An Agnostic Look at Bayesian Statistics and Econometrics

Bayesians and non-Bayesians, often called frequentists, seem to be perpetually at loggerheads on fundamental questions of statistical inference. This paper takes as agnostic a stand as is possible for a practising frequentist, and tries to elicit a Bayesian answer to questions of interest to frequentists. The argument is based on my presentation at a debate organised by the Rimini Centre for Economic Analysis, between me as the frequentist `àdvocate'', and Christian Robert on the Bayesian side.Bayesian methods; bootstrap; Bahadur-Savage result

STOCHASTIC DOMINANCE

The concept of stochastic dominance is defined, and its relation to welfare, poverty, and income inequality explained. A brief discussion is provided of how statistical inference may be performed for hypotheses relating to stochastic dominance.

Exploring the bootstrap discrepancy

Many simulation experiments have shown that, in a variety of circumstances, bootstrap tests perform better than current asymptotic theory predicts. Specifically, the discrepancy between the actual rejection probability of a bootstrap test under the null and the nominal level of the test appears to be smaller than suggested by theory, which in any case often yields only a rate of convergence of this discrepancy to zero. Here it is argued that the Edgeworth expansions on which much theory is based provide a quite inaccurate account of the finite-sample distributions of even quite basic statistics. Other methods are investigated in the hope that they may give better agreement with simulation evidence. They also suggest ways in which bootstrap procedures can be improved so as to yield more accurate inference.bootstrap discrepancy, bootstrap test, Edgeworth expansion

RELIABLE INFERENCE FOR THE GINI INDEX

Although attention has been given to obtaining reliable standard errors for the plugin estimator of the Gini index, all standard errors suggested until now are either complicated or quite unreliable. An approximation is derived for the estimator by which it is expressed as a sum of IID random variables. This approximation allows us to develop a reliable standard error that is simple to compute. A simple but effective bias correction is also derived. The quality of inference based on the approximation is checked in a number of simulation experiments, and is found to be very good unless the tail of the underlying distribution is heavy. Bootstrap methods are presented which alleviate this problem except in cases in which the variance is very large or fails to exist. Similar methods can be used to find reliable standard errors of other indices which are not simply linear functionals of the distribution function, such as Sen's poverty index and its modification known as the Sen-Shorrocks-Thonindex.Gini index, delta method, asymptotic inference, jackknife, boot-strap

An Agnostic Look at Bayesian Statistics and Econometrics

Bayesians and non-Bayesians, often called frequentists, seem to be perpetually at logger- heads on fundamental questions of statistical inference. This paper takes as agnostic a stand as is possible for a practising frequentist, and tries to elicit a Bayesian answer to questions of interest to frequentists. The argument is based on my presentation at a debate organised by the Rimini Centre for Economic Analysis, between me as the frequentist "advocate", and Christian Robert on the Bayesian side.Bayesian methods, bootstrap, Bahadur-Savage result

Bootstraping econometric models

The bootstrap is a statistical technique used more and more widely in econometrics. While it is capable of yielding very reliable inference, some precautions should be taken in order to ensure this. Two “Golden Rules” are formulated that, if observed, help to obtain the best the bootstrap can offer. Bootstrapping always involves setting up a bootstrap data-generating process (DGP). The main types of bootstrap DGP in current use are discussed, with examples of their use in econometrics. The ways in which the bootstrap can be used to construct confidence sets differ somewhat from methods of hypothesis testing. The relation between the two sorts of problem is discussed.Bootstrap, hypothesis test, confidence set

Inference on Income Distributions

This paper attempts to provide a synthetic view of varied techniques available for per- forming inference on income distributions. Two main approaches can be distinguished: one in which the object of interest is some index of income inequality or poverty, the other based on notions of stochastic dominance. From the statistical point of view, many techniques are common to both approaches, although of course some are specific to one of them. I assume throughout that inference about population quantities is to be based on a sample or samples, and, formally, all randomness is due to that of the sampling process. Inference can be either asymptotic or bootstrap-based. In principle, the bootstrap is an ideal tool, since in this paper I ignore issues of complex sampling schemes, and suppose that observations are IID. However both bootstrap inference, and, to a considerably greater extent, asymptotic inference can fall foul of difficulties associated with the heavy right-hand tails observed with many income distributions. I mention some recent attempts to circumvent these difficulties.Income distribution; delta method; asymptotic inference; bootstrap; influence function; empirical process

Testing for Restricted Stochastic Dominance: Some Further Results

Extensions are presented to the results of Davidson and Duclos (2007), whereby the null hypothesis of restricted stochastic non dominance can be tested by both asymptotic and bootstrap tests, the latter having considerably better properties as regards both size and power. In this paper, the methodology is extended to tests of higher-order stochastic dom- inance. It is seen that, unlike the first-order case, a numerical nonlinear optimisation prob- lem has to be solved in order to construct the bootstrap DGP. Conditions are provided for a solution to exist for this problem, and efficient numerical algorithms are laid out. The em- pirically important case in which the samples to be compared are correlated is also treated, both for first-order and for higher-order dominance. For all of these extensions, the boot- strap algorithm is presented. Simulation experiments show that the bootstrap tests perform considerably better than asymptotic tests, and yield reliable inference in moderately sized samples.Higher-order stochastic dominance, empirical likelihood, bootstrap test, corre- lated samples

Asymptotic and bootstrap inference for inequality and poverty measures

A random sample drawn from a population would appear to offer an ideal opportunity to use the bootstrap in order to perform accurate inference, since the observations of the sample are IID. In this paper, Monte Carlo results suggest that bootstrapping a commonly used index of inequality leads to inference that is not accurate even in very large samples, although inference with poverty indices is satisfactory. We find that the major cause is the extreme sensitivity of many inequality indices to the exact nature of the upper tail of the income distribution. This leads us to study two non-standard bootstraps, the m out of n bootstrap, which is valid in some situations where the standard bootstrap fails, and a bootstrap in which the upper tail is modelled parametrically. Monte Carlo results suggest that accurate inference can be achieved with this last method in moderately large samples.Income distribution; Poverty; Bootstrap inference

Wild bootstrap tests for IV regression

We propose a wild bootstrap procedure for linear regression models estimated by instrumental variables. Like other bootstrap procedures that we have proposed elsewhere, it uses efficient estimates of the reduced-form equation(s). Unlike them, it takes account of possible heteroskedasticity of unknown form. We apply this procedure to t tests, including heteroskedasticity-robust t tests, and to the Anderson-Rubin test. We provide simulation evidence that it works far better than older methods, such as the pairs bootstrap. We also show how to obtain reliable confidence intervals by inverting bootstrap tests. An empirical example illustrates the utility of these procedures.Instrumental variables estimation, two-stage least squares, weak instruments, wild bootstrap, pairs bootstrap, residual bootstrap, confidence intervals, Anderson-Rubin test