Model-free evaluation of directional predictability in foreign exchange markets, страница 32

[11] A simulation study in Hong and Chung (2006) shows that the proposed generalized cross-spectral test has reasonable size with good power against directional predictability under various plausible linear and nonlinear data-generating processes. Furthermore, in an empirical study, Hong and Lee (2003) find that the changes of most major foreign exchange rates are serially uncorrelated, but the generalized spectral tests significantly reject the null hypothesis of martingale difference sequences, revealing the advantages of a generalized spectral approach over traditional linear models or measures.

[12] See Bierens (1982) and Stinchcombe and White (1998) for more discussion in a related but different context. ¹⁴ For a modeling exercise of directional forecasts, we refer to Hong and Chung (2006), in which a class of autologistic models is considered for an out-of-sample test. In addition, a moving average technical trading rule is used in Hong and Lee (2003), who find the nonlinearity in conditional mean by applying the generalized spectral tests of Hong (1999). While their research interests are primarily in examining the predictability of exchange rate changes in mean, they also conduct forecasts on the direction of changes as an integral part of forecasting exchange rate changes.

[13] For the choice of p, Hong (1999, Theorem 2.2) proposes a data-driven method which minimizes an asymptotic integrated mean squared error criterion for the generalized spectral density estimator. It still involves the choice of a preliminary ‘pilot’ lag order p, but the impact of choosing p is much smaller.

[14] For notational simplicity, we use the same notation Z for the direction indicator of joint changes in this section.

[15] Note that our definition of returns may be nominal (rather than actual) values to study solely the direction of the changes in the underlying spot (or futures) price: relative rates of return on US dolloars comprise both (nominal) price changes and interest rate differentials.

[16] For the Canadian dolloar, we use the 3-month Treasury Bill rates.

[17] These stylized facts differ from those of the daily returns in stock markets. The returns in stock markets commonly exhibit leptokurtosis, fat tails and negative skewness (see, for example, Fama, 1965). For more discussion on the stylized facts and statistical properties of the daily returns from foreign exchange markets, we refer to Hsieh (1988) and de Vries (1994).

[18] In this study, we do not consider c higher than one; the sample frequency that price changes are higher than one sample standard deviation of fY_tg is relatively low, and so this may reduce the statistical power of the test. The price limits in the futures market make the use of higher threshold values even more undesirable. Brennan (1986) and Kodres (1993), using a sign test, point out that price limits are more likely clustered in the same direction. Therefore, it may lead to spurious findings when we consider the stochastic behavior of directional movements of significantly higher changes.

[19] McQueen et al. (1996) find evidence of different autocorrelations in returns between up and down stock markets.

[20] For robust results, we also use preliminary lag orders p from 11 to 51. The results are very similar, and for reasons of space we only report the results with p D 21.

[21] One may be also interested in the joint changes between spot and futures markets. ²⁷ All (unreported) results are available upon request from the authors.

[22] For simplicity, subscript k is ignored in threshold c.

[23] We recall that, as shown in Table II, the negative directions of AD, CD, BP and JY in the spot and futures markets (FAD, FCD, FBP and FJY) are significantly predictable using their past own returns.

[24] This argument is also valid for the case of positive changes.

[25] This finding is not generally documented in the futures market.

[26] In general, the statistical significances of these GCS tests are modestly weaker for the futures market.

[27] We obtain similar results for the directions of joint positive changes, which are available upon request.