Simulation-Based Finite and Large Sample Tests in Multivariate Regressions.

In the context of multivariate linear regression (MLR) models, it is well known that commonly employed asymptotic test criteria are seriously biased towards overrejection. In this paper, we propose a general method for constructing exact tests of possibly nonlinear hypotheses on the coefficients of MLR systems. For the case of uniform linear hypotheses, we present exact distributional invariance results concerning several standard test criteria. These include Wilks' likelihood ratio (LR) criterion as well as trace and maximum root criteria. The normality assumption is not necessary for most of the results to hold. Implications for inference are two-fold. First, invariance to nuisance parameters entails that the technique of Monte Carlo tests can be applied on all these statistics to obtain exact tests of uniform linear hypotheses. Second, the invariance property of the latter statistic is exploited to derive general nuisance-parameter-free bounds on the distribution of the LR statistic for arbitrary hypotheses. Even though it may be difficult to compute these bounds analytically, they can easily be simulated, hence yielding exact bounds Monte Carlo tests. Illustrative simulation experiments show that the bounds are sufficiently tight to provide conclusive results with a high probability. Our findings illustrate the value of the bounds as a tool to be used in conjunction with more traditional simulation-based test methods (e.g., the parametric bootstrap) which may be applied when the bounds are not conclusive.

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, outperforms the likelihood-based MC tests. The tests are applied to data used by Fischer (1993) to analyze the macroeconomic determinants of growth.

Nonparametric Instrumental Regression

The focus of the paper is the nonparametric estimation of an instrumental regression function P defined by conditional moment restrictions stemming from a structural econometric model : E[Y-P(Z)|W]=0 and involving endogenous variables Y and Z and instruments W. The function P is the solution of an ill-posed inverse problem and we propose an estimation procedure based on Tikhonov regularization. The paper analyses identification and overidentification of this model and presents asymptotic properties of the estimated nonparametric instrumental regression function.

Some Robust Exact Results on Sample Autocorrelations and Tests of Randomness

Ce Texte Presente Plusieurs Resultats Exacts Sur les Seconds Moments des Autocorrelations Echantillonnales, Pour des Series Gaussiennes Ou Non-Gaussiennes. Nous Donnons D'abord des Formules Generales Pour la Moyenne, la Variance et les Covariances des Autocorrelations Echantillonnales, Dans le Cas Ou les Variables de la Serie Sont Interchangeables. Nous Deduisons de Celles-Ci des Bornes Pour les Variances et les Covariances des Autocorrelations Echantillonnales. Ces Bornes Sont Utilisees Pour Obtenir des Limites Exactes Sur les Points Critiques Lorsqu'on Teste le Caractere Aleatoire D'une Serie Chronologique, Sans Qu'aucune Hypothese Soit Necessaire Sur la Forme de la Distribution Sous-Jacente. Nous Donnons des Formules Exactes et Explicites Pour les Variances et Covariances des Autocorrelations Dans le Cas Ou la Serie Est un Bruit Blanc Gaussien. Nous Montrons Que Ces Resultats Sont Aussi Valides Lorsque la Distribution de la Serie Est Spheriquement Symetrique. Nous Presentons les Resultats D'une Simulation Qui Indiquent Clairement Qu'on Approxime Beaucoup Mieux la Distribution des Autocorrelations Echantillonnales En Normalisant Celles-Ci Avec la Moyenne et la Variance Exactes et En Utilisant la Loi N(0,1) Asymptotique, Plutot Qu'en Employant les Seconds Moments Approximatifs Couramment En Usage. Nous Etudions Aussi les Variances et Covariances Exactes D'autocorrelations Basees Sur les Rangs des Observations.

Tests of Joint Hypotheses for Time Series Regression with a Unit Root

This Paper Studies Tests of Joint Hypotheses in Time Series Regression with a Unit Root in Which Weakly Dependent and Heterogeneously Distributed Innovations Are Allowed. We Consider Two Types of Regression: One with a Constant and Lagged Dependent Variable, and the Other with a Trend Added. the Statistics Studied Are the Regression \"F-Test\" Originally Analysed by Dickey and Fuller (1981) in a Less General Framework. the Limiting Distributions Are Found Using Functinal Central Limit Theory. New Test Statistics Are Proposed Which Require Only Already Tabulated Critical Values But Which Are Valid in a Quite General Framework (Including Finite Order Arma Models Generated by Gaussian Errors). This Study Extends the Results on Single Coefficients Derived in Phillips (1986A) and Phillips and Perron (1986).

Some Additional Specification Tests for Generalized Method of Moments Estimators with Macro-Economic Applications Part I : Theory

In This Paper Several Additional Gmm Specification Tests Are Studied. a First Test Is a Chow-Type Test for Structural Parameter Stability of Gmm Estimates. the Test Is Inspired by the Fact That \"Taste and Technology\" Parameters Are Uncovered. the Second Set of Specification Tests Are Var Encompassing Tests. It Is Assumed That the Dgp Has a Finite Var Representation. the Moment Restrictions Which Are Suggested by Economic Theory and Exploited in the Gmm Procedure Represent One Possible Characterization of the Dgp. the Var Is a Different But Compatible Characterization of the Same Dgp. the Idea of the Var Encompassing Tests Is to Compare Parameter Estimates of the Euler Conditions and Var Representations of the Dgp Obtained Separately with Parameter Estimates of the Euler Conditions and Var Representations Obtained Jointly. There Are Several Ways to Construct Joint Systems Which Are Discussed in the Paper. Several Applications Are Also Discussed.

Tests of Conditional Asset Pricing Models in the Brazilian Stock Market

In this paper, we test a version of the conditional CAPM with respect to a local market portfolio, proxied by the Brazilian stock index during the 1976-1992 period. We also test a conditional APT model by using the difference between the 30-day rate (Cdb) and the overnight rate as a second factor in addition to the market portfolio in order to capture the large inflation risk present during this period. The conditional CAPM and APT models are estimated by the Generalized Method of Moments (GMM) and tested on a set of size portfolios created from a total of 25 securities exchanged on the Brazilian markets. The inclusion of this second factor proves to be crucial for the appropriate pricing of the portfolios.

GLS Detrending, Efficient Unit Root Tests and Structural Change

We extend the class of M-tests for a unit root analyzed by Perron and Ng (1996) and Ng and Perron (1997) to the case where a change in the trend function is allowed to occur at an unknown time. These tests M(GLS) adopt the GLS detrending approach of Dufour and King (1991) and Elliott, Rothenberg and Stock (1996) (ERS). Following Perron (1989), we consider two models : one allowing for a change in slope and the other for both a change in intercept and slope. We derive the asymptotic distribution of the tests as well as that of the feasible point optimal tests PT(GLS) suggested by ERS. The asymptotic critical values of the tests are tabulated. Also, we compute the non-centrality parameter used for the local GLS detrending that permits the tests to have 50% asymptotic power at that value. We show that the M(GLS) and PT(GLS) tests have an asymptotic power function close to the power envelope. An extensive simulation study analyzes the size and power in finite samples under various methods to select the truncation lag for the autoregressive spectral density estimator. An empirical application is also provided.

Simulation-Based Finite-Sample Normality Tests in Linear Regressions

In the literature on tests of normality, much concern has been expressed over the problems associated with residual-based procedures. Indeed, the specialized tables of critical points which are needed to perform the tests have been derived for the location-scale model; hence reliance on available significance points in the context of regression models may cause size distortions. We propose a general solution to the problem of controlling the size normality tests for the disturbances of standard linear regression, which is based on using the technique of Monte Carlo tests.

Tests of Conditional Asset Pricing Models in the Brazilian Stock Market

In this paper, we test a version of the conditional CAPM with respect to a local market portfolio, proxied by the Brazilian stock index during the 1976-1992 period. We also test a conditional APT model by using the difference between the 30-day rate (Cdb) and the overnight rate as a second factor in addition to the market portfolio in order to capture the large inflation risk present during this period. the conditional CAPM and APT models are estimated by the Generalized Method of Moments (GMM) and tested on a set of size portfolios created from a total of 25 securities exchanged on the Brazilian markets. the inclusion of this second factor proves to be crucial for the appropriate pricing of the portfolios.

Exact Skewness-Kurtosis Tests for Multivariate Normality and Goodness-of-fit in Multivariate Regressions with Application to Asset Pricing Models

We study the problem of testing the error distribution in a multivariate linear regression (MLR) model. The tests are functions of appropriately standardized multivariate least squares residuals whose distribution is invariant to the unknown cross-equation error covariance matrix. Empirical multivariate skewness and kurtosis criteria are then compared to simulation-based estimate of their expected value under the hypothesized distribution. Special cases considered include testing multivariate normal, Student t; normal mixtures and stable error models. In the Gaussian case, finite-sample versions of the standard multivariate skewness and kurtosis tests are derived. To do this, we exploit simple, double and multi-stage Monte Carlo test methods. For non-Gaussian distribution families involving nuisance parameters, confidence sets are derived for the the nuisance parameters and the error distribution. The procedures considered are evaluated in a small simulation experi-ment. Finally, the tests are applied to an asset pricing model with observable risk-free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over five-year subperiods from 1926-1995.

Identification, Weak Instruments and Statistical Inference in Econometrics

We discuss statistical inference problems associated with identification and testability in econometrics, and we emphasize the common nature of the two issues. After reviewing the relevant statistical notions, we consider in turn inference in nonparametric models and recent developments on weakly identified models (or weak instruments). We point out that many hypotheses, for which test procedures are commonly proposed, are not testable at all, while some frequently used econometric methods are fundamentally inappropriate for the models considered. Such situations lead to ill-defined statistical problems and are often associated with a misguided use of asymptotic distributional results. Concerning nonparametric hypotheses, we discuss three basic problems for which such difficulties occur: (1) testing a mean (or a moment) under (too) weak distributional assumptions; (2) inference under heteroskedasticity of unknown form; (3) inference in dynamic models with an unlimited number of parameters. Concerning weakly identified models, we stress that valid inference should be based on proper pivotal functions —a condition not satisfied by standard Wald-type methods based on standard errors — and we discuss recent developments in this field, mainly from the viewpoint of building valid tests and confidence sets. The techniques discussed include alternative proposed statistics, bounds, projection, split-sampling, conditioning, Monte Carlo tests. The possibility of deriving a finite-sample distributional theory, robustness to the presence of weak instruments, and robustness to the specification of a model for endogenous explanatory variables are stressed as important criteria assessing alternative procedures.

Monte Carlo Tests with Nuisance Parameters: A General Approach to Finite-Sample Inference and Nonstandard Asymptotics

The technique of Monte Carlo (MC) tests [Dwass (1957), Barnard (1963)] provides an attractive method of building exact tests from statistics whose finite sample distribution is intractable but can be simulated (provided it does not involve nuisance parameters). We extend this method in two ways: first, by allowing for MC tests based on exchangeable possibly discrete test statistics; second, by generalizing the method to statistics whose null distributions involve nuisance parameters (maximized MC tests, MMC). Simplified asymptotically justified versions of the MMC method are also proposed and it is shown that they provide a simple way of improving standard asymptotics and dealing with nonstandard asymptotics (e.g., unit root asymptotics). Parametric bootstrap tests may be interpreted as a simplified version of the MMC method (without the general validity properties of the latter).

Exact Multivariate Tests of Asset Pricing Models with Stable Asymmetric Distributions

In this paper, we propose exact inference procedures for asset pricing models that can be formulated in the framework of a multivariate linear regression (CAPM), allowing for stable error distributions. The normality assumption on the distribution of stock returns is usually rejected in empirical studies, due to excess kurtosis and asymmetry. To model such data, we propose a comprehensive statistical approach which allows for alternative - possibly asymmetric - heavy tailed distributions without the use of large-sample approximations. The methods suggested are based on Monte Carlo test techniques. Goodness-of-fit tests are formally incorporated to ensure that the error distributions considered are empirically sustainable, from which exact confidence sets for the unknown tail area and asymmetry parameters of the stable error distribution are derived. Tests for the efficiency of the market portfolio (zero intercepts) which explicitly allow for the presence of (unknown) nuisance parameter in the stable error distribution are derived. The methods proposed are applied to monthly returns on 12 portfolios of the New York Stock Exchange over the period 1926-1995 (5 year subperiods). We find that stable possibly skewed distributions provide statistically significant improvement in goodness-of-fit and lead to fewer rejections of the efficiency hypothesis.