This paper considers a nonparametric conditional momenttest of stability of an econometric model againstthe alternative of instability. The alternativehypothesis allows for more than one structuralchange, although in this case it has to be fairlysmooth. This complements existing results forstability in a parametric setting. Also, it is shownthat the test is always consistent, unlike theavailable “parametric” tests, which normally rely onthe assumption of a correct specification of themodel, at least under the null hypothesis of nostructural instability. Moreover, we show that thetest has local power comparable to the parametricones; that is, its asymptotic efficiency is greaterthan zero. A Monte Carlo experiment about theperformance of our test is described.

In this paper a new class of tests for parameterstability, the moving-estimates (ME) test, isproposed. It is shown that in the standard situationthe ME test asymptotically equivalent to the maximallikelihood ratio test under the alternative of atemporary parameter shift. It is also shown that theasymptotic null distribution of the ME test isdetermined by the increments of a vector Brownianbridge and that under a broad class of alternativesthe ME test is consistent and has nontrivial localpower in general. Our simulations also demonstratethat the proposed test has power superior to othercompeting tests when parameters are temporarilyinstable.

The unknown structural parameters of acontinuous/discrete state space model are estimatedby maximum likelihood in the presence of irregularsampling, missing values, and cross-sections of timeseries (panel data). Exogenous (control) variablesare included, and the sampling scheme and missingdata pattern can be different for each variable andsystem. Furthermore, the derived non-linearoptimization algorithm with analytical scorefunction can be used for the discrete time case aswell.

A quasi-maximum likelihood estimator of the break dateis analyzed. Consistency of the estimator isdemonstrated under very general conditions, providedthat the data-generating process is not integrated.However, the asymptotic distribution of theestimator is quite different for time series thatare integrated of order one. In that case, whenthere is no break, the analyst can be spuriously ledto the estimation of a break near the middle of thetime series.

We obtain simple and generally applicable conditionsfor the existence of mixed momentsE([X′AX]″/[X′BX]U)of the ratio of quadratic forms T =X′AX/X′BXwhere A and B aren × n symmetricmatrices and X is a randomn-vector. Our principal theoremis easily stated when X has anelliptically symmetric distribution, which classincludes the multivariate normal andt distributions, whetherdegenerate or not. The result applies to the ratioof multivariate quadratic polynomials and can beexpected to remain valid in most situations in whichX is subject to linearconstraints.

If u ≤ v, the precisedistribution of X, and inparticular the existence of moments ofX, is virtually irrelevant to theexistence of the mixed moments ofT; if u >v, a prerequisite for existenceof the (u, v)thmixed moment is the existence of the2(u − v)thmoment of X WhenXis not degenerate, the principalfurther requirement for the existence of the mixedmoment is that B has rank exceeding2v.