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This paper develops a multi-period extension of the Lucas (1972) overlapping generations "island" model with endogenous monetary policy (based on the minimization of a loss function over inflation and output deviations) and stochastic realization of the "allocation" of the young people across the two islands. These allocation realizations are interpreted as output shocks (since only the young people produce). The paper examines two cases: the certainty case when the exact monetary policy is known to the young, and uncertainty case where the young receive only a mixed signal of the output shock and the monetary policy weights through the price (the signal extraction problem). In the certainty case, the neutrality result holds. In the uncertainty case, even monetary shocks have real effects as a result of the signal extraction problem. After characterizing the resulting price function by its constant elasticity to the signal, we derive values of this elasticity and the monetary policy weights such that hyperinflations will develop. We find that for certain weights, hyperinflations can develop even when the price function is concave in the signal. Finally, we formulate a particular convex case of the price function (making distributional assumptions) to analyze the price and monetary policy examples and statics as functions of the weights on the inflation and output deviation terms.

A procedure based on density estimation is suggested in the paper to discriminate trend stationary processes about local linear time trends from difference stationary processes. A 'rule of thumb' is constructed to detect the suitability of a segmented trend representation, and a regression analysis is used to identify the number and the dates of structural breaks. The U.S. series of nominal wages over the period 1900-1970 is analysed according to the assumption the dynamics are driven by exogenous shocks which occur infrequently. In a multivariate domain, implications of segmented trend modeling for cointegration theory are also briefly considered.

Sometimes, it is of interest to single out the fluctuations associated to a given frequency. We propose a new variant of SSA, Circulant SSA (CiSSA), that allows to extract the signal associated to any frequency specified beforehand. This is a novelty when compared with other SSA procedures that need to identify ex-post the frequencies associated to the extracted signals. We prove that CiSSA is asymptotically equivalent to these alternative procedures although with the advantage of avoiding the need of the subsequent frequency identification. We check its good performance and compare it to alternative SSA methods through several simulations for linear and nonlinear time series. We also prove its validity in the nonstationary case. We apply CiSSA in two different fields to show how it works with real data and find that it behaves successfully in both applications. Finally, we compare the performance of CiSSA with other state of the art techniques used for nonlinear and nonstationary signals with amplitude and frequency varying in time ; Financial support from the Spanish government, contract grants MINECO/FEDER ECO2015-70331-C2-1-R, ECO2015-66593-P, ECO2016-76818-C3-3-P, PID2019-107161GB-C32 and PID2019-108079GB-C22 is acknowledged