table 9.1 The sampling periods of the forecast study.

The data samples used for initialization, model training (in-sample), and testing (out-of-sample).

runs from December 1, 1986, to September 1, 1990 (46 months), and is our in-sample period. The second runs from September 3, 1990, to October 3, 1992 (25 months). This second period is pure post ex-ante testing-that is, data from this period were not used at all for building the model. These 25 months constitute our out-of-sample test.

9.4.3 Forecast Effectiveness in Intraday Horizons

The forecast horizons here are for 2, 4, and 8 hr. This choice was made for two reasons. First, there is almost no literature to study models for such short horizons. Second, the statistical significance of the findings can be enhanced due to a large number of observations within a few years at the intraday frequency.

The quality measures are computed for each time horizon at an interval of 1/12 of the horizon. As mentioned earlier, the forecast accuracy is always measured in the physical time scale because it is in this scale that the different forecasts are useful. The number of relevant points in the statistical computation varies depending on the horizon and on the number of missing data for the different currencies. For the 2 hr horizon it varies from around 70,000 tests to 140,000, for 4 hr from 35,000 to 70,000, and for 8 hr 24,000 to 37,000. These very large numbers ensure that our statistical results are highly significant. We currently have 41 currencies running on the Olsen Information System (OIS), but in order not to overwhelm this study with numbers, we only show results for the 10 most important FX rates against the USD and 10 of the most traded cross rates. For the other currencies the results are very similar. The direction quality and the signal correlation are given in percentage for the in and out-of-sample testing periods in Table 9.2 for the USD rates and in Table 9.3 for the cross rates. Juxtaposing both results show clearly that, except for the 2 hr in the USD rates and for GBP-USD, the quality achieved in-sample is in most cases maintained out-of-sample and sometimes even slightly improved.

In Table 9.4 we summarize the significance of these results. For each horizon and for each currency we write a "+" sign if both the direction quality and the signal correlation are above the significance levels computed using Equation 9.15 for the direction quality and 1.96/\fn for the signal correlation. If one of the two

TABLE 9.2 Direction quality and signal correlation for 10 USD rates.

Direction quality and signal correlation, in-sample and out-of-sample, for 9 FX rates and gold price against the USD. The numbers are expressed in percentage.

Hor. Direction Correlation

Hor. Direction Correlation

USD-DEM

USD-JPY

GBP-USD

LSD-CHE

USD-FRF

+2.0 / -0.0 + 1.9/+1.2 +0.1 /+3.2

+1.6/-1.3 +2.4/+ 1.4 + 1.9/+4.3

+3.0 / +3.5 +4.8/+5.2 +2.9/+6.0

+0.8 / +3.5 +2.7 / +6.5 +4.0/+6.1

+3.5/+2.3 +4.3 / +3.8 +3.9/+2.9

measures or both are below the significance level we write a "-" sign. Except for two USD rates (GBP-USD and USD-FRF), two cross rates (CAD-CHF and XAU-CHF), and the 2 hr horizon for the USD rates, which do not sustain conclusively the out-of-sample test, the other cases confirm the success of the model. The 2 hr cross-rates pass the out-of-sample test for 80% of the cases (only 40% of the cases for the USD rates) and the 4 hr for 80% and 8 hr for 90% of the cases. The USD rates for the 4 hr pass the test in 90% of the cases and for 8 hr in 80% of the cases.

In this chapter, we have shown that with the help of high-frequency data the statistical properties of FX rates can be better understood and that specifying forecasting models for very short-term horizons is possible. These models contain ingredients all designed to better capture the dynamics at work in the FX market. The most important characteristics of the models are as follows:

Univariate time analysis type of model but based on intraday nonhomo-geneous data,

Variable time scales to capture both the seasonal heteroskedasticity (t>-scale) and the autoregressive conditional heteroskedasticity (r-scale)

Linear combination of nonlinear indicators

Multiple linear regression with two modifications to avoid instabilities and to correct for the leptokurtic behavior of the returns

TABLE 9.3 Direction quality and signal correlation for 10 cross rates.

Direction quality and signal correlation, in-sample and out-of-sample, for 9 FX cross rates and gold price. The numbers are expressed in percentage.

Hor. Direction Correlation

Hor. Direction Correlation

JPY-DEM

2hr 4hr 8hr

52.5/51.7 51.8/51.0 52.0/51.0

+2.3/+1.0 +3.0 / +2.5 +4.0/+1.3

GBP-CHF 2hr 54.4/55.1 +7.4/+6.5 4hr 53.5/54.1 +6.2/+5.0 8hr 53.8/54.2 +6.4/+8.0

GBP-DEM 2hr 54.0/55.2 +6.1/+5.6 JPY-CHF 2hr 52.6/51.7 +3.1/+0.9

4hr 53.3/54.5 +5.3/+4.4 4hr 52.2/50.8 +3.5/+2.1

8hr 53.3/54.2 +4.7/+8.7 8hr 54.1/51.4 +9.1/+1.5

CHF-DEM

2hr 4hr 8hr

57.1 /55.5 54.9/54.6 55.2/54.4

+12.5/+8.1 +9.7 / +6.2 +10.0/+6.7

GBP-JPY 2hr 52.8/53.3 +4.5/+4.0 4hr 52.3/53.0 +4.6/+4.8 8hr 52.8/52.8 +6.5/+6.2

FRF-DEM 2hr 62.0/62.7 +20.4/+22.0 CAD-CHF 2hr 51.1/51.2 +1.9/-0.4

4hr 59.7/59.5 +15.0 / +16.1 4hr 52.0/51.7 +2.6/+0.4

8hr 57.6/56.3 +14.3/+13.2 8hr 52.3/51.8 +4.3/+1.6

DEM-AUD

2hr 4hr 8hr

51.4/51.1 51.2/51.1 50.5/51.2

+2.5/+1.2 +3.1/+2.1 +4.0 / +3.2

XAU-CHF 2hr 51.0/50.6 +1.0/-0.8 4hr 52.1/51.0 +3.2/-0.2 8hr 52.0/52.4 +3.0/+5.1

Continuous optimization of the model coefficients in a finite size, forecasting horizon-dependent sample.

The forecast quality of these models is evaluated on a very large sample with two different measures that avoid statistical problems arising from the nature of the FX rate time series. The rigorous separation of in and out-of-sample measures, the large number of observations, and the stringent significance levels mean that the statistical results of the forecast evaluation are convincing evidence for our models having beat the random walk for most of the 20 studied currencies and for the very short-term forecasting horizons. These results are also corroborated by those we obtain on the other 21 rates that run on the Olsen Information System (OIS).

What are the consequences of such results on the economic theory of market efficiency? We believe that they point to the extension and improvement of methods and tools for defining and analyzing market efficiency. The accepted theory was probably never conceived for such short horizons, and even more important, it takes an unrealistic view of market response to new information. Being developed only in a statistical framework, the theory assumes that economic actors integrate new price information instantaneously, and very little attention is paid to the time needed for a piece of information to be available to all market participants and to the diverse interpretation of that information. In the context of very short time horizons these factors play critical roles in market adjustments. It is reasonable