Nonparametric inference in hidden Markov models using P-splines

Hidden Markov models (HMMs) are flexible time series models in which the distributions of the observations depend on unobserved serially correlated states. The state-dependent distributions in HMMs are usually taken from some class of parametrically specified distributions. The choice of this class can be difficult, and an unfortunate choice can have serious consequences for example on state estimates, on forecasts and generally on the resulting model complexity and interpretation, in particular with respect to the number of states. We develop a novel approach for estimating the state-dependent distributions of an HMM in a nonparametric way, which is based on the idea of representing the corresponding densities as linear combinations of a large number of standardized B-spline basis functions, imposing a penalty term on non-smoothness in order to maintain a good balance between goodness-of-fit and smoothness. We illustrate the nonparametric modeling approach in a real data application concerned with vertical speeds of a diving beaked whale, demonstrating that compared to parametric counterparts it can lead to models that are more parsimonious in terms of the number of states yet fit the data equally well

Consistent tests of conditional moment restrictions

We propose two classes of consistent tests in parametric econometric models defined through multiple conditional moment restrictions. The first type of tests relies on nonparametric estimation, while the second relies on a functional of a marked empirical process. For both tests, a simulation procedure for obtaining critical values is shown to be asymptotically valid. Finite sample performances of the tests are investigated by means of several Monte-Carlo experiments.Publicad

Detecting gradual changes in locally stationary processes

In a wide range of applications, the stochastic properties of the observed time series change over time. The changes often occur gradually rather than abruptly: the properties are (approximately) constant for some time and then slowly start to change. In many cases, it is of interest to locate the time point where the properties start to vary. In contrast to the analysis of abrupt changes, methods for detecting smooth or gradual change points are less developed and often require strong parametric assumptions. In this paper, we develop a fully nonparametric method to estimate a smooth change point in a locally stationary framework. We set up a general procedure which allows us to deal with a wide variety of stochastic properties including the mean, (auto)covariances and higher moments. The theoretical part of the paper establishes the convergence rate of the new estimator. In addition, we examine its finite sample performance by means of a simulation study and illustrate the methodology by two applications to financial return data.Comment: Published at http://dx.doi.org/10.1214/14-AOS1297 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Two Procedures for Robust Monitoring of Probability Distributions of Economic Data Streams induced by Depth Functions

Data streams (streaming data) consist of transiently observed, evolving in time, multidimensional data sequences that challenge our computational and/or inferential capabilities. In this paper we propose user friendly approaches for robust monitoring of selected properties of unconditional and conditional distribution of the stream basing on depth functions. Our proposals are robust to a small fraction of outliers and/or inliers but sensitive to a regime change of the stream at the same time. Their implementations are available in our free R package DepthProc.Comment: Operations Research and Decisions, vol. 25, No. 1, 201

On clustering procedures and nonparametric mixture estimation

This paper deals with nonparametric estimation of conditional den-sities in mixture models in the case when additional covariates are available. The proposed approach consists of performing a prelim-inary clustering algorithm on the additional covariates to guess the mixture component of each observation. Conditional densities of the mixture model are then estimated using kernel density estimates ap-plied separately to each cluster. We investigate the expected L 1 -error of the resulting estimates and derive optimal rates of convergence over classical nonparametric density classes provided the clustering method is accurate. Performances of clustering algorithms are measured by the maximal misclassification error. We obtain upper bounds of this quantity for a single linkage hierarchical clustering algorithm. Lastly, applications of the proposed method to mixture models involving elec-tricity distribution data and simulated data are presented

Indirect Inference for Locally Stationary Models

We propose the use of indirect inference estimation to conduct inference in complex locally stationary models. We develop a local indirect inference algorithm and establish the asymptotic properties of the proposed estimator. Due to the nonparametric nature of locally stationary models, the resulting indirect inference estimator exhibits nonparametric rates of convergence. We validate our methodology with simulation studies in the confines of a locally stationary moving average model and a new locally stationary multiplicative stochastic volatility model. Using this indirect inference methodology and the new locally stationary volatility model, we obtain evidence of non-linear, time-varying volatility trends for monthly returns on several Fama-French portfolios