Table 12.10: Continued Long Short

Table 12.11a: Monthly consolidated returns 1987, 1993 and 1998

(Over Risk Free Rate 5%)

Table 12.11b: Consolidated returns 1995-1999

The general methodology introduced by Engle and Kozicki (1993) defines features and common features (co-features) as follows: F is a feature if for all time series x and y:

>- x has F=> a + bx has F;

>- x does not have F and does not have F => x + does not have F; >- x has F and does not have F => x + has F.

Standard examples of such features in time series include a trend (stochastic or deterministic), autocorrelation and conditional heteroscedasticity. A feature F is a common feature (or co-feature) of time series xx, . . ., xn if each of xx, . . ., xn has the feature F but there are real numbers ax, . . ., an such that - axxx + ... + anxn does not have F. Such a linear combination is called a co-feature combination and (ax, . . ., a„) is called the co-feature vector.

To illustrate these definitions, consider two returns series {x,} and {y,} with the representations

xt = w, + ext, y, = Xw, + eyt,

where ex, and eyt are independent i.i.d. processes. If the feature is one of autocorrelation then w, has an AR representation, and if the feature is volatility then w, has an ARCH representation. In either case the linear combination x - (1 /X)y will not have the feature, so the co-feature vector is (1, -1 ).

Testing for a common feature is quite straightforward. It consists of two stages: first the existence of the feature in each time series is established, then a linear combination of the time series is searched for that does not have the feature. The rest of this section describes some empirical applications of the tests for common autocorrelation and common volatility in securities market indices and FX markets.

12.6.1 Common Autocorrelation

Before looking for common autocorrelation it is necessary to establish that the individual series do have the feature of autocorrelation. Not all market returns are autocorrelated; it depends not only on the frequency of the return, but also on the time period of measurement and, of course, on the market itself (§11.3.2). However, daily returns to equity indices often do exhibit some autocorrelation. A possible cause of autocorrelation in equity indices is the news arrival process, where new information affects trading in some stocks before others. When daily returns are autocorrelated this may be caused by news arriving in the market during the afternoon, which affects only those stocks which are traded late in the day. The prices of other stocks in the index will not be affected until they are traded on the next or subsequent days. Important international news is likely to affect the stock indices of different countries in the same way. Thus common autocorrelation is a possible cofeature in international equity markets.

A possible cause of autocorrelation in equity indices is the news arrival process, where new information affects trading in some stocks before others

A Box-Pierce test may be used to establish that there is significant autocorrelation in the individual returns series (§11.3.2). Then a grid search is employed to find a linear combinations of the time series that do not have autocorrelation. Of course, autocorrelation is not an all-or-nothing matter, so if - axxx + ... + anxn does not have autocorrelation, neither will y - a[xx + . . . + anxn for small perturbations a[ of ar The linear combination that is chosen will be the one that minimizes the Box-Pierce test statistic for autocorrelation.

The methodology of common autocorrelation testing is illustrated using daily closing prices on the Financial Times equity indices for Canada, France, Germany, Japan, the Netherlands, United Kingdom and United States from 19 June 1987 to 4 April 1993. Since the data period covers both the 1987 Black Monday crash and the 1989 mini-crash it is possible that autocorrelation co-features will be found that are caused by common market behaviour around these times. The effect of the October 1987 crash was evident in all the equity indices but much less so in France, which had a long bull market in equities for almost two years afterwards. From the middle of 1990 neither the Japanese nor the German market performed well. At the beginning of 1990 a sharp rise in Japanese interest rates in response to inflationary pressures preceded the bear market in equities, and the consequent slowdown in investment led to many other economic problems. On the other hand, in Germany the poor performance of equities was a result rather than a cause of economic problems: interest rates were raised in 1990 as a consequence of the effects of German unification, and German equity prices suffered. Therefore it may be expected that these two markets had less in common with the others during the period under study. Let us see.

All returns data were first converted to excess returns over the risk-free 30-day Treasury bill rate. Then AR(1) models were estimated on each excess