Pooling Time-Series of Cross-Section Data
12.1 Fixed Effects and the Within Transformation.
a. Premultiplying (12.11) by Q one gets Qy = «Qint + QX" + QZpp + Qv
But PZp = Zp and QZp = 0. Also, PiNT = iNT and Qint = 0. Hence, this
transformed equation reduces to (12.12)
Qy = QX" + Qv
Now E(Qv) = QE(v) = 0 and var(Qv) = Q var(v)Q0 = o2Q, since var(v) = ov2Int
and Q is symmetric and idempotent.
b. For the general linear model y = X" + u with E(uu0) = Й, a necessary and sufficient condition for OLS to be equivalent to GLS is given by X0 fi_1PX where PX = I – PX and PX = X(X0X)_1 X0, see Eq.(9.7) of Chap.9. For Eq. (12.12), this condition can be written as
(X0Q)(Q/o2)P qx = 0
using the fact that Q is idempotent, the left hand side can be written as (X0Q)P qx/ov2
which is clearly 0, since PqX is the orthogonal projection of QX.
One can also use Zyskind’s condition PX^ = ^PX given in Eq. (9.8) of Chap. 9. For Eq. (12.12), this condition can be written as
Pqx(ov2Q) = (ov2Q)Pqx
B. H. Baltagi, Solutions Manual for Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-54548-1_12, © Springer-Verlag Berlin Heidelberg 2015
But, PqX = QX(X0QX)_1X0Q. Hence, PqxQ = Pqx andQPQX = Pqx and the condition is met. Alternatively, we can verify that OLS and GLS yield the same estimates. Note that Q = INT — P where P = IN <S> JT is idempo – tent and is therefore its own generalized inverse. The variance-covariance matrix of the disturbance v = Qv in (12.12) is E(vv0) = E(Qvv0Q) = o^Q with generalized inverse Q/o^. OLS on (12.12) yields
" = (X, QQX)_1X, QQy = (X0 QX)“1X, Qy
which is " given by (12.13). Also, GLS on (12.12) using generalized inverse yields
" = (X, QQQX)“1X, QQQy = (X0QX)_1 X0Qy = ".
c. The Frisch-Waugh-Lovell (FWL) theorem of Davidson and MacKinnon (1993, p. 19) states that for
y = X1P1 C X2P2 C u (12.1)
If we premultiply by the orthogonal projection of X2 given by
M2 = I — X2(x2X2)“1×2, then M2X2 = 0 and (1) becomes
M2y = M2X1P1 C M2u (12.2)
The OLS estimate of "1 from (2) is the same as that from (1) and the residuals from (1) are the same as the residuals from (2). This was proved in Sect. 7.3. Here we will just use this result. For the model in (12.11)
y = Z8 C C v
Let Z = X1 and Z^ = X2. In this case, M2 = I — Z^ZJjZ^ ZJ) = I — P = Q. In this case, premultiplying by M2 is equivalent to premultiplying by Q and Eq. (2) above becomes Eq. (12.12) in the text. By the FWL theorem, OLS on (12.12) which yields (12.13) is the same as OLS on
(12.11). Note that
Z = [tNT, X] and QZ = [0, QX] since QtNT = 0.