Advanced Econometrics Takeshi Amemiya
Swamy’s Model
Swamy’s model (1970) is a special case of the Kelejian-Stephan model obtained by putting
2A — 0 and 2e = 2 ® Ir,
where 2 = diag {a,a, . . . , a%). It is more restrictive than Hsiao’s model in the sense that there is no time-specific component in Swamy’s model, but it is more general in the sense that Swamy assumes neither the diagonality of XM nor the homoscedasticity of є like Hsiao.
Swamy proposed estimating XM and 2 in the following steps:
Step 1. Estimate a} by af = y'[I — X,(X-X,)- 1X/']y(/(T — K).
Step 2. Define b, = (x; Х()->Х;у(.
Step 3. Estimate 2„ by % = (N - ІГ'Х^А-ЛМЕ^А)*
(b, - AM 2f_A)' - N-y^SjiX'Xi)-'.
It is easy to show that aj and 2^ are unbiased estimators of a2 and XM, respectively.
Swamy proved that the FGLS estimator of fi using a2 and 2^ is asymptotically normal with the order N~in and asymptotically efficient under the normality assumption. Note that GLS is of the order of N~l/2 in Swamy’s model because, using (6.7.7), we have in Swamy’s model
(6.7.9)
= 0(N) ~ 0(N/T).
6.7.1 The Swamy-Mehta Model
The Swamy-Mehta model (1977) is obtained from the Kelejian-Stephan model by putting a2 = 0 but making the time-specific component more gen-
eral as [diag (Xf, XJ............... X£)]A* where EX* = 0 and EX*X*' =
 |
diag (Ir® Ir® 22,. . . , lT® 2дг). In their model, (6.7.7) is reduced to
as in Swamy’s model.
