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.     

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.     

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.     

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.     

A zero-inflated ordered probit model,with an application to modelling tobacco consumption

Data for discrete ordered dependent variables are often characterised by “excessive” zero observations which may relate to two distinct data generating processes. Traditional ordered probit models have limited capacity in explaining this preponderance of zero observations. We propose a zero-inflated ordered probit model using a double-hurdle combination of a split probit model and an ordered probit model. Monte Carlo results show favourable performance in finite samples. The model is applied to a consumer choice problem of tobacco consumption indicating that policy recommendations could be misleading if the splitting process is ignored.     

A new structural break model,with an application to Canadian inflation forecasting

This paper develops an efficient approach to modelling and forecasting time series data with an unknown number of change-points. Using a conjugate prior and conditioning on time-invariant parameters, the predictive density and the posterior distribution of the change-points have closed forms. Furthermore, the conjugate prior is modeled as hierarchical in order to exploit the information across regimes. This framework allows breaks in the variance, the regression coefficients, or both. The regime duration can be modelled as a Poisson distribution. A new, efficient Markov chain Monte Carlo sampler draws the parameters from the posterior distribution as one block. An application to a Canadian inflation series shows the gains in forecasting precision that our model provides.     

A new class of asymptotically efficient estimators for moment condition models

In this paper, we propose a new class of asymptotically efficient estimators for moment condition models. These estimators share the same higher order bias properties as the generalized empirical likelihood estimators and once bias corrected, have the same higher order efficiency properties as the bias corrected generalized empirical likelihood estimators. Unlike the generalized empirical likelihood estimators, our new estimators are much easier to compute. A simulation study finds that our estimators have better finite sample performance than the two-step GMM, and compare well to several potential alternatives in terms of both computational stability and overall performance.     

A smoothed maximum score estimator for the binary choice panel data model with an application to labour force participation

In a binary choice panel data model with individual effects and two time periods, Manski proposed the maximum score estimator based on a discontinuous objective function and proved its consistency under weak distributional assumptions. The rate of convergence is low ( N ^1/3) and its limit distribution cannot easily be used for statistical inference. In this paper we apply the idea of Horowitz to smooth Manski's objective function. The resulting smoothed maximum score estimator is consistent and asymptotically normal with a rate of convergence that can be made arbitrarily close to N ^1/2, depending on the strength of the smoothness assumptions imposed. The estimator can be applied to panels with more than two time periods and to unbalanced panels. We apply the estimator to analyze labour force participation of married Dutch females.     

Latent variable models for time series: A frequency domain approach with an application to the permanent income hypothesis

The theory of estimation and inference in a very general class of latent variable models for time series is developed by showing that the distribution theory for the finite Fourier transform of the observable variables in latent variable models for time series is isomorphic to that for the observable variables themselves in classical latent variable models. This implies that analytic work on classical latent variable models can be adapted to latent variable models for time series, an implication which is illustrated here in the context of a general canonical form. To provide an empirical example a latent variable model for permanent income is developed, its parameters are shown to be identified, and a variety of restrictions on these parameters implied by the permanent income hypothesis are tested.     

Panel data methods for fractional response variables with an application to test pass rates

We revisit the effects of spending on student performance using data from the state of Michigan. In addition to exploiting a dramatic change in funding in the mid-1990s and subsequent nonsmooth changes, we propose nonlinear panel data models that recognize the bounded nature of the pass rate. Importantly, we show how to estimate average partial effects, which can be compared across many different models (linear and nonlinear) under different assumptions and estimated using many different methods. We find that spending has nontrivial and statistically significant effects, although the diminishing effect is not especially pronounced.     

Maximum likelihood estimation and Lagrange multiplier tests for panel seemingly unrelated regressions with spatial lag and spatial errors: An application to hedonic housing prices in Paris

This paper proposes maximum likelihood estimators for panel seemingly unrelated regressions with both spatial lag and spatial error components. We study the general case where spatial effects are incorporated via spatial errors terms and via a spatial lag dependent variable and where the heterogeneity in the panel is incorporated via an error component specification. We generalize the approach of Wang and Kockelman (2007) and propose joint and conditional Lagrange multiplier tests for spatial autocorrelation and random effects for this spatial SUR panel model. The small sample performance of the proposed estimators and tests are examined using Monte Carlo experiments. An empirical application to hedonic housing prices in Paris illustrate these methods. The proposed specification uses a system of three SUR equations corresponding to three types of flats within 80 districts of Paris over the period 1990-2003. We test for spatial effects and heterogeneity and find reasonable estimates of the shadow prices for housing characteristics.     

A general class of estimators for estimating population mean using auxiliary information

Summary A general class of estimators for estimating the population mean of the character under study which make use of auxiliary information is proposed. Under simple random sampling without replacement (SRSWOR), the expressions of Bias and Mean Square Error (MSE), up to the first and the second degrees of approximation are derived. General conditions, up to the first order approximation, are also obtained under which any member of this class performs more efficiently than the mean per unit estimator, the ratio estimator and the product estimator. The class of estimators in its optimum case, under the first degree approximation, is discussed. It is shown that it is not possible to obtain optimum values of parameters “a”, “b” and “p”, that are independent of each other. However, the optimum relation among them is given by (b−a)p=ρ C _y/C _x. Under this condition, the expression of MSE of the class is that of the linear regression estimator.