Subsampling high frequency data

The main contribution of this paper is to propose a novel way of conducting inference for an important general class of estimators that includes many estimators of integrated volatility. A subsampling scheme is introduced that consistently estimates the asymptotic variance for an estimator, thereby facilitating inference and the construction of valid confidence intervals. The new method does not rely on the exact form of the asymptotic variance, which is useful when the latter is of complicated form. The method is applied to the volatility estimator of Aït-Sahalia et al. (2011) in the presence of autocorrelated and heteroscedastic market microstructure noise.     

Symmetry-based inference in an instrumental variable setting

We describe exact inference based on group-invariance assumptions that specify various forms of symmetry in the distribution of a disturbance vector in a general nonlinear model. It is shown that such mild assumptions can be equivalently formulated in terms of exact confidence sets for the parameters of the functional form. When applied to the linear model, this exact inference provides a unified approach to a variety of parametric and distribution-free tests. In particular, we consider exact instrumental variable inference, based on symmetry assumptions. The unboundedness of exact confidence sets is related to the power to reject a hypothesis of underidentification. In a multivariate instrumental variables context, generalizations of Anderson–Rubin confidence sets are considered.     

R-optimal designs for multi-factor models with heteroscedastic errors

In this paper, we consider the R-optimal design problem for multi-factor regression models with heteroscedastic errors. It is shown that a R-optimal design for the heteroscedastic Kronecker product model is given by the product of the R-optimal designs for the marginal one-factor models. However, R-optimal designs for the additive models can be constructed from R-optimal designs for the one-factor models only if sufficient conditions are satisfied. Several examples are presented to illustrate and check optimal designs based on R-optimality criterion.     

Estimation and testing in time-series regression models with heteroscedastic disturbances

We consider ARMAX models with heteroscedastic residuals. Consistent estimation of the regression coefficient allows the Bicker-White approach to heteroscedasticity to be extended to moving averages of heteroscedastic disturbances. Tests for the presence of a moving-average or of heteroscedasticity are developed and estimation of the moving-average parameters considered.     

Stochastic cointegration: estimation and inference

This paper considers the estimation of a stochastically cointegrating regression within the stochastic cointegration modelling framework introduced in McCabe et al. (Stochastic cointegration: testing, 2001). A stochastic cointegrating regression allows some or all of the variables to be conventionally or heteroscedastically integrated. This generalizes Hansen's (J. Econom. 54 (1992) 139) heteroscedastic cointegrating regression model, where the dependent variable is heteroscedastically integrated, but all the regressor variables are restricted to being conventionally integrated. In contrast to conventional and heteroscedastic cointegrating regression, ordinary least-squares (OLS) estimation is shown to be inconsistent, in general, in a stochastically cointegrating regression. As a solution, a new instrumental variables (IVs) estimator is proposed and is shown to be consistent. Under a suitable exogeneity assumption, standard asymptotic inference on the stochastic cointegrating vector can be carried out based on the IV estimator. The finite sample properties of the test statistics, including their robustness to the exogeneity assumption, are examined by simulation.     

Least absolute deviations estimation for the censored regression model

This paper proposes an alternative to maximum likelihood estimation of the parameters of the censored regression (or censored ‘Tobit’) model. The proposed estimator is a generalization of least absolute deviations estimation for the standard linear model, and, unlike estimation methods based on the assumption of normally distributed error terms, the estimator is consistent and asymptotically normal for a wide class of error distributions, and is also robust to heteroscedasticity. The paper gives the regularity conditions and proofs of these large-sample results, and proposes classes of consistent estimators of the asymptotic covariance matrix for both homoscedastic and heteroscedastic disturbances.     

Bootstrapping Financial Time Series

It is well known that time series of returns are characterized by volatility clustering and excess kurtosis. Therefore, when modelling the dynamic behavior of returns, inference and prediction methods, based on independent and/or Gaussian observations may be inadequate. As bootstrap methods are not, in general, based on any particular assumption on the distribution of the data, they are well suited for the analysis of returns. This paper reviews the application of bootstrap procedures for inference and prediction of financial time series. In relation to inference, bootstrap techniques have been applied to obtain the sample distribution of statistics for testing, for example, autoregressive dynamics in the conditional mean and variance, unit roots in the mean, fractional integration in volatility and the predictive ability of technical trading rules. On the other hand, bootstrap procedures have been used to estimate the distribution of returns which is of interest, for example, for Value at Risk (VaR) models or for prediction purposes. Although the application of bootstrap techniques to the empirical analysis of financial time series is very broad, there are few analytical results on the statistical properties of these techniques when applied to heteroscedastic time series. Furthermore, there are quite a few papers where the bootstrap procedures used are not adequate.     

Asymptotic and bootstrap inference for inequality and poverty measures

A random sample drawn from a population would appear to offer an ideal opportunity to use the bootstrap in order to perform accurate inference, since the observations of the sample are IID. In this paper, Monte Carlo results suggest that bootstrapping a commonly used index of inequality leads to inference that is not accurate even in very large samples, although inference with poverty indices is satisfactory. We find that the major cause is the extreme sensitivity of many inequality indices to the exact nature of the upper tail of the income distribution. This leads us to study two non-standard bootstraps, the m out of n bootstrap, which is valid in some situations where the standard bootstrap fails, and a bootstrap in which the upper tail is modelled parametrically. Monte Carlo results suggest that accurate inference can be achieved with this last method in moderately large samples.     

Empirical likelihood for heteroscedastic partially linear single-index models with growing dimensional data

In this paper, we propose a new approach to the empirical likelihood inference for the parameters in heteroscedastic partially linear single-index models. In the growing dimensional setting, it is proved that estimators based on semiparametric efficient score have the asymptotic consistency, and the limit distribution of the empirical log-likelihood ratio statistic for parameters \((\beta ^{\top },\theta ^{\top })^{\top }\) is a normal distribution. Furthermore, we show that the empirical log-likelihood ratio based on the subvector of \(\beta \) is an asymptotic chi-square random variable, which can be used to construct the confidence interval or region for the subvector of \(\beta \). The proposed method can naturally be applied to deal with pure single-index models and partially linear models with high-dimensional data. The performance of the proposed method is illustrated via a real data application and numerical simulations.     

Exact permutation tests for non-nested non-linear regression models

This paper proposes exact distribution-free permutation tests for the specification of a non-linear regression model against one or more possibly non-nested alternatives. The new tests may be validly applied to a wide class of models, including models with endogenous regressors and lag structures. These tests build on the well-known J test developed by Davidson and MacKinnon [1981. Several tests for model specification in the presence of alternative hypotheses. Econometrica 49, 781–793] and their exactness holds under broader assumptions than those underlying the conventional J test. The J-type test statistics are used with a randomization or Monte Carlo resampling technique which yields an exact and computationally inexpensive inference procedure. A simulation experiment confirms the theoretical results and also shows the performance of the new procedure under violations of the maintained assumptions. The test procedure developed is illustrated by an application to inflation dynamics.     

A heteroscedasticity-consistent covariance matrix estimator for time series regressions

This paper provides a covariance matrix estimator for the ordinary least squares coefficients of a linear time series model which is consistent even when the disturbances are heteroscedastic. This estimator does not require a formal model of the heteroscedasticity. One can also obtain a direct test of heteroscedasticity, although Monte Carlo experiments show that it may have low power.     

Partially generalized least squares and two-stage least squares estimators

A class of partially generalized least squares estimators and a class of partially generalized two-stage least squares estimators in regression models with heteroscedastic errors are proposed. By using these estimators a researcher can attain higher efficiency than that attained by the least squares or the two-stage least squares estimators without explicitly estimating each component of the heteroscedastic variances. However, the efficiency is not as high as that of the generalized least squares or the generalized two-stage least squares estimator calculated using the knowledge of the true variances. Hence the use of the term partial.     

A note on the exact transformation associated with the first-order moving average process

Exact analytical expressions for the transformation that can be used to transform a generalized regression problem into a simple regression problem are available for a variety of models. Such is the case, for instance, for purely heteroscedastic models, for the first-order Markov process and for error components models. For the first-order moving average process, on the other hand, the exact transformation has not yet been produced. This gap is filled in the present note.Implications for estimation and prediction are also considered.     

Regression density estimation using smooth adaptive Gaussian mixtures

We model a regression density flexibly so that at each value of the covariates the density is a mixture of normals with the means, variances and mixture probabilities of the components changing smoothly as a function of the covariates. The model extends the existing models in two important ways. First, the components are allowed to be heteroscedastic regressions as the standard model with homoscedastic regressions can give a poor fit to heteroscedastic data, especially when the number of covariates is large. Furthermore, we typically need fewer components, which makes it easier to interpret the model and speeds up the computation. The second main extension is to introduce a novel variable selection prior into all the components of the model. The variable selection prior acts as a self-adjusting mechanism that prevents overfitting and makes it feasible to fit flexible high-dimensional surfaces. We use Bayesian inference and Markov Chain Monte Carlo methods to estimate the model. Simulated and real examples are used to show that the full generality of our model is required to fit a large class of densities, but also that special cases of the general model are interesting models for economic data.     

A point optimal test for heteroscedastic disturbances

A new test for additive heteroscedasticity in the disturbances of the linear regression model is proposed. Power functions of various forms of the new test are compared empirically with those of currently favoured tests for a range of heteroscedastic models. The results highlight the power advantage of a test which is MP1 at a central point in the alternative hypothesis parameter space. The main conclusion is that the recommended version of the new test is generally more powerful than existing tests against medium and severe heteroscedasticity, whereas the King and Szroeter tests perform better against weak heteroscedasticity.     

Pre-testing for linear restrictions in a regression model with spherically symmetric disturbances

In this paper we derive the exact risk (under quadratic loss) of pre-test estimators of the prediction vector and of the error variance of a linear regression model with spherically symmetric disturbances. The pre-test in question is one of the validity of a set of exact linear restrictions on the model's coefficient vector. We demonstrate how the known results for the model with normal disturbances can be extended to this broader case. We also show that the critical value of unity results in a minimum of the risk of the pre-test estimator of the error variance. To illustrate the results we assume multivariate Student-t regression disturbances and numerically evaluate the derived expressions.     

An alternative quasi likelihood approach,Bayesian analysis and data-based inference for model specification

This paper studies an alternative quasi likelihood approach under possible model misspecification. We derive a filtered likelihood from a given quasi likelihood (QL), called a limited information quasi likelihood (LI-QL), that contains relevant but limited information on the data generation process. Our LI-QL approach, in one hand, extends robustness of the QL approach to inference problems for which the existing approach does not apply. Our study in this paper, on the other hand, builds a bridge between the classical and Bayesian approaches for statistical inference under possible model misspecification. We can establish a large sample correspondence between the classical QL approach and our LI-QL based Bayesian approach. An interesting finding is that the asymptotic distribution of an LI-QL based posterior and that of the corresponding quasi maximum likelihood estimator share the same “sandwich”-type second moment. Based on the LI-QL we can develop inference methods that are useful for practical applications under possible model misspecification. In particular, we can develop the Bayesian counterparts of classical QL methods that carry all the nice features of the latter studied in  White (1982). In addition, we can develop a Bayesian method for analyzing model specification based on an LI-QL.     

Saddlepoint approximations to the mean and variance of the extended hypergeometric distribution

Conditional inference on 2 x 2 tables with fixed margins and unequal probabilities is based on the extended hypergeometric distribution. If the support of the distribution is large, exact calculation of the conditional mean and variance of the table entry may be computationally demanding. This paper proposes a single‐saddlepoint approximation to the mean and variance. While the approximation achieves acceptable accuracy for ordinary practical purposes, an alternative saddlepoint approximation is provided that gives much closer to exact results. It improves the accuracy of current approximations to up to more than four powers of ten.     

Approximate Bayesian forecasting

Approximate Bayesian Computation (ABC) has become increasingly prominent as a method for conducting parameter inference in a range of challenging statistical problems, most notably those characterized by an intractable likelihood function. In this paper, we focus on the use of ABC not as a tool for parametric inference, but as a means of generating probabilistic forecasts; or for conducting what we refer to as ‘approximate Bayesian forecasting’. The four key issues explored are: (i) the link between the theoretical behavior of the ABC posterior and that of the ABC-based predictive; (ii) the use of proper scoring rules to measure the (potential) loss of forecast accuracy when using an approximate rather than an exact predictive; (iii) the performance of approximate Bayesian forecasting in state space models; and (iv) the use of forecasting criteria to inform the selection of ABC summaries in empirical settings. The primary finding of the paper is that ABC can provide a computationally efficient means of generating probabilistic forecasts that are nearly identical to those produced by the exact predictive, and in a fraction of the time required to produce predictions via an exact method.     

Generalised shape theory via SV decomposition I

This work finds in terms of zonal polynomials, the non isotropic noncentral elliptical shape distributions via singular value decomposition; it avoids the invariant polynomials and the open problems for their computation. The new shape distributions are easily computable and then the inference procedure is based on exact densities, instead of the published approximations and asymptotic distribution of isotropic models. An application of the technique is illustrated with a classical landmark data of Biology, for this, three Kotz type models are proposed (including Gaussian); then the best one is chosen by using a modified BIC criterion.