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).