On Serial Correlation

Abstract

Define \begin{equation*}\tag{1.1}P = U/V \equiv \sum^m_{i=1}\lambda_ix_i^2/\sum^r_{i=1}x_i^2,\quad 1 \leqq m \leqq r,\end{equation*} where the $x_i$'s are independent and identically distributed as $N(0, 1)$, and $0 < \lambda_1 \leqq \lambda_2 \leqq \cdots \leqq \lambda_m$. We will give various representations for the distribution of $P$, and show how a special case of this distribution is useful for testing for correlation in a time series. We will also consider independence of the errors in the normal regression problem. Let $n$ be the number of observations, and $k$ the number of coefficient parameters in a univariate linear regression model. Take $r = n - k$ and $m = r - 1$. Suppose the $x_i$'s are the linear functions of the observations of the dependent variable obtained by a "Theil transformation" (see Theil [15]). It will be shown in Section 2 that under these conditions, $P$ is an appropriate test statistic for independence. In Section 3, two different characteristic function representations are developed for $P$; one involves an infinite series of complex valued gamma functions, while the other involves a doubly infinite series of real valued gamma functions. Section 4 discusses the fact that for $r = m + 1, P$ is distributed as a linear combination of correlated beta variates. An appropriate beta distribution approximation is given for comparison. A numerical tabulation of the distribution of $P$ for the case of $r = m + 1$ is in preparation. The cdf. of $P$ is found by numerically inverting the characteristic function of a related linear combination of central chi-square variates.