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132

Univariate Common autocorrelation tests*

auto- (parentheses indicate the X that gives the

correlation minimum autocorrelation)

Q Germany Japan Netherlands UK US

Canada 8.06 0.87 (1.6) 0.50 (0.7) 0.00003 (1.9) 0.04 (1.0) 0.13 (1.3)

Germany 3.60 1.30 (0.7) 2.05 (1.8) 1.41 (1.0) 0.63 (0.8)

Japan 5.89 0.97 (0.7) 0.70 (0.5) 0.17 (1.1)

Netherlands 12.58 0.02 (0.9) 1.05 (0.6)

UK 20.29 1.05 (0.6)

US 6.48

France 1.10

*Sincc France shows no evidence of univariate autocorrelation, it is not included in these common autocorrelation tests.

return series, and the Box-Pierce -statistic from each of these is reported in the first column of Table 12.12. Since the 5% and 1% critical values of X2 with 1 degree of freedom are 3.84 and 6.63 respectively, it is clear that Canada, Japan, the Netherlands, the UK and the US all have significant autocorrelation.

There is strong common autocorrelation in international equity indices. This is probably due to the cross-listing of major stocks and the autocorrelated news arrival process

The columns on the right of Table 12.12 report the minimum value of the autocorrelation test statistic that is obtained in an AR(1) model on the linear combination (row - X x column). The X that gives this minimum value is reported in parentheses. All the test statistics fall well below the 10% value for X2 with 1 degree of freedom, which is 2.71. So the results indicate that there is strong common autocorrelation in international equity indices. This is probably due to the cross-listing of major stocks and the autocorrelated news arrival process.

12.6.2 Common Volatility

The existence of common volatility has important implications for derivatives trading. Even the most straightforward of volatility trades, an at-the-money straddle, is not necessarily easy to effect. Near to ATM options may be expensive, or hard to find, but derivatives on assets which share a common volatility factor may be substituted in an ATM straddle or other volatility trade.

Unlike autocorrelation, the existence of ARCH volatility features in asset returns cannot be questioned (§4.1). Volatility clusters are a well-documented feature of almost all financial markets, and the feature becomes particularly pronounced as the frequency of the returns increases. But although individual ARCH volatilities may be easy to determine, common ARCH volatility patterns seem more difficult to detect.

Table 12.12: Autocorrelation and common autocorrelation tests for equity indices



Tests for common volatility are just like common autocorrelation tests, but on squared returns. First the individual squared returns are tested for autocorrelation, and this may be done using a Lagrange multiplier test, such as the TR2 from an AR(j?) model or the Box-Pierce -statistic (§11.3.2). Then, assuming ARCH features are present in two return series, some linear combination of these series is sought that has no ARCH effects. The linear combination is found by a grid search, and the test for no ARCH is an autocorrelation test on its squares.

Common volatility patterns have been found between international equity markets (Engle and Susmel, 1993). There is less evidence of common volatility between international bond markets (Alexander, 1994).13 In an investigation of common volatility in FX markets, Alexander (1995) finds that good results are only obtained with weekly data. It was also found that exchange rate variability is not dominated by regional factors, in fact no common volatility patterns were found in the DEM and NLG dollar rates. The only common volatility patterns detected were in the GBP-USD and JPY-USD exchange rates; possibly these were caused by speculative dollar-dominated exchange flows.

13Daily data appeared to be too noisy for common volatility patterns to be detected.



-13-

Forecasting High-Frequency Data

There is much evidence to suggest that market prices are not completely random. Hseih (1989, 1991), Brock et al. (1990), LeBaron (1995), Abhyankar et al. (1997), de Lima (1998), Drunat et al. (1998) and Zhou (1998) have all tested for dependencies in financial returns, in many markets and with different frequencies. These results indicate that there are non-linear dependencies in financial returns, particularly in very high-frequency data. Lo and MacKinley (1988) and Goodhart and Figliouli (1991) also provide strong evidence that some stock prices do not follow a random walk when measured at the intra-day level.

These findings contradict the efficient markets hypothesis. The implication is that it is possible to predict market prices at high frequency, with some degree of accuracy, but because of the non-linear nature of the dependencies that have been demonstrated in high-frequency data, linear statistical models are not appropriate for high-frequency price prediction. Nevertheless there is a common perception that it is possible to develop time series models that forecast high-frequency data successfully, and a major focus of recent research has been the development of non-linear models for financial market behaviour.

It is possible to predict market prices at high frequency but because of the non-linear nature of the dependencies linear statistical models are not appropriate

Given the huge literature on forecasting high-frequency data, this chapter has been extremely selective in its coverage. The first section begins with an overview of data sources and the recent initiatives in forecasting high-frequency financial data. It describes a basic technique for high-frequency data filtering and examines the time series properties that one might expect from tic data on financial market prices. This section ends with a brief overview of some of the parametric models of high-frequency data that have been used to investigate market microstructure.

The two sections that follow cover two non-linear models that have been found useful for high-frequency financial data prediction. Section 13.2 explains the concepts that underpin the design and estimation of neural networks and surveys their applications to forecasting high-frequency financial data. My understanding of neural networks owes much to many discussions with Dr Peter Williams of Sussex University, an acknowledged expert in this field.



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