Nonlinear cross-spectrum analysis via the local Gaussian correlation

Spectrum analysis can detect frequency related structures in a time series $\{Y_t\}_{t\in\mathbb{Z}}$, but may in general be an inadequate tool if asymmetries or other nonlinear phenomena are present. This limitation is a consequence of the way the spectrum is based on the second order moments (auto and cross-covariances), and alternative approaches to spectrum analysis have thus been investigated based on other measures of dependence. One such approach was developed for univariate time series in Jordanger and Tj{\o}stheim (2017), where it was seen that a local Gaussian auto-spectrum $f_{v}(\omega)$, based on the local Gaussian autocorrelations $\rho_v(\omega)$ from Tj{\o}stheim and Hufthammer (2013), could detect local structures in time series that looked like white noise when investigated by the ordinary auto-spectrum $f(\omega)$. The local Gaussian approach in this paper is extended to a local Gaussian cross-spectrum $f_{kl:v}(\omega)$ for multivariate time series. The local cross-spectrum $f_{kl:v}(\omega)$ has the desirable property that it coincides with the ordinary cross-spectrum $f_{kl}(\omega)$ for Gaussian time series, which implies that $f_{kl:v}(\omega)$ can be used to detect non-Gaussian traits in the time series under investigation. In particular: If the ordinary spectrum is flat, then peaks and troughs of the local Gaussian spectrum can indicate nonlinear traits, which potentially might discover local periodic phenomena that goes undetected in an ordinary spectral analysis.Comment: 41 pages, 12 figure

Nonlinear spectral analysis: A local Gaussian approach

The spectral distribution $f(\omega)$ of a stationary time series $\{Y_t\}_{t\in\mathbb{Z}}$ can be used to investigate whether or not periodic structures are present in $\{Y_t\}_{t\in\mathbb{Z}}$, but $f(\omega)$ has some limitations due to its dependence on the autocovariances $\gamma(h)$. For example, $f(\omega)$ can not distinguish white i.i.d. noise from GARCH-type models (whose terms are dependent, but uncorrelated), which implies that $f(\omega)$ can be an inadequate tool when $\{Y_t\}_{t\in\mathbb{Z}}$ contains asymmetries and nonlinear dependencies. Asymmetries between the upper and lower tails of a time series can be investigated by means of the local Gaussian autocorrelations introduced in Tj{\o}stheim and Hufthammer (2013), and these local measures of dependence can be used to construct the local Gaussian spectral density presented in this paper. A key feature of the new local spectral density is that it coincides with $f(\omega)$ for Gaussian time series, which implies that it can be used to detect non-Gaussian traits in the time series under investigation. In particular, if $f(\omega)$ is flat, then peaks and troughs of the new local spectral density can indicate nonlinear traits, which potentially might discover local periodic phenomena that remain undetected in an ordinary spectral analysis.Comment: Version 4: Major revision from version 3, with new theory/figures. 135 pages (main part 32 + appendices 103), 11 + 16 figure

Estimation in nonlinear time series models

A general framework for analyzing estimates in nonlinear time series is developed. General conditions for strong consistency and asymptotic normality are derived both for conditional least squares and maximum likelihood types estimates. Ergodie strictly stationary processes are studied in the first part and certain nonstationary processes in the last part of the paper. Examples are taken from most of the usual classes of nonlinear time series models

Uniform Consistency for Nonparametric Estimators in Null Recurrent Time Series

This paper establishes a suite of uniform consistency results for nonparametric kernel density and regression estimators when the time series regressors concerned are nonstationary null-recurrent Markov chains. Under suitable conditions, certain rates of convergence are also obtained for the proposed estimators. Our results can be viewed as an extension of some well-known uniform consistency results for the stationary time series case to the nonstationary time series case.β-null recurrent Markov chain, nonparametric estimation, rate of convergence, uniform consistency

Estimation in threshold autoregressive models with a stationary and a unit root regime

This paper treats estimation in a class of new nonlinear threshold autoregressive models with both a stationary and a unit root regime. Existing literature on nonstationary threshold models have basically focused on models where the nonstationarity can be removed by differencing and/or where the threshold variable is stationary. This is not the case for the process we consider, and nonstandard estimation problems are the result. This paper proposes a parameter estimation method for such nonlinear threshold autoregressive models using the theory of null recurrent Markov chains. Under certain assumptions, we show that the ordinary least squares (OLS) estimators of the parameters involved are asymptotically consistent. Furthermore, it can be shown that the OLS estimator of the coefficient parameter involved in the stationary regime can still be asymptotically normal while the OLS estimator of the coefficient parameter involved in the nonstationary regime has a nonstandard asymptotic distribution. In the limit, the rate of convergence in the stationary regime is asymptotically proportional to n-1/4, whereas it is n-1 in the nonstationary regime. The proposed theory and estimation method are illustrated by both simulated data and a real data example.Autoregressive process; null-recurrent process; semiparametric model; threshold time series; unit root structure.

Poisson Autoregression

This paper considers geometric ergodicity and likelihood based inference for linear and nonlinear Poisson autoregressions. In the linear case the conditional mean is linked linearly to its past values as well as the observed values of the Poisson process. This also applies to the conditional variance, implying an interpretation as an integer valued GARCH process. In a nonlinear conditional Poisson model, the conditional mean is a nonlinear function of its past values and a nonlinear function of past observations. As a particular example an exponential autoregressive Poisson model for time series is considered. Under geometric ergodicity the maximum likelihood estimators of the parameters are shown to be asymptotically Gaussian in the linear model. In addition we provide a consistent estimator of the asymptotic covariance, which is used in the simulations and the analysis of some transaction data. Our approach to verifying geometric ergodicity proceeds via Markov theory and irreducibility. Finding transparent conditions for proving ergodicity turns out to be a delicate problem in the original model formulation. This problem is circumvented by allowing a perturbation of the model. We show that as the perturbations can be chosen to be arbitrarily small, the differences between the perturbed and non-perturbed versions vanish as far as the asymptotic distribution of the parameter estimates is concerned.generalized linear models; non-canonical link function; count data; Poisson regression; likelihood; geometric ergodicity; integer GARCH; observation driven models; asymptotic theory

Measuring asymmetries in financial returns : an empirical investigation using local gaussian correlation

A number of studies have provided evidence that financial returns exhibit asymmetric dependence, such as increased dependence during bear markets, but there seems to be no agreement as to how such asymmetries should be measured. We introduce the use of a new measure of local dependence to study this asymmetry. The central idea of the new approach is to approximate an arbitrary bivariate return distribution by a family of Gaussian bivariate distributions. At each point of the return distribution there is a Gaussian distribution that gives a good approximation at that point. The correlation of the approximating Gaussian distribution is taken as the local correlation in that neighbourhood. The new measure does not suffer from the selection bias of the conditional correlation for Gaussian data, and is able to capture nonlinear dependence. Analyzing several financial returns from the US, UK, German and French markets, we confirm and are able to explicitly quantify the asymmetry. Finally, we discuss a risk management application, and point out a number of possible extensions