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Solutions to Problems Posed in Volume 20(1) and 20(2): 04.1.1. A Hausman Test Based on the Difference between Fixed Effects Two-Stage Least Squares and Error Components Two-Stage Least Squares—Solution

Published online by Cambridge University Press:  31 March 2005

B.H. Baltagi
B.H. Baltagi
Affiliation:
Texas A&M University
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Extract

(a) Let , where Q is the fixed effects (FE) transformation given by Q = INT − P with P = IN [otimes] ιTιT′/T.

Information

Type
SOLUTIONS TO PROBLEMS
Information
Econometric Theory , Volume 21 , Issue 2 , April 2005 , pp. 483 - 484
Copyright
© 2005 Cambridge University Press

(a) Let

, where Q is the fixed effects (FE) transformation given by Q = INT − P with P = IN [otimes ] ιTιT′/T. Baltagi (1981) showed that

where PX = X(X′X)−1X′ with

and the asymptotic variance is given by

. Also, Baltagi (1981) showed that

where y1 = Py1, Z1 = PZ1, and σ1112 = Tσμ112 + σν112. Under H0,

and the asymptotic variance of

is given by

. Under H0, both

are consistent, with

. Using the expressions for

, one gets the asymptotic covariance

This makes use of the following expression:

, where n = NT and Ω11 = σν112Q + σ1112P. Also,

, because PQ = 0.

Hence

.

(b) Using the fact that

, one gets

because

. Therefore

Footnotes

An excellent solution has been proposed independently by Shiferaw Gurmu, Georgia State University.

References

REFERENCE

Baltagi, B.H. (1981) Simultaneous equations with error components. Journal of Econometrics 17, 189–200.Google Scholar