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03.1.2. Redundancy of Lagged Regressors in a Conditionally Heteroskedastic Time Series Regression

Published online by Cambridge University Press:  01 February 2003

Stanislav Anatolyev
Stanislav Anatolyev
Affiliation:
New Economic School, Moscow
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Extract

Consider the following stationary time series regression.

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Type
PROBLEMS AND SOLUTIONS
Information
Econometric Theory , Volume 19 , Issue 1 , February 2003 , pp. 225 - 226
Copyright
© 2003 Cambridge University Press

Consider the following stationary time series regression:

where all variables are scalars and the error et is conditionally heteroskedastic with the following form of heteroskedasticity:

The object of estimation is β. Usually, under conditional heteroskedasticity the use of lagged values of regressors as instruments increases the efficiency of generalized method of moments (GMM) estimation in comparison with ordinary least squares (OLS) estimation (see, e.g., Broze, Francq, and Zakoïan, 2001; West, 2001). This problem shows that it may not necessarily be so.

Assume that the regressor xt can be represented as

, where ηt's are independently and identically distributed standard normal. Also assume that the parameters are constrained so that all variables have finite fourth moments. Show that the OLS estimator is at least as efficient as any GMM estimator that uses an arbitrary fixed number of instruments from the list {xt,xt−1,xt−2,…}.

References

REFERENCES

Broze, L., C. Francq, & J.-M. Zakoïan (2001) Non-redundancy of high order moment conditions for efficient GMM estimation of weak AR processes. Economics Letters 71, 317–322.Google Scholar
West, K.D. (2001) On optimal instrumental variables estimation of stationary time series models. International Economic Review 42, 1043–1050.Google Scholar