Tice, Julian H. (1998) Using non-linear/chaotic dynamics for interest rate determination. PhD thesis, University of Warwick.

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Abstract

A new class of interest rate models is proposed where the main driving terms for the large
scale dynamics of the system are deterministic. As an example, an economically motivated two
factor model of the term structure is presented that generalises existing stochastic mean term
structure models.
By allowing a certain parameter to acquire dynamical behaviour the model is extended to
three factors. It is shown, that in a deterministic version, the model is equivalent to the Lorenz
system of differential equations. With reasonable parameter values the model exhibits chaotic
behaviour. It successfully emulates certain properties of interest rates including regime switching
and behaviour of a business cycle nature. Pricing and term structure issues are discussed.
Standard PCA techniques used to estimate HJM type models are observed to be equivalent to
dimensional estimates commonly applied to 'spatial data' in non-linear systems analysis. An
empirical investigation uncovers surprising structure consistent with the existence of a low
dimensional attractor. Issues of control of the chaotic system with reference to the underlying
economic model are discussed.
A heuristic approach is made to estimating the three factor model. Exploiting properties of the
term structure, the existence of noise, and the geometry of the system allows a variety of
methods for uncovering the parameters of the model. A better approach is found in the
application of the Kalman filter to the estimation problem. Lack of explicit solutions motivates an
investigation into the use of approximated forms for the term structure. The traditional Kalman
filter is seen to be unstable when applied to the chaotic three factor model. A stable variant, from
the class known as 'square-root' filters, is adopted. A new method is created for finding the
analytical derivatives of the log-likelihood function such that it is consistent with the 'square-root'
filter. Estimates for the empirical estimation of the models developed earlier in the thesis are
given.
It is concluded that there is much scope for expanding the literature within the new class of
models proposed. The particular three factor model developed has been shown to have realistic
properties and be amenable to bond and contingent claim pricing. The chaotic nature of the
model, underpinned by an economic derivation, opens up new methods for authorities to
control/stabilise the economy. An analysis of the underlying dynamical structure of UK money
market rates is consistent with a low dimensional deterministic driving force. Heuristic methods
employed to estimate the parameters of the model allow for an insight into exploiting the
geometry of the system. Application of the Kalman filter to estimation of non-linear models is
found to be problematic due to the linearisations/approximations that are necessary.
An outline of areas for future research is given, providing ideas for extending the economic
formulation, further investigating control of chaotic interest rate systems, testing empirical data
for evidence of chaotic invariants and methods for quantifying and improving the Kalman filtering
procedures for better handling non-linear models.

Item Type:	Thesis (PhD)
Subjects:	H Social Sciences > HB Economic Theory
Library of Congress Subject Headings (LCSH):	Interest rates -- Econometric models, Nonlinear theories
Official Date:	September 1998
Dates:	Date Event September 1998 Submitted
Institution:	University of Warwick
Theses Department:	Warwick Business School
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Webber, Nick
Sponsors:	Economic and Social Research Council (Great Britain) (ESRC) (R00429334359) ; Warwick Business School
Extent:	iii, 186 leaves
Language:	eng

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