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2. The Predictability of Asset Returns

lwundfojVR(?)-l is -1. But then the smallest algebraic value that the lest statistic (VR(?)-])/yF can take on is:

VR(,/)

(2.5..))

Suppose that q is set at two-thirds of the sample size T so that 7/r/=. This impies that the normalized test statistic VR(q)/\fV can never be less than - 1.73; hence the test will never reject the null hypothesis al the 95% level of significance, regardless of the data! Of course, the tcststatislic can still reject the null hypothesis by drawing from the right tail, but against alternative hypdlheses that imply variance ratios less than one for large </-such as the permanent/transitory components model (2.5.1)-the variance ratio lest will ujave very little power when ij/ T is not close to zero.

more explicit illustration of the problems that arise when ( 7 is large may jbe obtained by performing an alternate asymptotic analysis, one in which 9 grows with 7so that q(T) approaches some limit S strictly between zero and one. In this case, under RWl Richardson and Stock (1989) show that tjhe unnormalized variance ratio VR(r/) converges in distribution to the following:

VR(,7)

(r)dz

Xi(r) = (r) - B(t-S) -SB(\),

(2.5.10)

(2.5.11)

where B() is standard Brownian motion defined on the unit interval (see Section 9.1 in Chapter 9). Unlike the standard fixed-fl asymptolics, in this case VR(<7) does not converge in probability to one. Instead, it converges in distribution to a random variable that is a functional of Brownian motion. The expected value of this limiting distribution in (2.5.10) is

(2.5.12)

In our earlier example where qjT = i, the alternative asymptotic approximation (2.5.10) implies that E[VR(r/)] converges to ц, considerably less than one despite the fact that RWl holds.

These biases are not unexpected in light of the daunting demands we arc placing on long-horizon returns-without more specific economic structure, it is extremely difficult lo infer much about phenomena that spans a significant portion of the entire dalaset. This problem is closely related to one in spectral analysis: estimating the spectral density function near frequency zero. Frequencies near zero correspond to extremely long periods.

2.6. Tests Tor Long-Range Dependence

and ii is notoriously difficult lo draw inferences about periodicities that exceed i! >an of the data.17 We shall see explicit evidence of such difficulties in the empirical results of Section 2.8. However, in some cases long-horizon returns can yield important insights, especially when other economic variables such as the dividend-price ratio come into play-see Section 7.2.1 in Chapter 7 for further discussion.

2.6 Tests For Long-Range Dependence

There is one departure from the random walk hypothesis that is outside the statistical framework we have developed so far, and that is the phenomenon of lung-range dependence. Long-range-dependent time series exhibit an unusually high degree of persistence-in a sense to be made precise below-so that observations in the remote past are nontrivially correlated with observations in the distant future, even as ihe lime span between the two observations increases. Natures predilection towards long-range dependence lus been well-documented in the natural sciences such as hydrology, meteorology, and geophysics, and some have argued that economic time series are also long-range dependent. In the frequency domain, such time series exhibit power at the lowest frequencies, and this was thought to be so commonplace a phenomenon that Granger (1966) dubbed it the typical spectral shape of an economic variable. Mandelbrot and Wallis (1968) used the more colorful term Joseph Effect, a reference to the passage in the Book of Genesis (Chapter 41) in which Joseph foretold the seven years of plenty followed by the seven years of famine that Egypt was to experience.18

2.6.1 Examples of Long-Range Dependence

A typical example of long-range dependence is given by the fractionally differenced time series models of Granger (1980), Granger andjoyeux (1980), and Hosking (1981), in which p, satisfies the following difference equation:

(l -i)dp, = c (, ~ im(o,of), (2.6.1)

where L is the lag operator, i.e., Lp, = p,-\. Granger andjoyeux (1980) and Hosking (1981) show that when the quantity (1 - L)d is extended to noninteger powers of d in the mathematically natural way, the result is a

175ee the discussion and analysis in Set lion 2.6 ior further details.

IMTlus biblical analogy is not completely frivolous, since long-iange dependence has been documented in various bydtological studies, not the least ol which was Hursts (1951) seminal study on measuring the long-term storage capacity ot reservoirs. Indeed, much of Hursts research was motivated by his empirical observations ol the Nile, the very same river that played so prominent a role in Josephs tirophei ics.

well-defined nine series thai is said to he fractionally differenced of order d (or, equivillnilly, fractionally integrated of order -d). Briefly, this involves expanding the expression (1-/,) via the binomial theorem for noninteger powers:

d\ d(d - \)(d - 2) (d - к + i)

J</ k\

and then applying the expansion to />,:

no / ,\ 00

(2.0.2)

(2.Г..З)

where the autoregressive roeflitienls are often re-expressed in terms of the gamma function:

* = i-D = :*. (2.0.4,

VM гн)Г(П1)

may also be viewed as an infinite-order MA process since

A-O-U-W*/. ,. Д. = 1 + £ / (2.(,5)

It is not obvious that such a definition of fractional differencing might yield a useful stochastic process, but Granger (1980), Granger andjoyeux (1980), and Hosking (1981) show that the characteristics of fractionally differenced time series are interesting indeed. For example, they show that />, is stationary and invertible for r/e(-i, 5) (see Hosking [1981 ]) and exhibits a unique kind of dependence that is positive or negative depending on whether d is positive or negative, i.e., the autocorrelation coefficients of/;, are of the same sign as ,/. So slowly do the autocorrelations decay that when d is positive their sum diverges lo infinity, and collapses to zero when d is negative.1

To develop a sense for long-range dependence, compare the autocorrelations of a fractionally differenced [h with those of a stationary AR(1) in Table 2.3. Although both the AR( 1) and the fractionally differenced (d=\)

Mandclbtot and others have tailed the r/<0ease anliprrsistrncr, reserving the term long-mugr (tr/friittrittr lot the ,/>(! case. However, since both cases involve auiotorrelatious that decay much more slnwlv than those o! more < onvenlinnal lime series, we call both long-range dependent.

 l-ag P,(k) [rf=i] 1=- Л [AK(1),0 = .5] 0.500 -0.250 0.500 0.400 -0.071 0.250 0.350 -0.030 0.125 0.318 -0.022 0.063 0.295 -0.015 0.031 0.235 -0.005 0.001 0.173 -0.001 2.98 x 10- 0.137 -3.24 x 10- 8.88 x lO 16 100 0.109 -1.02 x 10 4 7.89 x 10Г

Comparison of autocorrelation functions of fractionally differenced time series (1 ~L)dpt fc= *, forrf = i,-j.with lhatofan AR(\) p, = фр,-1+е ф - .5. The variance of t, was chjosen to yield a unit variance for p, in all three cases.

series have first-order autocorrelations of 0.500, at lag 25 the AR(1) correlation is 0.000 whereas the fractionally differenced series has correlation 0.173, declining only to 0.109 at lag 100. In fact, the defining characteristic of long-range dependent processes has been taken by many to be this slow decay of the autocovariance function.

More generally, long-range dependent processes [/>,) may be defined lo be those processes with a.vtocovariance functions yp(k) such that

( * /,( for v 6 (-1.0) or, yJk) ~ { as к ~* 00, (2.6.6)

}-fc7i(*> for v e (-2,-1)

where f(k) -is any slowly varying function at infinity.20 Alternatively, ling-range dependence has also been defined as processes with spectral density functions s(A) such that

s(X) ~ \-af2(k) as X -+ 0, a 6 (-1, 1), (2.6.7)

where / (*) is a slowly varying function. For example, the autocovariance

A function /(*) is said to be slowly varying at oo if lim oo /(*) (*) = 1 for all ( € [a. oo). The function log x is an example of a slowly varying function at infinity.

Y (k) =

o?rtt-2d)r(k+d) Г~(г/)Г(1-(,)Г(/< + 1 -d)

с, к

as Л

,(A) = (I - е~л)~(1 - izK)~of

a; 1С

as A -> 0,

(2.6.8)

(2.6.9)

Where d(z (-j, Depending on whether ri is negative or positive, the spectral density of (2.6.1) at frequency zero will either he zero or infinite.

I 2.6.2 The Hurst-Mandelbrot Reseated Range Statistic

The importance of long-range dependence in asset markets was first studied by Mandelbrot (1971), who proposed using the range over standard deviation, or R/S, statistic, also called the reseated range, to detect long-range dependence in economic lime series. The R/S statistic was originally developed by the English hydrologist Harold Edwin Hurst (1951) in his studies of river discharges. The R/S statistic is the range of partial sums of deviations of a time series from its mean, rescaled by its standard deviation. Specifically, consider a sample of continuously compounded asset returns (rt. Га,..., r ) and let r denote the sample mean A r}. Then the classical rescaled-range statistic, which we shall call Q , is given by

(2.6.10)

- - >=t

where s is the usual (maximum likelihood) standard deviation estimator,

Г i 11/2

(2.6.11)

The first term in brackets in (2.6.10) is the maximum (over k) of the partial sums of the first к deviations of rt from the sample mean. Since the sum of all n deviations of tjs from their mean is zero, this maximum is always nonnegative. The second term in (2.6.10) is the minimum (over k) of this same sequence of partial sums, and hence it is always nonpositive. The difference of the two quantities, called the range for obvious reasons, is always nonnegative and hence Qj,>0.n

v,The behavior of may tie belter understood by considering its origins in hydrotogi< al studies of reservoir design. To accommodate seasonalities in riverflow, a rcseivoirs < ap.u itv

In several seminal papers Mandelbrot, Taqqu, and Wallis demonstrate the superiority of R/S analysis to more conventional methods of determining long-range dependence, such as analyzing autocorrelations, variance ratios, and spectral decompositions. For example, Mandelbrot and Wallis (1969b) show by Monte Carlo simulation that the R/S statistic can detect long-range dependence in highly non-Gaussian time series with large skewness and/or kuriosis. In fact, Mandelbrot (1972, 1975) reports the almost-sure convergence of the R/S statistic for stochastic processes with infinite variances, a distinct advantage over autocorrelations and variance ratios which need not be well-defined for infinite variance processes. Further aspects of the R/S statistics robustness are developed in Mandelbrot and Taqqu (1979). Mandelbrot (1972) also argues that, unlike spectral analysis which detects periodic cycles, R/S analysis can detect nonperiodic cycles, cycles with periods equal to or greater than the sample period.

Although these claims may all be contested to some degree, it is a well-established fact that long-range dependence can indeed be detected by the classical R/S statistic. However, perhaps the most important shortcoming of the rescaled range is its sensitivity lo short-range dependence, implying that any incompatibility between the data and the predicted behavior of the R/S statistic under the null hypothesis need not come from long-range dependence, but may merely be a symptom of short-tci in memory.

In particular l.o (1991) shows that under RWI the asymptotic distribution of (1/,/n) О/, is given by the random variable V, the range of a Brownian bridge, but under a stationary AR(1) specification with autoregressive coefficient ф the normalized R/S statistic converges lo £V where £ = %/(l+0)/(l- </ ) For weekly returns of some portfolios of common stock, ф is as large as 50%, implying that the mean of Q /\J7i may be biased up-

must be chosen to allow for fluctuations in the supply of water above the dam while still maintaining a relatively constant flow of water below the dam. Since dam construction costs are immense, the importance of estimating the reservoir capacity necessary to meet long-term storage needs is apparent. The range is an estimate of this quantity. If Xy is the riverflow (per unit time) above the dam and X is the desired riverflow below the dam, the bracketed quantity in (2.(5.10) is the capacity of the reservoir needed to ensure this smooth flow given the pattern of Hows in periods 1 through n. For example, suppose annual riverllows are assumed to be 100, 50, 100, and 50 in years I through 4. II a constant annual How of 75 below the dam is desired each year, a reservoir must have a minimum total capacity ol 25 since it must store 2ft units in years 1 and 3 U> provide for the relatively dry years 2 and 4. Now suppose instead that the natural pattern of riverflow is 100, 100, 50, 50 in years I through 4. To ensure a flowol 75 below the dam in this case, the minimum capacity must increase to 50 so as to accommodate the excess storage needed in years 1 ami 2 to supply water during the dry spell in years 3 and 4. Seen in this context, it is clear that an increase in persistence will increase the required storage capacity as measured by the range. Indeed, it was the apparent persistence of dry spells in Kgypl that sparked Hursts lifelong fascination with the Nile, leading eventually to his interest in die rescaled range.

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