Distribution of the Serial Correlation Coefficient in a Circularly Correlated Universe

Abstract

It is desired to find an approximate distribution of simple form for the statistic $\bar r = \frac{x_1x_2 + \cdots + x_Tx_1} {x_1^2 + \cdots + x_T^2}$ ($\bar r$ is an estimate of the serial correlation coefficient $\rho$ in a circular universe) in the case that $\rho \neq O$ in the universe. Such a distribution is obtained by smoothing the joint characteristic function of the numerator and denominator of the expression for $\bar r$. The first two moments are calculated; from these $\bar r$ is seen to be a consistent estimate of $\rho$. A graph of this distribution for sample size $T = 20$ and various values of $\rho$ is given. In addition, an approximate distribution for $p = x^2_1 + \cdots + x^2_T$ is derived which reduces to the exact $(\chi^2-)$ distribution if $\rho = 0$. From a formula which yields all moments, it is concluded that, at least up to the degree of approximation attained, $p/T$ is an unbiased and consistent extimate of $\sigma^2$.