
Russell DavidsonMcGill University | McGill · Department of Economics
Russell Davidson
PhD
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119
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August 2002 - present
Publications
Publications (119)
We study the finite-sample properties of tests for overidentifying restrictions in linear regression models with a single endogenous regressor and weak instruments. Under the assumption of Gaussian disturbances, we derive expressions for a variety of test statistics as functions of eight mutually independent random variables and two nuisance parame...
A major contention in this paper is that scientific models can be viewed as virtual realities, implemented, or rendered, by mathematical equations or by computer simulations. Their purpose is to help us understand the external reality that they model. In economics, particularly in econometrics, models make use of random elements, so as to provide q...
It is known that Efron’s bootstrap of the mean of a distribution in the domain of attraction of the stable laws with infinite variance is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the limiting bootstrap distribution is ra...
The most widely used measure of segregation is the so-called dissimilarity index. It is now well understood that this measure also reflects randomness in the allocation of individuals to units; that is, it measures deviations from evenness, not deviations from randomness. This leads to potentially large values of the segregation index when unit siz...
Economists are often interested in the coefficient of a single endogenous explanatory variable in a linear simultaneous equations model. One way to obtain a confidence set for this coefficient is to invert the Anderson-Rubin test. The "AR confidence sets" that result have correct coverage under classical assumptions. However, AR confidence sets als...
A jump robust positive semidefinite rank-based estimator for the daily covariance matrix based on high-frequency intraday returns is proposed. It disentangles covariance estimation into variance and correlation components. This allows us to account for ...
Teaching graduate econometrics means covering three different kinds of subject matter: a grounding in the theory of econometrics, a long laundry list of available econometric techniques, and an introduction to the fact that the practice of linking models and data is every bit as untidy as mathematical statistics is neat. I assign Econometric Theory...
Income distributions are usually characterized by a heavy right‐hand tail. Apart from any ethical considerations raised by the presence among us of the very rich, statistical inference is complicated by the need to consider distributions of which the moments may not exist. In extreme cases, no valid inference about expectations is possible until re...
An axiomatic approach is used to develop a one-parameter family of measures of divergence between distributions. These measures can be used to perform goodness-of-fit tests with good statistical properties. Asymptotic theory shows that the test statistics have well-defined limiting distributions which are however analytically intractable. A paramet...
We study several methods of constructing confidence sets for the coefficient of the single right-hand-side endogenous variable in a linear equation with weak instruments. Two of these are based on conditional likelihood ratio (CLR) tests, and the others are based on inverting t statistics or the bootstrap P values associated with them. We propose a...
The standard forms of bootstrap iteration are very computationally demanding. As a result, there have been several attempts to alleviate the computational burden by use of approximations. In this paper, we extend the fast double bootstrap of Davidson and MacKinnon (2007) to higher orders of iteration, and provide algorithms for their implemen-tatio...
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...
This paper attempts to provide a synthetic view of varied techniques available for performing 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 tec...
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 higherorder stochastic...
It is known that Efron's resampling bootstrap of the mean of random variables with common distribution in the domain of attraction of the stable laws with infinite variance is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the...
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 c...
Testing for a unit root in a series obtained by summing a stationary MA(1) process with a parameter close to -1 leads to serious size distortions under the null, on account of the near cancellation of the unit root by the MA component in the driving stationary series. The situation is analysed from the point of view of bootstrap testing, and an exa...
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 bootst...
A Bayesian method for estimation of a hazard term structure is presented in a functional data analysis framework. The hazard terms structure is designed to include the effects of changes in economic conditions, as well as trends in stock prices and accounting ...
It is known that Efron's nonparametric bootstrap of the mean of random variables with common distribution in the domain of attraction of the stable laws is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the limiting distributi...
We study several tests for the coefficient of the single right-hand-side endogenous variable in a linear equation estimated by instrumental variables. We show that writing all the test statistics--Student's t, Anderson--Rubin, the LM statistic of Kleibergen and Moreira (K), and likelihood ratio (LR)--as functions of six random quantities leads to a...
Although attention has been given to obtaining reliable standard errors for the plug-in 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 re...
Associated with every popular nonlinear estimation method is at least one "artificial" linear regression. We define an artificial regression in terms of three conditions that it must satisfy. Then we show how artificial regressions can be useful for numerical optimization, testing hypotheses, and computing parameter estimates. Several existing arti...
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 sam...
Summary We develop a method based on the use of polar coordinates to investigate the existence of moments for instrumental variables and related estimators in the linear regression model. For generalized IV estimators, we obtain familiar results. For JIVE, we obtain the new result that this estimator has no moments at all. Simulation results illus...
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, includ...
Two procedures are proposed for estimating the rejection probabilities (RPs) of bootstrap tests in Monte Carlo experiments without actually computing a bootstrap test for each replication. These procedures are only about twice as expensive (per replication) as estimating RPs for asymptotic tests. Then a new procedure is proposed for computing boots...
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 bootst...
This paper estimates the risk preferences of cotton farmers in Southern Peru, using the results from a multiple-price-list lottery game. Assuming that preferences conform to two of the leading models of decision under risk--Expected Utility Theory (EUT) and Cumulative Prospect Theory (CPT)--we find strong evidence of moderate risk aversion. Once we...
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.
We perform an extensive series of Monte Carlo experiments to compare the performance of two variants of the ‘jackknife instrumental variables estimator’, or JIVE, with that of the more familiar 2SLS and LIML estimators. We find no evidence to suggest that JIVE should ever be used. It is always more dispersed than 2SLS, often very much so, and it is...
We perform an extensive series of Monte Carlo experiments to compare the performance of two variants of the 'jackknife instrumental variables estimator', or JIVE, with that of the more familiar 2SLS and LIML estimators. We find no evidence to suggest that JIVE should ever be used. It is always more dispersed than 2SLS, often very much so, and it is...
We introduce the concept of the bootstrap discrepancy, which measures the di#erence in rejection probabilities between a bootstrap test based on a given test statistic and that of a (usually infeasible) test based on the true distribution of the statistic. We show that the bootstrap discrepancy is of the same order of magnitude under the null hypot...
Asymptotic and bootstrap tests are studied for testing whether there is a relation of stochastic dominance between two distributions. These tests have a null hypothesis of nondominance, with the advantage that, if this null is rejected, then all that is left is dominance. This also leads us to define and focus on restricted stochastic dominance, th...
Artificial regressions are developed, based on elementary zero functions, that ex-ploit the fact that the normal distribution is completely characterised by its first two moments. These artificial regressions can be used as the basis of numeri-cal algorithms for the maximum likelihood estimation of models with normally distributed random elements,...
It has been shown in previous work that bootstrapping the J test for nonnested linear regression models dramatically improves its nite-sample performance. We provide evidence that a more sophisticated bootstrap procedure, which we call the fast double bootstrap, produces a very substantial further improvement in cases where the ordinary bootstrap d...
We provide a joint treatment of three major issues that surround testing for a unit root in practice: uncertainty as to whether or not a linear deterministic trend is present in the data, uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not, and the possible presence of nonstationary volatility in the...
It has been shown in previous work that bootstrapping the J test for nonnested linear regression models dramatically improves its finite-sample performance. We provide evidence that a more sophisticated bootstrap procedure, which we call the fast double bootstrap, produces a very substantial further improvement in cases where the ordinary bootstrap...
We show that the power of a bootstrap test will generally be very close to the level-adjusted power of the asymptotic test on which it is based, provided the latter is calculated properly. Our result, when combined with previous results on approximating the rejection frequency of bootstrap tests, provides a way to simulate the power of both asympto...
We first propose procedures for estimating the rejection probabilities for bootstrap tests in Monte Carlo experiments without actually computing a bootstrap test for each replication. These procedures are only about twice as expensive (per replication) as estimating rejection probabilities for asymptotic tests. We then propose procedures for comput...
This paper investigates the relation between hypothesis testing and the construc-tion of confidence intervals, with particular regard to bootstrap tests. In practice, confidence intervals are almost always based on Wald tests, and consequently are not invariant under nonlinear reparametrisations. Bootstrap percentile-t confidence intervals are an i...
The wild bootstrap is studied in the context of regression models with heteroskedastic disturbances. We show that, in one very specific case, perfect bootstrap inference is possible, and a substantial reduction in the error in the rejection probability of a bootstrap test is available much more generally. However, the version of the wild bootstrap...
We derive the asymptotic sampling distribution of various estimators frequently used to order distributions in terms of poverty, welfare, and inequality. This includes estimators of most of the poverty indices currently in use, as well as estimators of the curves used to infer stochastic dominance of any order. These curves can be used to determine...
Various versions of the wild bootstrap are studied as applied to regression models with heteroskedastic errors. It is shown that some versions can be qualified as "tamed," in the sense that the statistic bootstrapped is asymptotically independent of the distribution of the wild bootstrap DGP. This can, in one very specific case, lead to perfect boo...
A geometrical setting is constructed, based on Hilbert space, in which the asymptotic properties of estimators can be studied. Estimators are defined in the context of parametrised models, which are treated as submanifolds of an underlying Hilbert manifold, on which a parameter-defining mapping is defined as a submersion on to a finite-dimensional...
The paper is part of the research programme of the TMR network Living Standards, Inequality and Taxation" [Contract No. ERBFMRXCT 980248] of the European Communities, whose nancial support is gratefully acknowleged. This research is also supported by a grant from the Social Sciences and Humanities Research Council of Canada. May, 2000 1. Introducti...
This paper discusses how to choose the number of bootstrap samples when performing bootstrap tests. There are two important issues that arise when the number of bootstraps is finite. One is bias in the estimation of bootstrap P values or critical values, and the second is loss of power. We discuss an easy way to avoid bias and thus obtain exact tes...
We propose a form of semi-nonparametric regression based on wavelet analysis. Traditional time series methods usually involve either the time or the frequency domain, but wavelets can combine the information from both of these. While wavelet transforms are typically restricted to equally spaced observations an integer power of 2 in number, we show...
In practice, bootstrap tests must use a finite number of bootstrap samples. This means that the outcome of the test will depend on the sequence of random numbers used to generate the bootstrap samples, and it necessarily results in some loss of power. We examine the extent of this power loss and propose a simple pretest procedure for choosing the n...
The primary aim of the paper is to place current methodological discussions in macroeconometric modeling contrasting the ‘theory first’ versus the ‘data first’ perspectives in the context of a broader methodological framework with a view to constructively appraise them. In particular, the paper focuses on Colander’s argument in his paper “Economist...
We provide a theoretical framework in which to study the accuracy of bootstrap P values, which may be based on a parametric or nonparametric bootstrap. In the parametric case, the accuracy of a bootstrap test will depend on the shape of what we call the critical value function. We show that, in many circumstances, the error in rejection probability...
Bootstrap testing of nonlinear models normally requires at least one nonlinear estimation for every bootstrap sample. We show how to reduce computational costs by performing only a fixed, small number of Newton or quasi-Newton steps for each bootstrap sample. The number of steps is smaller for likelihood ratio tests than for other types of classica...
This paper examines the properties of stationary-state general equilibrium in a monocentric city with durable housing. On the demand side, identical households choose location, housing quality and quantity (floor area), and other goods. On the supply side, developers choose the structural density and time path of quality (which depends on construct...
Associated with every popular nonlinear estimationmethod is at least ont "artificial" linear regression. We define an artificial regression in terms of three conditions that it must satisfy. Then we show how artificial regressions can be useful for numerical optimization, testing hypotheses, and computing paremeter estimates. Several existing artif...
In this paper we are interested in inference based on heteroskedasticity consistent covariance matrix estimators, for which the appropriate bootstrap is a version of the wild bootstrap. Simulation results, obtained by a new very efficient methos, show that all wild bootstraps tests exhibit substantial size distortion if the error terms are skewed a...
Much progress has been made in recent years in understanding the mechanistic details of atomistic processes in the early stages of thin-film growth. We present in this paper a “shorthand” theoretical method for assessing the statistical consequences of different assumptions on the mechanism of island growth, and we illustrate the generality of the...
We draw upon the theoretical methods developed in the preceding contribution to explore different mechanisms of island growth in the initial stages of the development of a thin film. In particular, we study the entropic consequences of assuming different sequences for generating a final nucleation pattern or island morphology on a finite, planar ar...
Simple techniques for the graphical display of simulation evidence concerning the size and power of hypothesis tests are developed and illustrated. Three types of figures--called P value plots, P value discrepancy plots, and size-power curves--are discussed. Some Monte Carlo experiments on the properties of alternative forms of the information matr...
A geometrical setting is constructed, based on Hilbert space, in which the asymptotic properties of estimators can be studied. Estimators are defined in the context of parametrised models, which are treated as submanifolds of an underlying Hilbert manifold, on which a parameter-defining mapping is defined as a submersion on to a finite-dimensional...
We propose a form of semi-nonparametric regression based on wavelet analysis. Traditional time series methods usually involve either the time or the frequency domain, but wavelets can combine the information from both of these. While wavelet transforms are typically restricted to equally spaced observations an integer power of 2 in number, we show...
Bootstrap tests are tests for which the significance level is calculated using some variant of the bootstrap, which may be parametric or nonparametric. We show that the power of a bootstrap test will generally be very close to the power of the asymptotic test on which it is based, provided that both tests are properly adjusted to have the correct s...
Bootstrap tests are tests for which the significance level is calculated by some sort of bootstrap procedure, which may be parametric or nonparametric. We show that, in many circumstances, the size distortion of a bootstrap P value for a test will be one whole order of magnitude smaller than that of the corresponding asymptotic P value. We also sho...
Bootstrap tests are tests for which the significance level is calculated by some sort of bootstrap procedure, which may be parametric or nonparametric. We provide a theoretical framework in which to study the size distorsions of bootstrap P values. We show that, in many circumstances, the size distorsion of a bootstrap test will be one whole order...
We propose a form of semi-nonparametric regression based on wavelet analysis. Traditional time series methods usually involve either the time or the frequency domain, but wavelets can combine the information from both of these. While wavelet transforms are typically restricted to equally spaced observations an integer power of 2 in number, we show...
We establish the asymptotic sampling distribution of general functions of quantile-based estimators computed from samples that are not necessarily independent. The results provide the statistical framework within which to assess the progressivity of taxes and benefits, their horizontal inequity, and the change in the inequality of income which they...
Distribution-free techniques of statistical inference are developed for the cumulative coefficients of variation of an income distribution, thus allowing one to test for inequality dominance when Lorenz curves cross. The full covariance structure of the cumulative sample means and variances is worked out. As an illustration, the procedures are appl...
In both theoretical and empirical research it is a common practice to partition the economy into (at least) two sectors in order to conduct partial-equilibrium analysis. One merely hopes that general-equilibrium consequences will not obviate all of the analysis of the sector or market in question. In this paper we consider market demand functions w...
Offering a unifying theoretical perspective not readily available in any other text, this innovative guide to econometrics uses simple geometrical arguments to develop students' intuitive understanding of basic and advanced topics, emphasizing throughout the practical applications of modern theory and nonlinear techniques of estimation. One theme o...
A new form of the information matrix test is developed for a wide variety of statistical models. The test is constructed against an explicit alternative with random parameter variation. It is computed using a double-length artificial regression instead of the more conventional outer-product-of-the-gradient regression, which is known to have very po...
Any artificial regression that can be used to compute Lagrange Multiplier tests can just as easily be used to compute C(fi) tests. This also makes it possible to compute Wald-like tests by means of artificial regressions
Associated with every popular nonlinear estimation method is at least one "artificial" linear regression. We define an artificial regression in terms of three conditions that it must satisfy. Then we show how artificial regressions can be useful for numerical optimization, testing hypotheses, and computing parameter estimates. Several existing arti...
This chapter discusses that the use and misuse of single-consumer results in a multi-consumer economy. Many economic models are first tested in the context of an economy with a single consumer or a single firm or both. It is a sensible modeling strategy to abstract from those issues that are not immediately germane to the question at hand. However...
Methods based on linear regression provide an easy way to use the information in control variates to improve the efficiency with which certain features of the distributions of estimators and test statistics are estimated in Monte Carlo experiments. We propose a new technique that allows these methods to be used when the quantities of interest are q...
We characterize a general form of separability which we call implicit separability. It is more general than all known types of separability and contains them as special cases. We provide complete characterisations in terms of the direct preference ordering, its dual expenditure function, the demand functions, and the Slutsky
The issue of the non-invariance of the Wald test under nonlinear reparametrisations of the restrictions under test is studied from a differential geometric viewpoint. Quantities that can be defined in purely geometrical terms are by construction invariant under reparametrisation, and various attempts are made to construct a Wald test out of such in...
We introduce a new hybrid approach to joint estimation of Value at Risk (VaR) and Expected Shortfall (ES) for high quantiles of return distributions. We investigate the relative performance of VaR and ES models using daily returns for sixteen stock market indices (eight from developed and eight from emerging markets) prior to and during the 2008 fi...
We develop a new form of the information matrix test for a wide variety of statistical models, and present full details for the special case of univariate nonlinear regression models. Chesher (1984) showed that the implicit alternative of the information matrix test is a model with random parameter variation. We exploit this fact by constructing th...
In the leasing industry, the risk of loss on sales at the end of the contract term, as well as pricing are critically impacted by the forecasted resale price of the asset (residual value). We apply the Hedonic methodology to European auto lease portfolios, in order to estimate the resale price distribution. The Hedonic approach estimates the price...
Artificial linear regressions often provide a convenient way to calculate test statistics and estimated covariance ma trices. This paper discusses one family of these regressions called d ouble length because the number of observations in the artificial reg ression is twice the actual number of observations. These double-leng th regressions can be...
We consider several issues related to what Hausman (1978) called "specification tests", namely tests designed to verify the consistency of parameter estimates. We first review a number of results about these tests in linear regression models, and present some new material on their distribution when the model being tested is false, and on a simple w...
The local power of test statistics is analyzed by considering sequences of data-generating processes (DGPs) that approach the null hypothesis without necessarily satisfying the alternative. The three classical test statistics-LR, Wald, and LM-are shown to tend asymptot ically to the same random variable under all such sequences. The powe r of these...
The asymptotic power of a statistical test depends on the model being tested, the (implicit) alternative against which the test is constructed, and the process which actually generated the data. The exact way in which it does so is examined for several classes of models and tests. First, we analyze the power of tests of nonlinear regression models...
The primary aim of the paper is to place current methodological discussions in macroeconometric modeling contrasting the ‘theory first’ versus the ‘data first’ perspectives in the context of a broader methodological framework with a view to constructively appraise them. In particular, the paper focuses on Colander’s argument in his paper “Economist...
In this paper we calculate exactly the lineshape for a model of an excited two‐level atom in interaction with a continuous spectrum of radiation for the problem of spontaneous emission. Specifically, for the case of a d=1 radiation field, we use the exact results reported in our earlier work [J. Math. Phys. 14, 414, 423 (1973)] for the probability...
We develop simple procedures to test for omitted variables and perform other tests in regression directions, which are asymptotically valid in the presence of heteroskedasticity of unknown form. We examine the asymptotic behaviour of these tests, and use Edgeworth approximations to study their approximate finite-sample performance. We also present...
Focuses on the determinants of non-centrality parameters, which in turn determine the power of test statistics when the sample is large and the truth is close to the null hypothesis. Tests of regression models are divided into two types: tests against regression directions and tests against non-regression directions. A number of useful specificatio...
We propose several Lagrange multiplier tests of logit and probit models, which may be inexpen- sively computed by means of artificial linear regressions. These maybe used to test for various forms of model inadequacy, including the omission of specified variables and heteroskedasticity of known form. We perform a number of sampling experiments, in...
This paper investigates how a durable-goods owner responds to a once-for-all unanticipated shock in a stationary state market. This problem is examined in the context of housing. The circumstances are determined under which a landlord will respond in each of the following five ways: (i) abandon his building immediately; (ii) run down his building o...
Many specification tests can be computed by means of artificial linear regressions. These are linear regressions designed to be used as calculating devices to obtain test statistics and other quantities of interest. In this paper, we discuss the general principles which underlie all artificial regressions, and the use of such regressions to compute...
In a recent paper, Plosser, Schwert and White (1982) proposed a general test for model misspecification based on a comparison of estimates of the model in levels and first-differences. We demonstrate that this test is equivalent to a certain F test for omitted variables. The latter test has several advantages over the former. For example, the new t...
This paper constructs a two-country (Home and Foreign) general equilibrium model of Schumpeterian growth without scale effects. The scale effects property is removed by introducing two distinct specifications in the knowledge production function: the permanent effect on growth (PEG) specification, which allows policy effects on long-run growth; and...
In this paper we propose a non-nested hypothesis test for testing the specification of a multivariate econometric model in the presence of an alternative model which purports to explain the same phenomenon. We demonstrate that the new test statistic tends to minus the same random variable as the CPD test statistic introduced by Pesaran and Deaton (...
The paper considers the problem of statistical inference with estimated Lorenz curves and income shares. The full variance-covariance
structure of the (asymptotic) normal distribution of a vector of Lorenz curve ordinates is derived and shown to depend only
on conditional first and second moments that can be estimated consistently without prior spe...
Previous work of the authors on a three‐level quantum system is extended to allow the radiation field in interaction with the system to have a continuous spectrum of possible frequencies. The limiting procedure involved in the passage from discrete to continuous spectra is complicated by the need to express sums over different discrete spectra as i...
In this paper we examine two explanations of the observed positive relationship between inflation rates and savings rates in Canada and the United States. Several models are estimated using quarterly time-series data from both countries, and the best of these are subjected to a variety of tests. One of the two explanations appears to be broadly con...