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 by f ( r ) = (β(½, ½( T + 1))) −1 (1 − r 2 ) ½( T − 1 )(1+ c 2 − 2 cr ) −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.