Non-nested Regression Models Research Articles

Non-nested hypothesis testing inference for GAMLSS models

Two or more regression models are said to be non-nested if neither can be obtained from the remaining models when parametric restrictions are imposed. Tests for choosing between linear non-nested regression models are found in literature, such as J and MJ tests. In this paper we propose variants of these two tests for the GAMLSS (Generalized Additive Models for Location, Scale and Shape) class of models. We report Monte Carlo evidence on finite sample behaviour of the proposed tests. Bootstrap-based testing inference is also considered. Overall, bootstrap MJ test had the best performance. An empirical application is presented and discussed.

A simple test for regression specification with non-nested alternatives

In this paper, I introduce a simple test for the presence of the data-generating process among several non-nested alternatives. The test is an extension of the classical J test for non-nested regression models. I also provide a bootstrap version of the test that avoids possible size distortions inherited from the J test.

Robust Non-nested Testing for Ordinary Least Squares Regression when Some of the Regressors are Lagged Dependent Variables*

The problem of testing non-nested regression models that include lagged values of the dependent variable as regressors is discussed. It is argued that it is essential to test for error autocorrelation if ordinary least squares and the associated J and F tests are to be used. A heteroskedasticity–robust joint test against a combination of the artificial alternatives used for autocorrelation and non-nested hypothesis tests is proposed. Monte Carlo results indicate that implementing this joint test using a wild bootstrap method leads to a well-behaved procedure and gives better control of finite sample significance levels than asymptotic critical values.

Waist to Height Ratio as a Predictor of Abdominal Fat Distribution in Men

The accumulation of fat in visceral (VA) and subcutaneous abdominal adipose tissue (SA) is highly correlated with the metabolic abnormalities that contribute to increased risk of diabetes mellitus and cardiovascular disease. The purpose of the study was to determine which of the four indices-waist circumference (WC), waist-to-hip ratio (WHR), waist to height ratio (WHtR) and body mass index (BMI)-was the best predictor of VA and SA in men. We studied 111 men with a wide range of BMI, WC, WHtR, WHR and BMI determined by standard methods. SA and VA were quantified using computed tomography. In univariate and multiple regression analyses, WHtR had the highest correlation with VA and SA. To assess the relative strength of these associations, we used non-nested regression models. WHtR was a stronger predictor of SA than BMI (P = 0.02), but the relative strength of WC and BMI in predicting SA did not differ significantly (P > 0.05). WHtR was a stronger predictor of VA than WC (P = 0.012), BMI (P < 0.001) or WHR (P < 0.001). In men, WHtR is a good anthropometric index which has a stronger correlation with the distribution of visceral and subcutaneous abdominal adipose tissue than BMI or WHR. Its association is closer to or stronger than WC.

Bootstrap and Pretest J-Type Non-Nested Tests for Orthogonal Regression Models: Some Monte Carlo Evidence

This paper compares the size and power of two J-type tests for weakly correlated or nearly orthogonal non-nested regression models: a bootstrap and a pretest test. The latter seems to outperform the former in terms of its size characteristics, especially when the alternative model has more non-nested regressors and the orthogonality between the two sets of regressors is severe. The bootstrap test does better in terms of power.

Waist circumference, waist-to-hip ratio and body mass index as predictors of adipose tissue compartments in men

The accumulation of fat in visceral and posterior subcutaneous adipose tissue compartments is highly correlated with the metabolic abnormalities that contribute to increased risk of diabetes mellitus and cardiovascular disease. To determine which of waist circumference (WC), waist-to-hip ratio (WHR) and body mass index (BMI) was the best predictor of intraperitoneal and posterior subcutaneous abdominal adipose tissue mass in men. We studied 59 free-living men with a wide range of BMI. WC, WHR and BMI were determined using standard methods. Intraperitoneal, retroperitoneal, anterior subcutaneous and posterior subcutaneous abdominal adipose tissue masses (IPATM, RPATM, ASAATM and PSAATM, respectively) were quantified using magnetic resonance imaging. In univariate regression analysis, WC, WHR and BMI were all significantly and positively correlated (all p < 0.05) with IPATM, RPATM, ASAATM and PSAATM. To assess the relative strength of these associations, we used non-nested regression models. There was no significant difference between WC and WHR in predicting IPATM and RPATM; WC was a stronger predictor of ASAATM (p < 0.001) and PSAATM (p < 0.001) than WHR; WC was also a stronger predictor of IPATM (p = 0.042) and RPATM (p = 0.045) than BMI, but the relative strengths of WC and BMI in predicting ASSATM and PSAATM did not different significantly (p > 0.05); there was no significant difference between BMI and WHR in predicting IPATM and RPATM (p>0.05), but BMI was a stronger predictor of ASAATM (p = 0.036) and PSAATM (p < 0.001) than WHR. In men WC is the anthropometric index that most uniformly predicts the distribution of adipose tissue among several fat compartments in the abdominal region, there apparently being little value in measuring WHR or BMI.

A Bayesian model selection method with applications

In this paper, we consider Bayesian model selection using the well-known Bayes factor. A method on the basis of path sampling for computing the ratio of two normalizing constants involved in the Bayes factor is proposed. The key idea is to construct a continuous path to link up the competing models, then the Bayes factor can be estimated efficiently by means of grids in [0,1] and observations simulated from the posterior distribution of the parameters. This method is applied to non-nested regression models, mixture models with an unknown number of components, and a general latent variable model with mixed continuous and polytomous variables. Analyses of some real data sets are presented to illustrate the efficiency and flexibility of the method.

Bootstrap J tests of nonnested linear regression models

The J test for nonnested regression models often overrejects very severely as an asymptotic test. We provide a theoretical analysis which explains why and when it performs badly. This analysis implies that, except in certain extreme cases, the J test will perform very well when bootstrapped. Using several methods to speed up the simulations, we obtain extremely accurate Monte Carlo results on the finite-sample performance of the bootstrapped J test. These results fully support the predictions of our theoretical analysis, even in contexts where the analysis is not strictly applicable.

Tests of non-nested regression models some results on small sample behaviour and the bootstrap

The most popular procedures for testing a regression model against a single non-nested alternative can be substantially oversized in small samples. Also, when a regression model is to be tested in the presence of several non-nested alternatives, the null model is sometimes accepted only if testing against each alternative in turn produces no significant outcomes. This approach leads to an implicit overall test with a significance level that is not known, even asymptotically. It is shown that the bootstrap can be used to control significance levels in both types of situation. Power estimates for various tests using bootstrap critical values are also obtained and compared.

Bootstrapping J-type tests for non-nested regression models

We propose a bootstrap procedure to approximate the finite sample distributions of the J and the JA test statistics. The proposed method also overcomes the problem of the substantial size distortion of the J test when the regressors are near orthogonal.

Comparing Non-Nested Regression Models

A method for comparing the fits of two non-nested models, based on a suggestion of Davidson and MacKinnon (1981), is developed in the context of linear and nonlinear regression with normal errors. Each model is regarded as a special case of an artificial "supermodel" and is obtained by restricting the value of a mixing parameter gamma to 0 or 1. To enable estimation and hypothesis testing for gamma, an approximate supermodel is used in which the fitted values from the individual models appear in place of the original parametrization. In the case of nested linear models, the proposed test essentially reproduces the standard F test. The calculations required are for the most part straight-forward (basically, linear regression through the origin). The test is extended to cover situations in which serious bias in the maximum likelihood estimate of gamma occurs, simple approximate bounds for the bias being given. Two real datasets are used illustratively throughout.

Semiparametric Specification Testing of Non-nested Econometric Models

We propose a specification test of a parametrically specified nonlinear model against a weakly specified non-nested alternative. We estimate the alternative model by using nonparametric regression (nearest neighbours). The test is based on the t-statistic of an artificial regression. MonteCarlo simulations suggest that the test has good power and size characteristics. In this paper we propose a specification test for non-nested regression models where the alternative hypothesis is nonparametric. Therefore, the regressors in the null and in the alternative are non-nested. The proposed test is based on an artificial nesting procedure in the spirit of the Davidson and MacKinnon (1981) tests. We provide a natural extension of their results in a semiparametric environment. The performance of the alternative under 'the truth of the null' forms the basis for a test of the latter. We use k-nearest neighbour (k-NN) regression to estimate that a possible misspecified parameterization will have on testing the validity of the null. The consistency of the proposed test statistic, under fixed alternatives, is derived under the type of regulatory conditions required when the alternative model is parametrically specified. Nonparametric functional estimation has been recently used in deriving various specification tests. Specification tests of nonparametric and semiparametric restrictions based on average derivatives of the regression function have been considered by Stoker (1989) and Robinson (1989). Cox et al. (1988) considered generalized spline models for testing a parametric null hypothesis against semiparametric alternatives. Eubank and Spiegelman (1990) proposed a spline based method for testing the goodness of fit of a linear model. Wooldridge (1992) derived a residual based test which allows for certain non-fixed nonparametric alternatives. A Davidson and MacKinnon (1981)type test is shown to be consistent using a sieve estimator for the alternatives. In the next section we discuss the proposed test. In Section 3 we investigate the small sample properties of our test by means of a small Monte Carlo. Finally we conclude. The proofs are in the Appendix.

Choosing among multiple nonlinear non-nested regression models with different dependent variables: An application to money demand

The linearly-embedded regression model approach of Davidson and MacKinnon (1981) is used to construct joint tests of seven non-nested, autoregressive money demand equations with different regressands. The results support the Holmes-Smyth (1972) hypothesis that taxes affect money demand.

Hierarchical vs. Non-Nested Tests For Contrasting expectancy-Valence Models: Some Effects of Cognitive Characteristics.

This study tests the appropriateness of multiplicative versus additive expectancy-valence models for 82 undergraduate students. Participants are grouped according to the model that best explains their motivational force decisions to exert effort in a series of differing hypothetical college courses. Two statistical tests are used for grouping participants: the traditional hierarchical test for contrasting nested regression models, and a more complex test for contrasting non-nested regression models. Arguments are offered in favor of the non-nested test. Force decisions are obtained from a within-subjects analysis. The grouping tests are applied between subjects. Discriminant analyses are then used to determine if four cognitive characteristics affected the choice of model. These characteristics are tolerance for ambiguity (TA), scholastic aptitude in mathematics (SATM), facility in mathematics required in major curriculum (FMR), and cognitive complexity (CC). It is hypothesized that participants who have higher TA, SATM, FMR, and CC process expectancy valence information multiplicatively while those who have lower values of these cognitive characteristics, process the information additively. The hypothesis is supported for FMR. No effect on choice of models is found for TA, SATM, or CC, while controlling for FMR; but univariate tests show multipliers to have higher TA and SATM than adders. Some implications and suggestions for future research are discussed.

Exact Tests of a Model Against Nonnested Alternatives

This note applies the results of Milliken and Graybill (1970) to non-nested regression models. A test proposed by Fisher and McAleer (1981) is shown to have the t-distribution in small samples. An extended version of that test is developed for testing a model against several non-linear non-nested alternatives, and is shown to have the central F-distribution.

The J-test as a Hausman specification test

The purpose of this note is to show that the J-test of non-nested regression models proposed by Davidson and MacKinnon (1981), can also be viewed as a test of misspecification along the lines put forward by Hausman (1978). We give a proof for the case of single-equation linear models.

Tests of non-nested regression models

Abstract It is known that the small sample significance levels of Cox-type tests of non-nested regression models can be much greater than the nominal level. Adjustments designed to overcome this problem are discussed and two tests are proposed. Monte Carlo evidence on the performance of the tests derived in this paper, the Davidson-MacKinnon J -test and the Fisher-McAleer test is presented. The F -test applied to the comprehensive model is also included in the simulation experiments.

Confidence contours for two test statistics for non-nested regression models

Confidence contours for two test statistics for non-nested regression models

Comparison of Local Power of Alternative Tests of Non-Nested Regression Models

The paper derives and compares the local power of three different methods of testing non-nested regression models that are available in the literature. It shows that the asymptotic power of the orthodox F test against local alternatives is strictly less than that of Cox's non-nested test or the J test recently proposed by Davidson and MacKinnon [5], unless the number of non-overlapping variables of the alternative hypothesis over the null hypothesis is unity, in which case all three tests are shown to be asymptotically equivalent. The final section of the paper gives results of Monte Carlo experiments designed to check the validity of the theoretical findings of the paper and also to shed light on the small sample properties of the Cox test and the orthodox test.

On the comprehensive method of testing non-nested regression models

The paper derives the specific form of the exponentially combined likelihood function of two competing multivariate non-linear regression models and shows that the application of the comprehensive approach to testing non-nested regression models will, in general, be indeterminate. It establishes that in the univariate case there exists a large number of tests of non-nested regression models which are consistent in addition to having the same asymptotic distribution under the null hypothesis. The paper then derives a set of conditions under which all these consistent tests are asymptotically equivalent not only under the null hypothesis but also under local alternatives. As an application of this latter result the paper establishes the asymptotic equivalence of the tests recently proposed by Davidson and MacKinnon, and Fisher and McAleer under local alternatives, and shows that within the class of tests considered in the paper these proposed tests possess maximum local power. The latter test has this property only when the number of explanatory variables of the ‘true’ model is not more than that of the ‘false’ model.