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Price of a вЂ˜standard houseвЂ™ from 1980 to 2000

0.80

0.60

0.40

0.20

0.00

-0.20

1980:1 1985:1 1990:1 1995:1 1999:4

Figure 13.3. Smoothed common price component. Conп¬Ѓdence intervals
are calculated for the 90% level.
XFGsssm6.xpl

[1,] "==========================================="
[2,] " Estimated hedonic coefficients "
[3,] "==========================================="
[4,] " Variable coeff. t-Stat. p-value "
[5,] " ----------------------------------------- "
[6,] " log lot size 0.2664 21.59 0.0000 "
[7,] " log floor area 0.4690 34.33 0.0000 "
[8,] " age -0.0061 -29.43 0.0000 "
[9,] "==========================================="

Table 13.4. Estimated hedonic coeп¬ѓcients ОІ. XFGsssm6.xpl

zero. Thus, it is not surprising that the Kalman smoother produces constant
estimates through time for these coeп¬ѓcients. In the Appendix 13.6.2 we give
a formal proof of this intuitive result.
302 13 A simple state space model of house prices

The estimated coeп¬ѓcient of log lot size implies that, as expected, the size of the
lot has an positive inп¬‚uence on the price. The estimated relative price increase
for an one percent increase in the lot size is about 0.27%. The estimated eп¬Ђect
of an increase in the п¬‚oor space is even larger. Here, a one percent increase in
the п¬‚oor space lets the price soar by about 0.48%. Finally, note that the price
of a houses is estimated to decrease with age.

13.6 Appendix

13.6.1 Procedure equivalence

We show that our treatment of missing values delivers the same results as the
procedure proposed by Shumway and Stoп¬Ђer (1982; 2000). For this task, let us
assume that the (N Г— 1) vector of observations t

yt = y1,t . y3,t . y5,t ... yN,t

has missing values. Here, observations 2 and 4 are missing. Thus, we have only
Nt < N observations. For Kalman п¬Ѓltering in XploRe, all missing values in yt
and the corresponding rows and columns in the measurement matrices dt , Zt ,
and Ht , are deleted. Thus, the adjusted vector of observations is

yt,1 = y1,t y3,t y5,t ... yN,t

where the subscript 1 indicates that this is the vector of observations used in the
XploRe routines. The procedure of Shumway and Stoп¬Ђer instead rearranges the
vectors in such a way that the п¬Ѓrst Nt entries are the observationsвЂ”and thus
given by yt,1 вЂ”and the last (N в€’ Nt ) entries are the missing values. However,
all missing values must be replaced with zeros.
For our proof, we use the following generalized formulation of the measurement
equation
Оµm
yt,1 d Z
= t,1 + t,1 О±t + t,1
Оµm
yt,2 dt,2 Zt,2 t,2

and
Оµm Ht,11 Ht,12
cov t,1 = .
Оµm Ht,12 Ht,22
t,2

yt,1 contains the observations and yt,2 the missing values. The procedure of
Shumway and Stoп¬Ђer employs the generalized formulation given above and sets
13.6 Appendix 303

yt,2 = 0, dt,2 = 0, Zt,2 = 0, and Ht,12 = 0 (Shumway and Stoп¬Ђer, 2000, p. 330).
We should remark that the dimensions of these matrices also depend on t via
(N в€’Nt ). However, keep notation simple we do not make this time dependency
explicit. It is important to mention that matrices with subscript 1 and 11 are
equivalent to the adjusted matrices of XploReвЂ™s п¬Ѓltering routines.
First, we show by induction that both procedures deliver the same results for
the Kalman п¬Ѓlter. Once this equivalence is established, we can conclude that
the smoother also delivers identical results.
PROOF:
Given Вµ and ОЈ, the terms a1|0 and P1|0 are the same for both procedures. This
follows from the simple fact that the п¬Ѓrst two steps of the Kalman п¬Ѓlter do not
depend on the vector of observations (see Subsection 13.5.3).
Now, given at|tв€’1 and Pt|tв€’1 , we have to show that also the п¬Ѓlter recursions

at = at|tв€’1 + Pt|tв€’1 Zt Ftв€’1 vt , Pt = Pt|tв€’1 в€’ Pt|tв€’1 Zt Ftв€’1 Zt Pt|tв€’1 (13.13)

deliver the same results. Using ss to label the results of the Shumway and
Stoп¬Ђer procedure, we obtain by using

Zt,1
def
Zt,ss =
0

that
Zt,1 Pt|tв€’1 Zt,1 0 Ht,11 0
Ft,ss = + .
0 Ht,22
0 0
The inverse is given by (SydsГ¦ter, StrГёm and Berck, 2000, 19.49)
в€’1
Ft,1 0
в€’1
Ft,ss = (13.14)
в€’1
0 Ht,22

where Ft,1 is just the covariance matrix of the innovations of XploReвЂ™s proce-
dure. With (13.14) we obtain that
в€’1
в€’1
Zt,ss Ft,ss = Zt,1 Ft,1 0

and accordingly for the innovations

vt,1
vt,ss = .
0
304 13 A simple state space model of house prices

We obtain immediately
в€’1 в€’1
Zt,ss Ft,ss vt,ss = Zt,1 Ft,1 vt,1 .

Plugging this expression into (13.13)вЂ”taking into account that at|tв€’1 and
Pt|tв€’1 are identicalвЂ”delivers

at,ss = at,1 and Pt,ss = Pt,1 .

This completes the п¬Ѓrst part of our proof.
The Kalman smoother recursions use only system matrices that are the same
for both procedures. In addition to the system matrices, the output of the
п¬Ѓlter is used as an input, see Subsection 13.5.5. But we have already shown
that the п¬Ѓlter output is identical. Thus the results of the smoother are the
same for both procedures as well.

13.6.2 Smoothed constant state variables

We want to show that the Kalman smoother produces constant estimates
through time for all state variables that are constant by deп¬Ѓnition. To proof
this result, we use some of the smoother recursions given in Subsection 13.5.5.
First of all, we rearrange the state vector such that the last k K variables
are constant. This allows the following partition of the transition matrix

T11,t+1 T12,t+1
Tt+1 = (13.15)
0 I

with the kГ—k identity matrix I. Furthermore, we deп¬Ѓne with the same partition
Лњ Лњ
P P12,t
Лњ def
Pt = Tt+1 Pt Tt+1 = Лњ11,t Лњ
P12,t P22,t

The п¬Ѓlter recursion for the covariance matrix are given as

Pt+1|t = Tt+1 Pt Tt+1 + Rt+1

where the upper left part of Rt+1 contains the covariance matrix of the dis-
turbances for the stochastic state variables. We see immediately that only the
Лњ
upper left part of Pt+1|T is diп¬Ђerent from Pt .
13.6 Appendix 305

Our goal is to show that for the recursions of the smoother holds

M11,t M12,t
Ptв€— = , (13.16)
0 I

where both M s stand for some complicated matrices. With this result at hand,
we obtain immediately
ak = ak k
t+1|T = aT (13.17)
t|T

for all t, where ak contains the last k elements of the smoothed state at|T .
t|T

Furthermore, it is possible to show with the same result that the lower right
partition of Pt|T is equal to the lower right partition of PT for all t. This lower
right partition is just the covariance matrix of ak . Just write the smoother
t|T
recursion
Pt|T = Pt (I в€’ Tt+1 Ptв€— ) + Ptв€— Pt+1|T Ptв€— .
Then check with (13.15) and (13.16) that the lower-right partition of the п¬Ѓrst
matrix on the right hand side is a kГ—k matrix of zeros. The lower-right partition
of the second matrix is given by the the lower-right partition of Pt+1|T .
PROOF:
Now we derive (13.16): We assume that the inverse of Tt+1 and T11,t+1 exist.
The inverses for our model exist because we assume that П†2 = 0. For the
partitioned transition matrix (SydsГ¦ter, StrГёm and Berck, 2000, 19.48) we
derive
в€’1 в€’1
T11,t+1 в€’T11,t+1 T12,t+1
в€’1
Tt+1 = . (13.18)
0 I
Now, it is easy to see that
в€’1 Лњ в€’1
Ptв€— = Tt+1 Pt Pt+1|t . (13.19)

We have (SydsГ¦ter, StrГёm and Berck, 2000, 19.49)

Лњ в€’1
Лњ
в€’в€†t P12,t P22,t
в€†t
в€’1
Pt+1|t = (13.20)
в€’1 Лњ Лњ в€’1 + P в€’1 P12,t в€†t P12,t P в€’1
Лњ Лњ Лњ Лњ Лњ
в€’P22,t P12,t в€†t P22,t 22,t 22,t

with в€†t as a known function of the partial matrices. If we multiply this matrix
Лњ
with the lower partition of Pt we obtain immediately [0 I]. With this result
and (13.18) we derive (13.16).
306 13 A simple state space model of house prices

Bibliography
Bailey, M. J., Muth, R. F. and Nourse, H.O. (1963). A regression method for
real estate price index construction, Journal of the American Statistical
Association 58: 933вЂ“942.

Baumol, W. (1959). Economic Dynamics, 2nd ed., Macmillan, New York.
Bera, A. K. and Jarque, C. M. (1982). Model Speciп¬Ѓcation Tests: a Simulta-
neous Approach, Journal of Econometrics 20: 59вЂ“82.
Cho, M. (1996). House price dynamics: a survey of theoretical and empirical
issues, Journal of Housing Research 7:2: 145вЂ“172.
Clapp, J. M. and Giaccotto, C. (1998). Price indices based on the hedonic
repeat-sales method: application to the housing market, Journal of Real
Estate Finance and Economics 16:1: 5вЂ“26.
Durbin, J. and Koopman, J. S. (2001). Time Series Analysis by State Space
Methods, Oxford University Press, Oxford.
Engle, R. F. and M. W. Watson (1981). A One-Factor Multivariate Time Series
Model of Metropolitan Wage Rates, Journal of the American Statistical
Association 76: 774вЂ“781.
Gourieroux, C. and Monfort, A. (1997). Time Series and Dynamic Models,
Cambridge University Press, Cambridge.
Greene, W. H. (2000). Econometric Analysis. Fourth Edition, Prentice Hall,
Hamilton, J. D. (1994). Time Series Analysis, Princeton University Press,
Princeton, New Jersey.

Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the
Kalman Filter, Cambridge University Press, Cambridge.
Harvey, A. C. (1993). Time Series Models, 2. edn, Harvester Wheatsheaf, New
York.
Hill, R. C., Knight, J. R. and Sirmans, C. F. (1997). Estimating Capital Asset
Price Indexes, Review of Economics and Statistics 79: 226вЂ“233.
13.6 Appendix 307

Koopman, S. J., Shepard, N. and Doornik, J. A. (1999). Statistical Algorithms
for Models in State Space Using SsfPack 2.2, Econometrics Journal 2: 107вЂ“
160.
PeЛњa, D., Tiao, G. C. and Tsay, R. S. (2001). A Course in Time Series Analysis,
n
Wiley, New York.
Schwann, G. M. (1998). A real estate price index for thin markets, Journal of
Real Estate Finance and Economics 16:3: 269вЂ“287.
Shiller, R. J. (1993). Macro Markets. Creating Institutions for Managing Soci-
etyвЂ™s Largest Economic Risks, Clarendon Press, Oxford.

Shumway, R. H. and Stoп¬Ђer, D. S. (1982). An approach to time series smoothing
and forecasting using the EM algorithm, Journal of Time Series Analysis
3: 253вЂ“264.
Shumway, R. H. and Stoп¬Ђer, D. S. (2000). Time Series Analysis and Its Ap-
plications, Springer, New York, Berlin.
SydsГ¦ter, K., StrГёm, A. and Berck, P. (2000). EconomistsвЂ™ Mathematical Man-
ual, 3. edn, Springer, New York, Berlin.
Strategy
Oliver Jim Blaskowitz and Peter Schmidt

14.1 Introduction
Long range dependence is widespread in nature and has been extensively doc-
umented in economics and п¬Ѓnance, as well as in hydrology, meteorology, and
geophysics by authors such as Heyman, Tabatabai and Lakshman (1991), Hurst
(1951), Jones and Briп¬Ђa (1992), Leland, Taqqu, Willinger and Wilson (1993)
and Peters (1994). It has a long history in economics and п¬Ѓnance, and has
remained a topic of active research in the study of п¬Ѓnancial time series, Beran
(1994).
Historical records of п¬Ѓnancial data typically exhibit distinct nonperiodical cycli-
cal patterns that are indicative of the presence of signiп¬Ѓcant power at low fre-
quencies (i.e. long range dependencies). However, the statistical investigations
that have been performed to test for the presence of long range dependence in
economic time series representing returns of common stocks have often become
sources of major controversies. Asset returns exhibiting long range dependen-
cies are inconsistent with the eп¬ѓcient market hypothesis, and cause havoc on
stochastic analysis techniques that have formed the basis of a broad part of
modern п¬Ѓnance theory and its applications, Lo (1991). In this chapter, we
examine the methods used in Hurst analysis, present a process exhibiting long
memory features, and give market evidence by applying HurstвЂ™s R/S analysis
and п¬Ѓnally sketch a trading strategy for German voting and nonвЂ“voting stocks.
310 14 Long Memory Eп¬Ђects Trading Strategy

14.2 Hurst and Rescaled Range Analysis
Hurst (1900вЂ“1978) was an English hydrologist, who worked in the early 20th
century on the Nile River Dam project. When designing a dam, the yearly
changes in water level are of particular concern in order to adapt the damвЂ™s
storage capacity according to the natural environment. Studying an Egyptian
847вЂ“year record of the Nile RiverвЂ™s overп¬‚ows, Hurst observed that п¬‚ood occur-
rences could be characterized as persistent, i.e. heavier п¬‚oods were accompanied
by above average п¬‚ood occurrences, while below average occurrences were fol-
lowed by minor п¬‚oods. In the process of this п¬Ѓndings he developed the Rescaled
Range (R/S) Analysis.
We observe a stochastic process Yt at time points t в€€ I = {0, . . . , N }. Let n
be an integer that is small relative to N , and let A denote the integer part of
N/n. Divide the вЂ˜intervalвЂ™ I into A consecutive вЂ˜subintervalsвЂ™, each of length
n and with overlapping endpoints. In every subinterval correct the original
datum Yt for location, using the mean slope of the process in the subinterval,
obtaining Yt в€’ (t/n) (Yan в€’ Y(aв€’1)n ) for all t with (a в€’ 1)n в‰¤ t в‰¤ an and for all
a = 1, . . . , A. Over the aвЂ™th subinterval Ia = {(a в€’ 1)n, (a в€’ 1)n + 1, . . . , an},
for 1 в‰¤ a в‰¤ A, construct the smallest box (with sides parallel to the coordinate
axes) such that the box contains all the п¬‚uctuations of Yt в€’(t/n) (Yan в€’Y(aв€’1)n )
that occur within Ia . Then, the height of the box equals
t
Yt в€’ (Yan в€’ Y(aв€’1)n )
Ra = max
n
(aв€’1)nв‰¤tв‰¤an

t
в€’ Yt в€’ (Yan в€’ Y(aв€’1)n )
min
n
(aв€’1)nв‰¤tв‰¤an

Figure 14.1 illustrates the procedure. Let Sa denote the empirical standard
error of the n variables Yt в€’ Ytв€’1 , for (a в€’ 1)n + 1 в‰¤ t в‰¤ an. If the process
Y is stationary then Sa varies little with a; in other cases, dividing Ra by
Sa corrects for the main eп¬Ђects of scale inhomogeneity in both spatial and
temporal domains.
The total area of the boxes, corrected for scale, is proportional in n to
A
R Ra
в€’1
:= A . (2.1)
S Sa
n
a=1

Л†
The slope H of the regression of log(R/S)n on log n, for k values of n, may be
taken as an estimator of the Hurst constant H describing long-range depen-
dence of the process Y , Beran (1994) and Peters (1994).
14.2 Hurst and Rescaled Range Analysis 311

X(t)-(t/n){X(an)-X((a-1)n)}
8
7.5
7

0 500 1000 1500 2000 2500
time t

Figure 14.1. The construction of the boxes in the R/S analysis.

If the process Y is stationary then correction for scale is not strictly necessary,
and we may take each Sa to be the constant 1. In that case the RвЂ“S statistic
Л†
H is a version of the box-counting estimator that is widely used in physical
science applications, Carter, Cawley and Mauldin (1988), Sullivan and Hunt
(1988) and Hunt (1990). The box-counting estimator is related to the capacity
deп¬Ѓnition of fractal dimension, Barnsley (1988) p. 172п¬Ђ, and the RвЂ“S estimator
may be interpreted in the same way. Statistical properties of the box-counting
estimator have been discussed by Hall and Wood (1993).
A more detailed analysis, exploiting dependence among the errors in the regres-
sion of log(R/S)n on log n, may be undertaken in place of RвЂ“S analysis. See
Kent and Wood (1997) for a version of this approach in the case where scale
correction is unnecessary. However, as Kent and Wood show, the advantages
of the approach tend to be asymptotic in character, and sample sizes may need
to be extremely large before real improvements are obtained.
Hurst used the coeп¬ѓcient H as an index for the persistence of the time series
considered. For 0.5 < H < 1, it is positively persistent and characterized
by вЂ˜long memoryвЂ™ eп¬Ђects, as described in the next section. A rather informal
interpretation of H used by practitioners is this: H may be interpreted as
the chance of movements with the same sign, Peters (1994). For H > 0.5,
it is more likely that an upward movement is followed by a movement of the
same (positive) sign, and a downward movement is more likely to be followed
312 14 Long Memory Eп¬Ђects Trading Strategy

by another downward movement. For H < 0.5, a downward movement is
more likely to be reversed by an upward movement thus implying the reverting
behavior.

14.3 Stationary Long Memory Processes
A stationary process X has the long memory property, if for its autocorrelation
function ПЃ(k) = Cov(Xi , Xi+k )/Var(X1 ) holds:
в€ћ
= в€ћ.
ПЃ(k) (14.1)
k=в€’в€ћ

That is, the autocorrelations decay to zero so slowly that their sum does not
converge, Beran (1994).
With respect to (14.1), note that the classical expression for the variance of the
ВЇ def n
sample mean, X = nв€’1 i=1 Xi , for independent and identically distributed
X1 , . . . , Xn ,
2
ВЇ = Пѓ with Пѓ 2 = Var(Xi )
Var(X) (14.2)
n
is not valid anymore. If correlations are neither zero and nor so small to be
ВЇ
negligible, the variance of X is equal to
nв€’1
Пѓ2 k
ВЇ 1в€’
Var(X) = 1+2 ПЃ(k) . (14.3)
n n
k=1

Thus, for long memory processes the variance of the sample mean converges to
zero at a slower rate than nв€’1 , Beran (1994). Note that long memory implies
positive long range correlations. It is essential to understand that long range
dependence is characterized by slowly decaying correlations, although nothing
is said about the size of a particular correlation at lag k. Due to the slow
decay it is sometimes diп¬ѓcult to detect non zero but very small correlations by
в€љ
looking at the В±2/ nвЂ“conп¬Ѓdence band. Beran (1994) gives an example where
в€љ
the correct correlations are slowly decaying but within the В±2/ nвЂ“band. So
even if estimated correctly we would consider them as non signiп¬Ѓcant.
Note that (14.1) holds in particular if the autocorrelation ПЃ(k) is approximately
c|k|в€’О± with a constant c and a parameter О± в€€ (0, 1). If we know the autocor-
14.3 Stationary Long Memory Processes 313

relations we also know the spectral density f (О»), deп¬Ѓned as
в€ћ
Пѓ2
ПЃ(k)eikО» .
f (О») = (14.4)
2ПЂ
k=в€’в€ћ

The structure of the autocorrelation then implies, that the spectral density is
approximately of the form cf |k|О±в€’1 with a constant cf as О» в†’ 0. Thus the
spectral density has a pole at 0.
To connect the long memory property with the Hurst coeп¬ѓcient, we introduce
self similar processes. A stochastic process Yt is called self similar with self
similarity parameter H, if for any positive stretching factor c, the rescaled
process cв€’H Yct has the same distribution as the original process Yt . If the
increments Xt = Yt в€’ Ytв€’1 are stationary, there autocorrelation function is
given by
1
|k + 1|2H в€’ 2|k|2H + |k в€’ 1|2H ,
ПЃ(k) =
2
Beran (1994). From a Taylor expansion of ПЃ it follows

ПЃ(k)
в†’ 1 for k в†’ в€ћ .
H(2H в€’ 1)k 2Hв€’2

This means, that for H > 0.5, the autocorrelation function ПЃ(k) is approxi-
mately H(2H в€’ 1)k в€’О± with О± = 2 в€’ 2H в€€ (0, 1) and thus Xt has the long
memory property.

14.3.1 Fractional Brownian Motion and Noise

In this section, we introduce a particular self similar process with station-
ary increments, namely the fractional Brownian motion (FBM) and fractional
Gaussian noise (FGN), Mandelbrot and van Ness (1968), Beran (1994).

DEFINITION 14.1 Let BH (t) be a stochastic process with continuous sam-
ple paths and such that

вЂў BH (t) is Gaussian
вЂў BH (0) = 0
вЂў E {BH (t) в€’ BH (s)} = 0
314 14 Long Memory Eп¬Ђects Trading Strategy

Пѓ2
|t|2H в€’ |t в€’ s|2H + |s|2H
вЂў Cov {BH (t), BH (s)} = 2

for any H в€€ (0, 1) and Пѓ 2 a variance scaling parameter. Then BH (t) is called
fractional Brownian motion.

Essentially, this deп¬Ѓnition is the same as for standard Brownian motion besides
that the covariance structure is diп¬Ђerent. For H = 0.5, deп¬Ѓnition 14.1 contains
standard Brownian motion as a special case but in general (H = 0.5), incre-
ments BH (t) в€’ BH (s) are not independent anymore. The stochastic process
resulting by computing п¬Ѓrst diп¬Ђerences of FBM is called FGN with parameter
H. The covariance at lag k of FGN follows from deп¬Ѓnition 14.1:

Cov {BH (t) в€’ BH (t в€’ 1), BH (t + k) в€’ BH (t + k в€’ 1)}
Оі(k) =
Пѓ2
|k + 1|2H в€’ 2|k|2H + |k в€’ 1|2H
= (14.5)
2

For 0.5 < H < 1 the process has long range dependence, and for 0 < H < 0.5
the process has short range dependence.
Figures 14.2 and 14.3 show two simulated paths of N = 1000 observations of
FGN with parameter H = 0.8 and H = 0.2 using an algorithm proposed by
Davies and Harte (1987). For H = 0.2, the FBM path is much more jagged
and the range of the yвЂ“axis is about ten times smaller than for H = 0.8 which
is due to the reverting behavior of the time series.
The estimated autocorrelation function (ACF) for the path simulated with
в€љ
H = 0.8 along with the В±2/ N вЂ“conп¬Ѓdence band is shown in Figure 14.4.
For comparison the ACF used to simulate the process given by (14.5) is su-
perimposed (dashed line). The slow decay of correlations can be seen clearly.

Applying R/S analysis we can retrieve the Hurst coeп¬ѓcient used to simulate
the process. Figure 14.5 displays the estimated regression line and the data
points used in the regression. We simulate the process with H = 0.8 and the
Л†
R/S statistic yields H = 0.83.
Finally, we mention that fractional Brownian motion is not the only stationary
process revealing properties of systems with long memory. Fractional ARIMA
processes are an alternative to FBM, Beran (1994). As well, there are non
stationary processes with inп¬Ѓnite second moments that can be used to model
long range dependence, Samrodnitsky and Taqqu (1994).
14.4 Data Analysis 315

Sim. FGN with N=1000, H=0.80
-2 0 2

0 250 500 750 1000
t

Sim. FBM with N=1000, H=0.80
200
0

0 250 500 750 1000
t
Figure 14.2. Simulated FGN with H = 0.8, N = 1000 and path of
corresponding FBM.

Sim. FGN with N=1000, H=0.20
4
0
-4

0 250 500 750 1000
t

Sim. FBM with N=1000, H=0.20
-5 0 5

0 250 500 750 1000
t
Figure 14.3. Simulated FGN with H = 0.2, N = 1000 and path of
corresponding FBM. XFGSimFBM.xpl

14.4 Data Analysis
A set of four pairs of voting and nonвЂ“voting German stocks will be subject to
our empirical analysis. More precisely, our data sample retrieved from the data
information service Thompson Financial Datastream, consists of 7290 daily
316 14 Long Memory Eп¬Ђects Trading Strategy

True (dashed) & Est. ACF of Sim. FGN: N=1000, H=0.80

1
0.5
ACF

0

0 20 40 60 80
lag k
Figure 14.4. Estimated and true ACF of FGN simulated with H = 0.8,
N = 1000. XFGSimFBM.xpl

HurstPlot: FGN(N=1000, H=0.80), est. H=0.83
2 2.5 3 3.5 4 4.5
log[E(R/S)]

3 3.5 4 4.5 5 5.5 6
log(k)
Figure 14.5. Hurst regression and estimated Hurst coeп¬ѓcient
Л†
(H = 0.83) of FBM simulated with H = 0.8, N = 1000.
XFGSimFBMHurst.xpl

closing prices of stocks of WMF, Dyckerhoп¬Ђ, KSB and RWE from January 01,
1973, to December 12, 2000.
Figure 14.6 shows the performance of WMF stocks in our data period. The
plot indicates an intimate relationship of both assets. Since the performance
of both kinds of stocks are inп¬‚uenced by the same economic underlyings, their
relative value should be stable over time. If this holds, the logвЂ“diп¬Ђerence Xt of
v nv
the pairs of voting (St ) and nonвЂ“voting stocks (St ),
def v nv
Xt = log St в€’ log St (14.6)
14.4 Data Analysis 317

should exhibit a reverting behavior and therefore an R/S analysis should yield
estimates of the Hurst coeп¬ѓcient smaller than 0.5. In order to reduce the num-
ber of plots we show only the plot of WMF stocks. One may start the quantlet
XFGStocksPlots.xpl to see the time series for the other companies as well.
First, we perform R/S analysis on both individual stocks and the voting/nonвЂ“
voting logвЂ“diп¬Ђerences. In a second step, a trading strategy is applied to all four
voting/nonвЂ“voting logвЂ“diп¬Ђerences.

Time Series of Voting(dashed) and Non Voting WMF Stocks
EUR
35

25

15

5

1973 1980 1990 2000
Time

Figure 14.6. Time series of voting and nonвЂ“voting WMF stocks.
XFGStocksPlots.xpl

Table 14.1 gives the R/S statistic of each individual stock and of the logвЂ“
Л†
diп¬Ђerence process of voting and nonвЂ“voting stocks. While H is close to 0.5
for each time series taken separately, we п¬Ѓnd for the log diп¬Ђerences a Hurst
coeп¬ѓcient indicating negative persistence, i.e. H < 0.5.

WMF Dyck. KSB RWE
nv v nv v nv v nv v
Stock 0.51 0.53 0.57 0.52 0.53 0.51 0.50 0.51
Diп¬Ђerences 0.33 0.37 0.33 0.41

Table 14.1. Estimated Hurst coeп¬ѓcients of each stock and of logвЂ“
diп¬Ђerences.
318 14 Long Memory Eп¬Ђects Trading Strategy

To test for the signiп¬Ѓcance of the estimated Hurst coeп¬ѓcients we need to know
the п¬Ѓnite sample distribution of the R/S statistic. Usually, if the probabilistic
behavior of a test statistic is unknown, it is approximated by its asymptotic
distribution when the number of observations is large. Unfortunately, as, for
example, Lo (1991) shows, such an asymptotic approximation is inaccurate in
the case of the R/S statistic. This problem may be solved by means of bootstrap
and simulation methods. A semiparametric bootstrap approach to hypothesis
testing for the Hurst coeп¬ѓcient has been introduced by Hall, HВЁrdle, Kleinow
a
and Schmidt (2000), In the spirit of this chapter we use Brownian motion
(H = 0.5) to simulate under the null hypothesis. Under the null hypothesis
the logвЂ“diп¬Ђerence process follows a standard Brownian motion and by Monte
Carlo simulation we compute 99%, 95% and 90% conп¬Ѓdence intervals of the
R/S statistic. The results are given in Table 14.2. While the estimated Hurst
coeп¬ѓcients for each individual stock are at least contained in the 99% conп¬Ѓdence
interval, we consider the R/S statistic for voting/nonвЂ“voting log diп¬Ђerences as
signiп¬Ѓcant.

N Mean 90% 95% 99%
7289 0.543 [0.510, 0.576] [0.504, 0.582] [0.491, 0.595]

Table 14.2. Simulated conп¬Ѓdence intervals for R/S statistic for Brown-
ian motion.

The data analysis conducted so far indicates a negative persistence (H < 0.5)
of the log diп¬Ђerences of pairs of voting and nonвЂ“voting stocks of a company. It
should be possible to take advantage of this knowledge. If we found a proп¬Ѓtable
trading strategy, we would interpret this result as a further indication for the
reverting behavior of voting/nonвЂ“voting logвЂ“diп¬Ђerences.
The average relationship between voting and nonвЂ“voting stocks in the sample
period may be expressed in the following way,
log(voting) = ОІ log(non-voting) + Оµ
where ОІ may be estimated by linear regression. If the logвЂ“diп¬Ђerences of voting
and nonвЂ“voting stocks are reverting as the R/S analysis indicates, negative
diп¬Ђerences, Xt < 0, are often followed by positive diп¬Ђerences and vice versa.
In terms of the Hurst coeп¬ѓcient interpretation, given a negative diп¬Ђerence, a
14.5 Trading the Negative Persistence 319

positive diп¬Ђerence has a higher chance to appear in the future than a negative
one and vice versa, implying voting stocks probably to become relatively more
expensive than their nonвЂ“voting counterparts. Thus, we go long the voting and
short the nonвЂ“voting stock. In case of the inverse situation, we carry out the
inverse trade (short voting and long nonвЂ“voting). When initiating a trade we
take a cash neutral position. That is, we go long one share of the voting and
sell short m shares of the nonвЂ“voting stock to obtain a zero cash п¬‚ow from this
action.
But how to know that a вЂ˜turning pointвЂ™ is reached? What is a signal for the
reverse? Naturally, one could think, the longer a negative diп¬Ђerence persisted,
the more likely the diп¬Ђerence is going to be positive. In our simulation, we cal-
culate the maximum and minimum diп¬Ђerence of the preceding M trading days
(for example M = 50, 100, 150). If the current diп¬Ђerence is more negative than
the minimum over the last M trading days, we proceed from the assumption
that a reverse is to come and that the diп¬Ђerence is going to be positive, thereby
triggering a long voting and short nonвЂ“voting position. A diп¬Ђerence greater
than the M day maximum releases the opposite position.
When we take a new position, we compute the cash п¬‚ow from closing the old
one. Finally, we calculate the total cash п¬‚ow, i.e. we sum up all cash п¬‚ows
without taking interests into account. To account for transaction costs, we
compute the total net cash п¬‚ow. For each share bought or sold, we calculate a
hypothetical percentage, say 0.5%, of the share price and subtract the sum of
all costs incurred from the total cash п¬‚ow. In order to compare the total net
cash п¬‚ows of our four pairs of stocks which have diп¬Ђerent levels of stock prices,
we normalize them by taking WMF stocks as a numeraire.
In Table 14.3 we show the total net cash п¬‚ows and in Table 14.4 the number
of trade reverses are given. It is clear that for increasing transaction costs the
performance deteriorates, a feature common for all 4 pairs of stocks. Moreover,
it is quite obvious that the number of trade reverses decreases with the number
of days used to compute the signal. An interesting point to note is that for
RWE, which is in the German DAX30, the total net cash п¬‚ow is worse in all
situations. A possible explanation would be that since the Hurst coeп¬ѓcient
is the highest, the logвЂ“diп¬Ђerences contain less вЂ˜reversionвЂ™. Thus, the strategy
designed to exploit the reverting behavior should perform rather poorly. WMF
and KSB have a smaller Hurst coeп¬ѓcient than RWE and the strategy performs
320 14 Long Memory Eп¬Ђects Trading Strategy

better than for RWE. Furthermore, the payoп¬Ђ pattern is very similar in all
situations. Dyckerhoп¬Ђ with a Hurst coeп¬ѓcient of H = 0.37 exhibits a payoп¬Ђ
structure that rather resembles the one of WMF/KSB.

Transaction M WMF Dyckerhoп¬Ђ KSB RWE
Costs H = 0.33 H = 0.37 H = 0.33 H = 0.41
0.00 50 133.16 197.54 138.68 39.93
100 104.44 122.91 118.85 20.67
150 71.09 62.73 56.78 8.80
0.005 50 116.92 176.49 122.32 21.50
100 94.87 111.82 109.26 12.16
150 64.78 57.25 51.86 2.90
0.01 50 100.69 155.43 105.96 3.07
100 85.30 100.73 99.68 3.65
в€’3.01
150 58.48 51.77 49.97

Table 14.3. Performance of Long Memory Strategies (TotalNetCash-

M WMF Dyckerhoп¬Ђ KSB RWE
50 120 141 132 145
100 68 69 69 59
150 47 35 41 42

Table 14.4. Number of Reverses of Long Memory Trades

Regarding the interpretation of the trading strategy, one has to be aware that
neither the cash п¬‚ows are adjusted for risk nor did we account for interest rate
eп¬Ђects although the analysis spread over a period of time of about 26 years.

Bibliography
Barnsley, M. (1988). Fractals everywhere., Boston, MA etc.: Academic Press,
Inc.
Beran, J. (1994). Statistics for Long Memory Processes, Chapman and Hall,
New York.
14.5 Trading the Negative Persistence 321

Carter, P., Cawley, R. and Mauldin, R. (1988). Mathematics of dimension mea-
surements of graphs of functions, in D. Weitz, L. Sander and B. Mandel-
brot (eds), Proc. Symb. Fractal Aspects of Materials, Disordered Systems,
pp. 183вЂ“186.
Davies, R. B. and Harte, D. S. (1987). Test for Hurst Eп¬Ђect, Biometrica 74: 95вЂ“
102.
Hall, P., HВЁrdle, W., Kleinow, T. and Schmidt, P. (2000). Semiparametric
a
bootstrap approach to hypothesis tests and conп¬Ѓdence intervals for the
hurst coeп¬ѓcient, Statistical Inference for stochastic Processes 3.
Hall, P. and Wood, A. (1993). On the performance of box-counting estimators
of fractal dimension., Biometrika 80(1): 246вЂ“252.
Heyman, D., Tabatabai, A. and Lakshman, T.V. (1993). Statistical analysis
and simulation of video teleconferencing in ATM networks, IEEE Trans.
Circuits. Syst. Video Technol., 2, 49вЂ“59.
Hunt, F. (1990). Error analysis and convergence of capacity dimension algo-
rithms., SIAM J. Appl. Math. 50(1): 307вЂ“321.
Hurst, H. E. (1951). Long Term Storage Capacity of Reservoirs, Trans. Am.
Soc. Civil Engineers 116, 770вЂ“799.
Jones, P.D. and Briп¬Ђa, K.R. (1992). Global surface air temperature variations
during the twentieth century: Part 1, spatial, temporal and seasonals
details, The Holocene 2, 165вЂ“179.
Kent, J. T. and Wood, A. T. (1997). Estimating the fractal dimension of a
locally self-similar Gaussian process by using increments., J. R. Stat. Soc.,
Ser. B 59(3): 679вЂ“699.
Leland, W.E., Taqqu, M.S., Willinger, W. and Wilson, D.V. (1993). Ethernet
traп¬ѓc is selfвЂ“similar: Stochastic modelling of packet traп¬ѓc data, preprint,
Bellcore, Morristown.
Lo, A.W. (1991). Long-term memory in stock market prices, Econometrica,
59, 1279вЂ“1313.
Mandelbrot, B.B. and van Ness, J.W. (1968). Fractional Brownian Motion,
fractional Noises and Applications, SIAM Rev.10, 4, 422вЂ“437.
Peters, E.E. (1994). Fractal Market Analysis: Applying Chaos Theory to In-
vestment and Economics, John Wiley & Sons, New York.
322 14 Long Memory Eп¬Ђects Trading Strategy

Samrodnitsky, G. and Taqqu, M.S. (1994). Stable non-Gaussian Random Pro-
cesses: Stochastic Models with inп¬Ѓnite variance, Chapman and Hall, New
York.
Sullivan, F. and Hunt, F. (1988). How to estimate capacity dimension, Nuclear
Physics B (Proc. Suppl.) pp. 125вЂ“128.
15 Locally time homogeneous time
series modeling
Danilo Mercurio

15.1 Intervals of homogeneity
An adaptive estimation algorithm for time series is presented in this chapter.
The basic idea is the following: given a time series and a linear model, we
select on-line the largest sample of the most recent observations, such that
the model is not rejected. Assume for example that the data can be well
п¬Ѓtted by a regression, an autoregression or even by a constant in an unknown
interval. The main problem is then to detect the time interval where the model
approximately holds. We call such an interval: interval of time homogeneity.
This approach appears to be suitable in п¬Ѓnancial econometrics, where an on-
line analysis of large data sets, like e.g. in backtesting, has to be performed. In
this case, as soon as a new observation becomes available, the model is checked,
the sample size is optimally adapted and a revised forecast is produced.
In the remainder of the chapter we brieп¬‚y present the theoretical foundations
of the proposed algorithm which are due to Liptser and Spokoiny (1999) and
we describe its implementation. Then, we provide two applications to п¬Ѓnancial
data. In the п¬Ѓrst one we estimate the possibly time varying coeп¬ѓcients of an
exchange rate basket, while in the second one the volatility of an exchange rate
time series is п¬Ѓtted to a locally constant model. The main references can be
found in HВЁrdle, Herwartz and Spokoiny (2001), Mercurio and Spokoiny (2000),
a
HВЁrdle, Spokoiny and Teyssi`re (2000) and Mercurio and Torricelli (2001).
a e
Let us consider the following linear regression equation:

Yt = Xt Оё + ПѓОµt , t = 1, . . . , T (15.1)
324 15 Locally time homogeneous time series modeling

where Yt is real valued, Xt = (X1,t . . . Xp,t ) and Оё = (Оё1 . . . Оёp ) are Rp
valued and Оµt is a standard normally distributed random variable. If the matrix
T
t=1 Xt Xt is nonsingular with inverse W , then the least squares estimator of
Оё is:
T
Оё=W Xt Yt . (15.2)
t=1

Deп¬Ѓne wkk as the k-th element on the diagonal of W and let О» be a positive
scalar. For nonrandom regressors,the following exponential probability bound
is easy to prove:
в€љ О»2
P(|Оёk в€’ Оёk | > О»Пѓ wkk ) в‰¤ 2eв€’ 2 , k = 1, . . . , p. (15.3)

Indeed, the estimation error Оёk в€’ Оёk is N(0, wkk Пѓ 2 ) distributed, therefore:
2

О»(Оёk в€’ Оёk ) О»2
в€’
в€љ
1 = E exp
Пѓ wkk 2

О»(Оёk в€’ Оёk ) О»2 в€љ
в‰Ґ E exp в€’ 1(Оёk в€’ Оёk > О»Пѓ wkk )
в€љ
Пѓ wkk 2
О»2 в€љ
в‰Ґ exp P(Оёk в€’ Оёk > О»Пѓ wkk ).
2

The result in (15.3) follows from the symmetry of the normal distribution.
Equation (15.3) has been generalized by Liptser and Spokoiny (1999) to the
case of nonrandom regressors. More precisely, they allow the Xt to be only con-
ditionally independent of Оµt , and they include lagged values of Yt as regressors.
In this case the bound reads roughly as follows:
в€љ 2
в€’О»
P(|Оёk в€’ Оёk | > О»Пѓ wkk ; W is nonsingular ) в‰¤ P(О»)e . (15.4)
2

Where P(О») is a polynomial in О». It must be noticed that (15.4) is not as sharp
as (15.3), furthermore, because of the randomness of W , (15.4) holds only on
the set where W is nonsingular, nevertheless this set has in many cases a large
probability. For example when Yt follows an ergodic autoregressive process and
the number of observations is at least moderately large. More technical details
are given in Section 15.4.
We now describe how the bound (15.4) can be used in order to estimate the
coeп¬ѓcients Оё in the regression equation (15.1) when the regressors are (possi-
bly) stochastic and the coeп¬ѓcients are not constant, but follow a jump process.
15.1 Intervals of homogeneity 325

Оёi,t 6

-
П„ в€’m П„
time
Figure 15.1. Example of a locally homogeneous process.

The procedure that we describe does not require an explicit expression of the
law of the process Оёt , but it only assumes that Оёt is constant on some unknown
time interval I = [П„ в€’ m, П„ ], П„ в€’ m > 0, П„, m в€€ N. This interval is referred
as an interval of time homogeneity and a model which is constant only on some
time interval is called locally time homogeneous.
Let us now deп¬Ѓne some notation. The expression ОёП„ will describe the (п¬Ѓltering)
estimator of the process (Оёt )tв€€N at time П„ ; that is to say, the estimator which
uses only observations up to time П„ . For example if Оё is constant, the recursive
estimator of the form:
в€’1
П„ П„
ОёП„ = Xs Xs Xs Ys ,
s=1 s=1

represents the best linear estimator for Оё. But, if the coeп¬ѓcients are not con-
stant and follow a jump process, like in the picture above a recursive estimator
cannot provide good results. Ideally, only the observations in the interval
I = [П„ в€’ m, П„ ] should be used for the estimation of ОёП„ . Actually, an estima-
tor of ОёП„ using the observation of a subinterval J вЉ‚ I would be less eп¬ѓcient,
while an estimator using the observation of a larger interval K вЉѓ I would be
biased. The main objective is therefore to estimate the largest interval of time
homogeneity. We refer to this estimator as I = [П„ в€’ m, П„ ]. On this interval I
326 15 Locally time homogeneous time series modeling

we estimate ОёП„ with ordinary least squares (OLS):
пЈ¶в€’1
пЈ«

ОёП„ = ОёI = пЈ­ Xs Xs пЈё Xs Ys . (15.5)
sв€€I sв€€I

In order to determine I we use the idea of pointwise adaptive estimation de-
scribed in Lepski (1990), Lepski and Spokoiny (1997) and Spokoiny (1998).
The idea of the method can be explained as follows.
Suppose that I is an interval-candidate, that is, we expect time-homogeneity in
I and hence in every subinterval J вЉ‚ I. This implies that the mean values of the
ОёI and ОёJ nearly coincide. Furthermore, we know on the basis of equation (15.4)
that the events
в€љ в€љ
|Оёi,I в€’ ОёП„ | в‰¤ ВµПѓ wii,I and |Оёi,J в€’ ОёП„ | в‰¤ О»Пѓ wii,J

occur with high probability for some suп¬ѓciently large constants О» and Вµ. The
adaptive estimation procedure therefore roughly corresponds to a family of
tests to check whether ОёI does not diп¬Ђer signiп¬Ѓcantly from ОёJ . The latter is
done on the basis of the triangle inequality and of equation (15.4) which assigns
a large probability to the event
в€љ в€љ
|Оёi,I в€’ Оёi,J | в‰¤ ВµПѓ wii,I + О»Пѓ wii,J

under the assumption of homogeneity within I, provided that Вµ and О» are
suп¬ѓciently large. Therefore, if there exists an interval J вЉ‚ I such that the
hypothesis Оёi,I = Оёi,J cannot be accepted, we reject the hypothesis of time
homogeneity for the interval I. Finally, our adaptive estimator corresponds to
the largest interval I such that the hypothesis of homogeneity is not rejected
for I itself and all smaller intervals.

Now we present a formal description. Suppose that a family I of interval
candidates I is п¬Ѓxed. Each of them is of the form I = [П„ в€’ m, П„ ], so that the
set I is ordered due to m. With every such interval we associate an estimate
Оёi,I of the parameter Оёi,П„ and the corresponding conditional standard deviation
в€љ
wii,I . Next, for every interval I from I, we suppose to be given a set J (I)
of testing subintervals J. For every J в€€ J (I), we construct the corresponding
в€љ
estimate Оёi,J from the observations for t в€€ J and compute wii,J . Now, with
15.1 Intervals of homogeneity 327

two constants Вµ and О», deп¬Ѓne the adaptive choice of the interval of homogeneity
by the following iterative procedure:

вЂў Initialization: Select the smallest interval in I
вЂў Iteration: Select the next interval I in I and calculate the corresponding
в€љ
estimate Оёi,I and the conditional standard deviation wii,I Пѓ
вЂў Testing homogeneity: Reject I, if there exists one J в€€ J (I), and i =
1, . . . , p such that
в€љ в€љ
|Оёi,I в€’ Оёi,J | > ВµПѓ wii,I + О»Пѓ wii,J . (15.6)

вЂў Loop: If I is not rejected, then continue with the iteration step by choos-
ing a larger interval. Otherwise, set I = вЂњthe latest non rejected IвЂќ.

The adaptive estimator ОёП„ of ОёП„ is deп¬Ѓned by applying the selected interval I:

Оёi,П„ = Оёi,I for i = 1, . . . , p.

As for the variance estimation, note that the previously described procedure re-
quires the knowledge of the variance Пѓ 2 of the errors. In practical applications,
Пѓ 2 is typically unknown and has to be estimated from the data. The regression
representation (15.1) and local time homogeneity suggests to apply a residual-
based estimator. Given an interval I = [П„ в€’ m, П„ ], we construct the parameter
estimate ОёI . Next the pseudo-residuals Оµt are deп¬Ѓned as Оµt = Yt в€’Xt ОёI . Finally
the variance estimator is deп¬Ѓned by averaging the squared pseudo-residuals:
1
Пѓ2 = Оµ2 .
t
|I|
tв€€I

15.1.2 A small simulation study

The performance of the adaptive estimator is evaluated with data from the
following process:

Yt = Оё1,t + Оё2,t X2,t + Оё3,t X3,t + ПѓОµt .

The length of the sample is 300. The regressors X2 and X3 are two independent
random walks. The regressor coeп¬ѓcients are constant in the п¬Ѓrst half of the
328 15 Locally time homogeneous time series modeling

1 в‰¤ t в‰¤ 150 151 в‰¤ t в‰¤ 300

large jump medium jump small jump

Оё1,t = 1 Оё1,t = .85 Оё1,t = .99 Оё1,t = .9995

Оё2,t = .006 Оё2,t = .0015 Оё2,t = .004 Оё2,t = .0055

Оё3,t = .025 Оё3,t = .04 Оё3,t = .028 Оё3,t = .0255

Table 15.1. Simulated models.

sample, then they make a jump after which they continue being constant until
the end of the sample. We simulate three models with jumps of diп¬Ђerent
magnitude. The values of the simulated models are presented in Table 15.1.
The error term Оµt is a standard Gaussian white noise, and Пѓ = 10в€’2 . Note that
the average value of Пѓ|Оµt | equals 10в€’2 2/ПЂ в‰€ 0.008, therefore the small jump
of magnitude 0.0005 is clearly not visible by eye. For each of the three models
above 100 realizations of the white noise Оµt are generated and the adaptive
estimation is performed.
In order to implement the procedure we need two parameters: Вµ and О», and two
sets of intervals: I and J (I). As far as the latter are concerned the simplest
proposal is to use a regular grid G = {tk } with tk = m0 k for some integer
m0 and with П„ = tkв€— belonging to the grid. We next consider the intervals
Ik = [tk , tkв€— [= [tk , П„ [ for all tk < tkв€— = П„ . Every interval Ik contains exactly
k в€— в€’ k smaller intervals J = [tk , tkв€— [. So that for every interval Ik = [tk , tkв€— [
and k : k < k < k в€— we deп¬Ѓne the set J (Ik ) of testing subintervals J by
taking all smaller intervals with right end point tkв€— : J = [tk , tkв€— [ and all
smaller intervals with left end point tk :J = [tk , tk [:

J (Ik ) = {J = [tk , tkв€— [ or J = [tk , tk [: k < k < k в€— }.

The testing interval sets I and J (I) are therefore identiп¬Ѓed by the parameter
m0 : the grid step.
We are now left with the choice of three parameters: О», Вµ and m0 . These
parameters act as the smoothing parameters in the classical nonparametric
estimation. The value of m0 determines the number of points at which the
time homogeneity is tested and it deп¬Ѓnes the minimal delay after which a jump
15.2 Estimating the coeп¬ѓcients of an exchange rate basket 329

can be discovered. Simulation results have shown that small changes of m0
do not essentially aп¬Ђect the results of the estimation and, depending on the
number of parameters to be estimated, it can be set between 10 and 50.
The choice of О» and Вµ is more critical because these parameters determine the
acceptance or the rejection of the interval of time homogeneity as it can be
seen from equation (15.6). Large values of О» and Вµ reduce the sensitivity of
the algorithm and may delay the detection of the change point, while small
values make the procedure more sensitive to small changes in the values of the
estimated parameters and may increase the probability of a type-I error.
For the simulation, we set: m0 = 30, О» = 2 and Вµ = 4, while a rule for the
selection of О» and Вµ for real application will be discussed in the next section.
Figure 15.2 shows the results of the simulation. The true value of the coeп¬ѓcients
is plotted (Оё1,t : п¬Ѓrst row, Оё2,t : second row, Оё3,t : third row) along with the
median, the maximum and the minimum of the estimates from all realizations
for each model at each time point. The simulation results are very satisfactory.
The change point is quickly detected, almost within the minimal delay of 30
periods for all three models, so that the adaptive estimation procedure show a
good performance even for the small jump model.

15.2 Estimating the coeп¬ѓcients of an exchange
In this section we compare the adaptive estimator with standard procedures
which have been designed to cope with time varying regressor coeп¬ѓcients. A
simple solution to this problem consists in applying a window estimator, i.e.
an estimator which only uses the most recent k observations:
в€’1
t t
Оёt = Xs Xs Xs Ys , (15.7)
s=tв€’k s=tв€’k

where the value of k is speciп¬Ѓed by the practitioner. Another, more reп¬Ѓned
technique, consists in describing the coeп¬ѓcients Оё as an unobserved stochastic
process: (Оёt )tв€€N , see Elliot, Aggoun and Moore (1995). Apart from the cases
when there is some knowledge about the data generating process of Оёt , the
most common speciп¬Ѓcation is as a multivariate random walk:

О¶t в€ј N(0, ОЈ).
Оёt = Оёtв€’1 + О¶t (15.8)
330 15 Locally time homogeneous time series modeling

LARGE JUMP MEDIUM JUMP SMALL JUMP

50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300
ALPHA_1 ALPHA_1 ALPHA_1

50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300
ALPHA_2 ALPHA_2 ALPHA_2

50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300
ALPHA_3 ALPHA_3 ALPHA_3

Figure 15.2. On-line estimates of the regression coeп¬ѓcients with jumps
of diп¬Ђerent magnitude. Median (thick dotted line), maximum and min-
imum (thin dotted line) among all estimates.

In this context, equations (15.8) and (15.1) can be regarded as a state space
model, where equation (15.8) is the state equation (the signal) and equation
(15.1) is the measurement equation and it plays the role of a noisy observation
of Оёt . A Kalman п¬Ѓlter algorithm can be used for the estimation, see Cooley
and Prescott (1973). The Kalman п¬Ѓlter algorithm requires the initialization of
15.2 Estimating the coeп¬ѓcients of an exchange rate basket 331

two variables: Оё0|0 and P0|0 = Cov(Оё0|0 ) and its recursions read as follows, see
Chui and Chen (1998):
пЈ±
пЈґ P0|0 = Cov(Оё0|0 )
пЈґ t|tв€’1 = Ptв€’1|tв€’1 + ОЈПѓ 2
пЈґ
пЈґP
пЈґ
= Pt|tв€’1 Xt (Xt Pt|tв€’1 Xt + Пѓ 2 )в€’1
пЈґ
пЈІG
t
= (I в€’ Gt Xt )Pt|tв€’1
пЈґ Pt|t
пЈґ
пЈґ t|tв€’1 = Оёtв€’1|tв€’1
пЈґОё
пЈґ
пЈґ
пЈі
= Оёt|tв€’1 + Gt (Yt в€’ Xt Оёt|tв€’1 ).
Оёt|t

The question of the initialization of the Kalman п¬Ѓlter will be discussed in
the next section together with the Thai Baht basket example. In the notation
above, the index t|tв€’1 denotes the estimate performed using all the observation
before time t (forecasting estimate), while t|t refers to the estimate performed
using all the observations up to time t (п¬Ѓltering estimate). The four estimators
described above: the adaptive, the recursive, the window and the Kalman п¬Ѓlter
Estimator are now applied to the data set of the Thai Baht basket. For deeper
analysis of these data see Christoп¬Ђersen and Giorgianni (2000) and Mercurio
and Torricelli (2001).

An exchange rate basket is a form of pegged exchange rate regime and it takes
place whenever the domestic currency can be expressed as a linear combination
of foreign currencies. A currency basket can be therefore expressed in the form
of equation (15.1), where: X1,t is set constantly equal to one and is taken as
numeraire, Yt represents the home currency exchange rate with respect to the
numeraire, and Xj,t is the amount of currency 1 per unit of currency j, i.e.
the cross currency exchange rate. The above relationship usually holds only on
the average, because the central bank cannot control the exchange rate exactly,
therefore the error term Оµt is added.
Because modern capital mobility enables the investors to exploit the interest
rate diп¬Ђerentials which may arise between the domestic and the foreign cur-
rencies, a pegged exchange rate regime can become an incentive to speculation
and eventually lead to destabilization of the exchange rate, in spite of the fact
that its purpose is to reduce exchange rate п¬‚uctuations, see Eichengreen, Mas-
son, Savastano and Sharma (1999). Indeed, it appears that one of the causes
which have led to the Asian crisis of 1997 can be searched in short term capital
investments.
332 15 Locally time homogeneous time series modeling

From 1985 until its suspension on July 2, 1997 (following a speculative attack)
the Bath was pegged to a basket of currencies consisting of ThailandвЂ™s main
trading partners. In order to gain greater discretion in setting monetary pol-
icy, the Bank of Thailand neither disclosed the currencies in the basket nor the
weights. Unoп¬ѓcially, it was known that the currencies composing the basket
were: US Dollar, Japanese Yen and German Mark. The fact that the public
was not aware of the values of the basket weights, also enabled the monetary
authorities to secretly adjust their values in order to react to changes in eco-
nomic fundamentals and/or speculative pressures. Therefore one could express
the USD/THB exchange rate in the following way:

YU SD/T HB,t = ОёU SD,t + ОёDEM,t XU SD/DEM,t + ОёJP Y,t XU SD/JP Y,t + ПѓОµt .

This exchange rate policy had provided Thailand with a good stability of the
exchange rate as it can be seen in Figure 15.3. During the same period, though,
the interest rates had maintained constantly higher than the ones of the coun-
tries composing the basket, as it is shown in Figure 15.4.
This facts suggest the implementation of a speculative strategy, which con-
sists in borrowing from the countries with a lower interest rate and lending
to the ones with an higher interest rate. A formal description of the problem
can be made relying on a mean-variance hedging approach, see Musiela and
в€— в€—
Rutkowski (1997). The optimal investment strategy Оѕ1 , . . . , Оѕp is obtained by
the minimization of the quadratic cost function below:
пЈ±пЈ« пЈј
пЈ¶2
p
пЈґ пЈґ
пЈІ пЈЅ
E пЈ­Yt+h в€’ Оѕj Xj,t+h пЈё Ft .
пЈґ пЈґ
j=1
пЈі пЈѕ

The solution is:
в€—
Оѕj = E(Оёj,t+h |Ft ) for j = 1, . . . , p.
It can be seen that, when the interest rates in Thailand (r0 ) are suп¬ѓciently
high with respect to the foreign interest rates (rj , j = 1, . . . , p) the following
inequality holds
p
в€’1
(1 + rj )в€’1 E(Оёj,t+h |Ft )Xj,t .
(1 + r0 ) Yt < (15.9)
j=1

This means that an investment in Thailand is cheaper than an investment
with the same expected revenue in the countries composing the basket. In the
15.2 Estimating the coeп¬ѓcients of an exchange rate basket 333

DEM/USD
1.7
1.6
Y
1.5
1.4

1992 1993 1994 1995 1996 1997
X

JPY/USD
130
120
110
Y
100
90
80

1992 1993 1994 1995 1996 1997
X

THB/USD
26
25.5
Y
25
24.5

1992 1993 1994 1995 1996 1997
X

Figure 15.3. Exchange rate time series. XFGbasket.xpl

empirical analysis we п¬Ѓnd out that the relationship (15.9) is fulп¬Ѓlled during
the whole period under investigation for any of the four methods that we use
to estimate the basket weights. Therefore it is possible to construct a mean
334 15 Locally time homogeneous time series modeling

1 month interest rates

10
Y*E-2
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