26,331 research outputs found
Multivariate Generalized Gaussian Distribution: Convexity and Graphical Models
We consider covariance estimation in the multivariate generalized Gaussian
distribution (MGGD) and elliptically symmetric (ES) distribution. The maximum
likelihood optimization associated with this problem is non-convex, yet it has
been proved that its global solution can be often computed via simple fixed
point iterations. Our first contribution is a new analysis of this likelihood
based on geodesic convexity that requires weaker assumptions. Our second
contribution is a generalized framework for structured covariance estimation
under sparsity constraints. We show that the optimizations can be formulated as
convex minimization as long the MGGD shape parameter is larger than half and
the sparsity pattern is chordal. These include, for example, maximum likelihood
estimation of banded inverse covariances in multivariate Laplace distributions,
which are associated with time varying autoregressive processes
Noncausal autoregressions for economic time series
This paper is concerned with univariate noncausal autoregressive models and their potential usefulness in economic applications. In these models, future errors are predictable, indicating that they can be used to empirically approach rational expectations models with nonfundamental solutions. In the previous theoretical literature, nonfundamental solutions have typically been represented by noninvertible moving average models. However, noncausal autoregressive and noninvertible moving average models closely approximate each other, and therefore,the former provide a viable and practically convenient alternative. We show how the parameters of a noncausal autoregressive model can be estimated by the method of maximum likelihood and derive related test procedures. Because noncausal autoregressive models cannot be distinguished from conventional causal autoregressive models by second order properties or Gaussian likelihood, a model selection procedure is proposed. As an empirical application, we consider modeling the U.S. inflation which, according to our results, exhibits purely forward-looking dynamics
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