Nonlinear stochastic dependence models

The project consists of 5 sub-projects: i) Nonlinearity in a nonstationary time series environment, ii) Nonlinear integer time series, iii) Nonlinear frequency analysis, iv) Nonlinear estimation and discrimination in high dimensions, v) Nonlinear principal components. A common theme for many of the sub-projects is the concept of local Gaussian correlation. This is a new measure of statistical dependence that I believe has large potential and many applications. It is being used or is about to be used in sub-projects iii) - iv). A detailed explanation of this measure has been given in earlier project reports. Using the local Gaussian distribution one is able to describe the increased dependence between financial variables when the market is going down or even when it is close to a crash with the local correlation close to one. The local Gaussian correlation can also be used to test for independence and to identify another measure of dependence called the copula. We now have published papers about both of these aspects as well as described financial contagion and variation in dependence over time. Finally we have published a computer package for computing the local Gaussian correlation. During the last year more work has been done on measuring serial dependence in time time series as well as measuring the dependence between two time series also taking the the time lag structure into account. Among other things the nonlinear dependence between a number of financial indices such as the SP, FTSE, DAX, CAC and the Norwegian index has been examined, in particular lag-lead relationships. Perhaps not surprisingly, the SP index is found to lead the FTSE index, particularly when the market is going down, whereas there is little evidence for an opposite effect. We have also completed two papers on statistical density estimation and conditional density estimation in high dimension. The global Gaussian properties have been exploited locally and an additivity device is used to avoid the curse of dimensionality. There are potential applications to Bayesian networks and causality estimation which we plan to explore in the future. Two of the PhD students that have been working on the project will in fact continue as post docs here in Bergen. Finally, I mention that we have completed a survey paper and a popular paper meant for an economic audience on local correlation. WE have also started to do a spectral or frequency analysis of time series using the local correlation over time. This means that we are able to find a frequency decomposition which depends on the scale, for example it may look different for extreme events than for events of moderate amplitude. This work will continue in cooperation with a PHD candidate. Unfortunately. we have not been able to start on the nonlinear principal component (sub-project v)) in the project period, but we have concrete plans to do this in the near future cooperating with the University of Stavanger and a recently hired post doc here in Bergen. Subproject i) of the project concerns nonlinearity in a nonstationary time series environment. The main focus has been on trying to generalize the so-called cointegration concept from the linear to the nonlinear case. Roughly speaking two nonstationary time series are linearly cointegrated if it is possible to find a linear combination of them that is stationary. This means that although they are nonstationary they move together in the long run. The concept has become very important in econometrics, and Clive Granger was awarded the nobel prize for it. It is known that for certain cases the linear model does not work well even as an approximation. But extending the concept of cointegration to a nonlinear system has turned out to be difficult. The next step up from a linear model is a model where you have region-wise linearity, so-called threshold models. We have extended linear cointegration to this nonlinear threshold case. This is described in two publications with my previous post doc Biqing Cai. He has now moved to HUST university in Wuhan, and we are continuing to work together on a paper concerning this. In addition there are three more published paper related to these problems, 2 of them published in the prestigous journal Annals of Statistics. A main collaborator is professor Jiti Gao, Monash University, Melbourne. We are continuing to work on these problems. Finally on subproject ii) on nonlinear integer time series, we have now virtually finished the work on on a new multivariate model. I have also started working with a group in Seoul, South Korea, concerning change points in such models.

We have specific goals within each of the 5 sub-projects mentioned above: i) Nonlinearity in a nonstationary time series environment: With colleagues in Bergen and with international colleagues I have made progress in the use of nonparametric methodolog y to nonlinear nonstationary time series. In the project I want to turn to parametric models. We have been able recently to overcome a boundedness condition that has been an obstacle and want to extend our work to nonlinear autoregressive processes, where it has not been possible to obtain general results so far. ii) Integer-valued time series: I have worked on this theme with Professor Kostas Fokianos, Cyprus, and Professor Paul Doukhan, Paris. But it has been limited to the one-dimensional case. We wou ld like to extend this to the multivariate case using the copula in the first stage of the modelling. iii) Nonlinear frequency analysis: The concept of local Gaussian correlation has recently been developed by myself and several of my colleagues. The id ea is to use the local Gaussian autocovariance function of a time series to define a local spectral density. This would make it possible to analyse periodicity for extreme values and more moderate values separately. iv) Nonlinear estimation and discrim ination in high dimension: In kernel density estimation the curse of dimensionality makes estimation impossible for high dimensions. We propose to use a simplified form of a multivariate Gaussian approximation to a general density to circumvent the curse of dimensionality. The next step is to use these densities in a nonparametric discrimination framework. v) Nonlinear principal components: Ordinary principal components are obtained by solving the eigenvalue problem for the covariance matrix. By doing t he same for the local Gaussian covariance matrix, local principal components are obtained. These can subsequently be used to estimate principal curves in a noisy environment and to a local reduction of dimension.