Decision Theory Applied to a Linear Panel Data Model

This paper applies some general concepts in decision theory to a linear panel data model. A simple version of the model is an autoregression with a separate intercept for each unit in the cross section, with errors that are independent and identically distributed with a normal distribution. There is a parameter of interest γ and a nuisance parameter τ, a N×K matrix, where N is the cross‐section sample size. The focus is on dealing with the incidental parameters problem created by a potentially high‐dimension nuisance parameter. We adopt a “fixed‐effects” approach that seeks to protect against any sequence of incidental parameters. We transform τ to (δ, ρ, ω), where δ is a J×K matrix of coefficients from the least‐squares projection of τ on a N×J matrix x of strictly exogenous variables, ρ is a K×K symmetric, positive semidefinite matrix obtained from the residual sums of squares and cross‐products in the projection of τ on x, and ω is a (N−J) ×K matrix whose columns are orthogonal and have unit length. The model is invariant under the actions of a group on the sample space and the parameter space, and we find a maximal invariant statistic. The distribution of the maximal invariant statistic does not depend upon ω. There is a unique invariant distribution for ω. We use this invariant distribution as a prior distribution to obtain an integrated likelihood function. It depends upon the observation only through the maximal invariant statistic. We use the maximal invariant statistic to construct a marginal likelihood function, so we can eliminate ω by integration with respect to the invariant prior distribution or by working with the marginal likelihood function. The two approaches coincide. Decision rules based on the invariant distribution for ω have a minimax property. Given a loss function that does not depend upon ω and given a prior distribution for (γ, δ, ρ), we show how to minimize the average—with respect to the prior distribution for (γ, δ, ρ)—of the maximum risk, where the maximum is with respect to ω. There is a family of prior distributions for (δ, ρ) that leads to a simple closed form for the integrated likelihood function. This integrated likelihood function coincides with the likelihood function for a normal, correlated random‐effects model. Under random sampling, the corresponding quasi maximum likelihood estimator is consistent for γ as N→∞, with a standard limiting distribution. The limit results do not require normality or homoskedasticity (conditional on x) assumptions.     

Parameters of a Dose‐Response Model Are on the Boundary: What Happens with BMDL?

It is well known that, under appropriate regularity conditions, the asymptotic distribution for the likelihood ratio statistic is χ². This result is used in EPA's benchmark dose software to obtain a lower confidence bound (BMDL) for the benchmark dose (BMD) by the profile likelihood method. Recently, based on work by Self and Liang, it has been demonstrated that the asymptotic distribution of the likelihood ratio remains the same if some of the regularity conditions are violated, that is, when true values of some nuisance parameters are on the boundary. That is often the situation for BMD analysis of cancer bioassay data. In this article, we study by simulation the coverage of one- and two-sided confidence intervals for BMD when some of the model parameters have true values on the boundary of a parameter space. Fortunately, because two-sided confidence intervals (size 1–2α) have coverage close to the nominal level when there are 50 animals in each group, the coverage of nominal 1−α one-sided intervals is bounded between roughly 1–2α and 1. In many of the simulation scenarios with a nominal one-sided confidence level of 95%, that is, α= 0.05, coverage of the BMDL was close to 1, but for some scenarios coverage was close to 90%, both for a group size of 50 animals and asymptotically (group size 100,000). Another important observation is that when the true parameter is below the boundary, as with the shape parameter of a log-logistic model, the coverage of BMDL in a constrained model (a case of model misspecification not uncommon in BMDS analyses) may be very small and even approach 0 asymptotically. We also discuss that whenever profile likelihood is used for one-sided tests, the Self and Liang methodology is needed to derive the correct asymptotic distribution.     

Testing for Regime Switching

We analyze use of a quasi‐likelihood ratio statistic for a mixture model to test the null hypothesis of one regime versus the alternative of two regimes in a Markov regime‐switching context. This test exploits mixture properties implied by the regime‐switching process, but ignores certain implied serial correlation properties. When formulated in the natural way, the setting is nonstandard, involving nuisance parameters on the boundary of the parameter space, nuisance parameters identified only under the alternative, or approximations using derivatives higher than second order. We exploit recent advances by Andrews (2001) and contribute to the literature by extending the scope of mixture models, obtaining asymptotic null distributions different from those in the literature. We further provide critical values for popular models or bounds for tail probabilities that are useful in constructing conservative critical values for regime‐switching tests. We compare the size and power of our statistics to other useful tests for regime switching via Monte Carlo methods and find relatively good performance. We apply our methods to reexamine the classic cartel study of Porter (1983) and reaffirm Porter's findings.     

Decision Theory Applied to an Instrumental Variables Model

This paper applies some general concepts in decision theory to a simple instrumental variables model. There are two endogenous variables linked by a single structural equation; k of the exogenous variables are excluded from this structural equation and provide the instrumental variables (IV). The reduced‐form distribution of the endogenous variables conditional on the exogenous variables corresponds to independent draws from a bivariate normal distribution with linear regression functions and a known covariance matrix. A canonical form of the model has parameter vector (ρ, φ, ω), where φis the parameter of interest and is normalized to be a point on the unit circle. The reduced‐form coefficients on the instrumental variables are split into a scalar parameter ρand a parameter vector ω, which is normalized to be a point on the (k−1)‐dimensional unit sphere; ρmeasures the strength of the association between the endogenous variables and the instrumental variables, and ωis a measure of direction. A prior distribution is introduced for the IV model. The parameters φ, ρ, and ωare treated as independent random variables. The distribution for φis uniform on the unit circle; the distribution for ωis uniform on the unit sphere with dimension k‐1. These choices arise from the solution of a minimax problem. The prior for ρis left general. It turns out that given any positive value for ρ, the Bayes estimator of φdoes not depend on ρ; it equals the maximum‐likelihood estimator. This Bayes estimator has constant risk; because it minimizes average risk with respect to a proper prior, it is minimax. The same general concepts are applied to obtain confidence intervals. The prior distribution is used in two ways. The first way is to integrate out the nuisance parameter ωin the IV model. This gives an integrated likelihood function with two scalar parameters, φand ρ. Inverting a likelihood ratio test, based on the integrated likelihood function, provides a confidence interval for φ. This lacks finite sample optimality, but invariance arguments show that the risk function depends only on ρand not on φor ω. The second approach to confidence sets aims for finite sample optimality by setting up a loss function that trades off coverage against the length of the interval. The automatic uniform priors are used for φand ω, but a prior is also needed for the scalar ρ, and no guidance is offered on this choice. The Bayes rule is a highest posterior density set. Invariance arguments show that the risk function depends only on ρand not on φor ω. The optimality result combines average risk and maximum risk. The confidence set minimizes the average—with respect to the prior distribution for ρ—of the maximum risk, where the maximization is with respect to φand ω.     

Testing When a Parameter is on the Boundary of the Maintained Hypothesis

This paper considers testing problems where several of the standard regularity conditions fail to hold. We consider the case where (i) parameter vectors in the null hypothesis may lie on the boundary of the maintained hypothesis and (ii) there may be a nuisance parameter that appears under the alternative hypothesis, but not under the null. The paper establishes the asymptotic null and local alternative distributions of quasi‐likelihood ratio, rescaled quasi‐likelihood ratio, Wald, and score tests in this case. The results apply to tests based on a wide variety of extremum estimators and apply to a wide variety of models. Examples treated in the paper are: (i) tests of the null hypothesis of no conditional heteroskedasticity in a GARCH(1, 1) regression model and (ii) tests of the null hypothesis that some random coefficients have variances equal to zero in a random coefficients regression model with (possibly) correlated random coefficients.     

Testing Parameters in GMM Without Assuming that They Are Identified

We propose a generalized method of moments (GMM) Lagrange multiplier statistic, i.e., the K statistic, that uses a Jacobian estimator based on the continuous updating estimator that is asymptotically uncorrelated with the sample average of the moments. Its asymptotic χ² distribution therefore holds under a wider set of circumstances, like weak instruments, than the standard full rank case for the expected Jacobian under which the asymptotic χ² distributions of the traditional statistics are valid. The behavior of the K statistic can be spurious around inflection points and maxima of the objective function. This inadequacy is overcome by combining the K statistic with a statistic that tests the validity of the moment equations and by an extension of Moreira's (2003) conditional likelihood ratio statistic toward GMM. We conduct a power comparison to test for the risk aversion parameter in a stochastic discount factor model and construct its confidence set for observed consumption growth and asset return series.     

Conditional Linear Combination Tests for Weakly Identified Models

We introduce the class of conditional linear combination tests, which reject null hypotheses concerning model parameters when a data‐dependent convex combination of two identification‐robust statistics is large. These tests control size under weak identification and have a number of optimality properties in a conditional problem. We show that the conditional likelihood ratio test of Moreira, 2003 is a conditional linear combination test in models with one endogenous regressor, and that the class of conditional linear combination tests is equivalent to a class of quasi‐conditional likelihood ratio tests. We suggest using minimax regret conditional linear combination tests and propose a computationally tractable class of tests that plug in an estimator for a nuisance parameter. These plug‐in tests perform well in simulation and have optimal power in many strongly identified models, thus allowing powerful identification‐robust inference in a wide range of linear and nonlinear models without sacrificing efficiency if identification is strong.     

Conditional Inference With a Functional Nuisance Parameter

This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite‐dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter in a Gaussian problem and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi‐likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to existing alternatives in two examples.     

Comment on: Threshold Autoregressions With a Unit Root

In this paper we revisit the results in Caner and Hansen (2001), where the authors obtained novel limiting distributions of Wald type test statistics for testing for the presence of threshold nonlinearities in autoregressive models containing unit roots. Using the same framework, we obtain a new formulation of the limiting distribution of the Wald statistic for testing for threshold effects, correcting an expression that appeared in the main theorem presented by Caner and Hansen. Subsequently, we show that under a particular scenario that excludes stationary regressors such as lagged dependent variables and despite the presence of a unit root, this same limiting random variable takes a familiar form that is free of nuisance parameters and already tabulated in the literature, thus removing the need to use bootstrap based inferences. This is a novel and unusual occurrence in this literature on testing for the presence of nonlinear dynamics.     

Estimating and Testing Structural Changes in Multivariate Regressions

This paper considers issues related to estimation, inference, and computation with multiple structural changes that occur at unknown dates in a system of equations. Changes can occur in the regression coefficients and/or the covariance matrix of the errors. We also allow arbitrary restrictions on these parameters, which permits the analysis of partial structural change models, common breaks that occur in all equations, breaks that occur in a subset of equations, and so forth. The method of estimation is quasi‐maximum likelihood based on Normal errors. The limiting distributions are obtained under more general assumptions than previous studies. For testing, we propose likelihood ratio type statistics to test the null hypothesis of no structural change and to select the number of changes. Structural change tests with restrictions on the parameters can be constructed to achieve higher power when prior information is present. For computation, an algorithm for an efficient procedure is proposed to construct the estimates and test statistics. We also introduce a novel locally ordered breaks model, which allows the breaks in different equations to be related yet not occurring at the same dates.     

Risk of Bayesian Inference in Misspecified Models,and the Sandwich Covariance Matrix

It is well known that, in misspecified parametric models, the maximum likelihood estimator (MLE) is consistent for the pseudo‐true value and has an asymptotically normal sampling distribution with “sandwich” covariance matrix. Also, posteriors are asymptotically centered at the MLE, normal, and of asymptotic variance that is, in general, different than the sandwich matrix. It is shown that due to this discrepancy, Bayesian inference about the pseudo‐true parameter value is, in general, of lower asymptotic frequentist risk when the original posterior is substituted by an artificial normal posterior centered at the MLE with sandwich covariance matrix. An algorithm is suggested that allows the implementation of this artificial posterior also in models with high dimensional nuisance parameters which cannot reasonably be estimated by maximizing the likelihood.     

Individual Prior Information in a Physiological Model of 2H8-Toluene Kinetics: An Empirical Bayes Estimation Strategy

Physiologically-based toxicokinetic (PBTK) models are widely used to quantify whole-body kinetics of various substances. However, since they attempt to reproduce anatomical structures and physiological events, they have a high number of parameters. Their identification from kinetic data alone is often impossible, and other information about the parameters is needed to render the model identifiable. The most commonly used approach consists of independently measuring, or taking from literature sources, some of the parameters, fixing them in the kinetic model, and then performing model identification on a reduced number of less certain parameters. This results in a substantial reduction of the degrees of freedom of the model. In this study, we show that this method results in final estimates of the free parameters whose precision is overestimated. We then compared this approach with an empirical Bayes approach, which takes into account not only the mean value, but also the error associated with the independently determined parameters. Blood and breath ²H₈-toluene washout curves, obtained in 17 subjects, were analyzed with a previously presented PBTK model suitable for person-specific dosimetry. Model parameters with the greatest effect on predicted levels were alveolar ventilation rate Q_PC, fat tissue fraction V_FC, blood-air partition coefficient K_b, fraction of cardiac output to fat Q_a/co and rate of extrahepatic metabolism V_max-p. Differences in the measured and Bayesian-fitted values of Q_PC, V_FC and K_b were significant (p < 0.05), and the precision of the fitted values V_max-p and Q_a/co went from 11 ± 5% to 75 ± 170% (NS) and from 8 ± 2% to 9 ± 2% (p < 0.05) respectively. The empirical Bayes approach did not result in less reliable parameter estimates: rather, it pointed out that the precision of parameter estimates can be overly optimistic when other parameters in the model, either directly measured or taken from literature sources, are treated as known without error. In conclusion, an empirical Bayes approach to parameter estimation resulted in a better model fit, different final parameter estimates, and more realistic parameter precisions.     

Nonlinear Regressions with Integrated Time Series

An asymptotic theory is developed for nonlinear regression with integrated processes. The models allow for nonlinear effects from unit root time series and therefore deal with the case of parametric nonlinear cointegration. The theory covers integrable and asymptotically homogeneous functions. Sufficient conditions for weak consistency are given and a limit distribution theory is provided. The rates of convergence depend on the properties of the nonlinear regression function, and are shown to be as slow as n^1/4 for integrable functions, and to be generally polynomial in n^1/2 for homogeneous functions. For regressions with integrable functions, the limiting distribution theory is mixed normal with mixing variates that depend on the sojourn time of the limiting Brownian motion of the integrated process.     

Subsampling Intervals in Autoregressive Models with Linear Time Trend

A new method is proposed for constructing confidence intervals in autoregressive models with linear time trend. Interest focuses on the sum of the autoregressive coefficients because this parameter provides a useful scalar measure of the long‐run persistence properties of an economic time series. Since the type of the limiting distribution of the corresponding OLS estimator, as well as the rate of its convergence, depend in a discontinuous fashion upon whether the true parameter is less than one or equal to one (that is, trend‐stationary case or unit root case), the construction of confidence intervals is notoriously difficult. The crux of our method is to recompute the OLS estimator on smaller blocks of the observed data, according to the general subsampling idea of Politis and Romano (1994a), although some extensions of the standard theory are needed. The method is more general than previous approaches in that it works for arbitrary parameter values, but also because it allows the innovations to be a martingale difference sequence rather than i.i.d. Some simulation studies examine the finite sample performance.     

A Small Sample Correction for the Test of Cointegrating Rank in the Vector Autoregressive Model

With the cointegration formulation of economic long–run relations the test for cointegrating rank has become a useful econometric tool. The limit distribution of the test is often a poor approximation to the finite sample distribution and it is therefore relevant to derive an approximation to the expectation of the likelihood ratio test for cointegration in the vector autoregressive model in order to improve the finite sample properties. The correction factor depends on moments of functions of the random walk, which are tabulated by simulation, and functions of the parameters, which are estimated. From this approximation we propose a correction factor with the purpose of improving the small sample performance of the test. The correction is found explicitly in a number of simple models and its usefulness is illustrated by some simulation experiments.     

A gradient Markov chain Monte Carlo algorithm for computing multivariate maximum likelihood estimates and posterior distributions: mixture dose-response assessment

Multivariate probability distributions, such as may be used for mixture dose‐response assessment, are typically highly parameterized and difficult to fit to available data. However, such distributions may be useful in analyzing the large electronic data sets becoming available, such as dose‐response biomarker and genetic information. In this article, a new two‐stage computational approach is introduced for estimating multivariate distributions and addressing parameter uncertainty. The proposed first stage comprises a gradient Markov chain Monte Carlo (GMCMC) technique to find Bayesian posterior mode estimates (PMEs) of parameters, equivalent to maximum likelihood estimates (MLEs) in the absence of subjective information. In the second stage, these estimates are used to initialize a Markov chain Monte Carlo (MCMC) simulation, replacing the conventional burn‐in period to allow convergent simulation of the full joint Bayesian posterior distribution and the corresponding unconditional multivariate distribution (not conditional on uncertain parameter values). When the distribution of parameter uncertainty is such a Bayesian posterior, the unconditional distribution is termed predictive. The method is demonstrated by finding conditional and unconditional versions of the recently proposed emergent dose‐response function (DRF). Results are shown for the five‐parameter common‐mode and seven‐parameter dissimilar‐mode models, based on published data for eight benzene–toluene dose pairs. The common mode conditional DRF is obtained with a 21‐fold reduction in data requirement versus MCMC. Example common‐mode unconditional DRFs are then found using synthetic data, showing a 71% reduction in required data. The approach is further demonstrated for a PCB 126‐PCB 153 mixture. Applicability is analyzed and discussed. Matlab^® computer programs are provided.     

Testing for Common Conditionally Heteroskedastic Factors

This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that Hansen's (1982) J‐test statistic is asymptotically distributed as the minimum of the limit of a certain random process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a fifty–fifty mixture of χ²_H−1 and χ²_H, where H is the number of moment conditions, as opposed to a χ²_H−1. With more than two assets, this distribution lies between the χ²_H−p and χ²_H (p denotes the number of parameters). These results show that ignoring the lack of first‐order identification of the moment condition model leads to oversized tests with a possibly increasing overrejection rate with the number of assets. A Monte Carlo study illustrates these findings.     

Semiparametric Power Envelopes for Tests of the Unit Root Hypothesis

This paper derives asymptotic power envelopes for tests of the unit root hypothesis in a zero‐mean AR(1) model. The power envelopes are derived using the limits of experiments approach and are semiparametric in the sense that the underlying error distribution is treated as an unknown infinite‐dimensional nuisance parameter. Adaptation is shown to be possible when the error distribution is known to be symmetric and to be impossible when the error distribution is unrestricted. In the latter case, two conceptually distinct approaches to nuisance parameter elimination are employed in the derivation of the semiparametric power bounds. One of these bounds, derived under an invariance restriction, is shown by example to be sharp, while the other, derived under a similarity restriction, is conjectured not to be globally attainable.     

Swapping the Nested Fixed Point Algorithm: A Class of Estimators for Discrete Markov Decision Models

This paper proposes a new nested algorithm (NPL) for the estimation of a class of discrete Markov decision models and studies its statistical and computational properties. Our method is based on a representation of the solution of the dynamic programming problem in the space of conditional choice probabilities. When the NPL algorithm is initialized with consistent nonparametric estimates of conditional choice probabilities, successive iterations return a sequence of estimators of the structural parameters which we call K–stage policy iteration estimators. We show that the sequence includes as extreme cases a Hotz–Miller estimator (for K=1) and Rust's nested fixed point estimator (in the limit when K→∞). Furthermore, the asymptotic distribution of all the estimators in the sequence is the same and equal to that of the maximum likelihood estimator. We illustrate the performance of our method with several examples based on Rust's bus replacement model. Monte Carlo experiments reveal a trade–off between finite sample precision and computational cost in the sequence of policy iteration estimators.     

GMM Estimation of Autoregressive Roots Near Unity with Panel Data

This paper investigates a generalized method of moments (GMM) approach to the estimation of autoregressive roots near unity with panel data and incidental deterministic trends. Such models arise in empirical econometric studies of firm size and in dynamic panel data modeling with weak instruments. The two moment conditions in the GMM approach are obtained by constructing bias corrections to the score functions under OLS and GLS detrending, respectively. It is shown that the moment condition under GLS detrending corresponds to taking the projected score on the Bhattacharya basis, linking the approach to recent work on projected score methods for models with infinite numbers of nuisance parameters (Waterman and Lindsay (1998)). Assuming that the localizing parameter takes a nonpositive value, we establish consistency of the GMM estimator and find its limiting distribution. A notable new finding is that the GMM estimator has convergence rate , slower than , when the true localizing parameter is zero (i.e., when there is a panel unit root) and the deterministic trends in the panel are linear. These results, which rely on boundary point asymptotics, point to the continued difficulty of distinguishing unit roots from local alternatives, even when there is an infinity of additional data.