This paper considers a modified CUSUM test, suggested by Dufour (1982) for parameter instability and structural change with an unknown change point in a linear model with serially correlated disturbances, in which a preliminary estimate of the autoregressive coefficient for the error process is obtained, and used to transform the data. Then the standard CUSUM statistic is calculated on the transformed data. This paper derives the asymptotic distribution of the modified CUSUM test. We show that the modified CUSUM test retains its asymptotic significance level, i.e., the modified CUSUM test has the same asymptotic distribution as the CUSUM test with serially uncorrelated errors.