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A note on the estimation of functional autoregressive models
Umeå University, Faculty of Social Sciences, Department of Statistics.
Umeå University, Faculty of Social Sciences, Department of Statistics.
(English)Manuscript (Other academic)
Abstract [en]

Consider situations where a real valued function is observed over time and has a dynamic dependence structure. Linear autoregressive models, which have been proven useful to model dynamics of "pointwise" time series, can be generalized to such a functional time series situation. We call such models functional autoregressive models. Their parameters are functions of a real valued argument (as the data) and we consider a two-step estimation procedure inspired by Fan and Zhang's (2000) proposal for functional linear models. The latter proposal is based on a first step where the ordinary least squares is used to estimate pointwise linear models for given values of the argument of the functions observed. The second step smoothes the first-step estimates, regressing the latter on the mentioned arguments. The second step does not only yield smooth estimates of the functional parameters but also provides less variable pointwise estimates at the price of a bias. We do not only contribute  by presenting an autoregressive model for functional data but also by proposing a two-stage estimator where the first step takes into account the contemporaneous correlation structure through a multivariate generalized least squares estimator. Some of the properties of the resulting two-step procedure are given. Financial functional data is used as an illustration.

National Category
Computer and Information Science
Research subject
Econometrics
Identifiers
URN: urn:nbn:se:umu:diva-18756OAI: oai:DiVA.org:umu-18756DiVA: diva2:174705
Available from: 2009-02-26 Created: 2009-02-24 Last updated: 2010-01-14Bibliographically approved
In thesis
1. Modeling financial volatility: A functional approach with applications to Swedish limit order book data
Open this publication in new window or tab >>Modeling financial volatility: A functional approach with applications to Swedish limit order book data
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is designed to offer an approach to modeling volatility in the Swedish limit order market. Realized quadratic variation is used as an estimator of the integrated variance, which is a measure of the variability of a stochastic process in continuous time. Moreover, a functional time series model for the realized quadratic variation is introduced. A two-step estimation procedure for such a model is then proposed. Some properties of the proposed two-step estimator are discussed and illustrated through an application to high-frequency financial data and simulated experiments.

In Paper I, the concept of realized quadratic variation, obtained from the bid and ask curves, is presented. In particular, an application to the Swedish limit order book data is performed using signature plots to determine an optimal sampling frequency for the computations. The paper is the first study that introduces realized quadratic variation in a functional context.

Paper II introduces functional time series models and apply them to the modeling of volatility in the Swedish limit order book. More precisely, a functional approach to the estimation of volatility dynamics of the spreads (differences between the bid and ask prices) is presented through a case study. For that purpose, a two-step procedure for the estimation of functional linear models is adapted to the estimation of a functional dynamic time series model.

Paper III studies a two-step estimation procedure for the functional models introduced in Paper II. For that purpose, data is simulated using the Heston stochastic volatility model, thereby obtaining time series of realized quadratic variations as functions of relative quantities of shares. In the first step, a dynamic time series model is fitted to each time series. This results in a set of inefficient raw estimates of the coefficient functions. In the second step, the raw estimates are smoothed. The second step improves on the first step since it yields both smooth and more efficient estimates. In this simulation, the smooth estimates are shown to perform better in terms of mean squared error.

Paper IV introduces an alternative to the two-step estimation procedure mentioned above. This is achieved by taking into account the correlation structure of the error terms obtained in the first step. The proposed estimator is based on seemingly unrelated regression representation. Then, a multivariate generalized least squares estimator is used in a first step and its smooth version in a second step. Some of the asymptotic properties of the resulting two-step procedure are discussed. The new procedure is illustrated with functional high-frequency financial data.

Place, publisher, year, edition, pages
Umeå: , 2009. 18 p.
Series
Statistical studies, ISSN 1100-8989 ; 39
Keyword
Financial data, functional time series, multivariate generalized least squares, seemingly unrelated autoregression
National Category
Computer and Information Science
Research subject
Econometrics
Identifiers
urn:nbn:se:umu:diva-18757 (URN)
Public defence
2009-04-03, Hörsal C, Umeå Universitet, Samhällsvetarhuset, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2009-03-13 Created: 2009-02-24 Last updated: 2009-03-13Bibliographically approved

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