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.