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The Sampling Distribution of the Liviatan Estimator of the Geometric Distributed Lag Parameter

Econometrica 1973 41(3), 503
THE USE OF the geometric distributed lag in economics is widespread. The Liviatan [6] method for estimating its parameters is simple and provides consistent estimates. Moreover, it can be used to provide initial estimates for more sophisticated techniques [2]. Not much is known about the statistical properties of these estimators, especially their small sample properties. We do know that their asymptotic efficiencies are inferior to most alternatives [1], and recently Nagar and Gupta [7] have provided approximations to the small sample biases. The purpose of this note is to derive a simple way of displaying the Liviatan estimators which makes their nature clear and which allows the small sample distribution of one of them to be easily deduced. The main result is that the estimator of the parameter which defines the geometric distributed lag is a ratio of two ordinary least squares estimators. With this and the assumption that the error terms form a sequence of independent, identically distributed normal variables, it is possible, using the work of Geary [4], to derive the small sample distribution of this estimator. This result forms the main part of this note, which concludes with a brief discussion of the problem of setting confidence limits along the lines suggested by Fieller [3].