Estimating Euler equations

In this paper we consider conditions under which the estimation of a log-linearized Euler equation for consumption yields consistent estimates of preference parameters. When utility is isoelastic and a sample covering a long time period is available, consistent estimates are obtained from the loglinearized Euler equation when the innovations to the conditional variance of consumption growth are uncorrelated with the instruments typically used in estimation. We perform a Montecarlo experiment, consisting in solving and simulating a simple life cycle model under uncertainty, and show that in most situations, the estimates obtained from the log-linearized equation are not systematically biased. This is true even when we introduce heteroscedasticity in the process generating income. The only exception is when discount rates are very high (e.g. 47% per year). This problem arises because consumers are nearly always close to the maximum borrowing limit: the estimation bias is unrelated to the linearization and estimates using nonlinear GMM are as bad. Across all our situations, estimation using a log-linearized Euler equation does better than nonlinear GMM despite the absence of measurement error. Finally, we plot life cycle profiles for the variance of consumption growth, which, except when the discount factor is very high, is remarkably flat. This implies that claims that demographic variables in log-linearized Euler equations capture changes in the variance of consumption growth are unwarranted

A Blow-Up Criterion for the 3D Euler Equations Via the Euler-Voigt Inviscid Regularization

We propose a new blow-up criterion for the 3D Euler equations of incompressible fluid flows, based on the 3D Euler-Voigt inviscid regularization. This criterion is similar in character to a criterion proposed in a previous work by the authors, but it is stronger, and better adapted for computational tests. The 3D Euler-Voigt equations enjoy global well-posedness, and moreover are more tractable to simulate than the 3D Euler equations. A major advantage of these new criteria is that one only needs to simulate the 3D Euler-Voigt, and not the 3D Euler equations, to test the blow-up criteria, for the 3D Euler equations, computationally

A Nonlinear Analysis of the Averaged Euler Equations

This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter $\alpha$; one interpretation is that they are obtained by ensemble averaging the Euler equations in Lagrangian representation over rapid fluctuations whose amplitudes are of order $\alpha$. The particle flows associated with these equations are shown to be geodesics on a suitable group of volume preserving diffeomorphisms, just as with the Euler equations themselves (according to Arnold's theorem), but with respect to a right invariant $H^1$ metric instead of the $L^2$ metric. The equations are also equivalent to those for a certain second grade fluid. Additional properties of the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden theorem) are also shown to hold. Using this nonlinear analysis framework, the limit of zero viscosity for the corresponding viscous equations is shown to be a regular limit, {\it even in the presence of boundaries}.Comment: 25 pages, no figures, Dedicated to Vladimir Arnold on the occasion of his 60th birthday, Arnold Festschrift Volume 2 (in press

Stochastic isentropic Euler equations

We study the stochastically forced system of isentropic Euler equations of gas dynamics with a $\gamma$-law for the pressure. We show the existence of martingale weak entropy solutions; we also discuss the existence and characterization of invariant measures in the concluding section

Estimating Euler equations

In this paper we consider conditions under which the estimation of a log-linearized Euler equation for consumption yields consistent estimates of preference parameters. When utility is isoelastic and a sample covering a long time period is available, consistent estimates are obtained from the log-linearized Euler equation when the innovations to the conditional variance of consumption growth are uncorrelated with the instruments typically used in estimation. We perform a Montecarlo experiment, consisting in solving and simulating a simple life cycle model under uncertainty, and show that in most situations, the estimates obtained from the log-linearized equation are not systematically biased. This is true even when we introduce heteroscedasticity in the process generating income. The only exception is when discount rates are very high (e.g. 47% per year). This problem arises because consumers are nearly always close to the maximum borrowing limit: the estimation bias is unrelated to the linearization and estimates using nonlinear GMM are as bad. Across all our situations, estimation using a log-linearized Euler equation does better than nonlinear GMM despite the absence of measurement error. Finally, we plot life cycle profiles for the variance of consumption growth, which, except when the discount factor is very high, is remarkably flat. This implies that claims that demographic variables in log- linearized Euler equations capture changes in the variance of consumption growth are unwarranted.