Exact optimal and adaptive inference in regression models under heteroskedasticity and non-normality of unknown forms

In this paper, we derive simple point-optimal sign-based tests in the context of linear and nonlinear regression models with fixed regressors. These tests are exact, distribution-free, robust against heteroskedasticity of unknown form, and they may be inverted to obtain confidence regions for the vector of unknown parameters. Since the point-optimal sign tests depend on the alternative hypothesis, we propose an adaptive approach based on split-sample techniques in order to choose an alternative such that the power of point-optimal sign tests is close to the power envelope. The simulation results show that when using approximately 10% of sample to estimate the alternative and the rest to calculate the test statistic, the power of point-optimal sign test is typically close to the power envelope. We present a Monte Carlo study to assess the performance of the proposed “quasi”-point-optimal sign test by comparing its size and power to those of some common tests which are supposed to be robust against heteroskedasticity. The results show that our procedures are superior

Finite-Sample Simulation-Based Tests in Seemingly Unrelated Regressions

In this paper, we propose finite and large sample likelihood based test procedures for possibly non-linear hypotheses on the coefficients of SURE systems. Two complementary approaches are described. First, we propose an exact Monte Carlo bounds test based on the standard likelihood ratio criterion. Second, we consider alternative Monte Carlo tests which can be run whenever the bounds are not conclusive. These include, in particular, quasi-likelihood ratio criteria based on non-maximum-likelihood estimators. Illustrative Monte Carlo experiments show that: (i) the bounds are sufficiently tight to yield conclusive results in a large proportion of cases, and (ii) the randomized procedures correct all the usual size distortions in such contexts. The procedures proposed are finally applied to test restrictions on a factor demand model.Multivariate Linear Regression, Seemingly Unrelated Regressions, Monte Carlo Test, Bounds Tests, Nonlinear Hypothesis, Finite-Sample Test, Exact Test, Bootstrap, Factor Demand, Cost Function

Finite-Sample Simulation-Based Tests in Seemingly Unrelated Regressions

In this paper, we propose finite and large sample likelihood based test procedures for possibly non-linear hypotheses on the coefficients of SURE systems. Two complementary approaches are described. First, we propose an exact Monte Carlo bounds test based on the standard likelihood ratio criterion. Second, we consider alternative Monte Carlo tests which can be run whenever the bounds are not conclusive. These include, in particular, quasi-likelihood ratio criteria based on non-maximum-likelihood estimators. Illustrative Monte Carlo experiments show that: (i) the bounds are sufficiently tight to yield conclusive results in a large proportion of cases, and (ii) the randomized procedures correct all the usual size distortions in such contexts. The procedures proposed are finally applied to test restrictions on a factor demand model.Multivariate linear regression, Seemingly unrelated regressions, Monte Carlo test, Bounds test, Nonlinear hypothesis, Finite-sample test, Exact test, Bootstrap, Factor demand, Cost function

Markovian Processes, Two-Sided Autoregressions and Finite-Sample Inference for Stationary and Nonstationary Autoregressive Processes

In this paper, we develop finite-sample inference procedures for stationary and nonstationary autoregressive (AR) models. The method is based on special properties of Markov processes and a split-sample technique. The results on Markovian processes (intercalary independence and truncation) only require the existence of conditional densities. They are proved for possibly nonstationary and/or non-Gaussian multivariate Markov processes. In the context of a linear regression model with AR(1) errors, we show how these results can be used to simplify the distributional properties of the model by conditioning a subset of the data on the remaining observations. This transformation leads to a new model which has the form of a two-sided autoregression to which standard classical linear regression inference techniques can be applied. We show how to derive tests and confidence sets for the mean and/or autoregressive parameters of the model. We also develop a test on the order of an autoregression. We show that a combination of subsample-based inferences can improve the performance of the procedure. An application to U.S. domestic investment data illustrates the method. Dans cet article, nous proposons des procédures d'inférence valides à distance finie pour des modèles autorégressifs (AR) stationnaires et non-stationnaires. La méthode suggérée est fondée sur des propriétés particulières des processus markoviens combinées à une technique de subdivision d'échantillon. Les résultats sur les processus de Markov (indépendance intercalaire, troncature) ne requièrent que l'existence de densités conditionnelles. Nous démontrons les propriétés requises pour des processus markoviens multivariés possiblement non-stationnaires et non-gaussiens. Pour le cas des modèles de régression linéaires avec erreurs autorégressives d'ordre un, nous montrons comment utiliser ces résultats afin de simplifier les propriétés distributionnelles du modèle en considérant la distribution conditionnelle d'une partie des observations étant donné le reste. Cette transformation conduit à un nouveau modèle qui a la forme d'une autorégression bilatérale à laquelle on peut appliquer les techniques usuelles d'analyse des modèles de régression linéaires. Nous montrons comment obtenir des tests et régions de confiance pour la moyenne et les paramètres autorégressifs du modèle. Nous proposons aussi un test pour l'ordre d'une autorégression. Nous montrons qu'une technique de combinaison de tests obtenus à partir de plusieurs sous-échantillons peut améliorer la performance de la procédure. Enfin la méthode est appliquée à un modèle de l'investissement aux États-Unis.Time series, Markov process, autoregressive process, autocorrelation, dynamic model, distributed-lag model, two-sided autoregression, intercalary independence, exact test, finite-sample test, Ogawara-Hannan, investment, Séries chronologiques, processus de Markov, processus autorégressif, autocorrélation, modèle dynamique, modèle à retards échelonnés, autorégression bilatérale, indépendance intercalaire, test exact, Ogawara-Hannan, investissement

Confidence Regions for Calibrated Parameters in Computable General Equilibrium Models

We consider the problem of assessing the uncertainty of calibrated parameters in computable general equilibrium (CGE) models through the construction of confidence sets (or intervals) for these parameters. We study two different setups under which this can be done. The first one extends earlier work from Abdelkhalek and Dufour (1998) and is based on a projection technique which allows the construction of confidence sets for calibrated parameters from confidence sets on the free parameters of a (deterministc) CGE model. We discuss in detail how this approach can be applied to CES (Armington-type) function parameters frequently used in CGE models and illustrate it on models of the Moroccan economy. The second method allows one to extend the usual deterministic specification of CGE models by adding stochastic disturbances to the equations of the model and then to construct corresponding confidence sets for calibrated parameters using simulation techniques. This method uses the classical concept of a pivotal function for a parameter. We discuss in detail how this method can be applied to the calibrated parameters of a Cobb-Douglas production function. Nous considérons le problème de la prise en compte de l'incertitude sur les paramètres calibrés de modèles calculables d'équilibre général (MCEG) en construisant des régions (ou des intervalles) de confiance pour ces paramètres. Nous étudions en détail deux méthodes qui permettent de ce faire. La première est une extension des travaux de Abdelkhalek et Dufour (1998) et repose sur une technique de projection qui permet de construire des régions de confiance pour les paramètres calibrés à partir de régions de confiance pour les paramètres libres d'un MCEG déterministe. Nous discutons en détail comment cette approche peut être appliquée aux paramètres d'une fonction CES (de type Armington) d'usage fréquent dans les MCEG et nous l'illustrons sur des modèles de l'économie marocaine. La seconde méthode permet de dépasser le cadre déterministe usuel des MCEG en ajoutant des perturbations aléatoires à certaines équations du modèle pour construire des régions de confiance pour les paramètres calibrés en utilisant des techniques de simulation. Cette méthode utilise aussi le concept classique de fonction pivotale d'un paramètre. Nous discutons en détail comment cette méthode peut être appliquée aux paramèrtes calibrés d'une fonction de production de type Cobb-Douglas.Computable general equilibrium models, calibration, sensitivity analysis, confidence set, confidence interval, projection, Morocco, Modèles calculables d'équilibre général, calibration, région de confiance, intervalle de confiance, projection, analyse de sensibilité, Maroc

Simulation-Based Finite-Sample Inference in Simultaneous Equations

In simultaneous equation (SE) contexts, nuisance parameter, weak instruments and identification problems severely complicate exact and asymptotic tests (except for very specific hypotheses). In this paper, we propose exact likelihood based tests for possibly nonlinear hypotheses on the coefficients of SE systems. We discuss a number of bounds tests and Monte Carlo simulation based tests. The latter involves maximizing a randomized p-value function over the relevant nuisance parameter space which is done numerically by using a simulated annealing algorithm. We consider limited and full information models. We extend, to non-Gaussian contexts, the bound given in Dufour (Econometrica, 1997) on the null distribution of the LR criterion, associated with possibly non-linear- hypotheses on the coefficients of one Gaussian structural equation. We also propose a tighter bound which will hold: (i) for the limited information (LI) Gaussian hypothesis considered in Dufour (1997) and for more general, possibly cross-equation restrictions in a non-Gaussian multi-equation SE system. For the specific hypothesis which sets the value of the full vector of endogenous variables coefficients in a limited information framework, we extend the Anderson-Rubin test to the non-Gaussian framework. We also show that Wang and Zivot's (Econometrica, 1998) asymptotic bounds-test may be seen as an asymptotic version of the bound we propose here. In addition, we introduce a multi-equation Anderson-Rubin-type test. Illustrative Monte Carlo experiments show that: (i) bootstrapping standard instrumental variable (IV) based criteria fails to achieve size control, especially (but not exclusively) under near non-identification conditions, and (ii) the tests based on IV estimates do not appear to be boundedly pivotal and so no size-correction may be feasible. By contrast, likelihood ratio based tests work well in the experiments performedSimultaneous Equation, Weak Instruments, Monte Carlo test, Identification

Exact optimal and adaptive inference in regression models under heteroskedasticity and non-normality of unknown forms

In this paper, we derive simple point-optimal sign-based tests in the context of linear and nonlinear regression models with fixed regressors. These tests are exact, distribution-free, robust against heteroskedasticity of unknown form, and they may be inverted to obtain confidence regions for the vector of unknown parameters. Since the point-optimal sign tests depend on the alternative hypothesis, we propose an adaptive approach based on split-sample techniques in order to choose an alternative such that the power of point-optimal sign tests is close to the power envelope. The simulation results show that when using approximately 10% of sample to estimate the alternative and the rest to calculate the test statistic, the power of point-optimal sign test is typically close to the power envelope. We present a Monte Carlo study to assess the performance of the proposed “quasi”-point-optimal sign test by comparing its size and power to those of some common tests which are supposed to be robust against heteroskedasticity. The results show that our procedures are superior.Sign test, Point-optimal test, Nonlinear model, Heteroskedasticity, Exact inference, Distribution-free, Power envelope, Split-sample, Adaptive method, Projection

Exact Tests for Contemporaneous Correlation of Disturbances in Seemingly Unrelated Regressions

This paper proposes finite-sample procedures for testing the SURE specification in multi-equation regression models, i.e. whether the disturbances in different equations are contemporaneously uncorrelated or not. We apply the technique of Monte Carlo (MC) tests [Dwass (1957), Barnard (1963)] to obtain exact tests based on standard LR and LM zero correlation tests. We also suggest a MC quasi-LR (QLR) test based on feasible generalized least squares (FGLS). We show that the latter statistics are pivotal under the null, which provides the justification for applying MC tests. Furthermore, we extend the exact independence test proposed by Harvey and Phillips (1982) to the multi-equation framework. Specifically, we introduce several induced tests based on a set of simultaneous Harvey/Phillips-type tests and suggest a simulation-based solution to the associated combination problem. The properties of the proposed tests are studied in a Monte Carlo experiment which shows that standard asymptotic tests exhibit important size distortions, while MC tests achieve complete size control and display good power. Moreover, MC-QLR tests performed best in terms of power, a result of interest from the point of view of simulation-based tests. The power of the MC induced tests improves appreciably in comparison to standard Bonferroni tests and in certain cases outperform the likelihood-based MC tests. The tests are applied to data used by Fischer (1993) to analyze the macroeconomic determinants of growth. Cet article propose des procédures exactes pour tester la spécification SURE (régressions empilées) dans le contexte des régressions linéaires multivariées, i.e. si les perturbations des différentes équations sont corrélées ou non. Nous appliquons la technique des tests de Monte Carlo (MC) [Dwass (1957), Barnard (1963)] pour obtenir des tests d'indépendance exacts fondés sur les critères du quotient de vraisemblance (LR) et du multiplicateur de Lagrange (LM). Nous suggérons aussi un critère du type quasi-quotient de vraisemblance (QLR) dérivé sur base des moindres carrés généralisés réalisables (FGLS). Nous démontrons que ces statistiques sont libres de paramètres de nuisance sous l'hypothèse nulle, ce qui justifie l'application des tests de Monte Carlo. Par ailleurs, nous généralisons le test exact proposé par Harvey et Phillips (1982) au contexte des équations multiples. En particulier, nous proposons plusieurs tests induits basés sur des tests de type Harvey-Phillips et nous suggérons une technique basée sur des simulations afin de résoudre le problème de combinaison de tests. Nous évaluons les propriétés des tests que nous proposons dans le cadre d'une étude de Monte Carlo. Nos résultats montrent que les tests asymptotiques usuels présentent de sérieuses distorsions de niveau, alors que les tests de MC contrôlent parfaitement le niveau et ont une bonne puissance. De plus, les tests QLR se comportent bien du point de vue de la puissance; ce résultat est intéressant vu que les tests (multivariés) que nous proposons sont basés sur des simulations. La puissance des tests de MC induits augmente sensiblement par rapport aux tests fondés sur l'inégalité de Bonferroni et, dans certains cas, dépasse la puissance des tests de MC fondés sur la vraisemblance. Nous appliquons les tests sur des données utilisées par Fischer (1993) pour analyser des modèles de croissance.Seemingly unrelated regressions, SURE system, multivariate linear regression, contemporaneous correlation, exact test, finite-sample test, Monte Carlo test, bootstrap, induced test, LM test, likelihood ratio test, specification test, macroeconomics, growth, Régressions empilées, système SURE, test d'indépendance, régression linéaire multivariée, corrélation contemporaine, test exact, test à distance finie, test de Monte Carlo, bootstrap, test induit, test LM, quotient de vraisemblance, test de spécification, macroéconomie, croissance

Exact Nonparametric Two-Sample Homogeneity Tests for Possibly Discrete Distributions

In this paper, we study several tests for the equality of two unknown distributions. Two are based on empirical distribution functions, three others on nonparametric probability density estimates, and the last ones on differences between sample moments. We suggest controlling the size of such tests (under nonparametric assumptions) by using permutational versions of the tests jointly with the method of Monte Carlo tests properly adjusted to deal with discrete distributions. We also propose a combined test procedure, whose level is again perfectly controlled through the Monte Carlo test technique and has better power properties than the individual tests which are combined. Finally, in a simulation experiment, we show that the technique suggested provides perfect control of test size and that the new tests proposed can yield sizeable power improvements. Dans ce texte, nous étudions plusieurs tests pour l'egalité de deux distributions inconnues. Deux de ces tests sont basés sur des fonctions de distribution empiriques, trois autres sur des estimateurs non-paramétriques de fonctions de densité, et les trois derniers sur des moments empiriques. Nous proposons de contrôler la taille des tests (sous des hypothèses non-paramétriques) en employant des versions permutationnelles de ces tests conjointement avec la méthode des tests de Monte Carlo ajustée pour tenir compte de la possibilité de distributions discontinues. Nous proposons aussi une méthode pour combiner plusieurs de ces tests, le niveau de ces procédures étant aussi contrôlé par la technique des tests de Monte Carlo, laquelle possède de meilleures propriétés de puissance que les tests individuels combinés. Finalement, nous montrons dans une étude de simulation que la technique suggérée contrôle parfaitement la taille des différents tests considérés et que les nouveaux tests proposés peuvent fournir de notables améliorations de puissance.Nonparametric methods, two-sample problem, discrete distribution, discontinuous distribution, goodness-of-fit test, Kolmogorov-Smirnov test, Cramér-von Mises, kernel density estimator, exact test, permutation test, Monte Carlo test, bootstrap, combined test procedure, induced test, Méthodes non-paramétriques, problème des deux échantillons, distribution discrète, distribution discontinue, test d'ajustement, test de Kolmogorov-Smirnov, estimateur à noyau pour une densité, test exact, test de permutations, test de Monte Carlo, bootstrap, test combiné, test induit