Herman K. van Dijk

A Bayesian analysis of the unit root in real exchange rates

We propose a posterior odds analysis of the hypothesis of a unit root in real exchange rates. From a Bayesian viewpoint the random walk hypothesis for real exchange rates is a posteriori as probable as a stationary AR(1) process for four out of eight time series investigated. The French franc/German mark is clearly stationary, while the Japanese yen/US dollar is most likely a random walk. In contrast, classical tests are unable to reject the unit root for any of these series.

Combined forecasts from linear and nonlinear time series models

Combined forecasts from a linear and a nonlinear model are investigated for time series with possibly nonlinear characteristics. The forecasts are combined by a constant coefficient regression method as well as a time varying method. The time varying method allows for a locally (non)linear model. The methods are applied to data from two kinds of disciplines: the Canadian lynx and sunspot series from the natural sciences, and Nelson-Plossers U.S. series from economics. It is shown that the combined forecasts perform well, especially with time varying coefficients. This result holds for out of sample performance for the sunspot and Canadian lynx number series, but it does not uniformly hold for economic time series.

On the Shape of the Likelihood/Posterior in Cointegration Models

A vector autoregressive (VAR) model is specified with equation system parameters, which directly reflect the possible cointegrating nature of the analyzed time series. By using a flat/diffuse prior, we show that the marginal posteriors of the parameters of interest (multipliers of the cointegrating vectors) may be nonintegrable and favor difference stationary models in an undesired way. To choose between stationary, cointegrated, and difference stationary models in a meaningful way, the Jeffreys prior principle is used. We investigate the sensitivity of the posterior results with respect to the construction of the Jeffreys prior. In this context, we also analyze the effect of fixed and stochastic initial values. The theoretical results are illustrated by using a VAR model for shortand long–term interest rates in the United States.

Bayesian specification analysis and estimation of simultaneous equation models using Monte Carlo methods

Bayesian procedures for specification analysis or diagnostic checking of modeling assumptions for structural equations of econometric models are developed and applied using Monte Carlo numerical methods. Checks on the validity of identifying restrictions, exogeneity assumptions and other specifying assumptions are performed using posterior distributions for discrepancy vectors and functions representing departures from specifying assumptions. Several mappings or functions of reduced form coefficients are defined and their posterior distributions are computed. A restricted reduced form approach is used to compute posterior distributions for structural parameters. These procedures are applied in analyses of two econometric models.

Efficient estimation of income distribution parameters

The parameters of several families of distributions are estimated by means of minimum χ2; use is made of random samples taken from Dutch income-earning groups in 1973. The numerical search routine used, is the Complex method due to Box. The χ2 function is evaluated by standard numerical integration procedures. The lognormal and the Gamma families are rejected because of a poor fit. The log t and the log Pearson IV families are introduced. This results in a considerable improvement of χ2 critical levels. The generalized Gamma and the Champernowne function describe the income distribution reasonably well in some cases.

Testing for integration using evolving trend and seasonals models: A Bayesian approach

In this paper, we make use of state space models to investigate the presence of stochastic trends in economic time series. A model is specified where such a trend can enter either in the autoregressive representation or in a separate state equation. Tests based on the former are analogous to Dickey-Fuller tests of unit roots, while the latter are analogous to KPSS tests of trend-stationarity. We use Bayesian methods to survey the properties of the likelihood function in such models and to calculate posterior odds ratios comparing models with and without stochastic trends. We extend these ideas to the problem of testing for integration at seasonal frequencies and show how our techniques can be used to carry out Bayesian variants of either the HEGY or Canova-Hansen test. Stochastic integration rules, based on Markov Chain Monte Carlo, as well as deterministic integration rules are used. Strengths and weaknesses of each approach are indicated.

Bayes Estimates of Markov Trends in Possibly Cointegrated Series: An Application to U.S. Consumption and Income

Stylized facts show that average growth rates of U.S. per capita consumption and income differ in recession and expansion periods. Because a linear combination of such series does not have to be a constant mean process, standard cointegration analysis between the variables to examine the permanent income hypothesis may not be valid. To model the changing growth rates in both series, we introduce a multivariate Markov trend model that accounts for different growth rates in consumption and income during expansions and recessions and across variables within both regimes. The deviations from the multivariate Markov trend are modeled by a vector autoregression (VAR) model. Bayes estimates of this model are obtained using Markov chain Monte Carlo methods. The empirical results suggest the existence of a cointegration relation between U.S. per capita disposable income and consumption, after correction for a multivariate Markov trend. This result is also obtained when per capita investment is added to the VAR.

An algorithm for the computation of posterior moments and densities using simple importance sampling

In earlier work (van Dijk (1984, Chapter 3)) one of the authors discussed the use of Monte Carlo integration methods for the computation of the multivariate integrals that are defined in the posterior moments and the liosterior densities of the parameters of interest of econometric models. In the present paper we describe the computational steps of one Monte Carlo method, mentioned In that work, which is known in the literature as importance sampling. Further, we have prepared a set of standard programs, which may be used for the implementation of a simple case of importance sampling. The computer programs have been written in Fortran.

Classical and Bayesian aspects of robust unit root inference

This paper has two themes. First, we classify some effects which outliers in the data have on unit root inference. We show that, both in a classical and a Bayesian framework, the presence of additive outliers moves ‘standard’ inference towards stationarity. Second, we base inference on an independent Student-t instead of a Gaussian likelihood. This yields results that are less sensitive to the presence of outliers. Application to several time series with outliers reveals a negative correlation between the unit root and degrees of freedom parameter of the Student-t distribution. Therefore, imposing normality may incorrectly provide evidence against the unit root.

Posterior moments computed by mixed integration

A flexible numerical integration method is proposed for the computation of moments of a multivariate posterior density with different tail properties in different directions. The method (called mixed integration) amounts to a combination of classical numerical integration and Monte Carlo integration. Mixed integration is parsimonious in the sense that is makes use of the same parameters as the more restrictive multivariate normal importance function. The method is applied in order to compute the posterior scores of three candidates for a professorship in operations research, taking into account four different decision criteria.