. Following Hemmerle and Hartley (1973), we get log "! 1 ρ 2 tr[z ΣZ ]# 1 (y!xβ)σz Z Σ(y!Xβ), 2σ

9 log "! 1 ρ 2 tr[z ΣZ ]# 1 (y!xβ)σz Z Σ(y!Xβ). (27) 2σ Therefore, a numerical solution by means of iteration is needed. The Fisher scoring procedure is used to estimate ρ and ρ. The partition of the information matrix corresponding to ρ and ρ is given by E! log ρ "1 2 E! log ρ ρ "1 2 (N ¹ ) (1#ρ ¹ #ρ N ¹ ), N ¹ (1#ρ ¹ #ρ N ¹ ), log E! ρ "1 (N!1)¹ 2 (1#ρ ¹ ) #1 ¹ 2 (1#ρ ¹ #ρ N ¹ ), (28) see Harville (1977). Starting with an initial value, the (r#1)th updated value of ρ and ρ is given by log ρ E log! ρ ρ, ρ( ρ( B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} " #E ρ(! log E! ρ ρ E log! ρ log ρ log ρ (29) where at each step, log /ρ and log /ρ are obtained from Eq. (27), βk and σ( are obtained from (25) and (26), the information matrix is obtained from Eq. (28). The subscript r means this is evaluated at the rth iteration. For a review of the advantages and disadvantages of MLE, see Harville (1977) Restricted maximum likelihood estimator Patterson and Thompson (1971) suggested a restricted maximum likelihood (REML) estimation method that takes into account the loss of degrees of freedom due to the regression coe$cients in estimating the variance components. REML is based on a transformation that partitions the likelihood function into two parts, one being free of the "xed regression coe$cients. Maximizing this part yields REML. Patterson and Thompson (1971) suggest the singular transformation y[c ΣX/σ ], where C"I!X(XX)X. Cy is distributed as N(0, CΣC/σ ), and from the fact that CX"0, it is independent of XΣy/σ which is also distributed as N(XΣXβ/σ, XΣX/σ ). It is clear that Cy does not depend on β. Since C is an idempotent matrix of rank m!k, there exists an (m!k)m matrix A such that AA"C, AA"I. (30)

10 366 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381 Using the Ay transformation instead of Cy, we get Ay XΣy/σ &N 0 XΣXβ/σ, σ AΣA 0 0 XΣX/σ. (31) Following Corbeil and Searle (1976), the log-likelihood function of Ay and XΣy/σ are given by log and log respectively:, m!k log log(2π)! "! m!k log(σ)! logaσa! 1 2σ y[a(aσa)a]y, log "! k 2 log(2π)!1 2 log(σ )!logxσx! 1 (y!xβ)σx(xσx)xσ(y!xβ). (32) 2σ Using the results of Hocking (1985) and Corbeil and Searle (1976), we obtain A(AΣA)A"Σ[I!X(XΣX)XΣ]"Σ(I!M), (33) where M"X(XΣX)XΣ. Using log which is free from β, the "rst-order derivatives of log with respect to σ, ρ and ρ are given by log "! m!k # 1 ya(aσa)ay σ 2σ 2σ "! m!k # 1 yσ(i!m)y, 2σ 2σ log "! 1 ρ 2 tr[z A(AΣA)AZ ] # 1 y[a(aσa)az Z A(AΣA)A]y, 2σ log "! 1 ρ 2 tr[z A(AΣA)AZ ] # 1 y[a(aσa)az Z A(AΣA)A]y. (34) 2σ Equating the equations in (34) to 0's yield the REML estimates. For example, solving log /σ "0 conditional on ρ and ρ, we obtain σ( "ya(aσk A)Ay/(m!k)"y[ΣK (I!M)]y/(m!k). (35)

11 But there are no closed-form solutions on ρ and ρ. Thus a numerical solution by means of iteration is needed. The Fisher scoring procedure is used to estimate ρ and ρ. Using the results of Harville (1977) and Eq. (33), the information matrix with respect to ρ and ρ is given by log E! ρ "1 2 tr[z Z Σ(I!M)Z Z Σ(I!M)], E! log ρ ρ "1 2 tr[z Z Σ(I!M)Z Z Σ(I!M)], log E! ρ "1 2 tr[z Z Σ(I!M)Z Z Σ(I!M)]. (36) The updated values of ρ and ρ can be obtained as in (29) MINQUE and MIVQUE Rao (1971a) proposed a general procedure for variance components estimation which requires no distributional assumptions other than the existence of the "rst four moments. This procedure yields MINQUE of the variance components. Under normality of the disturbances, MINQUE and minimum variance quadratic unbiased estimators (MIVQUE) are identical. Since we assume normality, we will focus on MIVQUE. Let and B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} R"Σ[I!X(XΣX)XΣ]/σ, (37) S"s "tr(< R< R), i, j"1, 2, 3 (38) u"u "yr< Ry, i"1, 2, 3, (39) where < "I, < "Z Z and < "Z Z. Rao (1971b) shows that the vector of MIVQUEs is given by θk "Su, (40) where θk "(σ(, σ(, σ( ). However, MIVQUE requires a priori values of the variance components. Therefore, MIVQUE is only &locally minimum variance', see LaMotte (1973a, b), and &locally best', see Harville (1969). Three priors of the MIVQUE estimator are considered in our Monte Carlo study: (1) the identity matrix, which we denote by MV1, and (2) all values of the variance components equal to 1, see Swallow and Searle (1978) which we denote by MV2, and (3) the ANOVA estimator of WK, which we denote by MV3. Note that the MIVQUE estimator can produce negative estimates of the variance components. In this case, we replace the negative variance estimate by 0.

12 368 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} Monte Carlo results 3.1. Design of the Monte Carlo study We consider the following simple regression equation: y "α#x β#u, i"1,2, M, j"1,2, N, t"1,2, ¹, (41) with u "μ #ν #ε. The exogenous variable x was generated by a similar method to that of Nerlove (1971). In fact, x "0.3t#0.8x #w, where w is uniformly distributed on the interval [!0.5, 0.5]. The initial values x were chosen as (100#250w ). Throughout the experiment α"5 and β"2. For generating the u disturbances, we let μ &IIN(0, σ), ν &IIN(0, σ) and ε &IIN(0, σ). We "x σ"σ#σ#σ"20 and de"ne γ "σ/σ and γ "σ/σ. These are varied over the set (0, 0.2, 0.4, 0.6, 0.8) such that (1!γ!γ ) is always positive. Extending a measure of unbalancedness given by Ahrens and Pincus (1981) to the unbalanced nested model, we de"ne c "M/NM (1/N ) where NM " N /M, c "M/¹M (1/¹ ) where ¹M "¹ /M, c "M/N¹ (1/N ¹ ) where N¹" N ¹ /M, (42) where c, c and c denote the measures of subgroup unbalancedness, observed time unbalancedness and group unbalancedness due to each group size. Note that c, c and c take the value 1 when the data are balanced but take smaller values than 1 as the data pattern gets more unbalanced. Table 1 gives the (N, ¹ ) pattern used along with the corresponding unbalancedness measures for M"10. The "rst parentheses gives the N pattern, while the second parentheses below it gives the corresponding ¹ pattern. For example, P observes the "rst grouping of eight individuals over six time periods and the last grouping of 12 individuals over four time periods. The sample size is "xed at 500 for every pattern. Two other values of M are used, M"6 and 15. For each experiment, 1000 replications are performed. For each replication, we calculate OLS, WTN, WH, SA, WK, HFB, ML, REML, MV1, MV2, MV3 and true GLS. The last estimator is obtained for comparison purposes A comparison of regression coezcient estimates Table 2 gives the mean square error (MSE) of the estimate of βk relative to that of true GLS for the case when M"10. From this table it is clear that OLS is Similar MSE tables for the regression coe$cients and the variance components estimates are generated for M"6 and 15, but they are not produced here to save space. These tables are available upon request from the authors.

13 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} Table 1 (N, ¹ ) patterns considered and their corresponding unbalancedness measures when M"10 Pattern (N, N,2, N ) c c c (¹, ¹,2, ¹ ) (8,8,8,10,10,10,10,12,12,12) (6,6,6,5,5,5,5,5,4,4) (6,6,6,10,10,10,10,12,12,12) (9,9,9,9,8,3,3,3,3,3) (5,5,5,10,10,10,10,11,11,11) (2,2,3,3,3,6,7,8,8,9) (4,4,4,5,5,9,9,10,10,10) (14,15,15,15,15,3,3,4,4,4) (3,3,3,3,3,8,8,8,8,8) (2,2,2,3,3,11,11,12,12,12) (2,2,6,6,6,10,10,10,13,13) (16,16,16,16,16,2,2,3,3,3) (2,2,2,10,10,10,10,13,13,13) (2,1,1,1,1,8,8,8,8,8) (20,20,15,15,15,3,3,3,2,2) (1,1,6,6,6,10,10,10,25,25) (16,16,16,16,16,2,2,2,2,2) (2,2,3,3,3,28,28,30,30,30) (20,20,20,20,20,2,2,2,2,1) (2,2,2,3,3,25,30,30,30,30) (1,1,1,1,5,5,25,25,25,25) (1,2,2,35,35,2,2,3,3,3) (1,1,1,1,5,5,30,30,30,30) (27,27,28,28,28,2,2,2,2,2) The "rst parentheses gives the N pattern, while the parentheses below it gives the corresponding ¹ pattern. inferior to true GLS, ML-type (ML, REML) estimators and all feasible GLStype estimators except when γ " γ "0. For all experiments, the e!ect of an increase in γ on the MSE of OLS is much larger than that of an increase in γ. This is because γ a!ects the primary group while γ a!ects only the nested subgroup. The WTN estimator performs poorly for small γ and γ values. The performance of WTN is in some cases worse than OLS if either γ or γ is 0. However, its performance improves as γ and γ increase and the unbalancedness pattern gets more severe. The ANOVA-type (WH, WK, SA and HFB)

14 370 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381 Table 2 MSE of βk relative to that of true GLS when M"10 γ γ OLS WTN WH WK SA HFB MLE REML MV1 MV2 MV

15 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} Table 2 (Continued) γ γ OLS WTN WH WK SA HFB MLE REML MV1 MV2 MV estimators, the ML-type and MV2 and MV3 estimators all perform well relative to true GLS. See also Fig. 1 which plots the MSE of βk relative to that of true GLS for the 12 unbalanced patterns for M"10 and γ " γ "0.4. It is important to point out that the simple ANOVA-type estimators compare well in

16 372 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381 Fig. 1. MSE of βk relative to that of true GLS when M"10, γ "γ "0.4. MSE performance to the ML- and MIVQUE-type estimators which are relatively more di$cult to compute. They have at most 9.1% higher MSE than true GLS, see the WK estimator for pattern when γ "γ "0. MV1 performs well for γ (0.4 and γ (0.4. This is to be expected since the prior values for MV1 are γ "0 and γ "0. As γ and γ increase or the degree of unbalancedness gets large, the performance of MV1 deteriorates relative to the other estimators, see Fig. 1. To summarize, for the regression parameters, the computationally simple ANOVA estimators compare well with the more complicated estimators such as ML, REML, MV2 and MV3 in terms of MSE criteria. Also, MV1 is not recommended when the primary group e!ects or nested subgroup e!ects are suspected to be large or the unbalanced pattern is severe A comparison of variance components estimates Tables 3}5 report the MSE of the variance components relative to that of MLE for the 12 unbalancedness patterns for M"10 and γ "γ "0.4. Similar tables for other values of M"6 and 15 and various values of γ and γ are available upon request from the authors. These tables are plotted in Figs. 2}4 for ease of comparison. For the estimation of σ, see Table 3 or Fig. 2, MLE ranks "rst; REML, MV2 and MV3 rank second and the ANOVA methods rank third by the MSE criteria. MV1 does well only when γ and γ are close to 0. For γ "γ "0.4, MV1 performs the worst. The ANOVA methods yield almost twice the MSE of

17 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} Table 3 MSE of σ( relative to that of MLE when M"10, γ "γ "0.4 WH WK SA HFB REML MV1 MV2 MV Table 4 MSE of σ( relative to that of MLE when M"10, γ "γ "0.4 WH WK SA HFB REML MV1 MV2 MV MLE for patterns, and. On the other hand, REML, MV2 and MV3 have MSE that are only 14}20% higher than that of MLE. MV1 has 2}3 times the MSE of MLE for these patterns and is not included in Fig. 2. For the estimation of σ, see Table 4 or Fig. 3, REML, MLE, MV2 and MV3 rank "rst by the MSE criteria. They are followed by the ANOVA methods with MV1 performing the worst. For patterns,, and, these ANOVA methods yield more than 2.5 times the MSE of MLE. In contrast, MV1 yields 13}23 times the MSE of MLE and is not included in Fig. 3. For the estimation of σ, there is not much di!erence among WK, SA, HFB, REML, MV2 and MV3. However, WH performs slightly worse than the other

18 374 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381 Table 5 MSE of σ( relative to that of MLE when M"10, γ "γ "0.4 WH WK SA HFB REML MV1 MV2 MV Fig. 2. MSE of σ( relative to that of MLE when M"10, γ "γ "0.4. ANOVA methods yielding 3.8}20% higher MSE than that of MLE. MV1 performs the worst yielding MSE that is 5}154 times that of MLE and is therefore not included in Fig. 4. This con"rms that if one is interested in the estimates of the variance components per se one is better o! with MLE, REML or MV2 and MV3-type estimators. The ANOVA methods suggested here are second best. MV1 is not recommended unless one suspects γ and γ are close to 0. However, for the estimation of the regression coe$cients, the ANOVA methods compare well and are recommended.

19 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} Fig. 3. MSE of σ( relative to that of MLE when M"10, γ "γ "0.4. Fig. 4. MSE of σ( relative to that of MLE when M"10, γ "γ "0.4. For ANOVA and MIVQUE-type estimators, negative estimates of σ or σ occur in about 50% of the replications when γ "0 or γ "0. When the negative estimates of variance components are replaced by 0, the corresponding estimator forfeits its unbiasedness property. But, replacing these negative estimates by 0 did not lead to much loss in e$ciency using the MSE criterion.

20 376 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381 Fig. 5. MSE of standard errors of βk relative to that of MLE when M"10, γ "γ "0.4. Finally, better estimates of the variance components by the MSE criterion, do not necessarily imply better estimates of the regression coe$cients. A similar result was obtained by Baltagi and Chang (1994) for the unbalanced one way model and by Taylor (1980) and Baltagi (1981) for the balanced error component model. However, MV1 has worse relative MSE performance than other ANOVA, ML, REML and MIVQUE-type estimators of the variance components when γ and γ are large and the pattern is severely unbalanced and this clearly translates into a corresponding worse relative MSE performance of the regression coe$cients. Similar conclusions can be drawn for M"6 and 15 and are not produced here to save space A comparison of standard errors of the regression coezcients Fig. 5 plots the MSE of the standard error of βk relative to that of MLE for the 12 unbalanced patterns for M"10 and γ "γ "0.4. Besides the relative e$ciency of the parameter estimates, one is also interested in proper inference on the parameter values. This is where the computationally involved estimators (like MV2, MV3 and REML) perform well producing a MSE for the standard error of βk that is close to that of MLE. The computationally simple ANOVA methods (WH, WK, SA, HFB) have MSE for the standard error of βk that are 2 times that of MLE for severely unbalanced patterns like, and. However, these ANOVA methods perform reasonably well in patterns } giving MSEs of the standard error of βk that are no more than 30% higher than that of MLE.

21 4. Empirical example B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} Baltagi and Pinnoi (1995) estimated a Cobb}Douglas production function investigating the productivity of public capital in each state's private output. This is based on a panel of 48 states over the period 1970}1986. The data were provided by Munnell (1990). These states can be grouped into nine regions with the Middle Atlantic region for example containing three states: New York, New Jersey and Pennsylvania and the Mountain region containing eight states: Montana, Idaho, Wyoming, Colorado, New Mexico, Arizona, Utah and Nevada. The primary group would be the regions, the nested group would be the states and these are observed over 17 years. The dependent variable y is the gross state product and the regressors include the private capital stock (K) computed by apportioning the Bureau of Economic Analysis (BEA) national estimates. The public capital stock is measured by its components: highways and streets (KH), water and sewer facilities (KW), and other public buildings and structures (KO), all based on the BEA national series. Labor ( ) is measured by the employment in nonagricultural payrolls. The state unemployment rate is included to capture the business cycle in a given state. See Munnell (1990) for details on the data series and their construction. All variables except the unemployment rate are expressed in natural logarithm y "α#β K #β KH #β KW #β KO # β #β Unemp #u, (43) where i"1, 2,2, 9 regions, j"1,2, N with N equaling 3 for the Middle Atlantic region and 8 for the Mountain region and t"1, 2,2, 17. The data is unbalanced only in the di!ering number of states in each region. The disturbances follow the nested error component speci"cation given by (2). Table 6 gives the OLS, WTN, ANOVA, MLE, REML and MIVQUE-type estimates using this unbalanced nested error component model. The OLS estimates show that the highways and streets and water and sewer components of public capital have a positive and signi"cant e!ect upon private output whereas that of other public buildings and structures is not signi"cant. Because OLS ignores the state and region e!ects, the corresponding standard errors and t-statistics are biased, see Moulton (1986). The within estimator shows that the e!ect of KH and KW are insigni"cant whereas that of KO is negative and signi"cant. The primary region and nested state e!ects are signi"cant using several LM tests developed in Baltagi et al. (1999). This justi"es the application of the feasible GLS, MLE and MIVQUE methods. For the variance components estimates, there are no di!erences in the estimate of σ. But estimates of σ and σ vary. σ( is as low as for SA and MLE and as high as for HFB. Similarly, σ( is as low as for SA and as high as for WK. This variation had little e!ect on estimates of the regression coe$cients or their

22 378 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381 Table 6 Cobb}Douglas production function estimates with unbalanced nested error components 1970}1986, Nine regions, 48 states Variable OLS WTN WH WK SA HFB MLE REML MV1 MV2 MV3 Intercept * (0.053) (0.152) (0.160) (0.144) (0.150) (0.154) (0.157) (0.152) (0.154) (0.156) K (0.011) (0.026) (0.021) (0.022) (0.020) (0.021) (0.021) (0.022) (0.021) (0.021) (0.021) (0.016) (0.030) (0.026) (0.027) (0.025) (0.026) (0.026) (0.026) (0.026) (0.026) (0.026) KH (0.015) (0.031) (0.023) (0.024) (0.022) (0.022) (0.023) (0.023) (0.023) (0.023) (0.023) KW (0.012) (0.015) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014) KO 0.009!0.115!0.095!0.102!0.094!0.096!0.100!0.101!0.095!0.098!0.100 (0.012) (0.018) (0.017) (0.017) (0.017) (0.017) (0.017) (0.017) (0.017) (0.017) (0.017) Unemp!0.007!0.005!0.006!0.006!0.006!0.006!0.006!0.006!0.006!0.006!0.006 (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) σ σ * * σ * * The dependent variable is log of gross state product. Standard errors are given in parentheses.

23 B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357} standard errors. For all estimators of the random e!ects model, the highways and streets and water and sewer components of public capital had a positive and signi"cant e!ect, while the other public buildings and structures had a negative and signi"cant e!ect upon private output. 5. Conclusion For the regression coe$cients of the nested unbalanced error component model, the simple ANOVA methods proposed in this paper performed well in Monte Carlo experiments as well as in the empirical example and are recommended. However, for the variance components estimates themselves, as well as the standard errors of the regression coe$cients, the computationally more demanding MLE, REML or MIVQUE (MV2 and MV3) estimators are recommended especially if the unbalanced pattern is severe. Further research should extend the unbalanced nested error component model considered in this paper to allow for endogeneity of the regressors, a dynamic speci"cation, ignorability of the sample selection and serial correlation in the disturbances. Acknowledgements The authors would like to thank two anonymous referees and an Associate Editor for their helpful comments and suggestions. A preliminary version of this paper was presented at the European meetings of the Econometric Society held in Santiago de Compostela, Spain, August, Also, at the University of Chicago, University of Pennsylvania and the University of Rochester. Baltagi would like to thank the Texas Advanced Research Program and the Bush Program in economics of public policy for their "nancial support. References Ahrens, H., Pincus, R., On two measures of unbalancedness in a one-way model and their relation to e$ciency. Biometric Journal 23, 227}235. Amemiya, T., The estimation of variances in a variance components model. International Economic Review 12, 1}13. Antweiler, W., Nested random e!ects estimation in unbalanced panel data. Working Paper, University of British Columbia. Baltagi, B.H., Pooling: an experimental study of alternative testing and estimation procedures in a two-way error components model. Journal of Econometrics 17, 21}49. Baltagi, B.H., Pooling cross-sections with unequal time series lengths. Economics Letters 18, 133}136. Baltagi, B.H., Useful matrix transformations for panel data analysis: a survey. Statistical Papers 34, 281}301.

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