Constructing panel data estimators by aggregation: A general moment estimator and a suggested synthesis

Discussion Papers No. 420, May 2005 Statistics Norway, Research Department

Erik Biørn

Constructing Panel Data Estimators by Aggregation:

A General Moment Estimator and a Suggested Synthesis

Abstract:

A regression equation for panel data with two-way random or fixed effects and a set of individual specific and period specific `within individual' and `within period', estimators of its slope coefficients are considered. They can be given Ordinary Least Squares (OLS) or Instrumental Variables (IV) interpretations. A class of estimators, obtained as an arbitrary linear combination of these

`disaggregate' estimators, is defined and an expression for its variance-covariance matrix is derived.

Nine familiar `aggregate' estimators which utilize the entire data set, including two between, three within, three GLS, as well as the standard OLS, emerge by specific choices of the weights. Other estimators in this class which are more robust to simultaneity and measurement error bias than the standard aggregate estimators and more efficient than the `disaggregate' estimators, are also considered. An empirical illustration of robustness and efficiency, relating to manufacturing productivity, is given.

Keywords: Panel data. Aggregation. Simultaneity. Measurement error. Method of moments. Factor productivity

JEL classification: C13, C23, C43.

Address: Erik Biørn, University of Oslo and Statistics Norway,Address for correspondence:

Department of Economics, P.O. Box 1095 Blindern, 0317 Oslo, Norway. E-mail:

erik.biorn@econ.uio.no

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1 Introduction

A primary reason for the substantial growth in the availability and use of panel data in econometrics during the last thirty years is the opportunity that such data give for identifying and controlling for unobserved heterogeneity which may affect the estima- tion of slope coefficients and other parameters of interest from cross-section data, time- series data, or repeated (non-overlapping) cross-sections. It is well known [seee.g., Bal- tagi (2001, chapters 2 and 3), Balestra (1996), and M´aty´as (1996)] (i) that the potential nuisance created by fixed (additive) individual heterogeneity in OLS estimation can be eliminated by measuring all variables from their individual means or taking individual differences over time, (ii) that the potential nuisance created by fixed (additive) time specific heterogeneity in OLS estimation can be eliminated by measuring all variables from their time specific means or taking time specific differences over individuals, and (iii) that efficient estimation in the presence of suitably structuredrandom individual or time specific heterogeneity, can be performed by (Feasible) Generalized Least Squares.

It is, however, possible to construct such aggregate estimators from disaggregate building-blocks. Approaching estimation in this way, is far from an algebraic exercise. It is illuminating primarily because we can utilize the fact that regression coefficients can be estimated consistently from parts of a panel data set in a large number of ways and that some disaggregate estimators are more robust to bias than others. We can, for instance apply all observations from one or two individuals or from one or two periods only. By combining an increasing number of individual specific or period specific estimators, we can include an increasing part of the observations until, at the limit, we utilize the full panel data set. Such an investigation is interesting on the one hand because several familiar estimators (within, between, generalized least squares etc.) for coefficients in panel data models can be interpreted as known linear combinations of elementary estimators, on the other hand because we get suggestions of other estimators along the way.

The paper is structured as follows. After describing the model and some ways of transforming it (Section 2), we first, in Section 3, define ‘disaggregate’ ‘within individ- ual’ and ‘within period’ estimators, each of which can be given either an OLS or an Instrumental Variables (IV) interpretation. In Section 4, a more general moment esti- mator, obtained by an arbitrary weighting of these elementary estimators as well as its variance-covariance matrix, is constructed. We next reconsider nine familiar estimators of a slope coefficient vector in a linear regression equation for panel data with two-way random or fixed effects, three of which are ‘within (group)’, two are ‘between (group)’

estimators related to individual or time variation, one is the standard OLS (Ordinary Least Squares) and three are Generalized Least Squares (GLS) estimators. We show that these‘aggregate’ estimators can all belong to this class and demonstrate that our

general estimator contains not only the nine estimators mentioned above, but also several others which are more robust to violation of the standard assumptions in random coeffi- cient models. In this process, some textbook results [Maddala (1977, section 14–2) and Hsiao (2003, section 2.2)] and some results in Biørn (1994, 1996) are generalized. Both a standard regression framework and situations with simultaneity (correlation between individual effects, period effects, and/or disturbances on the one hand and the regressor vector on the other) and situations with random measurement errors in the regressor vector are considered. Among the latter estimators we select estimators which are more robust to simultaneity and measurement errors and more efficient than the ‘disaggregate’

estimators. Finally, an empirical illustration of robustness and efficiency loss, relating to manufacturing productivity, is given.

2 Model, notation, and transformations

Consider a linear regression equation relatingy to a vector of K (stochastic) regressors x, with data set from a panel of N ( ≥2) individuals observed in T (≥2) periods:

y_it=k+x_itβ+²_it, ²_it =α_i+γ_t+u_it, i= 1, . . . , N; t= 1, . . . , T, (2.1)

wherey_it and x_it = (x_1it, . . . , x_Kit) are the values of y and x for individual i in period t, β = (β₁, . . . , β_K)⁰ is the coefficient vector, α_i and γ_t are random effects specific to individualiand periodt, respectively,u_it is a genuine disturbance, andkis an intercept term. It is, however, possible to interpretα_i andγ_tas fixed effects, see Section 5. At the moment, we make the standard assumptions for two-way random effects models,

u_it ∼IID(0, σ²), α_i ∼IID(0, σ²_α), γ_t∼IID(0, σ_γ²), i= 1, . . . , N; t= 1, . . . , T, (2.2)

u_it, α_i, γ_t, x_it are independently distributed for all iand t, (2.3)

which imply

E(²_it|X) = 0, E(²_it²_js|X) =δ_ijσ_α² +δ_tsσ_γ²+δ_ijδ_tsσ², i, j= 1, . . . , N, t, s= 1, . . . , T, (2.4)

whereδ_ij = 1 for i=j and = 0 fori6=j, and δ_ts = 1 fort=sand = 0 fort6=s, andX is the (N T ×K) matrix containing all the x_it’s. Some of the assumptions in (2.2) and (2.3) will be relaxed later on.

The individual specific vectors and matrices, of dimension (T ×1) and (T ×K), respectively, and the period specific vectors and matrices, of dimension (N ×1) and (N×K), respectively, are

y_i·=





 y_i1

... y_iT





, X_i·=





 x_i1

... x_iT





, y·t=





 y_1t

... y_{N t}





, X·t=





 x_1t

... x_{N t}





,

which are stacked into y=





 y₁·

... y_N·





, X =





 X₁·

... X_N·





, y_∗ =





 y·1

... y·T





, X_∗=





 X·1

... X·T





.

Further, let e_H be the (H×1) vector of ones, I_H the H-dimensional identity matrix, A_H = e_He_H⁰ /H, B_H = I_H −A_H and let α = (α₁, . . . , α_N)⁰ and γ = (γ₁, . . . , γ_T)⁰. Alternative ways of writing (2.1) are then

y_i·=e_Tk+X_i·β+²_i·, ²_i·=e_Tα_i+γ+u_i·, i= 1, . . . , N, (2.5)

y·t=e_Nk+X·tβ+²·t, ²·t=α+e_Nγ_t+u·t, t= 1, . . . , T, (2.6)

where²_i·,u_i·,²·t,u·tare defined in similar way asy_i·andy·t, and after deducting global means we obtain

y_i·−y¯= (X_i·−X)β¯ +²_i·−¯², ²_i·−¯²=e_T(α_i−α) +¯ B_Tγ+u_i·−u,¯ (2.7)

y·t−y¯_∗= (X·t−X¯_∗)β+²·t−¯²_∗, ²·t−¯²_∗ =B_Nα+e_N(γ_t−¯γ) +u·t−u¯_∗, (2.8)

where ¯α = (1/N)^P_iα_i, ¯γ = (1/T)^P_tγ_t, X¯ = (1/N)^P_iX_i·, X¯_∗ = (1/T)^P_tX·t,

¯

y = (1/N)^P_iy_i·, y¯_∗ = (1/T)^P_ty·t, , etc. Premultiplying (2.5) by B_T, (2.7) by A_T, (2.6) byB_N and (2.8) byA_N, give, respectively,

B_Ty_i· = B_TX_i·β+B_T²_i·,

A_T(y_i· −y) =¯ A_T(X_i· −X)β¯ +A_T(²_i· −¯²), (2.9)

B_Ny·t = B_NX·tβ+B_N²·t,

A_N(y·t−y¯_∗) = A_N(X·t−X¯_∗)β+A_N(²·t−¯²_∗).

(2.10)

We letW,V,B, andC, with appropriate subscripts, symbolize matrices containing within individual, within period, between individual, and between period (co)variation, respectively. Define individual specific and period specific cross-product matrices as follows:

W_XXij =X_i⁰·B_T X_j·=^PT_t=1(x_it−x¯_i·)⁰(x_jt−x¯_j·),

W_Xγi =X_i⁰·B_T γ=^PT_t=1(x_it−x¯_i·)⁰(γ_t−¯γ), i, j= 1, . . . , N, (2.11)

V_XXts =X·⁰tB_NX·s=^PN_i=1(x_it−x¯·t)⁰(x_is−x¯·s),

V_Xαt =X·⁰tB_Nα=^PN_i=1(x_it−x¯·t)⁰(α_i−α),¯ t, s= 1, . . . , T, (2.12)

B_XXii = (X_i· −X)¯ ⁰A_T(X_i· −X) =¯ T(¯x_i· −x)¯ ⁰(x¯_i· −x),¯

B_Xαii= (X_i· −X)¯ ⁰e_T(α_i−α) =¯ T(x¯_i· −x)¯ ⁰(α_i−α),¯ i= 1, . . . , N, (2.13)

C_XXtt= (X·t−X¯_∗)⁰A_N(X·t−X¯_∗) =N(¯x·t−x)¯ ⁰(x¯·t−x),¯

C_Xγtt= (X·t−X¯_∗)⁰e_N(γ_t−¯γ) =N(x¯·t−x)¯ ⁰(γ_t−γ),¯ t= 1, . . . , T, (2.14)

etc., wherex¯_i·= (e_T⁰/T)X_i·,x¯·t= (e_N⁰ /N)X·t,x¯= (e_{N T}⁰ /(N T))X= (e_{T N}⁰ /(T N))X_∗. These matrices have the following properties:

• W_XXij, which has full rank K if x_it contains no individual specific variables, is the (K×K) matrix ofwithin individual covariation in thex’s of individuals iand j, and V_XXts, which has full rankK ifx_it contains no period specific variables, is the (K×K) matrix of within period covariation in the x’s of periods t and s. If individual specific regressors occur,W_XXij has one zero column (and row) for each such variable, and if period specific regressors occur, V_XXts has one zero column (and row) for each such variable.

• B_XXii and C_XXtt, which have rank 1, are the (K×K) matrices of between indi- vidual cross-products and between period cross-products of the x’s of individual i and period t, respectively.

• W_Xγi is the (K×1) vector of within covariation of thex’s of individual iand the period specific effects, V_Xαt is the (K×1) vector of within covariation of thex’s of period tand the individual specific effects,B_Xαi is the (K×1) vector ofbetween cross-products of thex’s of individualiand its individual specific effects, andC_Xγt is the (K×1) vector ofbetween cross-products of thex’s of periodtand its period specific effects.

Premultiplying the two equations in (2.9) by X_i⁰·B_T and (X_i· −X)¯ ⁰A_T, respec- tively, and premultiplying the two equations in (2.10) by X·⁰tB_N and (X·t−X¯_∗)⁰A_N, respectively, while using (2.11)–(2.14), we get

W_{XY ij} =W_XXijβ+W_X²ij, W_X²ij =W_Xγi+W_{XU ij}, i, j= 1, . . . , N, (2.15)

B_{XY ii} =B_XXiiβ+B_X²ii, B_X²ii =B_Xαii+B_{XU ii}, i= 1, . . . , N, (2.16)

V_{XY ts}=V_XXtsβ+V_X²ts, V_X²ts=V_Xαt+V_{XU ts}, t, s= 1, . . . , T, (2.17)

C_{XY tt}=C_XXttβ+C_X²tt, C_X²tt=C_Xγtt+C_{XU tt}, t= 1, . . . , T.

(2.18)

These can be considered ‘moment versions’ of Eq. (2.1),

3 Base estimators and their properties

The fact that W_X²ij and have zero expectations when (2.2) and (2.3) are satisfied, in combination with (2.15) and (2.17) motivate the followingN² individual specific and T² period specific estimators ofβ:

βb_{W ij} =W⁻¹_XXijW_{XY ij} = (X_i⁰·B_TX_j·)⁻¹(X_i⁰·B_Ty_j·), i, j= 1, . . . , N, (3.1)

βb_{V ts}=V⁻¹_XXtsV_{XY ts}= (X·⁰tB_NX·s)⁻¹(X·⁰tB_Ny·s), t, s= 1, . . . , T.

(3.2)

We denote them asbase estimators, ordisaggregate estimators, ofβ. They can be given the following interpretations:

(i) β^b_{W ii} is the OLS estimator based on observations from individual i, and β^b_{W ij}, for j 6= i, is the IV estimator based on the ‘within variation’ of individual j, B_TX_j·, using the ‘within variation’ of individual iin X_i·, B_TX_i·, as IV matrix.

(ii) β^b_{V tt} is the OLS estimator based on observations from periodt, andβ^b_{V ts}, fors6=t, is the IV estimator based on the ‘within variation’ of period s, B_NX·s, using the

‘within variation’ of period tin X·t, B_NX·t, as IV matrix.

All theseN²+T²estimators exist if all elements ofx_itvary across individuals and periods, since this usually ensures thatW_XXij and V_XXts have rankK.

If individual specific variables occur, so that W_XXij contains one or more zero rows and columns, their coefficients cannot be estimated from (3.1), but estimators for the coefficients of the other, i.e., the two-dimensional or period specific variables, can be solved fromW_XXijβ^b_{W ij} =W_{XY ij}. Likewise, if period specific variables occur, so that V_XXtscontains one or more zero rows and columns, their coefficients cannot be estimated from (3.2), but estimators for the coefficients of the other, i.e., the two-dimensional or individual specific variables, can be solved fromV_XXtsβ^b_{V ts}=V_{XY ts}.

Since inserting for W_{XY ij} from (2.15) and for V_{XY ts} from (2.17) in (3.1) and (3.2) gives, respectively,

βb_{W ij}−β=W⁻¹_XXijW_X²ij=W⁻¹_XXij(W_Xγi+W_{XU ij}), i, j= 1, . . . , N, (3.3)

βb_{V ts}−β=V⁻¹_XXtsV_X²ts=V⁻¹_XXts(V_Xαt+V_{XU ts}), t, s= 1, . . . , T, (3.4)

and (2.2) and (2.3) imply

E(W_{XU ij}|X) =E(W_Xγi|X) =0_K1, i, j= 1, . . . , N, (3.5)

E(V_{XU ts}|X) =E(V_Xαt|X) =0_K1, t, s= 1, . . . , T, (3.6)

we know that β^b_{W ij} and β^b_{V ts} are unbiased estimators for β. Furthermore, β^b_{W ij} is consistent when T → ∞ (T-consistent, for short), since then plim(W_X²ij/T) = 0_K1, provided that plim(W_XXij/T) is non-singular, andβ^b_{V ts}is consistent whenN → ∞(N- consistent, for short), since then plim(V_X²ts/N) = 0_K1, provided that plim(V_XXts/N) is non-singular.

However, some of the base estimators may be consistent even if conditions (2.2)–(2.3) are weakened. The followingrobustness results hold:

[1 ]Since (3.3) does not contain α, all β^b_{W ij} are T-consistent even if α_i is treated as fixed or allowed to be correlated with x¯_i·, but if γ_t is correlated with x¯·t, all β^b_{W ij} are inconsistent. Symmetrically, since (3.4) does not contain γ, all β^b_{V ts} are N- consistent even ifγ_t is treated as fixed or allowed to be correlated withx¯·t, but ifα_i is correlated with x¯_i·, all β^b_{V ts} are inconsistent.

[2 ]Endogeneity of or random measurement error in (some components of) x_it may cause E(x_it⁰u_it) 6=0_K1. Then plim(W_{XU ii}/T)6=0_K1, so that the OLS estimators βb_{W ii} are inconsistent, but the IV estimators β^b_{W ij} (j 6= i) remain T-consistent.

Symmetrically, we then also have plim(V_{XU tt}/N)6=0_K1, so that the OLS estima- tors β^b_{V tt} are inconsistent, but the IV estimatorsβ^b_{V ts} (s6=t)remainN-consistent.

In Appendix A it is shown that when (2.2)–(2.3) hold, the matrices of covariances be- tween the individual specific and the period specific base estimators, respectively, can be expressed as

C(β^b_{W ij},β^b_{W kl}|X) =E[(β^b_{W ij}−β)(β^b_{W kl}−β)⁰|X]

(3.7)

= (σ_γ²+δ_jlσ²)W⁻¹_XXijW_XXikW⁻¹_XXlk, C(β^b_{V ts},β^b_{V pq}|X) =E[(β^b_{V ts}−β)(β^b_{V pq}−β)⁰|X]

(3.8)

= (σ_α² +δ_sqσ²)V⁻¹_XXtsV_XXtpV⁻¹_XXqp, C(β^b_{W ij},β^b_{V pq}|X) =E[(β^b_{W ij}−β)(β^b_{V pq}−β)⁰|X]

(3.9)

=σ²W⁻¹_XXij(x_iq−x¯_i·)⁰(x_jp−x¯·p)V⁻¹_XXqp, i, j, k, l= 1, . . . , N, t, s, p, q= 1, . . . , T.

Eq. (3.7) for (k, l) = (i, j) and (3.8) for (p, q) = (t, s) give in particular the variance- covariance matrices

V(β^b_{W ij}|X) = E[(β^b_{W ij}−β)(β^b_{W ij}−β)⁰|X]

(3.10)

= (σ²_γ+σ²)W⁻¹_XXijW_XXiiW⁻¹_XXji, i, j= 1, . . . , N, V(β^b_{V ts}|X) = E[(β^b_{V ts}−β)(β^b_{V ts}−β)⁰|X]

(3.11)

= (σ²_α+σ²)V⁻¹_XXtsV_XXttV⁻¹_XXst, t, s= 1, . . . , T.

When (2.2)–(2.3) hold,β^b_{W jj} and β^b_{V ss} are always more efficient than β^b_{W ij} (j 6=i) and βb_{V ts} (s 6= t), respectively, i.e., V(β^b_{W ij}|X)−V(β^b_{W jj}|X) for i 6= j and V(β^b_{V ts}|X)− V(β^b_{V ss}|X) for t6=sare positive (semi)definite matrices. The formal proof of this is

V(β^b_{W ij}|X)−V(β^b_{W jj}|X) = (σ_γ²+σ²)(W⁻¹_XXijW_XXiiW⁻¹_XXji−W⁻¹_XXjj)

= (σ²_γ+σ²)(A⁻¹_{W Xij}A⁻¹_{W Xji}−I_K)W⁻¹_XXjj, V(β^b_{V ts}|X)−V(β^b_{V ss}|X) = (σ²_α+σ²)(V⁻¹_XXtsV_XXttV⁻¹_XXst−V⁻¹_XXss)

= (σ_α² +σ²)(A⁻¹_{V Xts}A⁻¹_{V Xst}−I_K)V⁻¹_XXss, where

A_{W Xij} =W⁻¹_XXiiW_XXij, A_{V Xts}=V⁻¹_XXttV_XXts.

The latter are the (K×K) matrix of (sample) regression coefficients when regressing the block ofX relating toj,i.e.,X_j·, on the block of X relating to individual i, i.e., X_i·,

and when regressing the block of X relating to period s, i.e., X·s, on the block of X relating to periodt, i.e., X·t, respectively. Here all (A⁻¹_{W Xij}A⁻¹_{W Xji}−I_K), j 6= i, and all (A⁻¹_{V Xts}A⁻¹_{V Xst} −I_K), s 6= t, are positive (semi)definite matrices, provided that all variables inx_it are two-dimensional.

The structure of the variance-covariance matrices of the estimators is transparent in theone-regressor case, K= 1. Then (3.7) and (3.10) read

C(β^b_{W ij},β^b_{W kl}|X) = (σ²_γ+δ_jlσ²) W_XXik

W_XXijW_XXkl, V(β^b_{W ij}|X) = (σ_γ²+σ²)W_XXii W_XXij² , (3.12)

where W_XXik, β^b_{W ij}, etc. denote the scalar counterparts to W_XXik, β^b_{W ij}, etc. The coefficient of correlation between two arbitrary individual specific base estimators for the slope coefficient can therefore be written as

ρ(β^b_{W ij},β^b_{W kl}|X) = C(β^b_{W ij},β^b_{W kl}|X) [V(β^b_{W ij}|X)V(β^b_{W kl}|X)]^1/2 (3.13)

= σ²_γ+δ_jlσ² σ²_γ+σ²

W_XXik

(W_XXiiW_XXkk)^1/2 = ρ(²_jt, ²_lt)R_{W Xik}, where R_{W Xik} = W_XXik/(W_XXiiW_XXkk)^1/2 is the sample coefficient of correlation be- tween thex’s of individualsiandkandρ(²_jt, ²_lt) = (σ_γ²+δ_jlσ²)/(σ_γ²+σ²) is the coefficient of correlation between²_jt and ²_lt. If we therefore consider (2.5) as an N-equation model with one equation for each individual and with common slope coefficient,ρ(β^b_{W ij},β^b_{W kl}|X) is simply the product of the coefficient of correlation between two²disturbances from in- dividuals (equations)jandlin the same period, and the coefficient of correlation between the values of the regressor (instrument) for individuals (equations)iand k. This means thatρ(β^b_{W ij},β^b_{W kl}|X) has one equation specific component (j vs. l) and one instrument specific component (i vs.k). Forj=land for i=k(3.13) gives, respectively,

ρ(β^b_{W ij},β^b_{W kj}|X) =R_{W Xik}, for allj;i6=k [same equation (individual), different IV], ρ(β^b_{W ij},β^b_{W il}|X) = σ²_γ

σ_γ²+σ², for alli;j6=l [different equations (individuals), same IV].

Symmetrically, (3.8) and (3.11) for K= 1 give C(β^b_{V ts},β^b_{V pq}|X) = (σ_α² +δ_sqσ²) V_XXtp

V_XXtsV_XXpq, V(β^b_{V ts}|X) = (σ_α² +σ²)V_XXtt V_XXts² . (3.14)

The coefficients of correlation can therefore be written as ρ(β^b_{V ts},β^b_{V pq}|X) = C(β^b_{V ts},β^b_{V pq}|X)

[V(β^b_{V ts}|X)V(β^b_{V pq}|X)]^1/2 (3.15)

= σ²_α+δ_sqσ² σ²_α+σ²

V_XXtp

(V_XXttV_XXpp)^1/2 = ρ(²_is, ²_iq)R_{V Xtp},

whereR_{V Xtp} =V_XXtp/(V_XXttV_XXpp)^1/2 is the coefficient of correlation between thex’s in periodstand pandρ(²_is, ²_iq) = (σ²_α+δ_sqσ²)/(σ_α²+σ²) is the coefficient of correlation between²_isand²_iq. If we therefore consider (2.6) as aT-equation modelwith one equation for each period and with common slope coefficient,ρ(β^b_{V ts},β^b_{V pq}|X) is simply the product of the coefficient of correlation between two ² disturbances from periods (equations) s and q for the same individual, and the coefficient of correlation between the values of the regressor (instrument) in periodst and p. This means thatρ(β^b_{V ts},β^b_{V pq}|X) has one equation specific component (svs. q) and one instrument specific component (t vs. p).

Fors=q and t=p (3.15) gives, respectively,

ρ(β^b_{V ts},β^b_{V ps}|X) =R_{V Xtp}, for all s;t6=p [same equation (period), different IV], ρ(β^b_{V ts},β^b_{V tq}|X) = σ_α²

σ²_α+σ², for allt;s6=q [different equations (periods), same IV].

From (3.12) and (3.14) we find that theinefficiency when using the (within) variation of individualias IV for the (within) variation of individualj relative to performing OLS on the observations from individualjand when using the (within) variation of periodtas IV for the (within) variation of periodsrelative to performing OLS on the observations from periods, can be expressed simply as, respectively,

V(β^b_{W ij}|X)

V(β^b_{W jj}|X) = 1

A_{W Xij}A_{W Xji} = 1 R²_{W Xij}, (3.16)

V(β^b_{V ts}|X)

V(β^b_{V ss}|X) = 1

A_{V Xts}A_{V Xst} = 1 R²_{V Xts}. (3.17)

Hence,R⁻²_{W Xij} (≥1) andR⁻²_{V Xts} (≥1) measure, respectively, the loss of efficiency when using estimators which are robust to inconsistency caused by simultaneity or random measurement error in the regressor, (i) by estimating a relationship for individual j by using as IV observations from another individual,i, rather than using OLS, and (ii) by estimating a relationship for period sby using as IV observations from another period, t, rather than using OLS.

4 A class of moment estimators

Since each of theN²+T²base estimators of β,β^b_{W ij} andβ^b_{V ts}, only uses a minor part of the panel data set, they may not be considered real competitors to estimators constructed from the complete data set, when (2.2)–(2.3) are valid. And even if these assumptions are violated by correlation between x_it and u_it, between x¯_i· and α_i, and/or between

¯

x·t and γ_t, aggregate estimators which are more efficient than any of the IV estimators βb_{W ij} (j 6= i) and β^b_{V ts} (s 6= t) may exist. Yet, the insight provided by examining

these base estimators as we have done in th previous section is useful when constructing composite estimators ofβ, of which they can serve as building-blocks.

This motivates the construction ofa class of estimators of β by weighting the indi- vidual specific or period specific (co)variation inX andy, as defined in (2.11)–(2.12), as follows: Letθ = (θ_ts) be a (T×T) matrix andτ = (τ_ij) an (N×N) matrix of (positive, zero or negative) weights and define ageneral moment estimator as

b=b(θ,τ) =^³PT_t=1^PT_s=1θ_tsV_XXts+^PN_i=1^PN_j=1τ_ijW_XXij^´−1 (4.1)

× ^³PT_t=1^PT_s=1θ_tsV_{XY ts}+^PN_i=1^PN_j=1τ_ijW_{XY ij}^´. Using (3.1)–(3.2) it can be written as a weighted average of the base estimators:

b=^³PT_t=1^PT_s=1θ_tsV_XXts+^PN_i=1^PN_j=1τ_ijW_XXij^´−1 (4.2)

× ^³PT_t=1^PT_s=1θ_tsV_XXtsβ^b_{V ts}+^PN_i=1^PN_j=1τ_ijW_XXijβ^b_{W ij}^´, or, in simplified notation,

b=^PT_t=1^PT_s=1G_{V ts}β^b_{V ts}+^PN_i=1^PN_j=1G_{W ij}β^b_{W ij}, (4.3)

whereG_{V ts}andG_{W ij} are (K×K) weighting matrices,^P_t^P_sG_{V ts}+^P_i^P_jG_{W ij} =I_K, given by

G_{V ts}=Q⁻¹θ_tsV_XXts, t, s= 1, . . . , T, G_{W ij} =Q⁻¹τ_ijW_XXij, i, j= 1, . . . , N, Q=Q(θ,τ) =^PT_t=1^PT_s=1θ_tsV_XXts+^PN_i=1^PN_j=1τ_ijW_XXij.

(4.4)

None of the latter matrices are symmetric in general. If, however, θ_ts = θ_st for all t, s andτ_ij =τ_ji for alli, j, thenQ⁰=Q.

The estimator b is unbiased for any θ and τ when (2.2) and (2.3) hold, and in Appendix B it is shown that its variance-covariance matrix is (This formula in the special case whereK = 1 and σ_γ² = 0 is derived in Biørn (1994, Appendix A).)

V(b|X) =Q⁻¹P (Q⁻¹)⁰=Q(θ,τ)⁻¹P(θ,τ, σ², σ²_α, σ_γ²)(Q(θ,τ)⁻¹)⁰, (4.5)

where

P =P(θ,τ, σ², σ²_α, σ_γ²) =σ²(S_V +S_W +S_{V W}) +σ_α²Z_V +σ_γ²Z_W, (4.6)

S_V =S_V(θ) = XT

t=1

XT

p=1

V_XXtp à _T

X

s=1

θ_tsθ_ps

! ,

S_W =S_W(τ) = XN

i=1

XN

k=1

W_XXik



X^N

j=1

τ_ijτ_kj



,

S_{V W} =S_{V W}(θ,τ) = XT

t=1

XT

s=1

XN

i=1

XN

j=1

θ_tsτ_ij(x_is−x¯_i·)⁰(x_jt−x¯·t), Z_V =Z_V(θ) =

XT

t=1

XT

p=1

V_XXtp à _T

X

s=1

θ_ts

! Ã _T X

r=1

θ_pr

! ,

Z_W =Z_W(τ) = XN

i=1

XN

k=1

W_XXik



X^N

j=1

τ_ij



 Ã_N

X

l=1

τ_kl

! . (4.7)

It is easily seen that S_{W V} = 0 if either θ_ts = θ for all t, s or τ_ij = τ for all i, j, that Z_V =0if^PT_s=1θ_ts= 0 for allt, and thatZ_W =0if^PT_j=1τ_ij = 0 for alli. The standard estimators in fixed and random effects models have at least one of these properties, which will be shown in the next section.

By utilizing (4.5)–(4.7), we can estimate V(b|X) consistently from a panel data set for any weighting matrices θ and τ we may choose when consistent estimators of the variancesσ²,σ_α², and σ²_γ have been obtained.

5 Specific aggregate estimators

In this section, we consider specific members of the class of estimators described by (4.1).

Some of these are familiar, others less familiar.

Aggregate within and between estimators

The estimatorbcontains several familiar estimators for fixed effects models as particular members. We first establish the weighting system (θ,τ) for six such estimators and comment on other, less familiar estimators which are more robust to violation of the basic assumptions. The results below generalize those in Biørn (1994, section 3), where only one regressor is included (K= 1) and period specific effects are disregarded (γ_t= 0).

We define, in the usual way [see, e.g., Greene (2003, section 13.3.2)], the (K×K)- matrices ofoverall (aggregate)within individual and within period, (co)variation as

W_XX =^PN_i=1W_XXii =^PN_i=1^PT_t=1(x_it−x¯_i·)⁰(x_it−x¯_i·), (5.1)

V_XX =^PT_t=1V_XXtt =^PT_t=1^PN_i=1(x_it−x¯·t)⁰(x_it−x¯·t), (5.2)

etc. The correspondingoverall between individual, andbetween period (co)variation are B_XX =^PN_i=1B_XXii =T^PN_i=1(¯x_i· −x)¯ ⁰(x¯_i· −x) = (1/T¯ )^PT_t=1^PT_s=1V_XXts, (5.3)

C_XX =^PT_t=1C_XXtt =N^PT_t=1(x¯·t−x)¯ ⁰(x¯·t−x) = (1/N¯ )^PN_i=1^PN_j=1W_XXij, (5.4)