Abstract
Since the market inefficiency is equivalent to the existence of certain pattern in the price dynamics, the question of modeling financial time series becomes one of estimating the coefficients of a linear or non-linear estimator. The spectral analysis in the spirit of traditional Fourier transform does not preserve the time dependence of the patterns when the series is nonstationary. The linear nature of the least square regression that most empirical study of economics, finance and econometrics are based upon may be too simple to capture the complexities in the financial market dynamics. Other non-linear models have only been proposed as ad hoc estimators.
In this study we propose an approach to incorporate wavelets in regression analysis. The resulted estimator is a stochastic nonlinear wavelet-based estimator. The new estimator possesses three advantages: (1) Since the local spatio-frequency properties of a time series is preserved by the wavelet decomposition, the chosen wavelet transform should be a better suited non-linear approximation to the underlying functional relationship; (2) With a systematic and consistent estimation method, the wavelet-based regression estimator avoids the ad hoc nature imbedded in try-and-error approach for finding the right functional format; (3) The estimator is formulated in the state-space model to capture the dynamic and stochastic features of the stock market and its parameters could be updated over time through an optimal path defined by the Kalman filter.
Several issues of optimization algorithm are also discussed.
The suggested model is illustrated in an application to the monthly
observations of Dow Jones Industrial Averages.