Semiparametric stationarity and fractional unit roots tests based on data-driven multidimensional increment ratio statistics

In this paper, we show that the central limit theorem (CLT) satisfied by the data-driven Multidimensional Increment Ratio (MIR) estimator of the memory parameter d established in Bardet and Dola (2012) for d $\in$ (--0.5, 0.5) can be extended to a semiparametric class of Gaussian fractionally integrated processes with memory parameter d $\in$ (--0.5, 1.25). Since the asymptotic variance of this CLT can be estimated, by data-driven MIR tests for the two cases of stationarity and non-stationarity, so two tests are constructed distinguishing the hypothesis d \textless{} 0.5 and d $\ge$ 0.5, as well as a fractional unit roots test distinguishing the case d = 1 from the case d \textless{} 1. Simulations done on numerous kinds of short-memory, long-memory and non-stationary processes, show both the high accuracy and robustness of this MIR estimator compared to those of usual semiparametric estimators. They also attest of the reasonable efficiency of MIR tests compared to other usual stationarity tests or fractional unit roots tests. Keywords: Gaussian fractionally integrated processes; semiparametric estimators of the memory parameter; test of long-memory; stationarity test; fractional unit roots test.Comment: arXiv admin note: substantial text overlap with arXiv:1207.245

Two likelihood-based semiparametric estimation methods for panel count data with covariates

We consider estimation in a particular semiparametric regression model for the mean of a counting process with ``panel count'' data. The basic model assumption is that the conditional mean function of the counting process is of the form $E\{\mathbb{N}(t)|Z\}=\exp(\beta_0^TZ)\Lambda_0(t)$ where $Z$ is a vector of covariates and $\Lambda_0$ is the baseline mean function. The ``panel count'' observation scheme involves observation of the counting process $\mathbb{N}$ for an individual at a random number $K$ of random time points; both the number and the locations of these time points may differ across individuals. We study semiparametric maximum pseudo-likelihood and maximum likelihood estimators of the unknown parameters $(\beta_0,\Lambda_0)$ derived on the basis of a nonhomogeneous Poisson process assumption. The pseudo-likelihood estimator is fairly easy to compute, while the maximum likelihood estimator poses more challenges from the computational perspective. We study asymptotic properties of both estimators assuming that the proportional mean model holds, but dropping the Poisson process assumption used to derive the estimators. In particular we establish asymptotic normality for the estimators of the regression parameter $\beta_0$ under appropriate hypotheses. The results show that our estimation procedures are robust in the sense that the estimators converge to the truth regardless of the underlying counting process.Comment: Published in at http://dx.doi.org/10.1214/009053607000000181 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Long Memory Persistence in the Factor of Implied Volatility Dynamics

The volatility implied by observed market prices as a function of the strike and time to maturity form an Implied Volatility Surface (IV S). Practical applications require reducing the dimension and characterize its dynamics through a small number of factors. Such dimension reduction is summarized by a Dynamic Semiparametric Factor Model (DSFM) that characterizes the IV S itself and their movements across time by a multivariate time series of factor loadings. This paper focuses on investigating long range dependence in the factor loadings series. Our result reveals that shocks to volatility persist for a very long time, affecting significantly stock prices. For appropriate representation of the series dynamics and the possibility of improved forecasting, we model the long memory in levels and absolute returns using the class of fractional integrated volatility models that provide flexible structure to capture the slow decaying autocorrelation function reasonably well.Implied Volatility, Dynamic Semiparametric Factor Modeling, Long Memory, Fractional Integrated Volatility Models.

Semiparametric inference in mixture models with predictive recursion marginal likelihood

Predictive recursion is an accurate and computationally efficient algorithm for nonparametric estimation of mixing densities in mixture models. In semiparametric mixture models, however, the algorithm fails to account for any uncertainty in the additional unknown structural parameter. As an alternative to existing profile likelihood methods, we treat predictive recursion as a filter approximation to fitting a fully Bayes model, whereby an approximate marginal likelihood of the structural parameter emerges and can be used for inference. We call this the predictive recursion marginal likelihood. Convergence properties of predictive recursion under model mis-specification also lead to an attractive construction of this new procedure. We show pointwise convergence of a normalized version of this marginal likelihood function. Simulations compare the performance of this new marginal likelihood approach that of existing profile likelihood methods as well as Dirichlet process mixtures in density estimation. Mixed-effects models and an empirical Bayes multiple testing application in time series analysis are also considered

Trend stationarity versus long-range dependence in time series analysis

Empirically, it is difficult to offer unequivocal judgment as to whether many real economic variables are fractionally integrated or trend stationary. The objective of this paper is to study the effects of spurious detrending of a nonstationary fractionally integrated NFI(d), dE (1/2, 3/2). With respect to the performance of the traditional least squares estimators and tests we prove that the estimated time trend coefficient is consistent but that the corresponding t-Student test diverges. We also analyze a local version in the frequency domain of least squares. We are able to show the consistency of this estimator and that, after conveniently adjusting variance estimates, its t-ratio has a well-defined but nonstandard limiting distribution. Nonetheless, in this latter case it is possible to obtain a set of critical values giving rise to the correct size for any given dE (1/2, 3/2).Publicad

Approximate Bayesian inference in semiparametric copula models

We describe a simple method for making inference on a functional of a multivariate distribution. The method is based on a copula representation of the multivariate distribution and it is based on the properties of an Approximate Bayesian Monte Carlo algorithm, where the proposed values of the functional of interest are weighed in terms of their empirical likelihood. This method is particularly useful when the "true" likelihood function associated with the working model is too costly to evaluate or when the working model is only partially specified.Comment: 27 pages, 18 figure