We consider the first serial correlation coefficient under an $\operatorname{AR}(1)$ model where errors are not assumed to be Gaussian. In this case it is necessary to consider bootstrap approximations for tests based on the statistic since the distribution of errors is unknown. We obtain saddle-point approximations for tail probabilities of the statistic and its bootstrap version and use these to show that the bootstrap tail probabilities approximate the true values with given relative errors, thus extending the classical results of Daniels [Biometrika 43 (1956) 169–185] for the Gaussian case. The methods require conditioning on the set of odd numbered observations and suggest a conditional bootstrap which we show has similar relative error properties.