|Written by Mikhail Kopytine|
|Thursday, 10 December 2009 15:01|
Just like logarithmic returns can be defined and analyzed for daily close, they can be defined for daily high and low. Japanese candlestick charting techniques, believed to have predictive power, study patterns formed by open, low, high and close as the time series progresses. In this report I extend application of my newly developed forecasting figure of merit, Pearson correlation coefficient between the real and predicted logarithmic returns, to the daily high and low, taking another look at the dependence of the prediction quality on the magnitude of a forecasting parameter nicknamed Fred. As the prediction quality is seen to be much better for the next high and low than it is for next close, I am contemplating ways of improving quality for close.
Day scale data for AUD/USD, USD/CAD, USD/CHF, EUR/USD, GBP/USD, and USD/JPY, covering the time span from August 20, 2002 through August 21, 2009 are used in Fig.1. (There is no other selection criteria being applied when selecting the data for Fig.1 -- the forecast is made every day and every day is counted). The middle and half-width of the bands around the data points in Fig.1 represent the mean and standard deviation of the set of the individual Pearson correlation coefficients for the 6 major forex exchange rates, treated as independent measurements.
Any time point in the analysis is represented by a triplet of price-related quantities: the logarithmic returns of high, low, and close. Due to 24-hour nature of forex, the open is typically so tightly related to the previous close that it makes little sense to consider the open-close pair as independent variables, and an either one can be chosen. There is a certain lack of statistical independence in the logarithmic returns among the low, high and close: there is a constraint that next low be below next high and the close be between them. Therefore strictly speaking there is a certain amount of "trivial" redundancy built in. The triplets are analyzed jointly (in the same sense as different markets are analyzed jointly in the Step three algorithm). One way of taking advantage of the high and low predictions is to require that the natural relationship (low below close, close below high) holds. This is not done in the present version of the forecasting algorithm, but is not difficult to implement. There ought to be a certain non-negligible fraction of forecast events where close accidentally lands outside the low-high range, in which case bringing it back should help improve prediction quality for close.
Getting back to Fig.1, recall that the Pearson correlation coefficent has a known range from -1 (the quantities being total opposites) to 1 (total correlation) and that way there is a scale for comparison to know what is large and what is small. A null hypothesis expectation for a measurement like that is to lie around 0. If forex is efficient, there is no way to design a system capable of making predictions, since all available information is instantly discounted by the market, therefore yesterday's (and older) data are of no use to predict today's close: all yesterday's information has been discounted yesterday. Therefore, in such a hypothetic situation, the predicted price levels (or equally, their representatives in the analysis, predicted logarithmic returns) and the actual ones have only one choice -- to yield zero covariance and zero Pearson correlation coefficient after a proper construction of these measures for a long enough chunk of data. Such an outcome indeed takes place for daily close values in random Monte Carlo simulations of hypothetical efficient markets with volatilities of real ones.
As for the predictability of high and low, I have not done simulations but can not at the moment think of a reason why day-to-day increments of these quantities (expressed as logarithmic returns or whatnot) could be predictable "trivially", that is in the context of an efficient market.
As before, Fig.1 as such is free of bias -- it shows you all the possible Fred values. Absence of the "benefit of hindsight" is thus ensured on the stage of Fig.1 analysis: the statement that it is more likely for an arbitrarily chosen Fred value to result in a positive correlation between reality and forecast is based on no particular Fred value and thus no choice is made with the benefit of hindsight. The benefit of hindsight will enter the game once a particular value of Fred is chosen on the basis of Fig.1 or other form of past experience.
|Last Updated ( Monday, 19 April 2010 07:48 )|