In this paper, the MOSUM tests with recursive and OLS residuals are considered, and their asymptotic null distributions are characterized analytically. We show that the limiting processes of moving sums of recursive residuals is the increments of a standard Wiener process. For a particular choice of the size of moving sums, a formula representing the probability of this limiting process crossing constant boundaries is derived, from which correct asymptotic critical values are calculated and

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... lated. We also show that the limiting process of moving sums of OLS residuals is the increments of a Brownian bridge whose boundary-crossing probability is known in literature. We then prove that the MOSUM tests are consistent and have non-trivial local power under a general class of alternatives. Our simulation further indicates that the propsoed MOSUM tests can complement other CUSUM-type of tests when there is a single structural change and have power advantage when there are certain double structural changes.