An approximate distribution of a noncircular serial correlation coefficient

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SUMMARY The paper applies a smoothing technique to obtain an approximate sampling distribution of a noncircular serial correlation coefficient. This technique is similar to the one employed by Koopmans and Dixon for finding the circular counterpart and thus provides an immediate comparison with that of the circularly defined coefficient. In fact, it is shown that the latter is a special case of the present result which is obtained in a generalized Leipnik's form with a correction factor to account for noncircularity. Also, the equivalence of this method of smoothing with the saddle point technique adopted by Daniels is demonstrated. The need for an inquiry into the sampling distribution of a serial correlation coefficient, r, requires no special emphasis; for example, econometric applications of time series data make the need immediate and obvious. Tractability for these distributions is achieved by a number of ingenious devices. One is to assume a circular definition for r. Other equally elegant manipulations are described by Durbin & Watson (1950, 1951). While the circular definition helps to provide an exact distribution, it suffers from the possibility of becoming too distant from reality when, for instance, the length of the sample series is as small as often happens with economic variables or has to be kept so for the risk of running into incomparability due to changes in the underlying structure. At the same time, as Koopmans (1942) has shown, the exact distribution of a noncircular r involves hyperelliptic integrals. The method of Koopmans and those of others for the noncircular serial correlation coefficient are briefly surveyed for the null case in a recent paper by Durbin & Watson (1971, Appendix 1). The practical computation of the exact probabilities of the distribution of a noncircular r to any desired degree of accuracy has become possible by the methods of Imhof (1961) and Pan Jie-Jian (1968) coupled with the present day electronic computer facilities. Given the distribution function F(ro) = pr (r < ro),