A reconsideration of the Angrist-Krueger analysis on returns to education

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 educationon 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 ofcompleted 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 requirementsand/or compulsory schooling laws for children of age 5-7 than to concentrate on thedifferences in compulsory schooling laws for students of age 16-18.

Note on neural network sampling for Bayesian inference of mixture processes

In this paper we show some further experiments with neural network sampling,a class of sampling methods that make use of neural network approximationsto (posterior) densities, introduced by Hoogerheide et al. (2007). We considera method where a mixture of Student's t densities, which can be interpreted asa neural network function, is used as a candidate density in importance samplingor the Metropolis-Hastings algorithm. It is applied to an illustrative2-regime mixture model for the US real GNP growth rate. We explain thenon-elliptical shapes of the posterior distribution, and show that the proposedmethod outperforms Gibbs sampling with data augmentation and the griddy Gibbs sampler.

Essays on Neural Network Sampling Methods and Instrumental Variables

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.This thesis consists of two parts. In the first part a class of sampling methods, which can be used in Bayesian analysis to get insight into the posterior density of model parameters, is introduced and explored. These sampling methods, which make use of neural network approximations to posterior densities, can quickly simulate draws from posterior distributions in many models. In the second part of this thesis new results are given for instrumental variables (IV) regression models. Particular attention is paid to a well-known IV model of Angrist and Krueger (1991, Quarterly Journal of Economics), who use quarter of birth to form instrumental variables in order to estimate the monetary returns to education. Measuring the effect of education on income is relevant for many decision processes; for example, for government agencies responsible for compulsory schooling laws. It should be noted that there is a connection between the two parts of this thesis: the ex! posed neural network sampling methods can be especially useful if one desires to get insight into irregularly shaped posterior distributions, and such posteriors may occur in IV regression models

Neural network approximations to posterior densities: an analytical approach

In Hoogerheide, Kaashoek and Van Dijk (2002) the class of neural networksampling methods is introduced to sample from a target (posterior)distribution that may be multi-modal or skew, or exhibit strong correlationamong the parameters. In these methods the neural network is used as animportance function in IS or as a candidate density in MH. In this note wesuggest an analytical approach to estimate the moments of a certain (target)distribution, where `analytical' refers to the fact that no samplingalgorithm like MH or IS is needed.We show an example in which our analyticalapproach is feasible, even in a case where a `standard' Gibbs approach wouldfail or be extremely slow.Markov chain Monte Carlo;Bayesian inference;importance sampling;neural networks

Functional approximations to posterior densities: a neural network approach to efficient sampling

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.Markov chain Monte Carlo;Bayesian inference;importance sampling;neural networks

On the shape of posterior densities and credible sets in instrumental variable regression models with reduced rank: an application of flexible sampling methods using neural networks

Likelihoods and posteriors of instrumental variable regression models with strongendogeneity and/or weak instruments may exhibit rather non-elliptical contours inthe parameter space. This may seriously affect inference based on Bayesian crediblesets. When approximating such contours using Monte Carlo integration methods likeimportance sampling or Markov chain Monte Carlo procedures the speed of the algorithmand the quality of the results greatly depend on the choice of the importance orcandidate density. Such a density has to be `close' to the target density in order toyield 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 ofillustrative models. The results indicate the feasibility of the neural networkapproach.Markov chain Monte Carlo;Bayesian inference;credible sets;importance sampling;instrumental variables;neural networks;reduced rank

Natural conjugate priors for the instrumental variables regression model applied to the Angrist-Krueger data

We propose a natural conjugate prior for the instrumentalvariables regression model. The prior is a natural conjugate onesince the marginal prior and posterior of the structural parameterhave the same functional expressions which directly reveal theupdate from prior to posterior. The Jeffreys prior results from aspecific setting of the prior parameters and results in a marginalposterior of the structural parameter that has an identicalfunctional form as the sampling density of the limited informationmaximum likelihood estimator. We construct informative priors forthe Angrist-Krueger (1991) data and show that the marginalposterior of the return on education in the US coincides with themarginal posterior from the Southern region when we use theJeffreys prior. This result occurs since the instruments are thestrongest in the Southern region and the posterior using theJeffreys prior, identical to maximum likelihood, focusses on thestrongest available instruments. We construct informative priorsfor the other regions that make their posteriors of the return oneducation similar to that of the US and the Southern region. Thesepriors show the amount of prior information needed to obtaincomparable results for all regions.

The AdMit Package

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 anddifficult 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.importance sampling;R software;Bayesian;adaptive mixture;student-t distribution;independence chain Metropolis-Hasting algorithm

Neural network based approximations to posterior densities: a class of flexible sampling methods with applications to reduced rank models

Likelihoods and posteriors of econometric models with strong endogeneity and weakinstruments may exhibit rather non-elliptical contours in the parameter space.This feature also holds for cointegration models when near non-stationarity occursand determining the number of cointegrating relations is a nontrivial issue, and in mixture processes where the modes are relatively far apart. The performance ofMonte Carlo integration methods like importance sampling or Markov ChainMonte 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 targetdensity in order to yield numerically accurate results with efficient sampling. Neural networks seem to be natural importance or candidate densities, as they havea universal approximation property and are easy to sample from. That is, conditionallyupon the specification of the neural network, sampling can be done either directly orusing 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 approximatesthe target density accurately. The methods are tested on a set of illustrative modelswhich include a mixture of normal distributions, a Bayesian instrumental variable regression problem with weak instruments and near non-identification, a cointegrationmodel with near non-stationarity and a two-regime growth model for US recessionsand expansions. These examples involve experiments with non-standard, non-ellipticalposterior distributions. The results indicate the feasibility of theneural network approach.Markov chain Monte Carlo;Bayesian inference;neural networks;importance sample

Simulation based bayesian econometric inference: principles and some recent computational advances.

In this paper we discuss several aspects of simulation basedBayesian econometric inference. We start at an elementary level on basic concepts of Bayesian analysis; evaluatingintegrals by simulation methods is a crucial ingredientin Bayesian inference. Next, the most popular and well-knownsimulation techniques are discussed, the Metropolis-Hastingsalgorithm and Gibbs sampling (being the most popular Markovchain Monte Carlo methods) and importance sampling. After that, we discuss two recently developed samplingmethods: 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 ofinterest. Both methods are especially useful in cases wherethe posterior distribution is not well-behaved, in the senseof having highly non-elliptical shapes. The simulationtechniques are illustrated in several example models, suchas a model for the real US GNP and models for binary data ofa US recession indicator.