There are basically two approaches for constructing forecasting models. The foreign exchange market again serves as our main example. The first approach builds upon structural economic models testing various forms of market efficiency1 or the study of the issues such as the purchasing power parity model and the modeling of risk premia (see Baillie and McMahon, 1989; MacDonald and Taylor, 1992). Meese and Rogoff (1983) carried out the first comprehensive out-of-sample tests of these models, which they call structural models.

Models following the second approach are often called time series models and are based on information extracted from the past of the time series through various forms of linear and nonlinear statistical operators and prefiltering techniques. These types of models can be univariate or multivariate.2 In this chapter, we adopt the second approach and study univariate time series models by utilizing only past prices to forecast a given series. There are two main motivations for this approach. First, the absence of any theory for the short-term movements of the foreign exchange (FX) rates makes the structural models irrelevant for these horizons. Second, the availability of high-frequency data can capture many of the market effects that are relevant to the short-term movements, (e.g., the behavior of different market participants).

The forecasting models presented in this chapter are univariate where only one time series is predicted. They are univariate not only in the predicted target variable but also in the information set used. Multivariate forecasting as an important but complex subject is not discussed here, but Chapter 10 has some relevant discussions.

The models work with high-frequency data as described by Chapter 2 and take into account every tick in the market. The predicted quantity (e.g., the price or future realized volatility) is related to a time horizon, (e.g., the return of the next hour or the volatility of one full business day from now.) The use of high-frequency data allows us to make short-term forecasts for time intervals less than a day. This leads to a large number of observed forecast intervals and thus high statistical significance.

In principle, the knowledge of the "true" data-generating process in the sense of Chapter 8 should also lead to the "true" forecasting model. We have indeed used statistical processes to generate forecasts and measured the success of these statistical processes in terms of their forecasting quality in Chapter 8. In practice, the way from a statistical process to a good forecasting model is not as straightforward. Many otherwise popular statistical processes have serious shortcomings when looking at the intradaily and temporally aggregated behavior, as shown in Section 8.2. Moreover, the statitical processes of that section are volatility models. The price aspect of these models is trivial by having the current price as expectation value for future prices. When moving to forecasting models, we can be more ambitious by also constructing nontrivial price forecasts. We also introduce new

The reader may refer to Fama (1970,1991).

Granger and Newbold (1977) and Priestley (1989) are introductions to these types of models.

testing methods for forecasts. Thus the two topics, data-generating processes and forecasting, are only loosely related.

Forecasting models can be tested by comparing the forecasts to the actual values of the predicted variable. A possible test criterion is the standard deviation of the forecasts from the actual values. Different test criteria can be computed by statistical means, using a test data sample as discussed in Section 9.4. The test result consists of not only a quantitative quality measure but also a statistical significance measure of this quality. The test sample can also be used to optimize the forecasting model and its parameters. In that case, the resulting optimized model should be tested in another sample (i.e., out-of-sample). The test results of the original sample (in-sample) cannot be used as an unbiased measure; they only give an upper limit of the otherwise unknown forecasting model quality.

Two examples of univariate time series models are given: volatility forecasting models used for risk assessment in Section 9.2 and a large real-time price forecasting system with live data feeds in Sections 9.3 and 9.4.3.3

9.2 FORECASTING VOLATILITY FOR VALUE-AT-RISK

Risk can be measured by different means, for example, through an extreme value analysis as in Sections 5.4.2 and 5.4.3. Here we follow a simpler approach by regarding volatility as the variable that determines the risk. This is also the view of popular risk assessment methods. In these methods, the volatility value is inserted in a standard model to compute the Value-at-Risk (VaR): the expected loss of a portfolio after one business day corresponding to the 1 % quantile,4 (i.e., in a scenario that is worse than 99% of the expected cases and better than the remaining 1%). Inserting a volatility figure (computed from variances and covariances of returns of the portfolio assets) may not be enough to compute a reliable VaR. This is discussed by Dave and Stahl (1998), but is not the focus of interest here.

The required volatility value is in fact a volatility forecast for the period from "now" to "now plus one business day." In this section, we discuss univariate volatility forecasting models. Multivariate volatility models need separate treatment because they depend on the intradaily covariance or correlation between assets. This poses some problems as discussed in Chapter 10.

9.2.1 Three Simple Volatility Forecasting Models

Muller (2000) has a discussion of volatility forecasts based on time series operators as presented in Section 3.3. Following that paper, we consider three operator-based volatility forecasting methods of increasing sophistication and quality: (1) the volatility forecasts of RiskMetrics™ developed by J. P. Morgan (1996) as a

3 This forecasting model is running in real time as a part of the Olsen & Associates Information System (OIS).

4 Our scientific interest also extends to forecast intervals other than one business day and quantiles other than 1%, of course.

9.2 FORECASTING VOLATILITY FOR VALUE-AT-RISK

well-known example, (2) an improved version based on tick-by-tick data, and (3) a further improved multi-horizon version.

All of these volatility models can be seen as observations of volatility in the past (i.e., realized volatility measurements as Equations 3.8 or 3.68, for example). However, the models are intended to be applied to the future. The computed volatility values, although measured in the past, are estimates of the future volatility and thus measures of risk. Autoregressive heteroskedasticity as discussed in Section 5.6.1 is the stylized fact that justifies using a certain past volatility as an estimate of future volatility. Section 9.3 has another approach where the volatility forecast is no longer a realized volatility of the past, and Section 8.4.2 considers volatility forecasts directly derived from statistical processes.

The RiskMetrics methodology uses a well-known example of a simple volatility forecast based on an IGARCH process with the following conditional expectation of the squared return:

with [i - 0.94. This formula is evaluated only once per business day, at a given daytime; the resulting volatility value is valid until it is replaced by a new one, one business day later. The time scale / is thus a business time scale omitting weekends, with At = 1 business day. Equation 9.1 is an exponential moving average (EMA) iteration as explained in Section 3.3.5 and can be written as such, using the notation of Equation 3.51,

evaluated at discrete time points separated by At = 1 business day, with an EMA range of = /(1- ) = 15.67 business days. The EMA operator is explained in Section 3.3.5, but Equation 3.52 has to be replaced here by a version for discrete, homogeneous time series,

as explained by Miiller (1991). The only parameter, [i = 0.94, has been chosen to optimize the volatility forecasting quality of Equation 9.1 over a wide range of financial assets and test periods according to J. R Morgan (1996).

In Figure 9.1, two volatilities are presented. The difference between the two curves solely originates from the choice of daytime when the price x is sampled and the volatility is computed by Equation 9.1 or 9.2. One curve is sampled at 7 a.m. (Greenwich Mean Time) GMT which is a time in the late afternoon of East Asian time zones or a suitable daytime for the daily risk calculations of an East Asian risk manager. The other curve is sampled at 5 p.m. GMT, a suitable daytime for a risk manager in London.

The differences between the two curves are surprisingly large: up to 25%, an alarming uncertainty for risk managers. In our case, two risk managers measure

o-2(0 = iia2{t-At) + (1 - fi)[x(t)-x(t - At)]2

(9.1)

(9.2)