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Journal Article Note on Uncertainty and Planning Get access T. W. Hutchison T. W. Hutchison Bonn. Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 5, Issue 1, October 1937, Pages 72–74, https://doi.org/10.2307/2967582 Published: 01 October 1937

Journal Article A Note on Tautologies and the Nature of Economic Theory Get access T. W. Hutchison T. W. Hutchison Cambridge Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 2, Issue 2, February 1935, Pages 159–161, https://doi.org/10.2307/2967564 Published: 01 February 1935

The distributions of the LIML and TSLS estimates of the coefficient of an endogenous variable in a single equation can be approximated by asymptotic expansions. This paper relates the expansions in terms of the noncentrality parameter and the sample size going to infinity, the noncentrality parameter going to infinity with the sample size held fixed, and the standard deviation of the disturbance going to zero (small-o). 1. INTRODUCriON RECENTLY, ASYMPTOTIC EXPANSIONS of the distributions of estimates of coefficients of a single equation in a system of simultaneous equations have been made by Anderson [1], Anderson and Sawa [2], Mariano [6 and 7], and Sargan and Mikhail [11]. The expansions have usually been carried out on the basis that the sample size increases and that the effect of the exogenous variables (the noncentrality parameter) increases along with the sample size. In this paper we consider the case of the covariance matrix of the disturbances known and alternatively the case of the sample size fixed. We relate these three cases to the approach of letting the disturbance decrease (the small-o- approach). The estimates treated are two-stage least squares (TSLS) and limited information maximum likelihood (LIML).

An estimate of y obtained by applying the method of maximum likelihood under the assumption that ut is normally distributed is consistent and asymptotically normally distributed. The asymptotic standard deviation is given in this note. Although Kendall considers many estimates of the period in his publication, he does not use the maximum-likelihood estimate although it has desirable properties in large samples that several of the other estimates do not have.4 It is interesting to compare the numerical results of using this estimate with those Kendall applies to four artificial series generated by (1), each series with a different pair of coefficients a and j3.5 If the ut (t , 2, . . . , T) are assumed to be normally distributed and if x-, and x0 are assumed to be fixed, the estimate defined by the method of maximum likelihood is obtained by substituting in (2) the estimates of a and ,B found by the method of maximum likelihood under these assumptions [see equations (8)]. H. B. Mann and A. Wald6 have