We propose a simple method to help researchers develop quantitative models of economic fluctuations. The method rests on the insight that many models are equivalent to a prototype growth model with time-varying wedges that resemble productivity, labor and investment taxes, and government consumption. Wedges that correspond to these variables—efficiency, labor, investment, and government consumption wedges—are measured and then fed back into the model so as to assess the fraction of various fluctuations they account for. Applying this method to U.S. data for the Great Depression and the 1982 recession reveals that the efficiency and labor wedges together account for essentially all of the fluctuations; the investment wedge plays a decidedly tertiary role, and the government consumption wedge plays none. Analyses of the entire postwar period and alternative model specifications support these results. Models with frictions manifested primarily as investment wedges are thus not promising for the study of U.S. business cycles.

For such a model, we show that consistency of the quasi-maximum likelihood estimator for the population parameter values, on which consistency of the test is based, does not hold. We describe a condition that ensures consistency of the estimator and discuss the consistency of the test in the absence of consistency of the estimator. In Cho and White (2007), Testing for Regime Switching, the authors stud ied the asymptotic behavior of a statistic that tests the null hypothesis of one regime against the alternative of Markov switching between two regimes. A key insight is that a consistent test can be based on a quasi-likelihood that ignores the Markov structure of regime switching and treats the state variables that indicate regimes as a sequence of independent and identically distributed ran dom variables. Consistency of the test follows from consistency of the quasi maximum likelihood estimator (QMLE) under the alternative, which appears as Theorem 1(b) in Cho and White. Consistency of the QMLE requires that the expected quasi-log-likelihood attain a global maximum at the population parameter values. We show that this requirement does not hold for the au toregressive process analyzed in Cho and White. Thus, for models of regime switching in which the conditional mean contains autoregressive components, consistency of the test proposed by Cho and White has not been established. For the observable random variables {X, e Md}=1, d e N, the Markov regime-switching autoregressive process analyzed by Cho and White (Sec tion 3, p. 1697) is

BETWEEN 1940 AND 1950, Kaldor [12], Goodwin [9] and Hicks [11] showed that adequate models of business cycle have to be essentially nonlinear, as only nonlinear systems allow The study of their properties is useful besides the construction of models, for example, the stochastic stability of the system, when shocks and external perturbations occur. This problem, already complex for linear systems, becomes even more complex for nonlinear systems (Kushner [15], Astrom [5]). Klein and Preston [13] and Kosobud and O'Neil [14] obtained interesting results on stochastic stability of nonlinear models of business cycles. Another interesting aspect is the dependence of the business cycle on the parameters characterizing the system. This analysis can be a sort of framework for the control of business cycles. The of stability can provide useful tools to this end. The of stability is a fusion of the two concepts of stability and qualitative behavior in the sense of topological equivalence. Andronov and Pontriagin [3] considered differential equations in two variables in a closed domain. They said that a system X is rough if, by perturbating it slightly (in the Cl-sense), one gets a system Y equivalent to X (in the sense specified in Appendix 1). Later Lefshetz [17] translated rough to structural stable. Thom [24, 25] saw stability, broadly understood, as the preservation of qualitative features under small perturbations. Smale [21, 22], Peixoto [19], and Abraham and Robbin [1] developed the giving fundamental theorems. Sotomayor [23], Andronov et al. [4], Chafee [6], and Sattinger [20] started to give good basis for the so called theory of bifurcations. Points of bifurcation are, in a parameter space, points where the topological structure changes abruptly, that is where stability fails: the creation of limit cycles from a multiple focus (Hopf bifurcation), the creation of a closed trajectory from a multiple limit cycle .... The of bifurcation can provide new criteria to prove the existence of limit cycles, besides the classical ones of Poincare and Bendixon. For example, it is possible to prove the existence of a limit cycle, without resorting to the theorem of Bendixon-Poincare, as done by Chang and Smyth [7] or to the theorem of Levison and Smyth, as done by Ichimura [16] for the Kaldor model.