| Summary: | Forecasting financial time series is regarded as one of the most challenging applications of time series prediction due to their dynamic nature. However, it is the fundamental element of most investment activities thus attracting the attention of practitioners and researchers for many decades. The purpose of this research is to investigate and develop novel methods for the prediction of financial time series considering their dynamic nature. The predictive performance of asset prices time series themselves is exploited by applying digital signal processing methods to their historical observations. The novelty of the research lies in the design of predictive filters by maximising their spectrum flatness of forecast errors. The filters are then applied to forecast linear combinations of daily open, high, low and close prices of financial time series. Given the assumption that there are no structural breaks or switching regimes in a time series, the sufficient and necessary conditions that a time series can be predicted with zero errors by linear filters are examined. It is concluded that a band-limited time series can be predicted with zero errors by a predictive filter that has a constant magnitude response and constant group delay over the bandwidth of the time series. Because real world time series are not band-limited thus cannot be forecasted without errors, statistical tests of spectrum flatness which evaluate the departure of the spectral density from a constant value are introduced as measures of the predictability of time series. Properties of a time series are then investigated in the frequency domain using its spectrum flatness. A predictive filter is designed by maximising the error spectrum flatness that is equivalent to maximise the “whiteness” of forecast errors in the frequency domain. The focus is then placed on forecasting real world financial time series. By applying spectrum flatness tests, it is found that the property of the spectrum of a linear combination of daily open, high, low and close prices, which is called target prices, is different from that of a random walk process as there are much more low frequency components than high frequency ones in its spectrum. Therefore, an objective function is proposed to derive the target price time series from the historical observations of daily open, high, low and close prices. A predictive filter is then applied to obtain the one-step ahead forecast of the target prices, while profitable trading strategies are designed based on the forecast of target prices series. As a result, more than 70% success ratio could be achieved in terms of one-step ahead out-of-sample forecast of direction changes of the target price time series by taking the S&P500 index for example.
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