On Arnol'd's second nonlinear stability theorem for two-dimensional quasi-geostrophic flow

Arnol'd's second hydrodynamical stability theorem, proven originally for the two-dimensional Euler equations, can establish nonlinear stability of steady flows that are maxima of a suitably chosen energy-Casimir invariant. The usual derivations of this theorem require an assumption of zero disturbance circulation. In the present work an analogue of Arnol'd's second theorem is developed in the more general case of two-dimensional quasi-geostrophic flow, with the important feature that the disturbances are allowed to have non-zero circulation. New nonlinear stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expressed in terms of the initial disturbance fields. While Arnol'd's stability method relies on the second variation of the energy-Casimir invariant being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance circulations. A version of Andrews' theorem is also established for this problem.

A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids

In addition to the Hamiltonian functional itself, non-canonical Hamiltonian dynamical systems generally possess integral invariants known as ‘Casimir functionals’. In the case of the Euler equations for a perfect fluid, the Casimir functionals correspond to the vortex topology, whose invariance derives from the particle-relabelling symmetry of the underlying Lagrangian equations of motion. In a recent paper, Vallis, Carnevale & Young (1989) have presented algorithms for finding steady states of the Euler equations that represent extrema of energy subject to given vortex topology, and are therefore stable. The purpose of this note is to point out a very general method for modifying any Hamiltonian dynamical system into an algorithm that is analogous to those of Vallis etal. in that it will systematically increase or decrease the energy of the system while preserving all of the Casimir invariants. By incorporating momentum into the extremization procedure, the algorithm is able to find steadily-translating as well as steady stable states. The method is applied to a variety of perfect-fluid systems, including Euler flow as well as compressible and incompressible stratified flow.

ENO adaptive method for solving one-dimensional conservation laws

In this work an efficient third order non-linear finite difference scheme for solving adaptively hyperbolic systems of one-dimensional conservation laws is developed. The method is based oil applying to the solution of the differential equation an interpolating wavelet transform at each time step, generating a multilevel representation for the solution, which is thresholded and a sparse point representation is generated. The numerical fluxes obtained by a Lax-Friedrichs flux splitting are evaluated oil the sparse grid by an essentially non-oscillatory (ENO) approximation, which chooses the locally smoothest stencil among all the possibilities for each point of the sparse grid. The time evolution of the differential operator is done on this sparse representation by a total variation diminishing (TVD) Runge-Kutta method. Four classical examples of initial value problems for the Euler equations of gas dynamics are accurately solved and their sparse solutions are analyzed with respect to the threshold parameters, confirming the efficiency of the wavelet transform as an adaptive grid generation technique. (C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.

Boundary and internal conditions for adjoint fluid-flow problems

The ever-increasing robustness and reliability of flow-simulation methods have consolidated CFD as a major tool in virtually all branches of fluid mechanics. Traditionally, those methods have played a crucial role in the analysis of flow physics. In more recent years, though, the subject has broadened considerably, with the development of optimization and inverse design applications. Since then, the search for efficient ways to evaluate flow-sensitivity gradients has received the attention of numerous researchers. In this scenario, the adjoint method has emerged as, quite possibly, the most powerful tool for the job, which heightens the need for a clear understanding of its conceptual basis. Yet, some of its underlying aspects are still subject to debate in the literature, despite all the research that has been carried out on the method. Such is the case with the adjoint boundary and internal conditions, in particular. The present work aims to shed more light on that topic, with emphasis on the need for an internal shock condition. By following the path of previous authors, the quasi-1D Euler problem is used as a vehicle to explore those concepts. The results clearly indicate that the behavior of the adjoint solution through a shock wave ultimately depends upon the nature of the objective functional.

Lie algebra methods for the applications to the statistical theory of turbulence

Approximate Lie symmetries of the Navier-Stokes equations are used for the applications to scaling phenomenon arising in turbulence. In particular, we show that the Lie symmetries of the Euler equations are inherited by the Navier-Stokes equations in the form of approximate symmetries that allows to involve the Reynolds number dependence into scaling laws. Moreover, the optimal systems of all finite-dimensional Lie subalgebras of the approximate symmetry transformations of the Navier-Stokes are constructed. We show how the scaling groups obtained can be used to introduce the Reynolds number dependence into scaling laws explicitly for stationary parallel turbulent shear flows. This is demonstrated in the framework of a new approach to derive scaling laws based on symmetry analysis [11]-[13].

A search-theoretic explanation for the negative correlation between labor income and impatience

Lawrance (1991) has shown, through the estimation of consumption Euler equations, that subjective rates of impatience (time preference) in the U.S. are three to Öve percentage points higher for households with lower average labor incomes than for those with higher labor income. From a theoretical perspective, the sign of this correlation in a job-search model seems at Örst to be undetermined, since more impatient workers tend to accept wage o§ers that less impatient workers would not, thereby remaining less time unemployed. The main result of this paper is showing that, regardless of the existence of e§ects of opposite sign, and independently of the particular speciÖcations of the givens of the model, less impatient workers always end up, in the long run, with a higher average income. The result is based on the (unique) invariant Markov distribution of wages associated with the dynamic optimization problem solved by the consumers. An example is provided to illustrate the method.

The forward- and the equity-premium puzzles: two symptoms of the same illness?

In this paper we revisit the relationship between the equity and the forward premium puzzles. We construct return-based stochastic discount factors under very mild assumptions and check whether they price correctly the equity and the foreign currency risk premia. We avoid log-linearizations by using moments restrictions associated with euler equations to test the capacity of our return-based stochastic discount factors to price returns on the relevant assets. Our main finding is that a pricing kernel constructed only using information on American domestic assets accounts for both domestic and international stylized facts that escape consumption based models. In particular, we fail to reject the null hypothesis that the foreign currency risk premium has zero price when the instrument is the own current value of the forward premium.

The forward- and the equity-premium puzzles: two symptoms of the same illness?

We build a pricing kernel using only US domestic assets data and check whether it accounts for foreign markets stylized facts that escape consumption based models. By interpreting our stochastic discount factor as the projection of a pricing kernel from a fully specified model in the space of returns, our results indicate that a model that accounts for the behavior of domestic assets goes a long way toward accounting for the behavior of foreign assets. We address predictability issues associated with the forward premium puzzle by: i) using instruments that are known to forecast excess returns in the moments restrictions associated with Euler equations, and; ii) by pricing Lustig and Verdelhan (2007)'s foreign currency portfolios. Our results indicate that the relevant state variables that explain foreign-currency market asset prices are also the driving forces behind U.S. domestic assets behavior.

The forward- and the equity-premium puzzles: two symptoms of the same illness?

Using information on US domestic financial data only, we build a stochastic discount factor—SDF— and check whether it accounts for foreign markets stylized facts that escape consumption based models. By interpreting our SDF as the projection of a pricing kernel from a fully specified model in the space of returns, our results indicate that a model that accounts for the behavior of domestic assets goes a long way toward accounting for the behavior of foreign assets prices. We address predictability issues associated with the forward premium puzzle by: i) using instruments that are known to forecast excess returns in the moments restrictions associated with Euler equations, and; ii) by pricing Lustig and Verdelhan (2007)’s foreign currency portfolios. Our results indicate that the relevant state variables that explain foreign-currency market asset prices are also the driving forces behind U.S. domestic assets behavior.

The forward- and the equity-premium puzzles: two symptoms of the same illness?

Using information on US domestic financial data only, we build a stochastic discount factor—SDF— and check whether it accounts for foreign markets stylized facts that escape consumption based models. By interpreting our SDF as the projection of a pricing kernel from a fully specified model in the space of returns, our results indicate that a model that accounts for the behavior of domestic assets goes a long way toward accounting for the behavior of foreign assets prices. We address predictability issues associated with the forward premium puzzle by: i) using instruments that are known to forecast excess returns in the moments restrictions associated with Euler equations, and; ii) by comparing this out-of-sample results with the one obtained performing an in-sample exercise, where the return-based SDF captures sources of risk of a representative set of developed and emerging economies government bonds. Our results indicate that the relevant state variables that explain foreign-currency market asset prices are also the driving forces behind U.S. domestic assets behavior.

The forward and the equity-premium puzzles: a straightforward test of whether they are two symptoms of the same illness

We build a stochastic discount factor—SDF— using information on US domestic financial data only, and provide evidence that it accounts for foreign markets stylized facts that escape SDF’s generated by consumption based models. By interpreting our SDF as the projection of the pricing kernel from a fully specified model in the space of returns, our results indicate that a model that accounts for the behavior of domestic assets goes a long way toward accounting for the behavior of foreign assets prices. In our tests, we address predictability, a defining feature of the Forward Premium Puzzle—FPP— by using instruments that are known to forecast excess returns in the moments restrictions associated with Euler equations both in the equity and the foreign markets.

A note on the forward and the equity-premium puzzles: two symptoms of the same illness?

We build a stochastic discount factor—SDF— using information on US domestic financial data only, and provide evidence that it accounts for foreign markets stylized facts that escape SDF’s generated by consumption based models. By interpreting our SDF as the projection of the pricing kernel from a fully specified model in the space of returns, our results indicate that a model that accounts for the behavior of domestic assets goes a long way toward accounting for the behavior of foreign assets prices. In our tests, we address predictability, a defining feature of the Forward Premium Puzzle—FPP— by using instruments that are known to forecast excess returns in the moments restrictions associated with Euler equations both in the equity and the foreign markets.

Testing consumption optimality using aggregate data

The objective of this paper is to test for optimality of consumption decisions at the aggregate level (representative consumer) taking into account popular deviations from the canonical CRRA utility model rule of thumb and habit. First, we show that rule-of-thumb behavior in consumption is observational equivalent to behavior obtained by the optimizing model of King, Plosser and Rebelo (Journal of Monetary Economics, 1988), casting doubt on how reliable standard rule-of-thumb tests are. Second, although Carroll (2001) and Weber (2002) have criticized the linearization and testing of euler equations for consumption, we provide a deeper critique directly applicable to current rule-of-thumb tests. Third, we show that there is no reason why return aggregation cannot be performed in the nonlinear setting of the Asset-Pricing Equation, since the latter is a linear function of individual returns. Fourth, aggregation of the nonlinear euler equation forms the basis of a novel test of deviations from the canonical CRRA model of consumption in the presence of rule-of-thumb and habit behavior. We estimated 48 euler equations using GMM, with encouraging results vis-a-vis the optimality of consumption decisions. At the 5% level, we only rejected optimality twice out of 48 times. Empirical-test results show that we can still rely on the canonical CRRA model so prevalent in macroeconomics: out of 24 regressions, we found the rule-of-thumb parameter to be statistically signi cant at the 5% level only twice, and the habit ƴ parameter to be statistically signi cant on four occasions. The main message of this paper is that proper return aggregation is critical to study intertemporal substitution in a representative-agent framework. In this case, we fi nd little evidence of lack of optimality in consumption decisions, and deviations of the CRRA utility model along the lines of rule-of-thumb behavior and habit in preferences represent the exception, not the rule.

Testing consumption optimality using aggregate data

This paper tests the optimality of consumption decisions at the aggregate level taking into account popular deviations from the canonical constant-relative-risk-aversion (CRRA) utility function model-rule of thumb and habit. First, based on the critique in Carroll (2001) and Weber (2002) of the linearization and testing strategies using euler equations for consumption, we provide extensive empirical evidence of their inappropriateness - a drawback for standard rule- of-thumb tests. Second, we propose a novel approach to test for consumption optimality in this context: nonlinear estimation coupled with return aggregation, where rule-of-thumb behavior and habit are special cases of an all encompassing model. We estimated 48 euler equations using GMM. At the 5% level, we only rejected optimality twice out of 48 times. Moreover, out of 24 regressions, we found the rule-of-thumb parameter to be statistically significant only twice. Hence, lack of optimality in consumption decisions represent the exception, not the rule. Finally, we found the habit parameter to be statistically significant on four occasions out of 24.

Do equity and foreign currency risk premiums display common patterns?

In da Costa et al. (2006) we have shown how a same pricing kernel can account for the excess returns of the S&:P500 over the US short term bond and of the uncovered over the covered trading of foreign government bonds. In this paper we estimate and test the overidentifying restrictiom; of Euler equations associated with "ix different versions of the Consumption Capital Asset Pricing I\Iodel. Our main finding is that the same (however often unreasonable) values for the parameters are estimated for ali models in both nmrkets. In most cases, the rejections or otherwise of overidentifying restrictions occurs for the two markets, suggesting that success and failure stories for the equity premium repeat themselves in foreign exchange markets. Our results corroborate the findings in da Costa et al. (2006) that indicate a strong similarity between the behavior of excess returns in the two markets when modeled as risk premiums, providing empirical grounds to believe that the proposed preference-based solutions to puzzles in domestic financiaI markets can certainly shed light on the Forward Premium Puzzle.