We estimate asset pricing models with multiple risks: long-run growth, long-run volatility, habit, and a residual. The Bayesian estimation accounts for the entire likelihood of consumption, dividends, and the pricedividend ratio. We find that the residual represents at least 80% of the variance of the price-dividend ratio. Moreover, the residual tracks most recognizable features of stock market history such as the 1990's boom and bust. Long run risks and habit contribute primarily in crises. The dominance of the residual comes from the low correlation between asset prices and consumption growth moments. We discuss theories which are consistent with our results.

A Newton-Krylov method is an implementation of Newton's method in which a Krylov subspace method is used to solve approximately the linear systems that characterize steps of Newton's method. Newton-Krylov methods are often implemented in "matrix-free" form, in which the Jacobian-vector products required by the Krylov solver are approximated by finite differences. Here we consider using approximate function values in these finite differences. We first formulate a finite-difference Arnoldi process that uses approximate function values. We then outline a Newton-Krylov method that uses an implementation of the GMRES or Arnoldi method based on this process, and we develop a local convergence analysis for it, giving sufficient conditions on the approximate function values for desirable local convergence properties to hold. We conclude with numerical experiments involving particular function-value approximations suitable for nonlinear diffusion problems. For this case, conditions are given for meeting the convergence assumptions for both lagging and linearizing the nonlinearity in the function evaluation.

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The maximum a posteriori penalty function (MAP-PF) approach is applied to tracking the bearing and bearing rate of multiple wideband sources in unknown colored noise. The track estimation problem is formulated directly from the array data using the maximum a posteriori (MAP) estimation criterion. The penalty function (PF) method of nonlinear programming is used to obtain a tractable solution. A sequential update procedure is developed in which penalized maximum likelihood estimates of target bearings, spectra, and noise parameters are computed and then used as synthetic measurements in a set of Kalman filter trackers. The two steps are coupled via the penalty function. During parameter estimation, the penalty function prevents erroneous outlier estimates which can cause the tracker to lose track. During the tracking step, it determines the influence of the parameter estimates on the final track estimates by adaptively adjusting the measurement error variance. Performance is demonstrated on a typical sonar scenario.