Modelling the Dynamics of Cross-Sectional Price Functions: an Econometric Analysis of the
Bid and Ask Curves of an Automated Exchange

Clive G. Bowsher
Nuffield College, University of Oxford

Abstract:
Functional Signal plus Noise (FSN) time series models are introduced for the econometric 
analysis of the dynamics of a large cross-section of prices in which contemporaneous 
observations are functionally related. A semiparametric FSN model is developed in which a 
smooth, cubic spline signal function is used to approximate the price curve data. Estimation
may then be performed using quasi-maximum likelihood methods based on the Kalman filter. The 
model is used to provide one of the first studies of the dynamics of the bid and ask curves 
of an electronic limit order book and enables the comprehensive measurement of the dynamic 
determinants of traders' execution costs. It is found that the differences between the bid 
and ask curves and their intercepts (i.e. the immediate price impacts of market orders) are
well described by covariance stationary processes. The in-sample, 1-step ahead point 
predictions for these curves perform well and motivate the development of parametric FSN 
models that take into account the monotonicity of the price curves and can be used 
to form predictive distributions.

Keywords: functional time series, bid and ask curves, liquidity, electronic limit order book, 
cubic spline, state space form, Kalman filter, quasi-maximum likelihood.

JEL classification: C14, C32, C33, C51, G12.