Pitfalls of testing non-nested hypotheses by the lagrange multiplier method

In applied research in econometrics a general model determined from the current knowledge of economic theory often establishes a ‘natural’ method of embedding a number of otherwise non-nested hypotheses. Under these circumstances, significant tests of various hypotheses can be carried out within the classical framework, and tests of non-nested or separate families of hypotheses do not require development of new statistical methods. The application of some suitable variant of likelihood ratio testing procedure will be quite appropriate.There are, however, many ocassions in applied econometrics where the hypotheses under consideration are intended to provide genuine rival explanations of the same given phenomenon and the state of economic theory is not such as to furnish us with a general model that contains both of the rival hypotheses in a ‘natural’ and theoretically consistent manner. A number of investigators have advocated that even when a ‘natural’ comprehensive model containing both of the hypotheses under consideration cannot be obtained from theoretical considerations, it is still appropriate to base significant tests of non-nested hypotheses upon a combined model ‘artificially’ constructed from the rival alternatives. Moreover, in a recent paper on the application of Lagrange Multiplier (LM) tests to model specification, T.S. Breusch and A.R. Pagan (1980) have claimed that Cox's test statistic is connected to an LM or ‘score’ statistic derived from the application of the LM method to an exponentially combined model earlier employed by A.C. Atkinson (1970).Although the use of ‘artificially’ constructed comprehensive models fortesting separate families of hypotheses is analytically tempting, nevertheless it is subject to two major difficulties. Firstly, in many cases of interest in econometrics, the structural parameters under the combined hypothesis are not identified. Secondly, the log likelihood function of the artificially constructed model has singularities under both the null and alternative hypotheses.The paper firstly examines the derivation of LM statistics in the case of non-nested hypotheses and shows that Atkinson's general test statistic, or Breusch and Pagan's result, can be regarded as an LM test if the parameters of the alternative hypothesis are known. The paper also shows that unless all the parameters of the combined models are identified, no meaningful test of the separate families of the hypotheses by the artificial embedding procedure is possible, and in the identified case an expression for the LM statistic which avoids the problem of the singularity of the information matrix under the null and the alternative hypotheses is obtained.The paper concludes that none of the artificially embedding procedures are satisfactory for testing non-nested models and should be abandoned. It, however, emphasizes that despite these difficulties associated with the use of artificial embedding procedures, Cox's original statistic (which is not derived as an LM statistic and does not depend on any arbitrary synthetic combination of hypotheses) can still be employed as a useful procedure for testing the rival hypotheses often encountered in applied econometrics.     

Pitfalls of testing non-nested hypotheses by the lagrange multiplier method

The paper considers the problem of testing non-nested hypotheses by the application of the Lagrange Multiplier (LM) method to an exponentially combined likelihood function. It shows that the general test statistic derived by Atkinson (1970) and later by Breusch and Pagan (1980) can be regarded as an LM statistic only if it can be assumed that the parameters of the alternative model are known. When none of the parameters are known, the paper shows that a straightforward application of the LM method to testing non-nested hypotheses breaks down and it discusses the reasons for this failure.     

Modified lagrange multiplier tests for problems with one-sided alternatives

A modified Lagrange multiplier test statistic is proposed which takes explicit account of the one-sided nature of the alternative in problems where the null hypothesis specifies that the true value of the parameter vector lies on the boundary of the parameter space. Computation of this statistic requires only the constrained maximum likelihood estimator. Conditions for the consistency of tests based on this statistic are examined and it is shown that the distribution of the statistic is not affected of nuisance parameters are allowed to lie on the boundary of the parameter space.     

A simplified approach to M-estimation with application to two-stage estimators

This paper presents a simple version of the theory of M-estimation. It is argued that the theory is immediately applicable to almost all estimation schemes employed by econometricians. It is further argued that the great overlooked benefit of the theory is that is provides almost automatic asymptotic results, e.g., probability limits and asymptotic covariances. Thus one need not be a theoretical econometrician to invent and use specially tailored estimators. To illustrate its use the theory is applied to a variety of theoretical and applied problems. Particular attention is paid to two-stage estimators.     

The behaviour of the Lagrangian multiplier test in testing the orders of an ARMA-model

Summary The Lagrange multiplier test for testing the order of an ARMA-model is investigated. It is shown that it exhibits some pathologies stemming from the special properties of parameter spaces of ARMA-models. As a consequence one has to be careful in using the LM-test in this context. Furthermore results of this paper are used in a subsequent paper [Pötscher, 1983], to show the consistency of an order estimation procedure based on Lagrange multiplier tests, described there.     

Armax model specification testing,with an application to unemployment in the Netherlands

In this paper we present a consistent standard normal model specification test for ARMAX models. The null hypothesis is that the ARMAX model represents the conditional expectation of the dependent variable relative to the entire past of the economic vector time series process under review. This null is tested against the general alternative hypothesis that the null is false. The test is applied to testing the rational expectations-natural rate (RE-NR) hypothesis for the Netherlands according to the approach of Sargent (1976). It appears that RE-NR hypothesis has to be rejected.     

An application of nonparametric volatility estimators to option pricing

We discuss the impact of volatility estimates from high frequency data on derivative pricing. The principal purpose is to estimate the diffusion coefficient of an Itô process using a nonparametric Nadaraya–Watson kernel approach based on selective estimators of spot volatility proposed in the econometric literature, which are based on high frequency data. The accuracy of different spot volatility estimates is measured in terms of how accurately they can reproduce market option prices. To this aim, we fit a diffusion model to S&P 500 data, and successively, we use the calibrated model to price European call options written on the S&P 500 index. The estimation results are compared to well-known parametric alternatives available in the literature. Empirical results not only show that using intra-day data rather than daily provides better volatility estimates and hence smaller pricing errors, but also highlight that the choice of the spot volatility estimator has effective impact on pricing.     

GMM estimators with improved finite sample properties using principal components of the weighting matrix,with an application to the dynamic panel data model

GMM estimators have poor finite sample properties in highly overidentified models. With many moment conditions the optimal weighting matrix is poorly estimated. We suggest using principal components of the weighting matrix. This effectively drops some of the moment conditions. Our simulations, done in the context of the dynamic panel data model, show that the resulting GMM estimator has better finite sample properties than the usual two-step GMM estimator, in the sense of smaller bias and more reliable standard errors.     

A lemma on relative stability with an application to an economic case study

In the existing literature relative stability of trajectories is obtained under particular assumptions. In this paper a lemma is stated which provides a technique to prove relative stability in most general cases. Such a technique permits to replace the study of the stability of a balanced growth path by the study of the stability of a fixed point for an associated function. The lemma is applied to a case study, which could not be treated by means of usual techniques.     

Regularized conditional estimators of unit inefficiency in stochastic frontier analysis,with application to electricity distribution market

In stochastic frontier analysis, the conventional estimation of unit inefficiency is based on the mean/mode of the inefficiency, conditioned on the composite error. It is known that the conditional mean of inefficiency shrinks towards the mean rather than towards the unit inefficiency. In this paper, we analytically prove that the conditional mode cannot accurately estimate unit inefficiency, either. We propose regularized estimators of unit inefficiency that restrict the unit inefficiency estimators to satisfy some a priori assumptions, and derive the closed form regularized conditional mode estimators for the three most commonly used inefficiency densities. Extensive simulations show that, under common empirical situations, e.g., regarding sample size and signal-to-noise ratio, the regularized estimators outperform the conventional (unregularized) estimators when the inefficiency is greater than its mean/mode. Based on real data from the electricity distribution sector in Sweden, we demonstrate that the conventional conditional estimators and our regularized conditional estimators provide substantially different results for highly inefficient companies.

     

A flexible parametric GARCH model with an application to exchange rates

Many asset prices, including exchange rates, exhibit periods of stability punctuated by infrequent, substantial, often one‐sided adjustments. Statistically, this generates empirical distributions of exchange rate changes that exhibit high peaks, long tails, and skewness. This paper introduces a GARCH model, with a flexible parametric error distribution based on the exponential generalized beta (EGB) family of distributions. Applied to daily US dollar exchange rate data for six major currencies, evidence based on a comparison of actual and predicted higher‐order moments and goodness‐of‐fit tests favours the GARCH‐EGB2 model over more conventional GARCH‐t and EGARCH‐t model alternatives, particularly for exchange rate data characterized by skewness. Copyright © 2001 John Wiley & Sons, Ltd.     

A quasi-negative binomial regression with an application to medical care data

Quality & Quantity - This paper introduced a Quasi-Negative Binomial Regression as an extension of Quasi-Negative Binomial to handle response count datasets modulated with covariates. In some...     

A generalized correlated binomial distribution with application in multiple testing problems

A typical microarray experiment often involves comparisons of hundreds or thousands of genes. Since a large number of genes are compared, simple use of a significance test without adjustment for multiple comparison artifacts could lead to a large chance of false positive findings. In this context, Tsai et al. (Biometrics 59:1071–1081, 2003) have presented a model that studies the overall error rate when testing multiple hypotheses. This model involves the distribution of the sum of non-independent Bernoulli trials and this distribution is approximated by using a beta-binomial structure. Instead of using a beta-binomial model, in this paper, we derive the exact distribution of the sum of non-independent and non-identically distributed Bernoulli random variables. The distribution obtained is used to compute the conditional false discovery rates and the results are compared to those obtained, in Table 3, by Tsai et al. (Biometrics 59:1071–1081, 2003).     

A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation

This paper presents a Bayesian approach to bandwidth selection for multivariate kernel regression. A Monte Carlo study shows that under the average squared error criterion, the Bayesian bandwidth selector is comparable to the cross-validation method and clearly outperforms the bootstrapping and rule-of-thumb bandwidth selectors. The Bayesian bandwidth selector is applied to a multivariate kernel regression model that is often used to estimate the state-price density of Arrow–Debreu securities with the S&P 500 index options data and the DAX index options data. The proposed Bayesian bandwidth selector represents a data-driven solution to the problem of choosing bandwidths for the multivariate kernel regression involved in the nonparametric estimation of the state-price density pioneered by Aït-Sahalia and Lo [Aït-Sahalia, Y., Lo, A.W., 1998. Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53, 499, 547.]     

Semiparametric econometric estimators for a truncated regression model: a review with an extension

Econometric estimators for a truncated regression model are reviewed. For each estimator, the motivations, the key assumptions, the asymptotic distribution and estimates for the asymptotic variance matrix are presented; also a new estimator is suggested. We select five practical estimators among those, and compare them through a Monte Carlo study where the response variable is simulated but the covariates are drawn from a real data set. Some practical and computational issues are addressed as well.     

A nonlinear long memory model, with an application to US unemployment

Two important empirical features of US unemployment are that shocks to the series seem rather persistent and that it seems to rise faster during recessions than that it falls during expansions. To jointly capture these features of long memory and nonlinearity, we put forward a new time series model and evaluate its empirical performance. We find that the model describes the data rather well and that it outperforms related competitive models on various measures of fit.     

A comparison of two model averaging techniques with an application to growth empirics

Parameter estimation under model uncertainty is a difficult and fundamental issue in econometrics. This paper compares the performance of various model averaging techniques. In particular, it contrasts Bayesian model averaging (BMA) — currently one of the standard methods used in growth empirics — with a new method called weighted-average least squares (WALS). The new method has two major advantages over BMA: its computational burden is trivial and it is based on a transparent definition of prior ignorance. The theory is applied to and sheds new light on growth empirics where a high degree of model uncertainty is typically present.