http://repub.eur.nl/res/aut/263/ List of Publications en http://repub.eur.nl/static-eur/img/logo.png http://repub.eur.nl http://repub.eur.nl/res/pub/40160/ 2013-04-30T00:00:00Z http://repub.eur.nl/res/pub/39840/ 2013-04-08T00:00:00Z This paper presents the Matlab package DeCo (Density Combination) which is based on the paper by Billio et al. (2013) where a constructive Bayesian approach is presented for combining predictive densities originating from different models or other sources of information. The combination weights are time-varying and may depend on past predictive forecasting performances and other learning mechanisms. The core algorithm is the function DeCo which applies banks of parallel Sequential Monte Carlo algorithms to filter the time-varying combination weights. The DeCo procedure has been implemented both for standard CPU computing and for Graphical Process Unit (GPU) parallel computing. For the GPU implementation we use the Matlab parallel computing toolbox and show how to use General Purposes GPU computing almost effortless. This GPU implementation comes with a speed up of the execution time up to seventy times compared to a standard CPU Matlab implementation on a multicore CPU. We show the use of the package and the computational gain of the GPU version, through some simulation experiments and empirical applications. http://repub.eur.nl/res/pub/38911/ 2013-02-01T00:00:00Z The empirical support for features of a Dynamic Stochastic General Equilibrium model with two technology shocks is evaluated using Bayesian model averaging over vector autoregressions. The model features include equilibria, restrictions on long-run responses, a structural break of unknown date, and a range of lags and deterministic processes. We find support for a number of features implied by the economic model, and the evidence suggests a break in the entire model structure around 1984, after which technology shocks appear to account for all stochastic trends. Business cycle volatility seems more due to investment-specific technology shocks than neutral technology shocks. http://repub.eur.nl/res/pub/37738/ 2012-12-01T00:00:00Z A class of adaptive sampling methods is introduced for efficient posterior and predictive simulation. The proposed methods are robust in the sense that they can handle target distributions that exhibit non-elliptical shapes such as multimodality and skewness. The basic method makes use of sequences of importance weighted Expectation Maximization steps in order to efficiently construct a mixture of Student-t densities that approximates accurately the target distribution-typically a posterior distribution, of which we only require a kernel-in the sense that the Kullback-Leibler divergence between target and mixture is minimized. We label this approach Mixture of t by Importance Sampling weighted Expectation Maximization (MitISEM). The constructed mixture is used as a candidate density for quick and reliable application of either Importance Sampling (IS) or the Metropolis-Hastings (MH) method. We also introduce three extensions of the basic MitISEM approach. First, we propose a method for applying MitISEM in a sequential manner, so that the candidate distribution for posterior simulation is cleverly updated when new data become available. Our results show that the computational effort reduces enormously, while the quality of the approximation remains almost unchanged. This sequential approach can be combined with a tempering approach, which facilitates the simulation from densities with multiple modes that are far apart. Second, we introduce a permutation-augmented MitISEM approach. This is useful for importance or Metropolis-Hastings sampling from posterior distributions in mixture models without the requirement of imposing identification restrictions on the model's mixture regimes' parameters. Third, we propose a partial MitISEM approach, which aims at approximating the joint distribution by estimating a product of marginal and conditional distributions. This division can substantially reduce the dimension of the approximation problem, which facilitates the application of adaptive importance sampling for posterior simulation in more complex models with larger numbers of parameters. Our results indicate that the proposed methods can substantially reduce the computational burden in econometric models like DCC or mixture GARCH models and a mixture instrumental variables model. http://repub.eur.nl/res/pub/38747/ 2012-12-01T00:00:00Z Changing time series properties of US inflation and economic activity are analyzed within a class of extended Phillips Curve (PC) models. First, the misspecification effects of mechanical removal of low frequency movements of these series on posterior inference of a basic PC model are analyzed using a Bayesian simulation based approach. Next, structural time series models that describe changing patterns in low and high frequencies and backward as well as forward inflation expectation mechanisms are incorporated in the class of extended PC models. Empirical results indicate that the proposed models compare favorably with existing Bayesian Vector Autoregressive and Stochastic Volatility models in terms of fit and predictive performance. Weak identification and dynamic persistence appear less important when time varying dynamics of high and low frequencies are carefully modeled. Modeling inflation expectations using survey data and adding level shifts and stochastic volatility improves substantially in sample fit and out of sample predictions. No evidence is found of a long run stable cointegration relation between US inflation and marginal costs. Tails of the complete predictive distributions indicate an increase in the probability of disinflation in recent years. http://repub.eur.nl/res/pub/38198/ 2012-10-29T00:00:00Z We propose a Bayesian combination approach for multivariate predictive densities which relies upon a distributional state space representation of the combination weights. Several specifications of multivariate time-varying weights are introduced with a particular focus on weight dynamics driven by the past performance of the predictive densities and the use of learning mechanisms. In the proposed approach the model set can be incomplete, meaning that all models can be individually misspecified. A Sequential Monte Carlo method is proposed to approximate the filtering and predictive densities. The combination approach is assessed using statistical and utility-based performance measures for evaluating density forecasts. Simulation results indicate that, for a set of linear autoregressive models, the combination strategy is successful in selecting, with probability close to one, the true model when the model set is complete and it is able to detect parameter instability when the model set includes the true model that has generated subsamples of data. For the macro series we find that incompleteness of the models is relatively large in the 70's, the beginning of the 80's and during the recent financial crisis, and lower during the Great Moderation. With respect to returns of the S&P 500 series, we find that an investment strategy using a combination of predictions from professional forecasters and from a white noise model puts more weight on the white noise model in the beginning of the 90's and switches to giving more weight to the professional forecasts over time. http://repub.eur.nl/res/pub/37314/ 2012-09-21T00:00:00Z We discuss Bayesian inferential procedures within the family of instrumental variables regression models and focus on two issues: existence conditions for posterior moments of the parameters of interest under a flat prior and the potential of Direct Monte Carlo (DMC) approaches for efficient evaluation of such possibly highly onelliptical posteriors. We show that, for the general case of m endogenous variables under a flat prior, posterior moments of order r exist for the coefficients reflecting the endogenous regressors’ effect on the dependent variable, if the number of instruments is greater than m+r, even though there is an issue of local non-identification that causes non-elliptical shapes of the posterior. This stresses the need for efficient Monte Carlo integration methods. We introduce an extension of DMC that incorporates an acceptance-rejection sampling step within DMC. This Acceptance-Rejection within Direct Monte Carlo (ARDMC) method has the attractive property that the generated random drawings are independent, which greatly helps the fast convergence of simulation results, and which facilitates the evaluation of the numerical accuracy. The speed of ARDMC can be easily further improved by making use of parallelized computation using multiple core machines or computer clusters. We note that ARDMC is an analogue to the well-known 'Metropolis-Hastings within Gibbs' sampling in the sense that one 'more difficult' step is used within an 'easier' simulation method. We compare the ARDMC approach with the Gibbs sampler using simulated data and two empirical data sets, involving the settler mortality instrument of Acemoglu et al. (2001) and father's education's instrument used by Hoogerheide et al. (2012a). Even without making use of parallelized computation, an efficiency gain is observed both under strong and weak instruments, where the gain can be enormous in the latter case. http://repub.eur.nl/res/pub/37313/ 2012-09-20T00:00:00Z This paper presents the R package MitISEM, which provides an automatic and flexible method to approximate a non-elliptical target density using adaptive mixtures of Student-t densities, where only a kernel of the target density is required. The approximation can be used as a candidate density in Importance Sampling or Metropolis Hastings methods for Bayesian inference on model parameters and probabilities. The package provides also an extended MitISEM algorithm, ‘sequential MitISEM’, which substantially decreases the computational time when the target density has to be approximated for increasing data samples. This occurs when the posterior distribution is updated with new observations and/or when one computes model probabilities using predictive likelihoods. We illustrate the MitISEM algorithm using three canonical statistical and econometric models that are characterized by several types of non-elliptical posterior shapes and that describe well-known data patterns in econometrics and finance. We show that the candidate distribution obtained by MitISEM outperforms those obtained by ‘naive’ approximations in terms of numerical efficiency. Further, the MitISEM approach can be used for Bayesian model comparison, using the predictive likelihoods. http://repub.eur.nl/res/pub/37710/ 2012-09-13T00:00:00Z We propose new forecast combination schemes for predicting turning points of business cycles. The proposed combination schemes are based on the forecasting performances of a given set of models with the aim to provide better turning point predictions. In particular, we consider predictions generated by autoregressive (AR) and Markov-switching AR models, which are commonly used for business cycle analysis. In order to account for parameter uncertainty we consider a Bayesian approach for both estimation and prediction and compare, in terms of statistical accuracy, the individual models and the combined turning point predictions for the United States and the Euro area business cycles. http://repub.eur.nl/res/pub/32101/ 2012-03-20T00:00:00Z The empirical support for features of a Dynamic Stochastic General Equilibrium model with two technology shocks is valuated using Bayesian model averaging over vector autoregressions. The model features include equilibria, restrictions on long-run responses, a structural break of unknown date and a range of lags and deterministic processes. We find support for a number of features implied by the economic model and the evidence suggests a break in the entire model structure around 1984 after which technology shocks appear to account for all stochastic trends. Business cycle volatility seems more due to investment specific technology shocks than neutral technology shocks. http://repub.eur.nl/res/pub/32010/ 2012-01-01T00:00:00Z http://repub.eur.nl/res/pub/30684/ 2011-11-30T00:00:00Z We propose a multivariate combination approach to prediction based on a distributional state space representation of the weights belonging to a set of Bayesian predictive densities which have been obtained from alternative models. Several specifications of multivariate time-varying weights are introduced with a particular focus on weight dynamics driven by the past performance of the predictive densities and the use of learning mechanisms. In the proposed approach the model set can be incomplete, meaning that all models are individually misspecified. The approach is assessed using statistical and utility-based performance measures for evaluating density forecasts of US macroeconomic time series and surveys of stock market prices. For the macro series we find that incompleteness of the models is relatively large in the 70's, the beginning of the 80's and during the recent financial crisis; structural changes like the Great Moderation are empirically identified by our model combination and the predicted probabilities of recession accurately compare with the NBER business cycle dating. Model weights have substantial uncertainty attached and neglecting this may seriously affect results. With respect to returns of the S&P 500 series, we find that an investment strategy using a combination of predictions from professional forecasters and from a white noise model puts more weight on the white noise model in the beginning of the 90's and switches to giving more weight to the left tail of the professional forecasts during the start of the financial crisis around 2008. http://repub.eur.nl/res/pub/26507/ 2011-09-27T00:00:00Z A Direct Monte Carlo (DMC) approach is introduced for posterior simulation in the Instrumental Variables (IV) model with one possibly endogenous regressor, multiple instruments and Gaussian errors under a flat prior. This DMC method can also be applied in an IV model (with one or multiple instruments) under an informative prior for the endogenous regressor's effect. This DMC approach can not be applied to more complex IV models or Simultaneous Equations Models with multiple endogenous regressors. An Approximate DMC (ADMC) approach is introduced that makes use of the proposed Hybrid Mixture Sampling (HMS) method, which facilitates Metropolis-Hastings (MH) or Importance Sampling from a proper marginal posterior density with highly non-elliptical shapes that tend to infinity for a point of singularity. After one has simulated from the irregularly shaped marginal distri- bution using the HMS method, one easily samples the other parameters from their conditional Student-t and Inverse-Wishart posteriors. An example illustrates the close approximation and high MH acceptance rate. While using a simple candidate distribution such as the Student-t may lead to an infinite variance of Importance Sampling weights. The choice between the IV model and a simple linear model un- der the restriction of exogeneity may be based on predictive likelihoods, for which the efficient simulation of all model parameters may be quite useful. In future work the ADMC approach may be extended to more extensive IV models such as IV with non-Gaussian errors, panel IV, or probit/logit IV. http://repub.eur.nl/res/pub/26505/ 2011-09-01T00:00:00Z Patton and Timmermann (2011, 'Forecast Rationality Tests Based on Multi-Horizon Bounds', Journal of Business & Economic Statistics, forthcoming) propose a set of useful tests for forecast rationality or optimality under squared error loss, including an easily implemented test based on a regression that only involves (long-horizon and short-horizon) forecasts and no observations on the target variable. We propose an extension, a simulation-based procedure that takes into account the presence of errors in parameter estimates. This procedure can also be applied in the field of 'backtesting' models for Value-at-Risk. Applications to simple AR and ARCH time series models show that its power in detecting certain misspecifications is larger than the power of well-known tests for correct Unconditional Coverage and Conditional Coverage. http://repub.eur.nl/res/pub/23459/ 2011-05-02T00:00:00Z We summarize the general combination approach by Billio et al. [2010]. In the combination model the weights follow logistic autoregressive processes, change over time and their dynamics are possible driven by the past forecasting performances of the predictive densities. For illustrative purposes we apply it to combine White Noise and GARCH models to forecast the Amsterdam Exchange index and use the combined predictive forecasts in an investment asset allocation exercise. http://repub.eur.nl/res/pub/22330/ 2011-01-04T00:00:00Z Using a Bayesian framework this paper provides a multivariate combination approach to prediction based on a distributional state space representation of predictive densities from alternative models. In the proposed approach the model set can be incomplete. Several multivariate time-varying combination strategies are introduced. In particular, a weight dynamics driven by the past performance of the predictive densities is considered and the use of learning mechanisms. The approach is assessed using statistical and utility-based performance measures for evaluating density forecasts of US macroeconomic time series and of surveys of stock market prices. http://repub.eur.nl/res/pub/22332/ 2011-01-01T00:00:00Z A class of adaptive sampling methods is introduced for efficient posterior and predictive simulation. The proposed methods are robust in the sense that they can handle target distributions that exhibit non-elliptical shapes such as multimodality and skewness. The basic method makes use of sequences of importance weighted Expectation Maximization steps in order to efficiently construct a mixture of Student-t densities that approximates accurately the target distribution -typically a posterior distribution, of which we only require a kernel - in the sense that the Kullback-Leibler divergence between target and mixture is minimized. We label this approach Mixture of t by Importance Sampling and Expectation Maximization (MitISEM). We also introduce three extensions of the basic MitISEM approach. First, we propose a method for applying MitISEM in a sequential manner, so that the candidate distribution for posterior simulation is cleverly updated when new data become available. Our results show that the computational effort reduces enormously. This sequential approach can be combined with a tempering approach, which facilitates the simulation from densities with multiple modes that are far apart. Second, we introduce a permutation-augmented MitISEM approach, for importance sampling from posterior distributions in mixture models without the requirement of imposing identification restrictions on the model's mixture regimes' parameters. Third, we propose a partial MitISEM approach, which aims at approximating the marginal and conditional posterior distributions of subsets of model parameters, rather than the joint. This division can substantially reduce the dimension of the approximation problem. http://repub.eur.nl/res/pub/22334/ 2011-01-01T00:00:00Z Divergent priors are improper when defined on unbounded supports. Bartlett's paradox has been taken to imply that using improper priors results in ill-defined Bayes factors, preventing model comparison by posterior probabilities. However many improper priors have attractive properties that econometricians may wish to access and at the same time conduct model comparison. We present a method of computing well defined Bayes factors with divergent priors by setting rules on the rate of diffusion of prior certainty. The method is exact; no approximations are used. As a further result, we demonstrate that exceptions to Bartlett's paradox exist. That is, we show it is possible to construct improper priors that result in well defined Bayes factors. One important improper prior, the Shrinkage prior due to Stein (1956), is one such example. This example highlights pathologies with the resulting Bayes factors in such cases, and a simple solution is presented to this problem. A simple Monte Carlo experiment demonstrates the applicability of the approach developed in this paper. http://repub.eur.nl/res/pub/21335/ 2010-10-18T00:00:00Z Strategic choices for efficient and accurate evaluation of marginal likelihoods by means of Monte Carlo simulation methods are studied for the case of highly non-elliptical posterior distributions. A comparative analysis is presented of possible advantages and limitations of different simulation techniques; of possible choices of candidate distributions and choices of target or warped target distributions; and finally of numerical standard errors. The importance of a robust and flexible estimation strategy is demonstrated where the complete posterior distribution is explored. Given an appropriately yet quickly tuned adaptive candidate, straightforward importance sampling provides a computationally efficient estimator of the marginal likelihood (and a reliable and easily computed corresponding numerical standard error) in the cases investigated, which include a non-linear regression model and a mixture GARCH model. Warping the posterior density can lead to a further gain in efficiency, but it is more important that the posterior kernel be appropriately wrapped by the candidate distribution than that it is warped. http://repub.eur.nl/res/pub/19830/ 2010-06-01T00:00:00Z Strategic choices for efficient and accurate evaluation of marginal likelihoods by means of Monte Carlo simulation methods are studied for the case of highly non-elliptical posterior distributions. A comparative analysis is presented of possible advantages and limitations of different simulation techniques; of possible choices of candidate distributions and choices of target or warped target distributions; and finally of numerical standard errors. The importance of a robust and flexible estimation strategy is demonstrated where the complete posterior distribution is explored. Given an appropriately yet quickly tuned adaptive candidate, straightforward importance sampling provides a computationally efficient estimator of the marginal likelihood (and a reliable and easily computed corresponding numerical standard error) in the cases investigated in this paper, which include a non-linear regression model and a mixture GARCH model. Warping the posterior density can lead to a further gain in efficiency, but it is more important that the posterior kernel is appropriately wrapped by the candidate distribution than that is warped. http://repub.eur.nl/res/pub/19511/ 2010-05-17T00:00:00Z The empirical support for a real business cycle model with two technology shocks is evaluated using a Bayesian model averaging procedure. This procedure 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 cointegration, a range of deterministic processes, equilibrium restrictions and restrictions on long-run responses to technology shocks. We find support for a number of the features implied by the real business cycle model. For example, restricting long run responses to identify technology shocks has reasonable support and important implications for the short run responses to these shocks. Further, there is evidence that savings and investment ratios form stable relationships, but technology shocks do not account for all stochastic trends in our system. There is uncertainty as to the most appropriate model for our data, with thirteen models receiving similar support, and the model or model set used has signficant implications for the results obtained. http://repub.eur.nl/res/pub/18574/ 2010-03-01T00:00:00Z Several Bayesian model combination schemes, including some novel approaches that simultaneously allow for parameter uncertainty, model uncertainty and robust time-varying model weights, are compared in terms of forecast accuracy and economic gains using financial and macroeconomic time series. The results indicate that the proposed time-varying model weight schemes outperform other combination schemes in terms of predictive and economic gains. In an empirical application using returns on the S&P 500 index, time-varying model weights provide improved forecasts with substantial economic gains in an investment strategy including transaction costs. Another empirical example refers to forecasting US economic growth over the business cycle. It suggests that time-varying combination schemes may be very useful in business cycle analysis and forecasting, as these may provide an early indicator for recessions. http://repub.eur.nl/res/pub/16303/ 2009-07-16T00:00:00Z Several Bayesian model combination schemes, including some novel approaches that simultaneously allow for parameter uncertainty, model uncertainty and robust time varying model weights, are compared in terms of forecast accuracy and economic gains using financial and macroeconomic time series. The results indicate that the proposed time varying model weight schemes outperform other combination schemes in terms of predictive and economic gains. In an empirical application using returns on the S&P 500 index, time varying model weights provide improved forecasts with substantial economic gains in an investment strategy including transaction costs. Another empirical example refers to forecasting US economic growth over the business cycle. It suggests that time varying combination schemes may be very useful in business cycle analysis and forecasting, as these may provide an early indicator for recessions. http://repub.eur.nl/res/pub/18272/ 2009-04-15T00:00:00Z http://repub.eur.nl/res/pub/14943/ 2009-02-01T00:00:00Z Quadratic optimization for asset portfolios often leads to error maximization, with optimizers zooming in on large errors in the predicted inputs, that is, expected returns and risks. The consequence in most cases is a poor real-time performance. In this paper we show how to improve real-time performance of the popular equity momentum strategy with robust optimization in an empirical application involving 1500-2500 US stocks over the period 1963-2006. We also show that popular procedures like Bayes-Stein estimated expected returns, shrinking the covariance matrix and adding weight constraints fail in such a practical case http://repub.eur.nl/res/pub/14945/ 2009-02-01T00:00:00Z Important choices for efficient and accurate evaluation of marginal likelihoods by means of Monte Carlo simulation methods are studied for the case of highly non-elliptical posterior distributions. We focus on the situation where one makes use of importance sampling or the independence chain Metropolis-Hastings algorithm for posterior analysis. A comparative analysis is presented of possible advantages and limitations of different simulation techniques; of possible choices of candidate distributions and choices of target or warped target distributions; and finally of numerical standard errors. The importance of a robust and flexible estimation strategy is demonstrated where the complete posterior distribution is explored. In this respect, the adaptive mixture of Student-t distributions of Hoogerheide et al.(2007) works particularly well. Given an appropriately yet quickly tuned candidate, straightforward importance sampling provides the most efficient estimator of the marginal likelihood in the cases investigated in this paper, which include a non-linear regression model of Ritter and Tanner (1992) and a conditional normal distribution of Gelman and Meng (1991). A poor choice of candidate density may lead to a huge loss of efficiency where the numerical standard error may be highly unreliable. http://repub.eur.nl/res/pub/15073/ 2009-01-01T00:00:00Z This paper presents the R package AdMit which provides flexible functions to approximate a certain target distribution and to efficiently generate a sample of random draws from it, given only a kernel of the target density function. The core algorithm consists of the function AdMit which fits an adaptive mixture of Student-t distributions to the density of interest. Then, importance sampling or the independence chain Metropolis-Hastings algorithm is used to obtain quantities of interest for the target density, using the fitted mixture as the importance or candidate density. The estimation procedure is fully automatic and thus avoids the time-consuming and difficult task of tuning a sampling algorithm. The relevance of the package is shown in two examples. The first aims at illustrating in detail the use of the functions provided by the package in a bivariate bimodal distribution. The second shows the relevance of the adaptive mixture procedure through the Bayesian estimation of a mixture of ARCH model fitted to foreign exchange log-returns data. The methodology is compared to standard cases of importance sampling and the Metropolis-Hastings algorithm using a naive candidate and with the Griddy-Gibbs approach. http://repub.eur.nl/res/pub/16312/ 2009-01-01T00:00:00Z We estimate the empirical bimodal cross-section distribution of real Gross Domestic Product per capita of 120 countries over the period 1960–1989 by a mixture of a Weibull and a truncated normal density. The components of the mixture represent a group of poor and a group of rich countries, while the mixing proportion describes the distribution over poor and rich. This enables us to analyse the development of the mean and variance of both groups separately and the switches of countries between the two groups over time. Empirical evidence indicates that the means of the two groups are diverging in terms of levels, but that the growth rates of the means of the two groups over the period 1960–1989 are the same. http://repub.eur.nl/res/pub/16384/ 2009-01-01T00:00:00Z This note presents the package AdMit (Ardia et al., 2008, 2009), an R implementation of the adaptive mixture of Student-t distributions (AdMit) procedure developed by Hoogerheide (2006); see also Hoogerheide et al. (2007); Hoogerheide and van Dijk (2008). The AdMit strategy consists of the construction of a mixture of Student-t distributions which approximates a target distribution of interest. The fitting procedure relies only on a kernel of the target density, so that the normalizing constant is not required. In a second step, this approximation is used as an importance function in importance sampling or as a candidate density in the independence chain Metropolis-Hastings (M-H) algorithm to estimate characteristics of the target density. The estimation procedure is fully automatic and thus avoids the difficult task, especially for non-experts, of tuning a sampling algorithm. Typically, the target is a posterior distribution in a Bayesian analysis, where we indeed http://repub.eur.nl/res/pub/14049/ 2008-10-02T00:00:00Z A Bayesian model averaging procedure is presented that makes use of a finite mixture of many model structures within the class of vector autoregressive (VAR) processes. It is applied to two empirical issues. First, stability of the Great Ratios in U.S. macro-economic time series is investigated, together with the effect of permanent shocks on business cycles. Second, the linear 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 U.S.A. and Japan is evaluated. Although this risk found to be reasonably high, we find only mild evidence that the monetary policy transmission mechanism is different and that central banks consider the expected cost of a liquidity trap in policy setting. Posterior probabilities of different models are evaluated using Markov chain Monte Carlo techniques. http://repub.eur.nl/res/pub/14045/ 2008-09-30T00:00:00Z An efficient and accurate approach is proposed for forecasting Value at Risk [VaR] and Expected Shortfall [ES] measures in a Bayesian framework. This consists of a new adaptive importance sampling method for Quantile Estimation via Rapid Mixture of t approximations [QERMit]. As a first step the optimal importance density is approximated, after which multi-step `high loss' scenarios are efficiently generated. Numerical standard errors are compared in simple illustrations and in an empirical GARCH model with Student-t errors for daily S&P 500 returns. The results indicate that the proposed QERMit approach outperforms several alternative approaches in the sense of more accurate VaR and ES estimates given the same amount of computing time, or equivalently requiring less computing time for the same numerical accuracy. http://repub.eur.nl/res/pub/13055/ 2008-08-25T00:00:00Z Several lessons learnt from a Bayesian analysis of basic macroeconomic time series models are presented for the situation where some model parameters have substantial posterior probability near the boundary of the parameter region. This feature refers to near-instability within dynamic models, to forecasting with near-random walk models and to clustering of several economic series in a small number of groups within a data panel. Two canonical models are used: a linear regression model with autocorrelation and a simple variance components model. Several well-known time series models like unit root and error correction models and further state space and panel data models are shown to be simple generalizations of these two canonical models for the purpose of posterior inference. A Bayesian model averaging procedure is presented in order to deal with models with substantial probability both near and at the boundary of the parameter region. Analytical, graphical and empirical results using U.S. macroeconomic data, in particular on GDP growth, are presented. http://repub.eur.nl/res/pub/13053/ 2008-08-21T00:00:00Z This short note presents the R package AdMit which provides flexible functions to approximate a certain target distribution and it provides an efficient sample of random draws from it, given only a kernel of the target density function. The estimation procedure is fully automatic and thus avoids the time-consuming and difficult task of tuning a sampling algorithm. To illustrate the use of the package, we apply the AdMit methodology to a bivariate bimodal distribution. We describe the use of the functions provided by the package and document the ability and relevance of the methodology to reproduce the shape of non-elliptical distributions. http://repub.eur.nl/res/pub/17279/ 2008-08-01T00:00:00Z http://repub.eur.nl/res/pub/13978/ 2008-06-18T00:00:00Z This paper presents the R package AdMit which provides functions to approximate and sample from a certain target distribution given only a kernel of the target density function. The core algorithm consists in the function AdMit which fits an adaptive mixture of Student-t distributions to the density of interest via its kernel function. Then, importance sampling or the independence chain Metropolis- Hastings algorithm are used to obtain quantities of interest for the target density, using the fitted mixture as the importance or candidate density. The estimation procedure is fully automatic and thus avoids the time-consuming and difficult task of tuning a sampling algorithm. The relevance of the package is shown in two examples. The first aims at illustrating in detail the use of the functions provided by the package in a bivariate bimodal distribution. The second shows the relevance of the adaptive mixture procedure through the Bayesian estimation of a mixture of ARCH model fitted to foreign exchange log-returns data. The methodology is compared to standard cases of importance sampling and the Metropolis-Hastings algorithm using a naive candidate and with the Griddy-Gibbs approach. http://repub.eur.nl/res/pub/13675/ 2008-04-18T00:00:00Z Highly non-elliptical posterior distributions may occur in several econometric models, in particular, when the likelihood information is allowed to dominate and data information is weak. We explain the issue of highly non-elliptical posteriors in a model for the effect of education on income using data from the well-known Angrist and Krueger (1991) study and discuss how a so-called Information Matrix or Jeffreys' prior may be used as a `regularization prior' that in combination with the likelihood yields posteriors with desirable properties. We further consider an 8-dimensional bimodal posterior distribution in a 2-regime mixture model for the real US GNP growth. In order to perform a Bayesian posterior analysis using indirect sampling methods in these models, one has to find a good candidate density. In a recent paper - Hoogerheide, Kaashoek and Van Dijk (2007) - a class of neural network functions was introduced as candidate densities in case of non-elliptical posteriors. In the present paper, the connection between canonical model structures, non-elliptical credible sets, and more sophisticated neural network simulation techniques is explored. In all examples considered in this paper – a bimodal distribution of Gelman and Meng (1991) and posteriors in IV and mixture models - the mixture of Student's t distributions is clearly a much better candidate than a Student's t candidate, yielding far more precise estimates of posterior means after the same amount of computing time, whereas the Student's t candidate almost completely misses substantial parts of the parameter space. http://repub.eur.nl/res/pub/16386/ 2008-02-20T00:00:00Z http://repub.eur.nl/res/pub/16385/ 2008-01-01T00:00:00Z Several lessons learnt from a Bayesian analysis of basic macroeconomic time series models are presented for the situation where some model parameters have substantial posterior probability near the boundary of the parameter region. This feature refers to near-instability within dynamic models, to forecasting with near-random walk models and to clustering of several economic series in a small number of groups within a data panel. Two canonical models are used: a linear regression model with autocorrelation and a simple variance components model. Several well-known time series models like unit root and error correction models and further state space and panel data models are shown to be simple generalizations of these two canonical models for the purpose of posterior inference. A Bayesian model averaging procedure is presented in order to deal with models with substantial probability both near and at the boundary of the parameter region. Analytical, graphical and empirical results using U.S. macroeconomic data, in particular on GDP growth, are presented. http://repub.eur.nl/res/pub/18656/ 2008-01-01T00:00:00Z Jan Tinbergen was the first Nobel Laureate in economics in 1969. This article presents a brief survey of his many contributions to economics, in particular to macroeconometric modelling, business cycle analysis, economic policymaking, development economics, income distribution, international economic integration and the optimal regime. It further emphasizes his desire to contribute to the solution of urgent socio-economic problems and his passion for a more humane world. http://repub.eur.nl/res/pub/11189/ 2007-10-01T00:00:00Z Trends and cyclical components in economic time series are modeled in a Bayesian framework. This enables prior notions about the duration of cycles to be used, while the generalized class of stochastic cycles employed allows the possibility of relatively smooth cycles being extracted. The posterior distributions of such underlying cycles can be very informative for policy makers, particularly with regard to the size and direction of the output gap and potential turning points. From the technical point of view a contribution is made in investigating the most appropriate prior distributions for the parameters in the cyclical components and in developing Markov chain Monte Carlo methods for both univariate and multivariate models. Applications to US macroeconomic series are presented. http://repub.eur.nl/res/pub/10451/ 2007-07-26T00:00:00Z Several frequentist and Bayesian model averaging schemes, including a new one that simultaneously allows for parameter uncertainty, model uncertainty and time varying model weights, are compared in terms of forecast accuracy over a set of simulation experiments. Artificial data are generated, characterized by low predictability, structural instability, and fat tails, which is typical for many financial-economic time series. Sensitivity of results with respect to misspecification of the number of included predictors and the number of included models is explored. Given the set up of our experiments, time varying model weight schemes outperform other averaging schemes in terms of predictive gains both when the correlation among individual forecasts is low and the underlying data generating process is subject to structural locations shifts. In an empirical application using returns on the S&P 500 index, time varying model weights provide improved forecasts with substantial economic gains in an investment strategy including transaction costs. http://repub.eur.nl/res/pub/11158/ 2007-07-01T00:00:00Z Likelihoods and posteriors of instrumental variable (IV) regression models with strong endogeneity and/or weak instruments may exhibit rather non-elliptical contours in the parameter space. This may seriously affect inference based on Bayesian credible sets. When approximating posterior probabilities and marginal densities using Monte Carlo integration methods like importance sampling or Markov chain Monte Carlo procedures the speed of the algorithm and the quality of the results greatly depend on the choice of the importance or candidate density. Such a density has to be ‘close’ to the target density in order to yield accurate results with numerically efficient sampling. For this purpose we introduce neural networks which seem to be natural importance or candidate densities, as they have a universal approximation property and are easy to sample from. A key step in the proposed class of methods is the construction of a neural network that approximates the target density. The methods are tested on a set of illustrative IV regression models. The results indicate the possible usefulness of the neural network approach. http://repub.eur.nl/res/pub/11382/ 2007-07-01T00:00:00Z http://repub.eur.nl/res/pub/11132/ 2007-05-01T00:00:00Z We propose a natural conjugate prior for the instrumental variables regression model. The prior is a natural conjugate one since the marginal prior and posterior of the structural parameter have the same functional expressions which directly reveal the update from prior to posterior. The Jeffreys prior results from a specific setting of the prior parameters and results in a marginal posterior of the structural parameter that has an identical functional form as the sampling density of the limited information maximum likelihood estimator. We construct informative priors for the Angrist–Krueger [1991. Does compulsory school attendance affect schooling and earnings? Quarterly Journal of Economics 106, 979–1014] data and show that the marginal posterior of the return on education in the US coincides with the marginal posterior from the Southern region when we use the Jeffreys prior. This result occurs since the instruments are the strongest in the Southern region and the posterior using the Jeffreys prior, identical to maximum likelihood, focusses on the strongest available instruments. We construct informative priors for the other regions that make their posteriors of the return on education similar to that of the US and the Southern region. These priors show the amount of prior information needed to obtain comparable results for all regions. http://repub.eur.nl/res/pub/11379/ 2007-05-01T00:00:00Z http://repub.eur.nl/res/pub/10090/ 2007-04-30T00:00:00Z In this paper we show some further experiments with neural network sampling, a class of sampling methods that make use of neural network approximations to (posterior) densities, introduced by Hoogerheide et al. (2007). We consider a method where a mixture of Student's t densities, which can be interpreted as a neural network function, is used as a candidate density in importance sampling or the Metropolis-Hastings algorithm. It is applied to an illustrative 2-regime mixture model for the US real GNP growth rate. We explain the non-elliptical shapes of the posterior distribution, and show that the proposed method outperforms Gibbs sampling with data augmentation and the griddy Gibbs sampler. http://repub.eur.nl/res/pub/11133/ 2007-04-01T00:00:00Z Editorial http://repub.eur.nl/res/pub/9303/ 2007-03-25T00:00:00Z 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 e¤ects of permanent shocks. Measures on manifolds are employed in order to elicit uniform priors on subspaces defned 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 using Markov chain Monte Carlo techniques. http://repub.eur.nl/res/pub/8523/ 2007-01-31T00:00:00Z 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 Metropolis-Hastings 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. http://repub.eur.nl/res/pub/7945/ 2006-08-28T00:00:00Z Several lessons learned from a Bayesian analysis of basic economic time series models by means of the Gibbs sampling algorithm are presented. Models include the Cochrane-Orcutt model for serial correlation, the Koyck distributed lag model, the Unit Root model, the Instrumental Variables model and as Hierarchical Linear Mixed Models, the State-Space model and the Panel Data model. We discuss issues involved when drawing Bayesian inference on regression parameters and variance components, in particular when some parameter have substantial posterior probability near the boundary of the parameter region, and show that one should carefully scan the shape of the posterior density function. Analytical, graphical and empirical results are used along the way. http://repub.eur.nl/res/pub/7888/ 2006-07-26T00:00:00Z In this paper we reconsider the analysis of the effect of education on income by Angrist and Krueger (1991). In order to account for possible endogeneity of the education spell, these authors use quarter of birth to form valid instruments. Angristand Krueger apply a classical method, two-stage least-squares (2SLS), and consider results for data sets on individuals from all states of the US. In this paper the research by Angrist and Krueger is extended both in a methodological and an empirical way. Classical as well as Bayesian methods are used. Bayesian results under the Jeffreys prior are emphasized, as these results are valid in finite samples and because in the instrumental variables (IV) regression model the Jeffreys prior is in a certain sense, truly, non-informative. Further, it is considered how results vary between subsets of the data corresponding to regions of the US. Finally, some assumptions of Angrist and Krueger are investigated and it is examined if one could still obtain usable results if some assumptions are dropped. Our main findings are: (1) The Angrist-Krueger results on returns to education for the USA are almost completely determined by data from a few Southern states; (2) The conclusion of Bound, Jaeger and Baker (1995), that the instruments of Angrist and Krueger give hardly any usable information concerning the causal effect of education on wages, is too strong. A model of Angrist and Krueger (or a slightly modified version) can give usable information on the causal effect of education on income in the Southern region of the US; (3) The instruments for education that are based on quarter of birth are stronger for people with at most 8 or at least 14 years of education than for people with 9-13 years of education. This suggests that quarter of birth does not only affect the number of completed years of schooling for those who leave school as soon as the law allows for it, as these persons usually have completed 9-13 years of education. Therefore, if one intends to increase the understanding of the working of the quarter-of-birth instruments, it is a better idea to focus on differences between states in school entry requirements and/or compulsory schooling laws for children of age 5-7 than to concentrate on the differences in compulsory schooling laws for students of age 16-18. http://repub.eur.nl/res/pub/11136/ 2006-05-01T00:00:00Z The high ranking of the Econometric Institute, as listed in recent leading scientific journals, is examined for a 50-year period using similar standard measures. The distribution of the publications over different research areas is analyzed and a time-series model is specified to describe and forecast the publication pattern. http://repub.eur.nl/res/pub/7743/ 2006-03-21T00:00:00Z We present a road map for effective application of Bayesian analysis of a class of well-known dynamic econometric models by means of the Gibbs sampling algorithm. Members belonging to this class are the Cochrane-Orcutt model for serial correlation, the Koyck distributed lag model, the Unit Root model and as Hierarchical Linear Mixed Models, the State-Space model and the Panel Data model. We discuss issues involved when drawing Bayesian inference on equation parameters and variance components and show that one should carefully scan the shape of the criterion function for irregularities before applying the Gibbs sampler. Analytical, graphical and empirical results are used along the way. http://repub.eur.nl/res/pub/7580/ 2006-02-28T00:00:00Z Jan Tinbergen was the first Nobel Laureate in Economics in 1969. This paper presents a brief survey of his many contributions to economics, in particular to macro-econometric modelling, business cycle analysis, economic policy making, development economics, income distribution, international economic integration and the optimal regime. It further emphasizes his desire to contribute to the solution of urgent socio-economic problems and his passion for a more humane world. http://repub.eur.nl/res/pub/7452/ 2006-02-20T00:00:00Z This paper contains a list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005. http://repub.eur.nl/res/pub/7446/ 2006-02-03T00:00:00Z 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. http://repub.eur.nl/res/pub/7247/ 2006-01-24T00:00:00Z We propose a natural conjugate prior for the instrumental variables regression model. The prior is a natural conjugate one since the marginal prior and posterior of the structural parameter have the same functional expressions which directly reveal the update from prior to posterior. The Jeffreys prior results from a specific setting of the prior parameters and results in a marginal posterior of the structural parameter that has an identical functional form as the sampling density of the limited information maximum likelihood estimator. We construct informative priors for the Angrist-Krueger (1991) data and show that the marginal posterior of the return on education in the US coincides with the marginal posterior from the Southern region when we use the Jeffreys prior. This result occurs since the instruments are the strongest in the Southern region and the posterior using the Jeffreys prior, identical to maximum likelihood, focusses on the strongest available instruments. We construct informative priors for the other regions that make their posteriors of the return on education similar to that of the US and the Southern region. These priors show the amount of prior information needed to obtain comparable results for all regions. http://repub.eur.nl/res/pub/7244/ 2006-01-02T00:00:00Z The high ranking of the Econometric Institute, as listed in recent leading scientific journals, is examined for a fifty year period using similar standard measures. The distribution of the publications over different research areas is analyzed and a time-series model is specified to describe and forecast the publication pattern. http://repub.eur.nl/res/pub/7027/ 2005-11-07T00:00:00Z Bartlett's paradox has been taken to imply that using improper priors results in Bayes factors that are not well defined, preventing model comparison in this case. We use well understood principles underlying what is already common practice, to demonstrate that this implication is not true for some improper priors, such as the Shrinkage prior due to Stein (1956). While this result would appear to expand the class of priors that may be used for computing posterior odds, we warn against the straightforward use of these priors. Highlighting the role of the prior measure in the behaviour of Bayes factors, we demonstrate pathologies in the prior measures for these improper priors. Using this discussion, we then propose a method of employing such priors by setting rules on the rate of diffusion of prior certainty. http://repub.eur.nl/res/pub/6913/ 2005-07-25T00:00:00Z Trends and cyclical components in economic time series are modeled in a Bayesian framework. This enables prior notions about the duration of cycles to be used, while the generalized class of stochastic cycles employed allows the possibility of relatively smooth cycles being extracted. The posterior distributions of such underlying cycles can be very informative for policy makers, particularly with regard to the size and direction of the output gap and potential turning points. From the technical point of view a contribution is made in investigating the most appropriate prior distributions for the parameters in the cyclical components and in developing Markov chain Monte Carlo methods for both univariate and multivariate models. Applications to US macroeconomic series are presented. http://repub.eur.nl/res/pub/11377/ 2005-07-01T00:00:00Z http://repub.eur.nl/res/pub/2007/ 2005-03-31T00:00:00Z Likelihoods and posteriors of instrumental variable regression models with strong endogeneity and/or weak instruments may exhibit rather non-elliptical contours in the parameter space. This may seriously affect inference based on Bayesian credible sets. When approximating such contours using Monte Carlo integration methods like importance sampling or Markov chain Monte Carlo procedures the speed of the algorithm and the quality of the results greatly depend on the choice of the importance or candidate density. Such a density has to be `close' to the target density in order to yield accurate results with numerically efficient sampling. For this purpose we introduce neural networks which seem to be natural importance or candidate densities, as they have a universal approximation property and are easy to sample from. A key step in the proposed class of methods is the construction of a neural network that approximates the target density accurately. The methods are tested on a set of illustrative models. The results indicate the feasibility of the neural network approach. http://repub.eur.nl/res/pub/1915/ 2005-03-11T00:00:00Z The purpose of this paper is to survey and critically assess the Bayesian cointegration literature. In one sense, Bayesian analysis of cointegration is straightforward. The researcher can combine the likelihood function with a prior and do Bayesian inference with the resulting posterior. However, interesting and empirically important issues of global and local identification (and, as a result, prior elicitation) arise from the fact that the matrix of long run parameters is potentially of reduced rank. As we shall see, these identification problems can cause serious problems for Bayesian inference. For instance, a common noninformative prior can lead to a posterior distribution which is improper (i.e. is not a valid p.d.f. since it does not integrate to one) thus precluding valid statistical inference. This issue was brought forward by Kleibergen and Van Dijk (1994, 1998). The development of the Bayesian cointegration literature reflects an increasing awareness of these issues and this paper is organized to reflect this development. In particular, we begin by discussing early work, based on VAR or Vector Moving Average (VMA) representations which ignored these issues. We then proceed to a discussion of work based on the ECM representation, beginning with a simple specification using the linear normalization and normal priors before moving onto the recent literature which develops methods for sensible treatment of the identification issues. http://repub.eur.nl/res/pub/2133/ 2005-01-01T00:00:00Z Economists have long sought to learn the effect of a "treatment" on some outcome of interest, just as doctors do with their patients. A central practical objective of research on treatment response is to provide decision makers with information useful in choosing treatments. Often the decision maker is a social planner who must choose treatments for a heterogeneous population--for example, a physician choosing medical treatments for diverse patients or a judge choosing sentences for convicted offenders. But research on treatment response rarely provides all the information that planners would like to have. How then should planners use the available evidence to choose treatments? This book addresses key aspects of this broad question, exploring and partially resolving pervasive problems of identification and statistical inference that arise when studying treatment response and making treatment choices. Charles Manski addresses the treatment-choice problem directly using Abraham Wald's statistical decision theory, taking into account the ambiguity that arises from identification problems under weak but justifiable assumptions. The book unifies and further develops the influential line of research the author began in the late 1990s. It will be a valuable resource to researchers and upper-level graduate students in economics as well as other social sciences, statistics, epidemiology and related areas of public health, and operations research. Charles F. Manski is Board of Trustees Professor of Economics at Northwestern University. He is the author of four books, including Partial Identification of Probability Distributions and Identification Problems in the Social Sciences. http://repub.eur.nl/res/pub/11191/ 2004-12-01T00:00:00Z Adaptive radial-based direction sampling (ARDS) algorithms are specified for Bayesian analysis of models with non-elliptical, possibly, multimodal target distributions. A key step is a radial-based transformation to directions and distances. After the transformation a Metropolis-Hastings method or, alternatively, an importance sampling method is applied to evaluate generated directions. Next, distances are generated from the exact target distribution. An adaptive procedure is applied to update the initial location and covariance matrix in order to sample directions in an efficient way. The ARDS algorithms are illustrated on a regression model with scale contamination and a mixture model for economic growth of the USA. http://repub.eur.nl/res/pub/1798/ 2004-11-05T00:00:00Z Cyclical components in economic time series are analysed in a Bayesian framework, thereby allowing prior notions about periodicity to be used. The method is based on a general class of unobserved component models that encompasses a range of dynamics in the stochastic cycle. This allows for instance relatively smooth cycles to be extracted from time series. Posterior densities of parameters and estimated components are obtained using Markov chain Monte Carlo methods, which we develop for both univariate and multivariate models. Features such as time-varying amplitude may be studied by examining different functions of the posterior draws for the cyclical component and parameters. The empirical application illustrates the method for annual US real GDP over the last 130 years. http://repub.eur.nl/res/pub/1288/ 2004-05-21T00:00:00Z Economic policy decisions are often informed by empirical analysis based on accurate econometric modeling. However, a decision-maker is usually only interested in good estimates of outcomes, while an analyst must also be interested in estimating the model. Accurate inference on structural features of a model improves policy analysis as it improves estimation, inference and forecast efficiency. In this paper a Bayesian inferential procedure is presented which allows for unconditional inference on structural features of vector autoregressive (VAR) processes. We employ measures on manifolds in order to elicit uniform priors on subspaces defined by particular structural features of VARs. The features considered are cointegration, exogeneity, deterministic processes and overidentification. Posterior probabilities of these features are used in a model averaging approach for forecasting and impulse response analysis. The methods are applied to three empirical economic issues: stability of Australian money demand; relative weights of permanent and transitory shocks in a US real business cycle model; and possible evidence on an inflationary oil price shock and a liquidity trap in a UK macroeconomic model. The results obtained illustrate the feasibility of the proposed methods. http://repub.eur.nl/res/pub/1277/ 2004-05-19T00:00:00Z A sensible Bayesian model selection or comparison strategy implies selecting the model with the highest posterior probability. While some improper priors have attractive properties such as, e.g., low frequentist risk, 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 this latter result is not generally true and expand the class of priors that may be used for computing posterior odds to include some improper priors. Our approach is to give a new representation of the issue of undefined Bayes factors and, from this representation, develop classes of improper priors from which well defined Bayes factors may be derived. This approach involves either augmenting or normalising the prior measure for the parameters. One of these classes of priors includes the well known and commonly employed shrinkage prior. Estimation of Bayes factors is demonstrated for a reduced rank model. http://repub.eur.nl/res/pub/1281/ 2004-05-19T00:00:00Z Likelihoods and posteriors of econometric models with strong endogeneity and weak instruments may exhibit rather non-elliptical contours in the parameter space. This feature also holds for cointegration models when near non-stationarity occurs and determining the number of cointegrating relations is a nontrivial issue, and in mixture processes where the modes are relatively far apart. The performance of Monte Carlo integration methods like importance sampling or Markov Chain Monte Carlo procedures greatly depends in all these cases on the choice of the importance or candidate density. Such a density has to be `close' to the target density in order to yield numerically accurate results with efficient sampling. Neural networks seem to be natural importance or candidate densities, as they have a universal approximation property and are easy to sample from. That is, conditionally upon the specification of the neural network, sampling can be done either directly or using a Gibbs sampling technique, possibly using auxiliary variables. A key step in the proposed class of methods is the construction of a neural network that approximates the target density accurately. The methods are tested on a set of illustrative models which include a mixture of normal distributions, a Bayesian instrumental variable regression problem with weak instruments and near non-identification, a cointegration model with near non-stationarity and a two-regime growth model for US recessions and expansions. These examples involve experiments with non-standard, non-elliptical posterior distributions. The results indicate the feasibility of the neural network approach. http://repub.eur.nl/res/pub/1181/ 2004-03-08T00:00:00Z Using annual data on real Gross Domestic Product per capita of seventeen industrialized nations in the twentieth century the empirical relevance of shocks, trends and cycles is investigated. A class of neural network models is specified as an extension of the class of vector autoregressive models in order to capture complex data patterns for different countries and subperiods. Empirical evidence indicates nonlinear positive trends in the levels of real GDP per capita, time varying growth rates, switching behavior of individual countries with respect to their position in the distribution of real GDP per capita levels over time and club behavior with respect to convergence. Such evidence presents great challenges for economic modelling, forecasting and policy analysis in the long run. http://repub.eur.nl/res/pub/11190/ 2004-01-01T00:00:00Z Using annual data on real Gross Domestic Product per capita of seventeen industrialized nations in the twentieth century the empirical relevance of shocks, trends and cycles is investigated. A class of neural network models is specified as an extension of the class of vector autoregressive models in order to capture complex data patterns for different countries and subperiods. Empirical evidence indicates nonlinear positive trends in the levels of real GDP per capita, time varying growth rates, switching behavior of individual countries with respect to their position in the distribution of real GDP per capita levels over time. Such evidence presents challenges for economic modelling, forecasting and policy analysis for the long run. http://repub.eur.nl/res/pub/11375/ 2004-01-01T00:00:00Z http://repub.eur.nl/res/pub/11376/ 2004-01-01T00:00:00Z On April 12, 2003 it was hundred years ago that Jan Tinbergen was born. In 1969 he received, together with Ragnar Frisch, the first Nobel Prize in Economics 'for having developed and applied dynamic models for the analysis of economic processes'. In this issue of De Economist, which commemorates Tinbergen's 100th anniversary, three other Nobel laureates, viz. Paul Samuelson, Lawrence Klein and Robert Solow, give their views on this and other contributions by Tinbergen to economic science. In addition this issue contains six articles giving present-day views on topics which were high on Tinbergen's research agenda, as well as an overview of articles Tinbergen wrote for De Economist. http://repub.eur.nl/res/pub/11369/ 2003-12-01T00:00:00Z http://repub.eur.nl/res/pub/11199/ 2003-10-01T00:00:00Z 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. http://repub.eur.nl/res/pub/1045/ 2003-08-07T00:00:00Z In this short paper we summarize the computational steps of Adaptive Radial-Based Direction Sampling (ARDS), which can be used for Bayesian analysis of ill behaved target densities. We consider one simulation experiment in order to illustrate the good performance of ARDS relative to the independence chain MH algorithm and importance sampling. http://repub.eur.nl/res/pub/1047/ 2003-08-07T00:00:00Z In Hoogerheide, Kaashoek and Van Dijk (2002) the class of neural network sampling methods is introduced to sample from a target (posterior) distribution that may be multi-modal or skew, or exhibit strong correlation among the parameters. In these methods the neural network is used as an importance function in IS or as a candidate density in MH. In this note we suggest an analytical approach to estimate the moments of a certain (target) distribution, where `analytical' refers to the fact that no sampling algorithm like MH or IS is needed.We show an example in which our analytical approach is feasible, even in a case where a `standard' Gibbs approach would fail or be extremely slow. http://repub.eur.nl/res/pub/1080/ 2003-08-07T00:00:00Z A flexible decomposition of a time series into stochastic cycles under possible non-stationarity is specified, providing both a useful data analysis tool and a very wide model class. A Bayes procedure using Markov Chain Monte Carlo (MCMC) is introduced with a model averaging approach which explicitly deals with the uncertainty on the appropriate number of cycles. The convergence of the MCMC method is substantially accelerated through a convenient reparametrization based on a hierarchical structure of variances in a state space model. The model and corresponding inferential procedure are applied to simulated data and to economic time series like industrial production, unemployment and real exchange rates. We derive the implied posterior distributions of model parameters and some relevant functions thereof, shedding light on a wide range of key features of each economic time series. http://repub.eur.nl/res/pub/1722/ 2003-08-06T00:00:00Z Adaptive radial-based direction sampling (ARDS) algorithms are specified for Bayesian analysis of models with nonelliptical, possibly, multimodal target distributions. A key step is a radial-based transformation to directions and distances. After the transformations a Metropolis-Hastings method or, alternatively, an importance sampling method is applied to evaluate generated directions. Next, distances are generated from the exact target distribution by means of the numerical inverse transformation method. An adaptive procedure is applied to update the initial location and covariance matrix in order to sample directions in an efficient way. Tested on a set of canonical mixture models that feature multimodality, strong correlation, and skewness, the ARDS algorithms compare favourably with the standard Metropolis-Hastings and importance samplers in terms of flexibility and robustness. The empirical examples include a regression model with scale contamination and a mixture model for economic growth of the USA. http://repub.eur.nl/res/pub/1717/ 2003-06-17T00:00:00Z Economic policy decisions are often informed by empirical economic analysis. While the decision-maker is usually only interested in good estimates of outcomes, the analyst is interested in estimating the model. Accurate inference on the structural features of a model, such as cointegration, can improve policy analysis as it can improve estimation, inference and forecast efficiency from using that model. However, using a model does not guarantee good estimates of the object of interest and, as it assigns a probability of one to a model and zero to near-by models, takes extreme zero-one account of the "weight of evidence" in the data and the resarcher's uncertainty. By using the uncertainty associated with the structural features in a model set, one obtains policy analysis that is not conditional on the structure of the model and can improve efficiency if the features are appropriately weighted. In this paper tools are presented to allow for unconditional inference on the vector autoregressive (VAR) model. In particular, we employ measures on manifolds to elicit priors on subspaces defined by particular features of the VAR model. The features considered are cointegration, exogeneity, deterministic processes and overidentification. Two applications -- money demand in Australia, and a macroeconomic model of the UK proposed by Garratt, Lee, Persaran, and Shin (2002) are used to illustrate the feasibility of the proposed methods. http://repub.eur.nl/res/pub/1707/ 2003-03-26T00:00:00Z In this paper a potential solution is given to the conflict in Bayesian inference between the desire to employ diffuse priors to represent ignorance and the desire to report proper posterior probabilities for alternative models. Using the concept of Stiefel manifolds, diffuse priors are specified on dimension and direction of subspaces of parameter spaces within the context of a linear regression model and a cointegration model. The approach is illustrated using a CAPM and a term structure of interest rates model. http://repub.eur.nl/res/pub/11201/ 2003-01-01T00:00:00Z Bayesian model selection with posterior probabilities and no subjective prior information is generally not possible because of the Bayes factors being ill-defined. Using careful consideration of the parameter of interest in cointegration analysis and a re-specification of the triangular model of Phillips (Econometrica, Vol. 59, pp. 283-306, 1991), this paper presents an approach that allows for Bayesian comparison of models of cointegration with 'ignorance' priors. Using the concept of Stiefel and Grassman manifolds, diffuse priors are specified on the dimension and direction of the cointegrating space. The approach is illustrated using a simple term structure of the interest rates model. http://repub.eur.nl/res/pub/1727/ 2002-12-31T00:00:00Z The performance of Monte Carlo integration methods like importance sampling or Markov Chain Monte Carlo procedures greatly depends on the choice of the importance or candidate density. Usually, such a density has to be "close" to the target density in order to yield numerically accurate results with efficient sampling. Neural networks seem to be natural importance or candidate densities, as they have a universal approximation property and are easy to sample from. That is, conditional upon the specification of the neural network, sampling can be done either directly or using a Gibbs sampling technique, possibly using auxiliary variables. A key step in the proposed class of methods is the construction of a neural network that approximates the target density accurately. The methods are tested on a set of illustrative models which include a mixture of normal distributions, a Bayesian instrumental variable regression problem with weak instruments and near-identification, and two-regime growth model for US recessions and expansions. These examples involve experiments with non-standard, non-elliptical posterior distributions. The results indicate the feasibility of the neural network approach. http://repub.eur.nl/res/pub/538/ 2002-12-04T00:00:00Z Stylized facts show that average growth rates of US per capita consumption and income differ in recession and expansion periods. Since 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, which 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 autoregressive 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 US per capita disposable income and consumption, after correction for a multivariate Markov trend. This results is also obtained when per capita investment is added to the vector autoregression. http://repub.eur.nl/res/pub/11339/ 2002-12-01T00:00:00Z Neural networks are fitted to real exchange rates of several industrialized countries. The size and topology of the networks is found through the use of multiple correlation coefficients, principal component analysis of residuals and graphical analysis of network output per hidden layer cell and input layer cell. These pruned neural networks are good approximations to varying non-linear trends in real exchange rates. Non-linear dynamic analysis shows that the long-term equilibrium values of several European currencies correspond to the actual values within the European Monetary System. Based on its long-term equilibrium value, the Euro appears to be undervalued vis-à-vis the US dollar at the introduction of the Euro on 1 January 1999. http://repub.eur.nl/res/pub/540/ 2002-11-14T00:00:00Z Cyclical components in economic time series are analysed in a Bayesian framework, thereby allowing prior notions about periodicity to be used. The method is based on a general class of unobserved component models that allow relatively smooth cycles to be extracted. Posterior densities of parameters and smoothed cycles are obtained using Markov chain Monte Carlo methods. An application to estimating business cycles in macroeconomic series illustrates the viability of the procedure for both univariate and bivariate models. http://repub.eur.nl/res/pub/555/ 2002-09-17T00:00:00Z Adaptive Polar Sampling (APS) algorithms are proposed for Bayesian analysis of models with nonelliptical, possibly, multimodal posterior distributions. A location-scale transformation and a transformation to polar coordinates are used. After the transformation to polar coordinates, a Metropolis-Hastings method or, alternatively, an importance sampling method is applied to sample directions and, conditionally on these, distances are generated by inverting the cumulative distribution function. A sequential procedure is applied to update the initial location and scaling matrix in order to sample directions in an efficient way. Tested on a set of canonical mixture models that feature multimodality, strong correlation, and skewness, the APS algorithms compare favourably with the standard Metropolis-Hastings and importance samplers in terms of flexibility and robustness. APS is applied to several econometric and statistical examples. The empirical results for a regression model with scale contamination, an ARMA-GARCH-Student t model with near cancellation of roots and heavy tails, a mixture model for economic growth, and a nonlinear threshold model for industrial production growth confirm the practical flexibility and robustness of APS. http://repub.eur.nl/res/pub/11338/ 2002-07-30T00:00:00Z 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 modeling. The methods are applied to three data sets: Canadian lynx and sunspot series, US annual macro-economic time series — used by Nelson and Plosser (J. Monetary Econ., 10 (1982) 139) — and US monthly unemployment rate and production indices. 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 series, the Canadian lynx number series and the monthly series, but it does not uniformly hold for the Nelson and Plosser economic time series. http://repub.eur.nl/res/pub/577/ 2002-05-01T00:00:00Z Econometric issues that are considered fundamental in the development of Bayesian structural inference within a Simultaneous Equation Model are surveyed. The difficulty of specifying prior information which is of interest to economists and which yields tractable posterior and predictive distributions has started this line of research. A major issue is the nonstandard shape of the likelihood due to reduced rank restrictions. It implies that existence of structural posterior moments under vague prior information is a nontrivial issue. The problem is illustrated through simple examples using artificially generated data in a so-called limited information framework where the connection with the problem of weak instruments in classical econometrics is also described. A positive development is Bayesian inference of implied characteristics, in particular, dynamic features of a Simultaneous Equation Model. The potential of Bayesian structural inference, using a predictive approach for prior specification and using Monte Carlo simulation techniques for computational purposes, is illustrated by means of a prior and posterior analysis of the US business cycle in the period of the depression. A structural prior is elicited through investigation of the implied predictive features. http://repub.eur.nl/res/pub/6835/ 2001-11-20T00:00:00Z The failure to describe the time series behaviour of most real exchange rates as temporary deviations from fixed long-term means may be due to time variation of the equilibria themselves, see Engel (2000). We implement this idea using an unobserved components model and decompose the observations on real exchange rates in long-term components, which capture the time-variation of the mean and in medium and short-term components which measure temporary deviations. A simulation-based Bayesian analysis is introduced to compute the posterior distribution of (functions) of the model parameters. A stationarity test in this setup indicates that the mean is slowly time-varying. Subsequently, we use our flexible model to derive the implied distributions of some key features of real exchange rates. Most notably, the half-life of deviations from the mean, which is a measure of persistence, is lowered. This provides a possible explanation for the PPP puzzle. http://repub.eur.nl/res/pub/1702/ 2001-11-13T00:00:00Z The failure to describe the time series behaviour of most real exchange rates as temporary deviations from fixed long-term means may be due to time variation of the equilibria themselves, see Engel (2000). We implement this idea using an unobserved components model and decompose the observations on real exchange rates in long-term components, which capture the time-variation of the mean and in medium and short-term components which measure temporary deviations. A simulation-based Bayesian analysis is introduced to compute the posterior distribution of (functions) of the model parameters. A stationarity test in this setup indicates that the mean is slowly time-varying. Subsequently, we use our flexible model to derive the implied distributions of some key features of real exchange rates. Most notably, the half-life of deviations from the mean, which is a measure of persistence, is lowered. This provides a possible explanation for the PPP puzzle. http://repub.eur.nl/res/pub/1670/ 2001-02-19T00:00:00Z The flexibility of neural networks to handle complex data patterns of economic variables is well known. In this survey we present a brief introduction to a neural network and focus on two aspects of its flexibility . First, a neural network is used to recover the dynamic properties of a nonlinear system, in particular, its stability by making use of the Lyapunov exponent. Second, a two-stage network is introduced where the usual nonlinear model is combined with time transitions, which may be handled by neural networks. The connection with time-varying smooth transition models is indicated. The procedures are illustrated using three examples: a structurally unstable chaotic model, nonlinear trends in real exchange rates and a time-varying Phillips curve using US data from 1960-1997. http://repub.eur.nl/res/pub/1669/ 2001-02-12T00:00:00Z In this paper we discuss the similarity between the Anderson-Rubin test for overidentification in a Simultaneous Equations Model and the Johansen test for cointegration in a Vector Autoregressive model. The similar structure of the two models is shown to be important in this respect. An alternative procedure for computing the Anderson-Rubin test is given, which appears to be faster than the conventional method. The derivation of the likelihood ratio test for the hypothesis of reduced rank is given for the general case. Both the Anderson-Rubin test and the Johansen test are shown to be monotonically increasing functions of the singular values of a scaled version of the unrestricted least-squares estimator of the matrix upon which the rank restriction is imposed. http://repub.eur.nl/res/pub/6877/ 2001-02-08T00:00:00Z Internationally operating firrns naturally face the decision whether or not to hedge the currency risk implied by foreign investments. In a recent paper, Bos, Mahieu and van Dijk (2000) evaluate the returns from optimal and alternative currency hedging strategies, for a series of 7 models, using Bayesian inference and decision analysis. The models differ in the way time-varying means, variances or the unconditional error distributions are incorporated. In this extension, we compare the hedging decisions and financial returns and utilities as they result from the modelling assumptions and the attitudes towards risk. http://repub.eur.nl/res/pub/6878/ 2001-02-08T00:00:00Z We construct models which enable a decision-maker to analyze the implications of typical time series patterns of daily exchange rates for currency risk management. Our approach is Bayesian where extensive use is made of Markov chain Monte Carlo methods. The effects of several model characteristics (unit roots, GARCH, stochastic volatility, heavy tailed disturbance densities) are investigated in relation to the hedging strategies. Consequently, we can make a distinction between statistical relevance of model specifications, and the economic consequences from a risk management point of view. We compute payoffs and utilities from several alternative hedge strategies. The results indicate that modelling time varying features of exchange rate returns may lead to improved hedge behaviour within currency overlay management. http://repub.eur.nl/res/pub/1653/ 2000-11-09T00:00:00Z Internationally operating firms naturally face the decision whether or not to hedge the currency risk implied by foreign investments. In a recent paper, Bos, Mahieu and van Dijk evaluate the returns from optimal and alternative currency hedging strategies, for a series of 7 models, using Bayesian inference and decision analysis. The models differ in the way time-varying means, variances or the unconditional error distributions are incorporated. In this extension, we compare the hedging decisions and financial returns and utilities as they result from the modelling assumptions and the attitudes towards risk. http://repub.eur.nl/res/pub/1661/ 2000-10-25T00:00:00Z The flexibility of neural networks to handle complex data patterns of economic variables is well known. In this survey we present a brief introduction to a neural network and focus on two aspects of its flexibility . First, a neural network is used to recover the dynamic properties of a nonlinear system, in particular, its stability by making use of the Lyapunov exponent. Second, a two-stage network is introduced where the usual nonlinear model is combined with time transitions, which may be handled by neural networks. The connection with time-varying smooth transition models is indicated. The procedures are illustrated using three examples: a structurally unstable chaotic model, nonlinear trends in real exchange rates and a time-varying Phillips curve using US data from 1960-1997. http://repub.eur.nl/res/pub/1657/ 2000-08-30T00:00:00Z We construct models which enable a decision-maker to analyze the implications of typical time series patterns of daily exchange rates for currency risk management. Our approach is Bayesian where extensive use is made of Markov chain Monte Carlo methods. The effects of several model characteristics (unit roots, GARCH, stochastic volatility, heavy tailed disturbance densities) are investigated in relation to the hedging strategies. Consequently, we can make a distinction between statistical relevance of model specifications, and the economic consequences from a risk management point of view. We compute payoffs from several alternative hedge strategies. These payoffs indicate that modelling time-varying features of exchange rate returns may lead to improved hedge behaviour within currency overlay management. http://repub.eur.nl/res/pub/11332/ 2000-01-01T00:00:00Z 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. http://repub.eur.nl/res/pub/11335/ 2000-01-01T00:00:00Z We construct models which enable a decision maker to analyse the implications of typical time series patterns of daily exchange rates for currency risk management. Our approach is Bayesian where extensive use is made of Markov chain Monte Carlo methods. The effects of several model characteristics (unit roots, GARCH, stochastic volatility, heavy-tailed disturbance densities) are investigated in relation to the hedging strategies. Consequently, we can make a distinction between statistical relevance of model specifications and the economic consequences from a risk management point of view. We compute payoffs and utilities from several alternative hedge strategies. The results indicate that modelling time-varying features of exchange rate returns may lead to improved hedge behaviour within currency overlay management. http://repub.eur.nl/res/pub/11336/ 2000-01-01T00:00:00Z http://repub.eur.nl/res/pub/1621/ 1999-12-08T00:00:00Z 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-Plosser's 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. http://repub.eur.nl/res/pub/7700/ 1999-11-30T00:00:00Z 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-Plosser's 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. http://repub.eur.nl/res/pub/7712/ 1999-10-21T00:00:00Z Adaptive Polar Sampling (APS) is proposed as a Markov chain Monte Carlo method for Bayesian analysis of models with ill-behaved posterior distributions. In order to sample efficiently from such a distribution, a location-scale transformation and a transformation to polar coordinates are used. After the transformation to polar coordinates, a Metropolis-Hastings algorithm is applied to sample directions and, conditionally on these, distances are generated by inverting the CDF. A sequential procedure is applied to update the location and scale. Tested on a set of canonical models that feature near non-identifiability, strong correlation, and bimodality, APS compares favourably with the standard Metropolis-Hastings sampler in terms of parsimony and robustness. APS is applied within a Bayesian analysis of a GARCH-mixture model which is used for the evaluation of the Value-at-Risk of the return of the Dow Jones stock index. http://repub.eur.nl/res/pub/1603/ 1999-10-13T00:00:00Z 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. http://repub.eur.nl/res/pub/1605/ 1999-10-13T00:00:00Z Exchange rates typically exhibit time-varying patterns in both means and variances. The histograms of such series indicate heavy tails. In this paper we construct models which enable a decision-maker to analyze the implications of such time series patterns for currency risk management. Our approach is Bayesian where extensive use is made of Markov chain Monte Carlo methods. The effects of several model characteristics (unit roots, GARCH, stochastic volatility, heavy tailed disturbance densities) are investigated in relation to the hedging decision strategies. Consequently, we can make a distinction between statistical relevance of model specifications, and the economic consequences from a risk management point of view. The empirical results suggest that econometric modelling of heavy tails and time-varying means and variances pays off compared to a efficient markets model. The different ways to measure persistence and changing volatilities appear to strongly influence the hedging decision the investor faces. http://repub.eur.nl/res/pub/1569/ 1999-03-31T00:00:00Z In this paper neural networks are fitted to the real exchange rates of seven industrialized countries. The size and topology of the used networks is found by reducing the size of the network through the use of multiple correlation coefficients, principal component analysis of residuals and graphical analysis of network output per hidden layer cell and input layer cell. http://repub.eur.nl/res/pub/11328/ 1999-01-01T00:00:00Z The effect which the oil price time series has on the long run properties of Vector AutoRegressive (VAR) models for price levels and import demand is investigated. As the oil price variable is assumed to be weakly exogenous for the long run parameters, a cointegration testing procedure allowing for weakly exogenous variables is developed using a LU decomposition of the long run multiplier matrix. The likelihood based cointegration test statistics, Wald, Likelihood Ratio and Lagrange Multiplier, are constructed and their limiting distributions derived. Using these tests, we find that incorporating the oil price in a model for the domestic or import price level of seven industrialized countries decreases the long run memory of the inflation rate. Second, we find that the results for import demand can be classified with respect to the oil importing or exporting status of the specific country. The result for Japan is typical as its import price is not influenced by GNP in the long run, which is the case for all other countries. http://repub.eur.nl/res/pub/1523/ 1998-12-31T00:00:00Z A major problem in applying neural networks is specifying the size of the network. Even for moderately sized networks the number of parameters may become large compared to the number of data. In this paper network performance is examined while reducing the size of the network through the use of multiple correlation coefficients, principal component analysis of residuals and graphical analysis of network output per hidden layer cell and input layer cell. http://repub.eur.nl/res/pub/1550/ 1998-07-02T00:00:00Z Adaptive Polar Sampling is proposed as an algorithm where random drawings are directly generated from the target function (posterior) in all-but-one directions of the parameter space. The method is based on the mixed integration technique of Van Dijk, Kloek & Boender (1985) but extends this one by replacing the one-dimensional quadrature step by Monte Carlo simulation from this one-dimensional distribution function. The method is particularly suited for the analysis of ill-behaved surfaces. An illustrative example shows the feasibility of the algorithm. http://repub.eur.nl/res/pub/2030/ 1998-07-01T00:00:00Z We estimate the empirical bimodal cross-sectional distribution of real Gross Domestic Product per capita of 120 countries over the period 1960–1989 by a mixture of a Weibull and a truncated normal density. The components of the mixture represent a group of poor and a group of rich countries, while the mixing proportion describes the distribution over poor and rich. This enables us to analyse the development of the mean and variance of both groups separately and the switches of countries between the two groups over time. Empirical evidence indicates that the means of the two groups are diverging in terms of levels, but that the growth rates of the means of the two groups over the period 1960–1989 are the same. http://repub.eur.nl/res/pub/7799/ 1997-05-08T00:00:00Z 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. In addition, we extend these ideas to the problem of testing for integration at seasonal frequencies and show how techniques can be used to carry out Bayesian variants of HEGY test or the Canova-Hansen test. http://repub.eur.nl/res/pub/1414/ 1997-01-01T00:00:00Z Diffuse priors lead to pathological posterior behavior when used in Bayesian analyses of Simultaneous Equation Models (SEMs). This results from the local nonidentification of certain parameters in SEMs. When this, a priori known, feature is not captured appropriately, an a posteriori favor for certain specific parameter values results which is not the consequence of strong data information but of local nonidentification. We show that a proper consistent Bayesian analysis of a SEM explicitly has to consider the reduced form of the SEM as a standard linear model on which nonlinear (reduced rank) restrictions are imposed, which result from a singular value decomposition. The priors/posteriors of the parameters of the SEM are therefore proportional to the priors/posteriors of the parameters of the linear model under the condition that the restrictions hold. This leads to a framework for constructing priors and posteriors for the parameters of SEMs. The framework is used to construct priors and posteriors for one, two and three structural equation SEMs. These examples jointly with a theorem, which states that the reduced forms of SEMs accord with sets of reduced rank restrictions on standard linear models, show how Bayesian analyses of generally specified SEMs are conducted. http://repub.eur.nl/res/pub/1418/ 1997-01-01T00:00:00Z The effect which the oil price time series has on the long run properties of Vector AutoRegressive (VAR) models for price levels and import demand is investigated. As the oil price variable is assumed to be weakly exogenous for the long run parameters, a cointegration testing procedure allowing for weakly exogenous variables is developed using a LU\\ decomposition of the long run multiplier matrix. The likelihood based cointegration test statistics, Wald, Likelihood Ratio and Lagrange Multiplier, are constructed and their limiting distributions derived. Using these tests, we find that incorporating the oil price in a model for the domestic or import price level of seven industrialized countries decreases the long run memory of the inflation rate. Second, we find that the results for import demand can be classified with respect to the oil importing or exporting status of the specific country. The result for Japan is typical as its import price is not influenced by gnp in the long run, which is the case for all other countries. http://repub.eur.nl/res/pub/11316/ 1996-01-01T00:00:00Z http://repub.eur.nl/res/pub/11310/ 1995-01-01T00:00:00Z 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. http://repub.eur.nl/res/pub/11296/ 1994-01-01T00:00:00Z Abstract An error correction model is specified having only exact identified parameters, some of which reflect a possible departure from a cointegration model. Wald, likelihood ratio, and Lagrange multiplier statistics are derived to test for the significance of these parameters. The construction of the Wald statistic only involves linear regression, and under certain conditions the limiting distribution of the Wald statistic differs from the limiting distributions of the likelihood ratio and Lagrange multiplier statistics. A special ordering of the variables is recommended so that equal limiting distributions of the three different test statistics are obtained. The applicability of the derived testing procedures is illustrated using real demand for money, real GNP, and bond and deposit interest rates from Denmark. http://repub.eur.nl/res/pub/11299/ 1994-01-01T00:00:00Z http://repub.eur.nl/res/pub/11248/ 1993-01-01T00:00:00Z First, the non-stationarity properties of the conditional variances in the GARCH(1,1) model are analysed using the concept of infinite persistence of shocks. Given a time sequence of probabilities for increasing/decreasing conditional variances, a theoretical formula for quasi-strict non-stationarity is defined. The resulting conditions for the GARCH(1,1) model are shown to differ from the weak stationarity conditions mainly used in the literature. Bayesian statistical analysis using Monte Carlo integration is applied to analyse both stationarity concepts for the conditional variances of the US 3-month treasury bill rate. Interest rates are known for their weakly non-stationary conditional variances but, using a quasi-strict stationarity measure, it is shown that the conditional variances are likely to be stationary. Second, the level of the treasury bill rate is analysed for non-stationarity using Bayesian unit root methods. The disturbances of the GARCH model for the treasury bill rate are t-distributed. It is shown that the unit root parameter is negatively correlated with the degrees-of-freedom parameter. Imposing normally distributed disturbances leads therefore to underestimation of the non-stationarity in the level of the treasury bill rate. http://repub.eur.nl/res/pub/11250/ 1993-01-01T00:00:00Z http://repub.eur.nl/res/pub/11240/ 1991-01-01T00:00:00Z 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. http://repub.eur.nl/res/pub/11241/ 1991-01-01T00:00:00Z Comments on an article on the modelling of relative price variability and aggregate inflation in Great Britain. Advantage of the general to specific modeling approach; Steps in the analysis of macroeconomic time series; Information on the PC-GIVE menu-driven program. http://repub.eur.nl/res/pub/11243/ 1991-01-01T00:00:00Z This paper is a comment on P. C. B. Phillips, `To criticise the critics: an objective Bayesian analysis of stochastic trends' [Phillips, (1991)]. Departing from the likelihood of an univariate autoregressive model different routes that lead to a posterior odds analysis of the unit root hypothesis are explored, where the differences in routes are due to the different choices of the prior. Improper priors like the uniform and the Jeffreys prior are less suited for Bayesian inference on a sharp null hypothesis as the unit root. A proper normal prior on the mean of the process is analysed and empirical results using extended Nelson-Plosser data are presented. http://repub.eur.nl/res/pub/11238/ 1988-01-01T00:00:00Z 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. http://repub.eur.nl/res/pub/11234/ 1987-01-01T00:00:00Z 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 densities of the parameters of interest of econometric models. In the present paper we describe the computational steps of one Monte Carlo method, which is known in the literature as importance sampling. Further, a set of standard programs is available, which may be used for the implementation of a simple case of importance sampling. The computer programs have been written in FORTRAN 77. http://repub.eur.nl/res/pub/11235/ 1987-01-01T00:00:00Z http://repub.eur.nl/res/pub/9271/ 1985-08-01T00:00:00Z In this paper we apply Maximum Likelihood and Bayesian methods to explain differences in floorspace productivity among retail establishments in the grocery trade. The model we develop is a switching model where sales are either supply-determined or demand-determined. Under excess supply the model allows for so-called ‘trading-down’, i.e., an increase in the share of selling area, and, thereby, a decrease in service level. To estimate our model we employ a cross-section of observations on individual shops. We present maximum likelihood results, and also study the shape of the likelihood surface by means of Monte Carlo numerical integration methods. With a uniform prior we obtain marginal posterior density functions both of the parameters of interest and of the average probability of the excess supply regime in the sample. The average probability of excess supply is 0.23, with a standard deviation of 0.06. This shows that, according to our estimates, excess demand is the rule and excess supply the exception in the sample that we analyse. http://repub.eur.nl/res/pub/11231/ 1985-01-01T00:00:00Z http://repub.eur.nl/res/pub/11232/ 1985-01-01T00:00:00Z 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. http://repub.eur.nl/res/pub/11229/ 1983-01-01T00:00:00Z The posterior distribution of a small-scale illustrative econometric model is used to compare symmetric simple importance sampling with asymmetric simple importance sampling. The numerical results include posterior first and second order moments, numerical error estimates of the first order moments, posterior modes, univariate marginal posterior densities and bivariate marginal posterior densities plotted in three-dimensional figures. http://repub.eur.nl/res/pub/11227/ 1980-01-01T00:00:00Z An earlier paper [Kloek and Van Dijk (1978)] is extended in three ways. First, Monte Carlo integration is performed in a nine-dimensional parameter space of Klein's model I [Klein (1950)]. Second, Monte Carlo is used as a tool for the elicitation of a uniform prior on a finite region by making use of several types of prior information. Third, special attention is given to procedures for the construction of importance functions which make use of nonlinear optimization methods. *1 This paper started as a revision of Van Dijk and Kloek (1978). In the course of the work our ideas developed to such an extent that the final result is an almost completely new paper. We are indebted to a referee for a number of very useful suggestions. We also wish to thank A.S. Louter and G. den Broeder of the Econometric Institute for their help in preparing the necessary computer programs. http://repub.eur.nl/res/pub/11228/ 1980-01-01T00:00:00Z This paper discusses inferential procedures for the family of stable distributions, when the data are tabulated in the form of interval frequencies. The estimation criteria used are minimum chi-square and multinomial maximum likelihood. In evaluating the theoretical probabilities corresponding to the intervals, use is made of the inversion theorem for characteristic functions. Chi-square tail probabilities for independent samples are pooled by means of theKolmogorov statistic. As an illustration, the methods are applied to Dutch and Australian income data. http://repub.eur.nl/res/pub/11224/ 1978-01-01T00:00:00Z Monte Carlo (MC) is used to draw parameter values from a distribution defined on the structural parameter space of an equation system. Making use of the prior density, the likelihood, and Bayes' Theorem it is possible to estimate posterior moments of both structural and reduced form parameters. The MC method allows a rather liberal choice of prior distributions. The number of elementary operations to be preformed need not be an explosive function of the number of parameters involved. The method overcomes some existing difficulties of applying Bayesian methods to medium size models. The method is applied to a small scale macro model. The prior information used stems from considerations regarding short and long run behavior of the model and form extraneous observations on empirical long term ratios of economic variables. Likelihood contours for several parameter combinations are plotted, and some marginal posterior densities are assessed by MC. http://repub.eur.nl/res/pub/11225/ 1978-01-01T00:00:00Z 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.