Quasi-likelihood ratio tests for autoregressive moving-average (ARMA) models are examined. The ARMA models are stationary and invertible with white-noise terms that are not restricted to be normally distributed. The white-noise terms are instead subject to the weaker assumption that they are independently and identically distributed with an unspecified distribution. Bootstrap methods are used to improve control of the finite sample significance levels. The bootstrap is used in two ways: first, to approximate a Bartlett-type correction; and second, to estimate the p-value of the observed test statistic. Some simulation evidence is provided. The bootstrap p-value test emerges as the best performer in terms of controlling significance levels. Copyright 2007 The Authors Journal compilation 2007 Blackwell Publishing Ltd.