The Error Components Model

The regression model is still the same, but it now has double subscripts

yit = a + Xite + uit (12.1)

B. H. Baltagi, Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-20059-5_12, © Springer-Verlag Berlin Heidelberg 2011

where i denotes cross-sections and t denotes time-periods with i = 1,2,…,N, and t = 1,2, .. .,T. a is a scalar, в is K x 1 and Xit is the it-th observation on K explanatory variables. The observations are usually stacked with i being the slower index, i. e., the T observations on the first household followed by the T observations on the second household, and so on, until we get to the N-th household. Under the error components specification, the disturbances take the form

Uit = Hi + Vit (12.2)

where the Hi’s are cross-section specific components and vit are remainder effects. For example, Hi may denote individual ability in an earnings equation, or managerial skill in a production function or simply a country specific effect. These effects are time-invariant.

In vector form, (12.1) can be written as

y = aiNT + Хв + u = ZS + u (12.3)

where y is NT x 1, X is NT x K, Z = [iNT, X], S’ = (а’,в’), and iNT is a vector of ones of dimension NT. Also, (12.2) can be written as

u = Z^h + v (12.4)

where u’ = (uii,…, ит, U21,…, U2T,…, uni, …, unt) and Z^ = In Z іт. In is an identity

matrix of dimension N, іт is a vector of ones of dimension T, and Z denotes Kronecker product defined in the Appendix to Chapter 7. Z^ is a selector matrix of ones and zeros, or simply the ma­trix of individual dummies that one may include in the regression to estimate the Hi’s if they are assumed to be fixed parameters. h’ = (H1, …, HN) and v’ = (v 11,…, v1T,…, vN 1,…, vNT).

Note that Z^Z’^ = InZJT where JT is a matrix of ones of dimension T, and P = Z^(Z’^Z^)-1Z’^, the projection matrix on ZM, reduces to P = In ZJt where JT = JT/T. P is a matrix which aver­ages the observation across time for each individual, and Q = Int — P is a matrix which obtains the deviations from individual means. For example, Pu has a typical element u^ = ^2T=1 uit/T repeated T times for each individual and Qu has a typical element (uit — щ). P and Q are (i) symmetric idempotent matrices, i. e., P’ = P and P2 = P. This means that the rank (P) = tr(P) = N and rank (Q) = tr(Q) = N(T — 1). This uses the result that rank of an idempotent matrix is equal to its trace, see Graybill (1961, Theorem 1.63) and the Appendix to Chapter 7. Also, (ii) P and Q are orthogonal, i. e., PQ = 0 and (iii) they sum to the identity matrix P + Q = Int. In fact, any two of these properties imply the third, see Graybill (1961, Theorem 1.68).

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