15.9 Practical Considerations

Previous studies of long-memory and fractional integration in time series are numerous. Barkoulas, Baum, and Oguz (1999a), Barkoulas, Baum, and Oguz (1999b) studied the long run dynamics of long term interest rates and currencies. Recent studies of stock prices include Cheung and Lai (1995), Lee and Robinson (1996), Andersson and Nydahl (1998). Batten, Ellis, and Hogan (1999) worked with credit spreads of bonds. Wilson and Okunev (1999) searched for long term co-dependence between stock and property markets. While the results on the level of returns are mixed, but there is general consensus that the unconditional distribution is non-normal and there is long-memory in squared and absolute returns. The followings are some issues. Though not mutually exclusive, they are separated by headings for easier discussions:


15.9.1 Risk and Volatility

Standard deviation is a statistical measure of variability and it has been called the measure of investment risk in the finance literature. Balzer (1995) argues that standard deviation is a measure of uncertainty and it is only a candidate, among many others, for a risk measure. Markowitz (1959) and Murtagh (1995) both found that portfolio selection based on semi-variance tend to produce better performance than those based on variance.

A normal distribution is completely characterised by its first two statistical moments, namely, the mean and standard deviation. However, once nonlinearity is introduced, investment returns distribution is likely to become markedly skewed away from a normal distribution. In such cases, higher order moments such as skewness and kurtosis are required to specify the distribution. Standard deviation, in such a context, is far less meaningful measure of investment risk and not likely to be a good proxy for risk. While recent developments are interested in the conditional volatility and long memory in squared and absolute returns, most practitioners continue to think in terms of unconditional variance and continue to work with unconditional Gaussian distribution in financial applications. Recent publications are drawing attention to the issue of distribution characteristics of market returns, especially in emerging markets , which cannot be summarized by a normal distribution (Bekaert et al.; 1998).


15.9.2 Estimating and Forecasting of Asset Prices

Earlier perception was that deseasonalised time series could be viewed as consisting of two components, namely, a stationary component and a non-stationary component. It is perhaps more appropriate to think of the series consisting of both a long and a short memory components. A semiparametric estimate d can be the first step in building a parametric time series model as there is no restriction of the spectral density away from the origin. Fractional ARIMA, or ARFIMA, can be use for forecasting although the debates on the relative merits of using this class of models are still inconclusive (Hauser, Pötscher, and Reschenhofer; 1999), (Andersson; 1998). Lower risk bounds and properties of confidence sets of so called ill-posed problems associated with long-memory parameters are also discussed in Potscher (1999). The paper casts doubts on the used on statistical tests in some semiparametric models on the ground that a priori assumptions regarding the set of feasible data generating processes have to be imposed to achieve uniform convergence of the estimator.


15.9.3 Portfolio Allocation Strategy

The results of Porterba and Summers (1988) and Fama and French (1988) provided the evidence that stock prices are not truly random walk. Based on these observations, Samuelson (1992) has deduced on some rational basis that it is appropriate to have more equity exposure with long investment horizon than short horizon. Optimal portfolio choice under processes other than white noise can also suggest lightening up on stocks when they have risen above trend and loading up when they have fallen behind trend. This coincides with the conventional wisdom that long-horizon investors can tolerate more risk and thereby garner higher mean returns. As one grows older, one should have less holding of equities and more assets with lower variance than equities. This argues for ``market timing'' asset allocation policy and against the use of ``strategic'' policy by buying and holding as implied by the random walk model.

Then there is the secondary issue of short-term versus long-horizon tactical asset allocation. Persistence or a more stable market calls for buying and holding after market dips. This would likely to be a mid to long-horizon strategy in a market trending upwards. In a market that exhibits antipersistence, asset prices tend to reverse its trend in the short term thus creating short-term trading opportunities. It is unclear, taking transaction costs into account, whether trading the assets would yield higher risk adjusted returns. This is an area of research that may be of interest to practitioners.


15.9.4 Diversification and Fractional Cointegration

If assets are not close substitutes for each other, one can reduce risk by holding such substitutable assets in the portfolio. However, if the assets exhibit long-term relationship (e.g., to be co-integrated over the long-term), then there may be little gain in terms of risk reduction by holding such assets jointly in the portfolio. The finding of fractional cointegration implies the existence of long-term co-dependence, thus reducing the attractiveness of diversification strategy as a risk reduction technique. Furthermore, portfolio diversification decisions in the case of strategic asset allocation may become extremely sensitive to the investment horizon if long-memory is present. As Cheung and Lai (1995) and Wilson and Okunev (1999) have noted, there may be diversification benefits in the short and medium term, but not if the assets are held together over the long term if long-memory is presence.


15.9.5 MMAR and FIGARCH

The recently developed MMAR (multifractal model of asset returns) of Mandelbrot, Fisher and Calvet (1997) and FIGARCH process of Baillie, Bollerslev, and Mikkelsen (1996) incorporate long-memory and thick-tailed unconditional distribution. These models account for most observed empirical characteristics of financial time series, which show up as long tails relative to the Gaussian distribution and long-memory in the volatility (absolute return). The MMAR also incorporates scale-consistency, in the sense that a well-defined scaling rule relates return over different sampling intervals.