Empirical Likelihood Intervals for Conditional Value‐at‐Risk in Heteroscedastic Regression Models

Abstract. Non‐parametric regression models have been studied well including estimating the conditional mean function, the conditional variance function and the distribution function of errors. In addition, empirical likelihood methods have been proposed to construct confidence intervals for the conditional mean and variance. Motivated by applications in risk management, we propose an empirical likelihood method for constructing a confidence interval for the pth conditional value‐at‐risk based on the non‐parametric regression model. A simulation study shows the advantages of the proposed method.     

Inferences in dynamic logit models in semi-parametric setup for repeated binary data

Binary dynamic fixed and mixed logit models are extensively studied in the literature. These models are developed to examine the effects of certain fixed covariates through a parametric regression function as a part of the models. However, there are situations where one may like to consider more covariates in the model but their direct effect is not of interest. In this paper we propose a generalization of the existing binary dynamic logit (BDL) models to the semi-parametric longitudinal setup to address this issue of additional covariates. The regression function involved in such a semi-parametric BDL model contains (i) a parametric linear regression function in some primary covariates, and (ii) a non-parametric function in certain secondary covariates. We use a simple semi-parametric conditional quasi-likelihood approach for consistent estimation of the non-parametric function, and a semi-parametric likelihood approach for the joint estimation of the main regression and dynamic dependence parameters of the model. The finite sample performance of the estimation approaches is examined through a simulation study. The asymptotic properties of the estimators are also discussed. The proposed model and the estimation approaches are illustrated by reanalysing a longitudinal infectious disease data.     

Moment-based estimation of nonlinear regression models with boundary outcomes and endogeneity,with applications to nonnegative and fractional responses

In this article, we suggest simple moment-based estimators to deal with unobserved heterogeneity in a special class of nonlinear regression models that includes as main particular cases exponential models for nonnegative responses and logit and complementary loglog models for fractional responses. The proposed estimators: (i) treat observed and omitted covariates in a similar manner; (ii) can deal with boundary outcomes; (iii) accommodate endogenous explanatory variables without requiring knowledge on the reduced form model, although such information may be easily incorporated in the estimation process; (iv) do not require distributional assumptions on the unobservables, a conditional mean assumption being enough for consistent estimation of the structural parameters; and (v) under the additional assumption that the dependence between observables and unobservables is restricted to the conditional mean, produce consistent estimators of partial effects conditional only on observables.     

Conditional Akaike Information Criteria for a Class of Poisson Mixture Models with Random Effects

Focusing on the model selection problems in the family of Poisson mixture models (including the Poisson mixture regression model with random effects and zero‐inflated Poisson regression model with random effects), the current paper derives two conditional Akaike information criteria. The criteria are the unbiased estimators of the conditional Akaike information based on the conditional log‐likelihood and the conditional Akaike information based on the joint log‐likelihood, respectively. The derivation is free from the specific parametric assumptions about the conditional mean of the true data‐generating model and applies to different types of estimation methods. Additionally, the derivation is not based on the asymptotic argument. Simulations show that the proposed criteria have promising estimation accuracy. In addition, it is found that the criterion based on the conditional log‐likelihood demonstrates good model selection performance under different scenarios. Two sets of real data are used to illustrate the proposed method.     

Generalised quasi‐likelihood inference in a semi‐parametric binary dynamic mixed logit model

There exists a recent study where dynamic mixed‐effects regression models for count data have been extended to a semi‐parametric context. However, when one deals with other discrete data such as binary responses, the results based on count data models are not directly applicable. In this paper, we therefore begin with existing binary dynamic mixed models and generalise them to the semi‐parametric context. For inference, we use a new semi‐parametric conditional quasi‐likelihood (SCQL) approach for the estimation of the non‐parametric function involved in the semi‐parametric model, and a semi‐parametric generalised quasi‐likelihood (SGQL) approach for the estimation of the main regression, dynamic dependence and random effects variance parameters. A semi‐parametric maximum likelihood (SML) approach is also used as a comparison to the SGQL approach. The properties of the estimators are examined both asymptotically and empirically. More specifically, the consistency of the estimators is established and finite sample performances of the estimators are examined through an intensive simulation study.     

Extending conditional likelihood in models for stratified binary data

The conditional likelihood is widely used in logistic regression models with stratified binary data. In particular, it leads to accurate inference for the parameters of interest, which are common to all strata, eliminating stratum-specific nuisance parameters. The modified profile likelihood is an accurate approximation to the conditional likelihood, but has the advantage of being available for general parametric models. Here, we propose the modified profile likelihood as an ideal extension of the conditional likelihood in generalized linear models for binary data, with generic link function. An important feature is that for the implementation we only need standard outputs of routines for generalized linear models. The accuracy of the method is supported by theoretical properties and is confirmed by simulation results.This research was supported by MIUR COFIN 2001-2003.     

Estimation and specification testing in female labor participation models: parametric and semiparametric methods

Female labor participation models have been usually studied through probit and logit specifications. Little attention has been paid to verify the assumptions that are used in these sort of models, basically distributional assumptions and homoskedasticity. In this paper we apply semiparametirc methods in order to test the previous hypothesis. We also estimate a Spanish female labor participation model using both parametric and semiparametirc approaches. The parametirc model includes fixed and random coefficients probit specification. The estimation procedures are parametric maximum likelihood for both probit and logit models, and semiparametric quasi maximum likelihood following Klein and Spady (1993). The results depend cricially in the assumed model.     

Testing for Granger-causality in quantiles

This paper proposes a consistent parametric test of Granger-causality in quantiles. Although the concept of Granger-causality is defined in terms of the conditional distribution, most articles have tested Granger-causality using conditional mean regression models in which the causal relations are linear. Rather than focusing on a single part of the conditional distribution, we develop a test that evaluates nonlinear causalities and possible causal relations in all conditional quantiles, which provides a sufficient condition for Granger-causality when all quantiles are considered. The proposed test statistic has correct asymptotic size, is consistent against fixed alternatives, and has power against Pitman deviations from the null hypothesis. As the proposed test statistic is asymptotically nonpivotal, we tabulate critical values via a subsampling approach. We present Monte Carlo evidence and an application considering the causal relation between the gold price, the USD/GBP exchange rate, and the oil price.     

Missing covariates in generalized linear models when the missing data mechanism is non-ignorable

We propose a method for estimating parameters in generalized linear models with missing covariates and a non-ignorable missing data mechanism. We use a multinomial model for the missing data indicators and propose a joint distribution for them which can be written as a sequence of one-dimensional conditional distributions, with each one-dimensional conditional distribution consisting of a logistic regression. We allow the covariates to be either categorical or continuous. The joint covariate distribution is also modelled via a sequence of one-dimensional conditional distributions, and the response variable is assumed to be completely observed. We derive the E- and M-steps of the EM algorithm with non-ignorable missing covariate data. For categorical covariates, we derive a closed form expression for the E- and M-steps of the EM algorithm for obtaining the maximum likelihood estimates (MLEs). For continuous covariates, we use a Monte Carlo version of the EM algorithm to obtain the MLEs via the Gibbs sampler. Computational techniques for Gibbs sampling are proposed and implemented. The parametric form of the assumed missing data mechanism itself is not `testable' from the data, and thus the non-ignorable modelling considered here can be viewed as a sensitivity analysis concerning a more complicated model. Therefore, although a model may have `passed' the tests for a certain missing data mechanism, this does not mean that we have captured, even approximately, the correct missing data mechanism. Hence, model checking for the missing data mechanism and sensitivity analyses play an important role in this problem and are discussed in detail. Several simulations are given to demonstrate the methodology. In addition, a real data set from a melanoma cancer clinical trial is presented to illustrate the methods proposed.     

Conditional Likelihood Estimators for Hidden Markov Models and Stochastic Volatility Models

ABSTRACT.  This paper develops a new contrast process for parametric inference of general hidden Markov models, when the hidden chain has a non-compact state space. This contrast is based on the conditional likelihood approach, often used for ARCH-type models. We prove the strong consistency of the conditional likelihood estimators under appropriate conditions. The method is applied to the Kalman filter (for which this contrast and the exact likelihood lead to asymptotically equivalent estimators) and to the discretely observed stochastic volatility models.     

Testing generalized linear and semiparametric models against smooth alternatives

We propose goodness-of-fit tests for testing generalized linear models and semiparametric regression models against smooth alternatives. The focus is on models having both continous and factorial covariates. As a smooth extension of a parametric or semiparametric model we use generalized varying-coefficient models as proposed by Hastie and Tibshirani. A likelihood ratio statistic is used for testing. Asymptotic expansions allow us to write the estimates as linear smoothers which in turn guarantees simple and fast bootstrapping of the test statistic. The test is shown to have √ n -power, but in contrast with parametric tests it is powerful against smooth alternatives in general.     

Maximum Likelihood Estimation and Inference in Multivariate Conditionally Heteroscedastic Dynamic Regression Models With Student t Innovations

We provide numerically reliable analytical expressions for the score, Hessian, and information matrix of conditionally heteroscedastic dynamic regression models when the conditional distribution is multivariatet. We also derive one-sided and two-sided Lagrange multiplier tests for multivariate normality versus multivariate t based on the first two moments of the squared norm of the standardized innovations evaluated at the Gaussian pseudo-maximum likelihood estimators of the conditional mean and variance parameters. Finally, we illustrate our techniques through both Monte Carlo simulations and an empirical application to 26 U.K. sectorial stock returns that confirms that their conditional distribution has fat tails.     

An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.     

Kalman filter-based modelling and forecasting of stochastic volatility with threshold

We propose a parametric nonlinear time-series model, namely the Autoregressive-Stochastic volatility with threshold (AR-SVT) model with mean equation for forecasting level and volatility. Methodology for estimation of parameters of this model is developed by first obtaining recursive Kalman filter time-update equation and then employing the unrestricted quasi-maximum likelihood method. Furthermore, optimal one-step and two-step-ahead out-of-sample forecasts formulae along with forecast error variances are derived analytically by recursive use of conditional expectation and variance. As an illustration, volatile all-India monthly spices export during the period January 2006 to January 2012 is considered. Entire data analysis is carried out using EViews and matrix laboratory (MATLAB) software packages. The AR-SVT model is fitted and interval forecasts for 10 hold-out data points are obtained. Superiority of this model for describing and forecasting over other competing models for volatility, namely AR-Generalized autoregressive conditional heteroscedastic, AR-Exponential GARCH, AR-Threshold GARCH, and AR-Stochastic volatility models is shown for the data under consideration. Finally, for the AR-SVT model, optimal out-of-sample forecasts along with forecasts of one-step-ahead variances are obtained.     

Sieve Empirical Likelihood and Extensions of the Generalized Least Squares

The empirical likelihood cannot be used directly sometimes when an infinite dimensional parameter of interest is involved. To overcome this difficulty, the sieve empirical likelihoods are introduced in this paper. Based on the sieve empirical likelihoods, a unified procedure is developed for estimation of constrained parametric or non-parametric regression models with unspecified error distributions. It shows some interesting connections with certain extensions of the generalized least squares approach. A general asymptotic theory is provided. In the parametric regression setting it is shown that under certain regularity conditions the proposed estimators are asymptotically efficient even if the restriction functions are discontinuous. In the non-parametric regression setting the convergence rate of the maximum estimator based on the sieve empirical likelihood is given. In both settings, it is shown that the estimator is adaptive for the inhomogeneity of conditional error distributions with respect to predictor, especially for heteroscedasticity.     

Linear calibration in functional regression models

This paper discusses calibration in functional regression models. Classical and inverse type estimators are considered. First order approximation to the bias and to the mean squared error (MSE) of the estimators are considered. Numerical comparisons seem to indicate that the classical estimator obtained via maximum likelihood estimation performs better than the other estimators considered.     

Quantile smoothing in financial time series

Various parametric models have been designed to analyze volatility in time series of financial market data. For maximum likelihood estimation these parametric methods require the assumption of a known conditional distribution. In this paper we examine the conditional distribution of daily DAX returns with the help of nonparametric methods. We use kernel estimators for conditional quantiles resulting from a kernel estimation of conditional distributions. This work was financially supported by the Deutsche Forschungsgemeinschaft     

On conditional risk estimation considering model risk

Usually, parametric procedures used for conditional variance modelling are associated with model risk. Model risk may affect the volatility and conditional value at risk estimation process either due to estimation or misspecification risks. Hence, non-parametric artificial intelligence models can be considered as alternative models given that they do not rely on an explicit form of the volatility. In this paper, we consider the least-squares support vector regression (LS-SVR), weighted LS-SVR and Fixed size LS-SVR models in order to handle the problem of conditional risk estimation taking into account issues of model risk. A simulation study and a real application show the performance of proposed volatility and VaR models.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

Small-sample properties of covariance-adjusted survivorship data tests for treatment effect

Monte Carlo methods are used to examine the small-sample properties of 11 test statistics that can be used for comparing several treatments with respect to their mortality experiences while adjusting for covariables. The test statistics are investigated from three distinct models: the parametric, semiparametric and rank analysis of covariance (Quade, 1967) models. Four tests (likelihood ratio, Wald, conditional and unconditional score tests) from each of the first two models and three tests (based on rank scores) from the last model are discussed. The empirical size and power of the tests are investigated under a proportional hazards model in three situations: (1) the baseline hazard is correctly assumed to be Exponential, (2) the baseline hazard is incorrectly assumed to be Exponential, and (3) a treatment-covariate interaction is omitted from the analysis.