the historical deal frequencies and the mean simulated deal frequencies remain large. Therefore, the examination of the GARCH(1,1) process with the maximum drawdown and the deal frequency indicates that the historical realizations of these two measures stay outside of the 5% level of simulated distributions of these two performance measures.

The mean simulated deal frequency for the USD-DEM is 2.39 trades per week. In annual terms, this is approximately 124.28 deals per year. The half spread for the USD-DEM series is about 0.00025 and this yields 3.11% when multiplied with the number of deals per year. The -3.11% return would be the annual transaction cost of the model. For the model to be profitable, it should yield more than 3.11% per year. Table 11.9 indicates that the RTT model generates an excess annual return of 9.63%, whereas the mean of the annualized return from the GARCH(1,1) process stays at the -3.27% level.

The multi-horizon examination of the equity curve with the Xeg and Reg performance measures indicates that the GARCH(1,1) process as a candidate for the data generation mechanism is strongly rejected at all horizons from a 7-day horizon to a horizon as long as 301 days. The overall picture coming out of the test is not very different for the GARCH( 1,1) than that of the random walk process.

AR(4)-GARCH(I, I) Process A further direction is to investigate whether a conditional mean dynamics with GARCH( 1,1) innovations would be a more successful characterization of the dynamics of the high-frequency foreign exchange returns. The conditional means of the foreign exchange returns are estimated with four lags of these returns. The additional lags did not lead to substantial increases in the likelihood value.

The results of the AR(4)-GARCH(1,1) optimization are presented in Table 11.10. The numbers in parentheses are the robust standard errors and all four lags are statistically significant at the 5% level. The negative autocorrelation is large and highly significant for the first lag of the returns. This is consistent with the high-frequency behavior of the foreign exchange returns and is also observed in Dacorogna et al. (1993). The Ljung-Box statistics still indicate serial correlation in the normalized residuals. The variances of the normalized residuals are near one. There is no evidence of skewness but the excess kurtosis remains large for the residuals.

The -values of the annualized returns are presented in Table 11.6. They are 0.1, 3.7,0.3, and 0.5% for the USD-DEM, USD-CHF, USD-FRF, and DEM-JPY. The results indicate that the AR(4)-GARCH(1,1) process is also rejected under the RTT model as a representative data generating process of foreign exchange returns. Here again, a possible explanation of this failure is the relationship between the dealing frequency of the model and the frequency of the simulated data. The AR(4)-GARCH(1,1) process is generated at the 5-min frequency but the model dealing frequency is between one or two deals per week. Therefore, the model picks up the high-frequency serial correlation as noise and this serial correlation works against the process. This cannot be treated as a failure of the RTT model.

TABLE 11.10 AR(4)-GARCH( 1,1) parameter estimates.

The sample is 5-min return from 1990-1996. ceo values are 10~9. The numbers in parentheses are the standard errors. The standard errors of ceo are 10~". LL is the average log likelihood value. g(12) refer to the Ljung-Box portmanteau test for serial correlation and it is distributed x2 with 12 degrees of freedom. The Xqo() s 21.03. ea2, esk ard *ku are the variance, skewness, and the excess kurtosis of the residuals.

Rather, this strong rejection is evidence of the failure of the temporal aggregation properties of the AR(4)-GARCH(1,1) process at lower frequencies.

The rejection of the AR(4)-GARCH(1,1) process with the Xeg and Reg is even stronger and very much in line with the results for the random walk and the GARCH(U). The p-values of the Xeff and Reff are 0.1, 0.0 percent for USD-DEM, 1.9, 2.3% for USD-CHF, 0.2,0.1 % for USD-FRF, and 0.1,0.1 % for DEM-JPY. The /j-values remain low at all horizons for the Xeg and Reg. The p-values of the maximum drawdown and the deal frequency also indicate that in almost all replications the AR(4)-GARCH(1,1) generates higher maximum drawdowns and deal frequencies.

Conclusions This extensive analysis of real-time trading models with high-frequency data suggests two main conclusions. First, technical trading models can generate excess returns, which are explained neither by traditional theoretical processes nor by luck. Second, the foreign exchange rates contain conditional mean dynamics that are neither present in the random walk nor GARCH( 1,1), and AR-GARCH( 1,1) processes.

The dealing frequency of the model is approximately between one and two per week although the data feed is at the 5-min frequency. Because the models trading frequency is less than two deals per week, it does not pick up the 5-min level heteroskedastic structure at the weekly frequency. Overall, the results presented in this section have a general message to the standard paradigm in econometrics. It is

TABLE 11.11 AR(4)-GARCH(I,I) simulations for USD-DEM.

not sufficient to develop sophisticated statistical processes and choose an arbitrary data frequency (e.g., 1 week, 1 month, annual) claiming afterward that this particular process does a "good job" of capturing the dynamics of the data-generating process. Tn financial markets, the data generating process is a complex network of layers where each layer corresponds to a particular frequency. A successful characterization of such data generating processes should be estimated with models whose parameters are functions of intra and mter-frequency dynamics. In other fields, such as in signal processing, paradigms of this sort are already in place. Our understanding of financial markets would be increased with the incorporation of such paradigms into financial econometrics. Our trading model, within this perspective, helps us to observe this subtle structure as a diagnostic tool.

11.7 trading model portfolios

In the previous sections we have described what trading models are and how we can optimize and test them. In this section we will briefly study the combination of different trading model strategies into portfolios and discuss the particular case of currency risk hedging.

Any trading strategy is based on some specific indicators and decision rules and then will perform better in some market conditions. To reduce the risk implied by the use of such trading models, it is common to combine various trading strategies, which provide different trading signals for the same asset, in a