Time Series

Time series analysis is a main area within mathematical statistics, data analysis, stochastic finance and econometrics. A main component of the research of the statistics group in Bergen is undertaken in this area.

A time series is a series of observations taken over time. Examples are from many disciplines:

• Prices of shares, Exchange rates, price indices
• Time development of biological populations
• Seismic observations from earthquakes and/or explosions
• Development of year classes of species of fish, such as arctic cod in the Barents Sea
• Geological varve data (depositional data from the last ice age)
• Climate time series

All of these data examples with accompanying problems have been studied by members of the statistics group.

The goal of time series analysis depends on what kind of data are available, but generally it can be subdivided into three parts:

• Forecasting: Usually for economic variables, but population data is another example.
• Characterization: By using certain parameters one can, for example, be able to discriminate between the seismic signal from an earthquake and the corresponding signal from an underground nuclear explosion. Similar techniques are being used in voice recognition.
• Modeling of relationships: In the analysis of large systems of equations of economic variables one seeks to isolate variables which has a causal effect on others (control variables or exogenous variables) and variables which do not have this property. These systems of equations are usually linear, but can also be non-linear. Another example is environmental variables and climate variables that may influence abundance of fish.

To try to solve the problems sketched above, one has to build up a statistical model. In time series analysis these have predominantly been linear and of a regression type. They are then not always realistic, and recently non-linear models have increasingly been used. The model has to be estimated using the given data, either parametrically or nonparametrically. Of course this is more difficult to do in the nonlinear case. Prior to the estimation phase there is an identification phase, where it is determined what kind of model should be used and how long memory this model should possess. At last there is a diagnostic checking phase to check whether the model works satisfactorily. One possibility is to look at forecast errors and the distribution of these.

Time series analysis plays a significant role for mathematical economics and finance, especially through the concept of volatility. Mathematically speaking volatility is the conditional variance of future observations given observations until today's date. The volatility is difficult to estimate, but it is important to determine, because it expresses the economic risk. In the formula for option pricing it enters, and has to be estimated from the available data. One type of model that has been much used is the so-called ARCH model. Modifications of this exist and has been proposed to handle transactions in real time (tick by tick data) on stock markets.

The modeling and computational aspects constitute essential parts of modern time series analysis. Usually, extensive analysis and simulations on a PC will be required, but for students that are primarily interested in mathematical problems and challenges there are also plenty of possibilities. Theoretically, time series modeling is in many ways mathematically equivalent to the study of stochastic difference equations. For non-linear models there are many unsolved problems when it comes to stability of solutions. Relatively advanced Markov chain theory is used. In the nonstationary case the concept of null recurrence is of interest, because it can be used to describe nonlinear relationships between nonstationary processes. An alternative approach to studying the asymptotics of such systems is to use theory based on the local time of Brownian motion.

Research

The field of time series analysis consists in analysing observations taken sequentially in time. One tries to uncover structures which describe underlying dependencies between the observations. The methods are applied in econometrics, but are also used in many other contexts, for example in the analysis of geological, biological and climate data series.

Time series analysis has been an active area of research within the group of statistics in Bergen for a long time, and one has obtained a good international reputation. Roughly speaking, two permanent members of the staff have worked with these problems together with a number of doctoral and master degree students. The research has been conducted, and is being conducted, in close contact with research groups abroad, such as the groups at the Humboldt University, Berlin, Universidad Carlos III, Madrid, University of California, San Diego, Université Paul Sabatier, Toulouse, London School of Economics and the University of Western Australia.

Over time a relatively large selection of research topics has been studied, but especially nonlinear modeling and nonparametric methods have been much looked at. Thus, nonparametric tests of linearity and independence have been constructed, which have subsequently been followed up by other groups. Recently there has been a collaboration with researchers from Madrid and the University of Michigan on testing and estimation of additive models using the so-called marginal integration method (developed in Bergen and by a couple of other groups 5-6 years ago. There are applications to production modeling in economics. Work is in progress on extensions to nonstationary time series and spatial data. Both extensions offer challenging theoretical problems.

An important theoretical project the last couple of years has been to seek to establish a nonparametric theory of estimation for nonstationary processes. The split method for Markov chains has played a decisive role, and so-called beta-recurrent Markov processes have been found suitable as a model class. This work has strong connections to econometrics and to the theory of nonlinear cointegration.

The time series group has started a cooperation with the Institute for Marine Research, formally regulated through a subsidiary position. Mainly this work has consisted in analysing time series for arctic cod and haddock obtained from research surveys in the Barents Sea over a period of 20 years. The goal is to understand the population dynamics in a better way, and to exploit the results to obtain better abundance estimates. Recently the cooperation has been extended to an analysis of the influence of the climate on abundance, in particular of plankton. It is the plan to integrate this in a joint effort with the Institute of geophysics on the statistical analysis of climate data.

Related to the fish population data is the study of panels of time series. Here the work has been concentrated on panels consisting of intercorrelated time series. Several traditional estimates from the univariate realm then have to be modified. Burg type estimates, where end effects are taken care of, are becoming much more important, because one is often in a situation where one has many but short time series. This is true in particular for micro array data. One of our doctoral students is working in this area. The intercorrelated-time-series methodology is currently being extended to nonlinear and additive models in cooperations with research groups in Heidelberg and Chicago.