Approximate sampling distribution of the serial correlation coefficient for small samples

The probability density function for the sample serial correlation coefficient r can be approximated byf(r) = (β(½, ½(T + 1)))⁻¹(1 − r²)^{½(T− 1})(1+ c² − 2cr)^−½(T), whereβ is the Beta function, T= n− 2, c = ρ − [(1 + ρ)/(n − 3)], n is the number of observations, and ρ is the population lag one serial correlation. This distribution is derived from a large Monte Carlo study at points between ρ= −0.9 and ρ = 0.9 and for n =10, 20, and 30.