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

image488(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*’ =

image489

diag (Ir® Ir® 22,. . . , lT® 2дг). In their model, (6.7.7) is reduced to

as in Swamy’s model.

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