This paper considers tests for parameter instability and structural change with unknown change point. The results apply to a wide class of parametric models that are suitable for estimation by generalized method of moments procedures. The paper considers Wald, Lagrange multiplier, and likelihood ratio-like tests. Each test implicitly uses an estimate of a change point. The change point may be completely unknown or it may be known to lie in a restricted interval. Tests of both "pure" and "partial" structural change are discussed. The asymptotic distributions of the test statistics considered here are nonstandard because the change point parameter only appears under the alternative hypothesis and not under the null. The asymptotic null distributions are found to be given by the supremum of the square of a standardized tied-down Bessel process of order $p \geqslant 1$, as in D. L. Hawkins (1987). Tables of critical values are provided based on this asymptotic null distribution. As tests of parameter instability, the tests considered here are shown to have nontrivial asymptotic local power against all alternatives for which the parameters are nonconstant. As tests of one-time structural change, the tests are shown to have some weak asymptotic local power optimality properties for large sample size and small significance level. The tests are found to perform quite well in a Monte Carlo experiment reported elsewhere.