Scattering in one dimension: The coupled Schroedinger equation, threshold behaviour and Levinson's theorem

We formulate scattering in one dimension due to the coupled Schr\"{o}dinger equation in terms of the $S$ matrix, the unitarity of which leads to constraints on the scattering amplitudes. Levinson's theorem is seen to have the form $\eta(0) = \pi (n_b + 1/2 n - 1/2 N)$, where $\eta(0)$ is the phase of the $S$ matrix at zero energy, $n_b$ the number of bound states with nonzero binding energy, $n$ the number of half-bound states, and $N$ the number of coupled equations. In view of the effects due to the half-bound states, the threshold behaviour of the scattering amplitudes is investigated in general, and is also illustrated by means of particular potential models.Comment: to appear in Journal of Mathematic Physics, RevTex, 16 pages, 3 figures (PostScript

Bayesian Model Averaging in Vector Autoregressive Processes with an Investigation of Stability of the US Great Ratios and Risk of a Liquidity Trap in the USA, UK and Japan.

A Bayesian model averaging procedure is presented within the class of vector autoregressive (VAR) processes and applied to two empirical issues. First, stability of the Great Ratios in U.S. macro-economic time series is investigated, together with the presence and effects of permanent shocks. Measures on manifolds are employed in order to elicit uniform priors on subspaces defined by particular structural features of linear VARs. Second, the VAR model is extended to include a smooth transition function in a (monetary) equation and stochastic volatility in the disturbances. The risk of a liquidity trap in the USA, UK and Japan is evaluated, together with the expected cost of a policy adjustment of central banks. Posterior probabilities of different models are evaluated usingMarkov chainMonte Carlo techniques.

Evidence on a DSGE Business Cycle model subject to Neutral and Investment-Specific Technology Shocks using Bayesian Model Averaging

The empirical support for a DSGE type of real business cycle model with two technology shocks is evaluated using a Bayesian model averaging procedure that makes use of a finite mixture of many models within the class of vector autoregressive (VAR) processes. The linear VAR model is extended to permit equilibrium restrictions and restrictions on long-run responses to technology shocks apart from having a range of lag structures and deterministic processes. These model features are weighted as posterior probabilites and computed using MCMC and analytical methods. Uncertainty exists as to the most appropriate model for our data, with five models receiving significant support. The model set used has substantial implications for the results obtained. We do find support for a number of features implied by the real business cycle model. Business cycle volatility seems more due to investment specific technology shocks than neutral technology shocks and this result is robust to model specification. These techonolgy schocks appear to account for all stochastic trends in our system after 1984. we provide evidence on the uncertainty bands associated with these results.

Range-based covariance estimation using high-frequency data: The realized co-range

We introduce the realized co-range, utilizing intraday high-lowprice ranges to estimate asset return covariances. Using simulationswe find that for plausible levels of bid-ask bounce and infrequentand non-synchronous trading the realized co-range improves upon therealized covariance, which uses cross-products of intraday returns.One advantage of the co-range is that in an ideal world it is fivetimes more efficient than the realized covariance when sampling atthe same frequency. The second advantage is that the upward bias dueto bid-ask bounce and the downward bias due to infrequent andnon-synchronous trading partially offset each other. In a volatilitytiming strategy for S\\&P500, bond and gold futures we find that theco-range estimates are less noisy as exemplified by lowertransaction costs and also higher Sharpe ratios when using moreweight on recent data for predicting covariances.bias-correction;market microstructure noise;high-frequency date;realized co-range;realized covariance

Model Uncertainty and Bayesian Model Averaging in Vector Autoregressive Processes

Economic forecasts and policy decisions are often informed by empirical analysis based on econometric models. However, inference based upon a single model, when several viable models exist, limits its usefulness. Taking account of model uncertainty, a Bayesian model averaging procedure is presented which allows for unconditional inference within the class of vector autoregressive (VAR) processes. Several features of VAR process are investigated. Measures on manifolds are employed in order to elicit uniform priors on subspaces defined by particular structural features of VARs. The features considered are the number and form of the equilibrium economic relations and deterministic processes. Posterior probabilities of these features are used in a model averaging approach for forecasting and impulse response analysis. The methods are applied to investigate stability of the "Great Ratios" in U.S. consumption, investment and income, and the presence and effects of permanent shocks in these series. The results obtained indicate the feasibility of the proposed method.Posterior probability; Grassman manifold; Orthogonal group; Cointegration; Model averaging; Stochastic trend; Impulse response; Vector autoregressive model.

Improper priors with well defined Bayes Factors

While some improper priors have attractive properties, it is generally claimed that Bartlett’s paradox implies that using improper priors for the parameters in alternative models results in Bayes factors that are not well defined, thus preventing model comparison in this case. In this paper we demonstrate, using well understood principles underlying what is already common practice, that this latter result is not generally true and so expand the class of priors that may be used for computing posterior odds to two classes of improper priors: the shrink age prior; and a prior based upon a nesting argument. Using a new representation of the issue of undefined Bayes factors, we develop classes of improper priors from which well defined Bayes factors result. However, as the use of such priors is not free of problems, we include discussion on the issues with using such priors for model comparison.Improper prior; Bayes factor; marginal likelihood; shrinkage prior; measure

Simulation based Bayesian econometric inference: principles and some recent computational advances

In this paper we discuss several aspects of simulation based Bayesian econometric inference. We start at an elementary level on basic concepts of Bayesian analysis; evaluating integrals by simulation methods is a crucial ingredient in Bayesian inference. Next, the most popular and well-known simulation techniques are discussed, the MetropolisHastings algorithm and Gibbs sampling (being the most popular Markov chain Monte Carlo methods) and importance sampling. After that, we discuss two recently developed sampling methods: adaptive radial based direction sampling [ARDS], which makes use of a transformation to radial coordinates, and neural network sampling, which makes use of a neural network approximation to the posterior distribution of interest. Both methods are especially useful in cases where the posterior distribution is not well-behaved, in the sense of having highly non-elliptical shapes. The simulation techniques are illustrated in several example models, such as a model for the real US GNP and models for binary data of a US recession indicator.