Peter Thomson: conference and seminar abstracts

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Seasonal Time series

Transformation and seasonal adjustment

Many time series, particularly monthly economic and official time series, are both non-linear and seasonal. Seasonal adjustment is concerned with the identification and removal of seasonal variation from such time series. A main objective is to provide data free of seasonal variation which might otherwise confound simple estimates of trend. In practice simple power transformations are often used to transform such series to additive linear models and standard seasonal-trend decomposition procedures applied.

This paper considers the effects of seasonal adjustment on transformed time series which are then transformed back to provide seasonally adjusted series in the original scale of the observations. It is shown that this approach can lead to ambiguities in terms of the definition of trend, particularly where there is significant variation about the trend, due to either or both of the seasonal and irregular components. General adjustment and bias correction methods are given. Results are illustrated by simulation and with reference to NZ official time series.

This work is joint with Professor Tohru Ozaki, Institute of Statistical Mathematics, Tokyo, Japan.

A dynamic nonlinear model for multiplicative seasonal time series

A dynamic nonlinear model is introduced for multiplicative seasonal time series that follows and extends the X-11 paradigm where the observed time series is a product of trend, seasonal and irregular factors. A selection of standard seasonal and trend component models used in additive dynamic time series models are adapted for the multiplicative framework and a nonlinear filtering procedure is proposed. The results are illustrated and compared to X-11 and log-additive models using real data. In particular it is shown that the new procedures do not suffer from the trend bias present in log-additive models.

This work is joint with Professor Tohru Ozaki, Institute of Statistical Mathematics, Tokyo, Japan.

On a family of moving-average trend filters for the ends of series

Many seasonal adjustment procedures decompose time series into trend, seasonal, irregular and other components using non-seasonal moving-average trend filters. This paper is concerned with the extension of the central moving-average trend filter used in the body of the series to the ends where there are missing observations.

For any given central moving-average trend filter, a family of end filters is constructed using a minimum revisions criterion and a local dynamic model operating within the span of the central filter. These end filters are equivalent to evaluating the central filter with unknown observations replaced by constrained optimal linear predictors. Two prediction methods are considered; best linear unbiased prediction (BLUP) and best linear biased prediction where the bias is time invariant (BLIP). The BLIP end filters generalise those developed by Musgrave for the central X-11 Henderson filters and include the BLUP end filters as a special case.

The properties of these end filters are determined both in theory and practice. In particular, they are compared to the Musgrave end filters used by X-11 and to the case where the central filter is evaluated with unknown observations predicted by global ARIMA models. The latter parallels the forecast extension method used in X-11-ARIMA.

This work is joint with Mr Alistair Gray, Statistics New Zealand.

Financial Time Series

Modelling interest rate time series

Markov diffusion processes are widely used in the finance literature to model interest rates which are typically assumed to be mean reverting and stationary. Such series also exhibit variation over a wide variety of time scales, from daily through to decadal.

A selection of the more commonly used models are reviewed and their time series properties explored. Particular issues addressed include stationarity and the specification of the volatility or diffusion function. Parametric and non-parametric estimation methods are also applied to both simulated data and New Zealand interest rate data. The results of this analysis are used to highlight areas for further research.

This work is joint with Mr Darren Upton (Mathematical and Computing Sciences) and Dr Martin Lally (Economics and Finance) both from the Victoria University of Wellington.

On CEV models and option pricing

In the option pricing literature, Cox's constant elasticity of variance (CEV) model for share price evolution was first introduced in 1975 as an alternative to geometric Brownian motion. The CEV model incorporates stochastic volatility since its instantaneous variance is directly proportional to a power of the share price. This model and its associated option pricing formula are better able to describe observed share and option prices than geometric Brownian motion which is a special case of the CEV model.

The stochastic properties of the CEV model are reviewed together with various methods of estimation. In particular, a semi-parametric maximum likelihood estimation method is proposed and its properties explored through simulation. This method is then used to fit CEV models to a selection of Australian stock price series. The results of this exploratory analysis are reported together with other work in progress.

This work is joint with Mr John Randal (Statistics and Operations Research) and Dr Martin Lally (Economics and Finance) both from the Victoria University of Wellington.

Geophysical Time Series

Fitting hidden semi-Markov models to rainfall precipitation data

Many meteorological data sets are mixtures of components that correspond to particular physical phenomena, the accurate identification of which are important from a meteorological standpoint. In particular, rainfall is generated by at least two processes, one convection and the other frontal systems, each of which is characterised by its own distribution of rain rates and durations. Breakpoint rainfall precipitation data records the timings of rain rate changes and the steady rates between changes. This data has only recently become available and promises to better capture the information needed to model rainfall phenomena.

A hidden semi-Markov model for breakpoint rainfall data is proposed which builds on and extends the seminal work of Ferguson (1980) on variable duration models for speech. For the breakpoint data the transformed observations are modelled as mixtures of normals within unobserved states where the states evolve over time according to a semi-Markov process. For the latter, parametric forms need to be specified for the state transition probabilities and dwell-time distributions.

Recursions for constructing the likelihood are developed and the EM algorithm used to fit the parameters of the model. The choice of dwell-time distribution is discussed with a mixture distribution over disjoint ranges providing a flexible choice. The methods are also extended to deal with censored data. An application of the model to large-scale bivariate data sets of breakpoint rainfall measurements at selected New Zealand locations is discussed.

This work is joint with Dr John Sansom, National Institute of Water and Atmospheric Research, New Zealand.

Spectral estimation for jittered time series

This paper considers spectral estimation for a zero-mean, band-limited, stationary process that has been sampled at time points jittered from a regular, equi-interval, sampling scheme. The case of interest is where the sampling scheme is near regular so that the jitter standard deviation is small compared to the sampling interval. Such situations occur with many time series collected in the physical sciences including, in particular, oceanographic profiles.

Spectral estimation procedures are developed for the case of independent jitter and autocovariance estimation procedures for both independent and dependent jitter. These are typically modifications of general estimation procedures proposed elsewhere, but tailored to the particular jittered sampling scheme considered. However the results available for dependent jitter are mixed. These issues will be discussed as well as possible multivariate extensions of this work to oceanographic array data.

Other

An investigation of load duration curve forecasting

In 1995 the Electricity Corporation of New Zealand Ltd (ECNZ) and Transpower New Zealand Ltd jointly contracted Peter Thomson to guide the research and development of better forecasting methods for weekly load duration curves. The object was to improve the existing methodology used by ECNZ and Transpower, particularly around the time of peak demand, and to identify approaches that might lead to better forecasts. Specific concerns were to identify models that were more robust and transparent (defensible) than current methods; were suitable for medium term (up to 5 years) forecasting with the capability of including econometric variables to allow forecasts to reflect predicted sectoral/economic shifts; and to identify models which have the capacity to incorporate forecast and demand uncertainties.

The findings of this exploratory research are reported and promising new forecasting methods identified. However further research and development remains to be done to refine these methods to the point where they can be used operationally in practice.

This work is joint with Mr Simon Jurke (Core Management Systems Ltd, New Zealand) and Dr Jonathan Lermit (Transpower New Zealand Ltd).

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