Tests of parameter instabilities are likely to have low power when change-points occur towards the end of the sample. This paper considers various modifications of existing tests and introduces new statistics designed to have high power in such circumstances. The properties of both Wald-type tests of a one-time shift in the parameters and locally most powerful (LMP) tests against the hypothesis of random walk coefficients are examined. It is proposed to take functionals of the Wald and LMP statistics such that either the set of possible change-points is restricted to the last part of the sample or the occurrence of change-points is given increasing weight throughout the sample. For the case of an unknown end-of-sample change-point, the LMP-type tests appear to have, in general, better properties than Wald-type tests, even against the hypothesis of a one-time shift in the parameters. Empirical illustrations describe the use of the tests for detecting structural changes at the time of the 'Great Recession'.
