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92 9.3 FORECASTING RETURNS OVER MULTIPLE TIME HORIZONS This section examines the forecasting model of Dacorogna et al. (1996). This model supports several forecast intervals. Hourly returns are predicted as well as daily, weekly, monthly, and quarterly returns. The forecasting model of returns relies on an underlying volatility forecast. Both the volatility and return forecasts use the same methodology. Volatility is treated with the help of an alternative time scale, the Intrinsic time of the time series. 9.3.1 Intrinsic Time The foreign exchange returns exhibit conditional heteroskedasticity which can be treated through a change of time scale. This is the second layer of our forecasting model on top of the business time scale. Some literature followed a similar approach to treat the conditional heteroskedasticity, such as Stock (1988), who uses two types of time deformation, one based on the time series itself and one on business cycle variables.5 In our approach, we also use the underlying time scries to construct a time deformation. It is based on the scaling law defined in Equation 6.2 and on the price volatility: r(/c) = ( ) + (9.7) Av where tc is the current time, the price difference is taken on the same interval as A I?, and E and are the scaling law inverse exponent and factor, respectively. The constant is a calibration factor dependent on the particular time series. Its role is to keep r in line with physical time in the long run. This relationship is in fact the reverse of the scaling law for a particular return taken on a constant #timc interval size. This second new time scale, the rscale, does not directly use the physical time i, and does not need to have fundamental information about the behavior of the series. The only information needed to define the scale are the values of the time series themselves. Thus we have chosen to call this time scale intrinsic time. The consequence of using such a scale is to expand periods of high volatility and contract those of low volatility, thus better capturing the relative importance of events to the market. Any moving average based on the intrinsic time r dynamically adapts its range to market events. Therefore a forecasting model based on the scale has a dynamic memory of the price history. There is, however, a problem when using such a time scale. The intrinsic time r is only known for the past, contrary to the business time scale which is known also for the future, because it is based on average behavior. Thus a forecasting model for the price actually needs to be composed of two forecasting models, one for the intrinsic time and one for the price. The first requires forecasting of the size (not the direction) as time cannot flow backward. 5 In the same paper Stock (1988) indicates how this approach can be compared to the ARCH models.
9.3.2 Model Structure The price generating process is far from stationary in physical time. In Section 9.3.1, the geographic seasonality and conditional heteroskedasticity are modeled through successive changes of the underlying time scales. After these transformations, the remaining structure and the dynamics of the transformed series can be analyzed. The model presented in this section captures the dynamics through the computation of nonlinear indicators. Because the model is on the businesstime scale i7, all equations are written in terms of this new scale. The relation to the physical time scale is given by Equation 6.17. 9.3.3 A Linear Combination of Nonlinear Indicators The model equations are based on nonlinear indicators, which are modeled with moving averages. Indicators for market prices come conceptually from simple trading systems used in practice by market participants.6 Those trading systems yield buy and sell signals by evaluating an indicator function. The crossing of a certain threshold by the indicator on the positive side is regarded as a buy signal, on the negative side as a sell signal. An indicator is thus used as a predictor of a variable or its change, for instance, a price change (i.e., a return). Finding an ideal indicator, if it exists at all, would be enough to make a good price forecast. We, however, have no ideal indicators. Therefore there is need to combine different indicators appropriately to optimize their respective influence. Partly, the forecasting models presented here are based on a linear combination of price indicators zx where the relative weights are estimated by multiple linear regression. For a fixed forecasting horizon Aft (corresponding to a At/ in physical time), the price forecast / is computed with where xc is the current price, and m is the number of indicators used in the model (from two to five per horizon). All the indicators are estimated in intrinsic time scale (rscale). The coefficients cxj(Aftf) are estimated with a multiple linear regression model. Af/ in Equation 9.8, the forecasting horizon expressed in intrinsic time, is not yet defined. This quantity must be computed from its own forecasting model, which is similar to that in Equation 9.8. The forecasting horizon, , can be written as an intrinsic time forecast, (9.8) (9.9) 6 See for instance Dunis and Fceny (1989); Murphy (1986).
where the forecasting model is computed in #scale. The coefficients cTj(A&f) are estimated through a multiple linear regression and ,/( # , #c) are the intrinsic time indicators. Contrary to most traditional forecasting models, this model does not rely on a fixed basic time interval but is designed with a concept of continuous time. In fact, the time when a price is recorded in the database is unequally spaced in time. Moreover, the use of the rscale implies that our forecasting models must be computed simultaneously over several fixed time horizons Given a forecasting horizon in physical time At/ and the price history until xc, one can compute A#y with Equation 6.17 and rc with Equation 9.7. With a sufficiently large set of indicators, zTj(A#y, #c), zxj (Atf, rc), and coefficients, cTj(Auf) and cxj(Adf), the price forecast can be computed by choosing the appropriate f with Equation 9.9 and substituting it into Equation 9.8. In the next two sections, we define the indicators and study how to compute the coefficients 9.3.4 Moving Averages, Momenta, and Indicators In Equations 9.8 and 9.9, the indicators are based on momenta, which are based on moving averages. In particular, we work with exponential moving averages (EMA) because they may be conveniently expressed in terms of recursion formulae (see Chapter 3). The momentum indicator is which compares the most recent price to its own exponential moving average (EMA), using the notation in Section 3.3.5 and computed by the recursion formula of Equation 3.51. This is done by using intrinsic time as the time scale. It is also possible to define the first and the second momenta. The first momentum, mx\ is the difference of two exponential moving averages (or momenta) with different ranges. This can be considered as the first derivative of ( ). The second momentum, mx , is the linear combination of three exponential moving averages (or momenta) with different ranges, which provide information on the curvature of the series over a certain past history. In section 9.3.3 we introduce the concept of indicators. Here we want to define those that are used in our forecasting models. There are a large number of technical indicators (Murphy, 1986; Dunis and Feeny, 1989) and momenta indicators are widely used in technical trading systems. We limit our focus on momenta type indicators. Following Equation 9.8, let an indicator for returns with a range Arr, zx(Atv, rc) be defined as follows: ( , ) = xc  [ ; x](rc) (9.10) "1 p zx(ATr, rc) = m\ ( , tc) (9.11)
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