It is argued that P-values and the tests based upon them give unsatisfactory results, especially in large samples. It is shown that, in regression, when there are many candidate independent variables, standard variable selection procedures can give very misleading results. Also, by selecting a single model, they ignore model uncertainty and so underestimate the uncertainty about quantities of interest. The Bayesian approach to hypothesis testing, model selection and accounting for model uncertainty is presented. Implementing this is straightforward using the simple and accurate BIC approximation, and can be done using the output from standard software. Specific results are presented for most of the types of model commonly used in sociology. It is shown that this approach overcomes the difficulties with P values and standard model selection procedures based on them. It also allows easy comparison of non-nested models, and permits the quantification of the evidence for a null hypothesis...

We consider the problem of accounting for model uncertainty in linear regression models. Conditioning on a single selected model ignores model uncertainty, and thus leads to the underestimation of uncertainty when making inferences about quantities of interest. A Bayesian solution to this problem involves averaging over all possible models (i.e., combinations of predictors) when making inferences about quantities of

In contrast to a posterior analysis given a particular sampling model, posterior model probabilities in the context of model uncertainty are typically rather sensitive to the specification of the prior. In particular, “diffuse” priors on model-specific parameters can lead to quite unexpected consequences. Here we focus on the practically relevant situation where we need to entertain a (large) number of sampling models and we have (or wish to use) little or no subjective prior information. We aim at providing an “automatic” or “benchmark” prior structure that can be used in such cases. We focus on the Normal linear regression model with uncertainty in the choice of regressors. We propose a partly noninformative prior structure related to a Natural Conjugate g-prior specification, where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g0j. The consequences of different choices for g0j are examined. We investigate theoretical properties, such as consistency of the implied Bayesian procedure. Links with classical information criteria are provided. More importantly, we examine the finite sample implications of several choices of g0j in a simulation study. The use of the MC3 algorithm of Madigan and York (1995), combined with efficient coding in Fortran, makes it feasible to conduct large simulations. In addition to posterior criteria, we shall also compare the predictive performance of different priors. A classic example concerning the economics of crime will also be provided and contrasted with results in the literature. The main findings of the paper will lead us to propose a “benchmark” prior specification in a linear regression context with model uncertainty.

We consider the problems of variable selection and accounting for model uncertainty in linear regression models. Conditioning on a single selected model ignores model uncertainty, and thus leads to the underestimation of uncertainty when making inferences about quantities of interest. The complete Bayesian solution to this problem involves averaging over all possible models when making inferences about quantities of interest. This approach is often not practical. In this paper we offer two alternative approaches. First we describe a Bayesian model selection algorithm called "Occam's "Window" which involves averaging over a reduced set of models. Second, we describe a Markov chain Monte Carlo approach which directly approximates the exact solution. Both these model averaging procedures provide better predictive performance than any single model which might reasonably have been selected. In the extreme case where there are many candidate predictors but there is no relationship between any of them and the response, standard variable selection procedures often choose some subset of variables that yields a high R² and a highly significant overall F value. We refer to this unfortunate phenomenon as "Freedman's Paradox" (Freedman, 1983). In this situation, Occam's vVindow usually indicates the null model as the only one to be considered, or else a small number of models including the null model, thus largely resolving the paradox.

In this article we take an empirical cross-country perspective to investigate the robustness and causality of the link between income inequality and crime rates. First, we study the correlation between the Gini index and, respectively, homicide and robbery rates along different dimensions of the data (within and between countries). Second, we examine the inequality-crime link when other potential crime determinants are controlled for. Third, we control for the likely joint endogeneity of income inequality in order to isolate its exogenous impact on homicide and robbery rates. Fourth, we control for the measurement error in crime rates by modelling it as both unobserved country-specific effects and random noise. Lastly, we examine the robustness of the inequality-crime link to alternative measures of inequality. The sample for estimation consists of panels of non-overlapping 5-year averages for 39 countries over 1965-95 in the case of homicides, and 37 countries over 1970-1994 in the case of robberies. We use a variety of statistical techniques, from simple correlations to regression analysis and from static OLS to dynamic GMM estimation. We find that crime rates and inequality are positively correlated (within each country and, particularly, between countries), and it appears that this correlation reflects causation from inequality to crime rates, even controlling for other crime determinants.

The paper investigates whether there is a causal link between poverty or low education and participation in terrorist activities. After presenting a discussion of theoretical issues, we review evidence on the determinants of hate crimes, which are closely related to terrorism. This literature finds that the occurrence of hate crimes is largely independent of economic conditions. Next we analyze data on support for terrorism from public opinion polls conducted in the West Bank and Gaza Strip. These polls indicate that support for terrorism does not decrease among those with higher education and higher living standards. The core contribution of the paper is a statistical analysis of the determinants of participation in Hezbollah terrorist activities in Lebanon in the late 1980s and early 1990s. The evidence that we have assembled suggests that having a living standard above the poverty line or a secondary school or higher education is positively associated with participation in terrorism. Although our results are tentative and exploratory, they suggest that neither poverty nor education have a direct, causal impact on terrorism. The conclusion speculates on why economic conditions and education are largely unrelated to participation in, and support for, terrorism. *This paper was prepared for the World Bank's Annual Bank Conference on Development Economics, April 2002. We thank Claude Berrebi for excellent research assistance, Eli Hurvitz, Ayoub Mustafa, Adib Nehmeh and Zeina el Khalil for providing data, and Joshua Angrist, Guido Imbens and Elie Tamer for helpful discussions.

Model averaging provides an alternative to model selection. An algorithm ARM rooted in information theory is proposed to combine different regression models/methods. A simulation is conducted in the context of linear regression to compare its performance with familiar model selection criteria AIC and BIC, and also with some Bayesian model averaging (BMA) methods. The simulation suggests

Abstract—The labor market prospects of young, unskilled men fell dramatically in the 1980s and improved in the 1990s. Crime rates show a reverse pattern: increasing during the 1980s and falling in the 1990s. Because young, unskilled men commit most crime, this paper seeks to establish a causal relationshi p between the two trends. Previous work on the relationship between labor markets and crime focused mainly on the relationshi p between the unemployment rate and crime, and found inconclusive results. In contrast, this paper examines the impact of both wages and unemployment on crime, and uses instrumental variables to establish causality. We conclude that both wages and unemployment are signi �-cantly related to crime, but that wages played a larger role in the crime trends over the last few decades. These results are robust to the inclusion of deterrence variables, controls for simultaneity, and controlling for individual and family characteristics.

The need to explore model uncertainty in linear regression models with many predictors has motivated improvements in Markov chain Monte Carlo sampling algorithms for Bayesian variable selection. Currently used sampling algorithms for Bayesian variable selection may perform poorly when there are severe multicollinearities among the predictors. This article describes a new sampling method based on an analogy with the Swendsen-Wang algorithm for the Ising model, and which can give substantial improvements over alternative sampling schemes in the presence of multicollinearity. In linear regression with a given set of potential predictors we can index possible models by a binary parameter vector that indicates which of the predictors are included or excluded. By thinking of the posterior distribution of this parameter as a binary spatial field, we can use auxiliary variable methods inspired by the Swendsen-Wang algorithm for the Ising model to sample from the posterior where dependence among parameters is reduced by conditioning on auxiliary variables. Performance of the method is described for both simulated and real data.

As the title of this essay suggests, I believe there are both positive and negative effects of inequality. On the positive side, differential rewards provide incentives for individuals to work hard, invest and innovate. On the negative side, differences in rewards that are unrelated to productivity – due to racial discrimination, for example – are corrosive to civil society and cause resources to be misallocated. Even if discrimination did not exist, however, income inequality would be problematic in a democratic society if those who are privileged use their economic muscle to curry favor in the political arena and thereby secure monopoly rents or other advantages. Moreover, for several reasons discussed in the next section, poverty and income inequality create negative externalities. Consequently, it can be in the interest of the wealthy as well as the poor to raise the incomes of the poor, especially by using education and training as a means for redistribution. The term inequality is often used rather loosely, and can be a lightning rod. 2 Some have argued that only extreme poverty is a concern. Others have argued that the gap in income or wealth between the well off and the poor is a concern. Yet others have argued that the