http://repub.eur.nl/res/aut/2001/ List of Publications en http://repub.eur.nl/static-eur/img/logo.png http://repub.eur.nl http://repub.eur.nl/res/pub/39847/ 2014-04-15T00:00:00Z Accurate prediction of risk measures such as Value at Risk (VaR) and Expected Shortfall (ES) requires precise estimation of the tail of the predictive distribution. Two novel concepts are introduced that offer a specific focus on this part of the predictive density: the censored posterior, a posterior in which the likelihood is replaced by the censored likelihood; and the censored predictive likelihood, which is used for Bayesian Model Averaging. We perform extensive experiments involving simulated and empirical data. Our results show the ability of these new approaches to outperform the standard posterior and traditional Bayesian Model Averaging techniques in applications of Value-at-Risk prediction in GARCH models. 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/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/38059/ 2012-07-01T00:00:00Z Education is a well-known driver of (entrepreneurial) income. The measurement of its influence, however, suffers from endogeneity suspicion. For instance, ability and occupational choice are mentioned as driving both the level of (entrepreneurial) income and of education. Using instru-mental variables can provide a way out. However, two questions remain: whether endogeneity is really present and whether it matters for the size of the estimated relationship. Using Bayesian methods, we find that the relationship between education and entrepreneurial income is indeed en-dogenous and that the impact of endogeneity on the estimated relationship between education and income is sizeable. Implications of our findings for research and practice are discussed. http://repub.eur.nl/res/pub/32155/ 2012-03-16T00:00:00Z The validity of family background variables instrumenting education in income regressions has been much criticized. In this paper, we use data from the 2004 German Socio-Economic Panel and Bayesian analysis to analyze to what degree violations of the strict validity assumption affect the estimation results. We show that, in case of moderate direct effects of the instrument on the dependent variable, the results do not deviate much from the benchmark case of no such effect (perfect validity of the instrument's exclusion restriction). In many cases, the size of the bias is smaller than the width of the 95% posterior interval for the effect of education on income. Thus, a violation of the strict validity assumption does not necessarily lead to results which are strongly different from those of the strict validity case. This finding provides confidence in the use of family background variables as instruments in income regressions. The paper analyzes to what degree violations of the perfect validity of the exclusion restriction for family background variables in income regression affect the estimation results. ► In case of moderate direct effects of the instrument on the dependent variable, the results do not deviate much from the benchmark case of no such effect (perfect validity of the instrument's exclusion restriction). ► The finding provides confidence in the use of family background variables as instruments in income regressions. http://repub.eur.nl/res/pub/32010/ 2012-01-01T00:00:00Z 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/37789/ 2011-06-01T00:00:00Z Abstract: Education is argued to be an important driver of the decision to start a business. However, the measurement of its influence is difficult since it is considered to be an endogenous variable. This study accounts for this endogeneity by using an instrumental variables approach and a dataset of more than 10,000 individuals from 27 European countries and the USA. The effect of education on the decision to become self-employed is found to be strongly positive, much higher than the estimated effect in case no instrumental variables are used. That is, the higher the respondent’s level of education, the greater the likelihood that they will start a business. Implications for entrepreneurship research and practice are discussed. 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/22344/ 2011-01-01T00:00:00Z Using well-known GARCH models for density prediction of daily S&P 500 and Nikkei 225 index returns, a comparison is provided between frequentist and Bayesian estimation. No significant difference is found between the qualities of the forecasts of the whole density, whereas the Bayesian approach exhibits significantly better left-tail forecast accuracy. 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/20281/ 2010-07-01T00:00:00Z The validity of family background variables instrumenting education in income regressions has been much criticized. In this paper, we use data of the 2004 German Socio-Economic Panel and Bayesian analysis in order to analyze to what degree violations of the strong validity assumption affect the estimation results. We show that, in case of moderate direct effects of the instrument on the dependent variable, the results do not deviate much from the benchmark case of no such effect (perfect validity of the instrument). The size of the bias is in many cases smaller than the standard error of education’s estimated coefficient. Thus, the violation of the strict validity assumption does not necessarily lead to strongly different results when compared to the strict validity case. This provides confidence in the use of family background variables as instruments in income regressions. 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/19380/ 2010-04-27T00:00:00Z This paper proposes an up-to-date review of estimation strategies available for the Bayesian inference of GARCH-type models. The emphasis is put on a novel efficient procedure named AdMitIS. The methodology automatically constructs a mixture of Student-t distributions as an approximation to the posterior density of the model parameters. This density is then used in importance sampling for model estimation, model selection and model combination. The procedure is fully automatic which avoids difficult and time consuming tuning of MCMC strategies. The AdMitIS methodology is illustrated with an empirical application to S&P index log-returns. Several non-nested GARCH-type models are estimated and combined to predict the distribution of next-day ahead log-returns. 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/18349/ 2010-02-26T00:00:00Z Education is a well-known driver of (entrepreneurial) income. The measurement of its influence, however, suffers from endogeneity suspicion. For instance, ability and occupational choice are mentioned as driving both the level of (entrepreneurial) income and of education. Using instrumental variables can provide a way out. However, three questions remain: whether endogeneity is really present, whether it matters and whether the selected instruments make sense. Using Bayesian methods, we find that the relationship between education and entrepreneurial income is indeed endogenous and that the impact of endogeneity on the estimated relationship between educa-tion and income is sizeable. We do so using family background variables and show that relaxing the strict validity assumption of these instruments does not lead to strongly different results. This is an important finding because family background variables are generally strongly correlated with education and are available in most datasets. Our approach is applicable beyond the field of returns to education for income. It applies wherever endogeneity suspicion arises and the three questions become relevant. http://repub.eur.nl/res/pub/19377/ 2010-01-01T00:00:00Z This note presents the R package bayesGARCH (Ardia, 2007) which provides functions for the Bayesian estimation of the parsimonious and effective GARCH(1,1) model with Student-t innovations. The estimation procedure is fully automatic and thus avoids the tedious task of tuning a MCMC sampling algorithm. The usage of the package is shown in an empirical application to exchange rate logreturns. http://repub.eur.nl/res/pub/17090/ 2009-10-01T00:00:00Z Education is argued to be an important driver of the decision to start a business. The measurement of its influence, however, is difficult since it is considered to be an endogenous variable. This study is the first to account for this endogeneity by using an instrumental variables approach. The effect of education on the decision to become self-employed is found to be strongly positive, much higher than the estimated effect in case no instrumental variables are used. That is, the higher the respondent's level of education, the greater the likelihood that he or she starts a business. Implications for method and practice are discussed. 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/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/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/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/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/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/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/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/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/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/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/7847/ 2006-06-29T00:00:00Z De laatste decennia zijn voor allerlei economische processen complexe modellen afgeleid, zoals voor de groei van het Bruto Binnenlands Product (BBP). In deze modellen zijn in sommige gevallen geavanceerde methoden nodig om kansen te berekenen, bijvoorbeeld de kans op een naderende recessie. In zijn proefschrift Essays on Neural Network Sampling Methods and Instrumental Variables vergelijkt Lennart Hoogerheide een nieuwe, op neurale netwerken gebaseerde, methode met verschillende bekende methoden. De nieuwe methode blijkt betrouwbaar en snel te zijn. Tevens bekritiseert Hoogerheide een beroemd artikel van Angrist en Krueger uit 1991. Zij concludeerden dat in de Verenigde Staten ieder extra jaar onderwijs - gemiddeld genomen - later leidt tot een inkomensstijging van ongeveer 10 procent. Dit resultaat werd echter volledig bepaald door data van maar drie zuidelijke staten, en is dus niet representatief voor de gehele Verenigde Staten. Het meten van het effect van het genoten onderwijs van mensen op hun verdiende inkomen is van belang voor het vaststellen van onderwijsbeleid. Om dit effect te meten wordt een model met zogenaamde instrumentele variabelen gebruikt. 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/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/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/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/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/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.