0.04

-0.03 J-1

Figure 13.2 Correlogram of 1-hour returns.

autocorrelation, as shown by their correlogram in Figure 13.2. If the price series were based on transacted prices then the observed negative autocorrelation in 1-hr returns could be due to a bid-ask bounce (the transaction price is either the bid or the ask). But since the data are filtered averages of bid and ask quotes, it is probably due to heterogeneous agents responding to market events in different ways and at different times.

The autocorrelation statistics for squared returns show an entirely different pattern. There is excessively high positive autocorrelation in the 1-hr squared return data. That is, ARCH effects are quite enormous in very high-frequency data. This observation has motivated much research into GARCH volatility models for very high-frequency data (§4.2.5).

13.1.4 Parametric Models of High-Frequency Data

High-frequency data provide a rich source of information on the micro-structure of financial markets, and excellent surveys of high-frequency data analysis in financial markets are found in Goodhart and OHara (1997) and Engle (2000a). There is a clear positive relationship between trading volume and volatility (Tauchen and Pitts, 1983; DeGennaro and Shrieves, 1995; Jones et al.. 1994). Figure 13.3 shows smoothed volume data on the USD-DEM spot exchange rate, and the corresponding GARCH volatility, for tic data that are bucketed into hourly intervals.

The positive relationship between volume and volatility has motivated the analysis of high-frequency data that are not sampled at regular intervals. An excellent overview of the regularly spaced and time transformation frameworks for analysing high-frequency data is given by Giot (2001). Time can be deformed by mapping calendar time to an operational time that is determined by the volume of transactions (Ghysels and Jasiek, 1995a, 1995b; Ghysels et al..

-Volatility - Volume] Figure 13.3 Volume and volatility.

The ACD model uses a parametric model for the time between trades that has much in common with the GARCH model; in the ACD model the expected duration between trades depends on past durations

1998) or the volatility (Zhou, 1996). The price quotes (or trades) may be sampled at irregularly spaced intervals such that the total volume of quotes (or trades) in each interval is constant. When the sampling intervals for high-frequency data are transformed using volume durations in this way, they become more amenable to standard modelling techniques (Ane and Geman, 2000).

An alternative approach to time transformation is to model the price durations, that is, the interval between trades, or between price changes that are larger than a given threshold. The autoregressive conditional duration (ACD) model was developed by Engle and Russell (1997) on high-frequency exchange rate data and by Engle and Russell (1998) on high-frequency stock market data. The ACD model uses a parametric model for the time between trades that has much in common with the GARCH model; in the ACD model the expected duration between trades depends on past durations. Since then a number of different ACD models have been developed, notably by Bauwens and Giot (1998) and Dufour and Engle (2000); the latter have applied the ACD model to the important area of liquidity risk by measuring the price impact of trades.

Parametric models have also been used to investigate the effect of news arrival and other public information on market activity. Almeida et al. (1997) show that the effect of macroeconomic news on the USD-DEM exchange rate is very short-lived, lasting little more than a few hours in most cases. Goodhart et al. (1991) use very high-frequency data on the GBP-USD exchange rate to investigate the effect of news announcements on GARCH volatility estimates. In line with their results, Low and Muthuswamy (1996) find strong evidence that news activity increases the volatility of returns and the volatility of the bid-ask spread. And the mechanisms by which news is carried around the world are investigated by Engle and Susmel (1994), who examine the volatility spillovers between international equity markets on an hourly basis.

When there is a common feature in financial assets there will be Granger causality; that is, there will be a lead-lag relationship between asset returns (§12.6). De Jong and Nijman (1997) find strong evidence for a lead-lag relationship between S&P 500 spot and futures returns when measured at high frequency. Low and Muthuswamy (1996) also find evidence of a lead-lag structure from the USD-DEM and USD-JPY to the DEM-JPY exchange rate.

13.2 Neural Networks

It is clear that dependencies between high-frequency returns are non-linear. There are significant linear dependencies between squared returns, but it is often more important to be able to predict the sign of a return than the magnitude of a return, so this finding is not very informative from the point of view of predictive modelling. Neural networks are comprehensive and powerful non-linear statistical models that have found applications in many disciplines, and there has been much research on the use of neural networks for modelling non-linear dependencies in financial markets (Refenes et al., 1996) - notably for exchange rates (Kuan and Liu, 1995; Nabney et al., 1996; Bolland et al, 1998). Neural networks have also been used to develop prediction algorithms for financial asset prices, such as the technical trading rules for stocks and commodities in Fishman et al. (1991), Shih (1991), Katz (1992), Kean (1992), Swales and Yoon (1992), Wong (1992), Baestaens et al. (1996), White (1988) and Min (1999).

This section gives a brief overview of the structure of basic neural networks and the propagation algorithms that are used to estimate them. More details may be found in many text books; Azoff (1994) in particular provides an accessible introduction to the subject and some examples of financial applications. Much commercial software is available for designing and implementing neural networks, and they can now be found in many statistical packages. Useful numerical recipes for neural networks can be found in Blum (1992) and Masters (1993).

Neural networks are universal approximators in the sense that they can fit any non-linear function with any degree of accuracy. In fact neural networks have much in common with linear models (for example, the method of parameter estimation is usually some form of maximum likelihood), and neural networks have applications to every situation where linear models are used. The key idea in any neural network is a non-linear twist, without which the network would collapse into a standard linear statistical model. Neural networks are designed to model arbitrary non-linear effects. It is not necessary to specify any underlying model for the process. On the other hand, a greater emphasis is placed on the preprocessing of the input data than is the case in standard linear models.

Neural networks are universal approximators in the sense that they can fit any non-linear function with an\ degree of accuracy