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.