Analysis and forecasting of self-similar financial time series
Time series of prices of MSFT ticker are considered. Results on selfsimilarity of this time series are presented. A method of prediction from FARIMA model for long-range dependent time series is described. This method is used for prediction of MSFT time series of prices that exhibits long-range de...
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| Zitieren: | Analysis and forecasting of self-similar financial time series / M. Moklyachuk, A. Zrazhevsky // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 114–122. — Бібліогр.: 10 назв.— англ. |
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| citation_txt | Analysis and forecasting of self-similar financial time series / M. Moklyachuk, A. Zrazhevsky // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 114–122. — Бібліогр.: 10 назв.— англ. |
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| description | Time series of prices of MSFT ticker are considered. Results on selfsimilarity
of this time series are presented. A method of prediction from FARIMA model for long-range dependent time series is described. This method is used for prediction of MSFT time series of prices that exhibits
long-range dependence with the Hurst parameter close to 1.
|
| first_indexed | 2025-11-30T10:26:02Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 114–122
MIKHAIL MOKLYACHUK AND ALEKSEY ZRAZHEVSKY
ANALYSIS AND FORECASTING OF SELF-SIMILAR
FINANCIAL TIME SERIES
Time series of prices of MSFT ticker are considered. Results on self-
similarity of this time series are presented. A method of prediction from
FARIMA model for long-range dependent time series is described. This
method is used for prediction of MSFT time series of prices that exhibits
long-range dependence with the Hurst parameter close to 1.
1. Introduction
Many methods of analysis of time series of various types and origin are based
on self-similarity and long-range dependence of time series. The long-range
dependent processes provide a good description of many highly persistent
financial time series. The predominant way to quantify long-range depen-
dence and self-similarity is the value of the Hurst parameter. Methods of
estimating the Hurst parameter are described in [1], [3], [4], [9], [10]. The
problem of prediction of time series is one of the important problems of the
theory of stochastic processes. Prediction the cash flows, constructing the
arbitration strategy for prices of indexes on securities markets are examples
of the problem. Different methods are proposed in literature for prediction
of time series (see, for example, [5], [6]). Most of the proposed methods of
prediction of long-range dependent time series are based on the choice of
an appropriate stochastic model. Examples of such model are FARIMA,
SEMIFAR, FIGARCH, GARCH [5].
The self-similarity of time series of prices of MSFT ticker and the cor-
responding derived series (the returns of MSFT ticker) is analyzed in [1].
It is shown that for prices of MSFT ticker the Hurst parameter is close to
1, so the prices may include trend and the data is predictable. This time
series can be simulated with the help of self-similar processes. For the re-
turns of MSFT ticker the Hurst parameter is close to 0.5, so the returns are
uncorrelated, and it is impossible to predict them.
This paper deals with the prediction of long-range dependent time series
based on the FARIMA model. On the basis long-range dependence of time
series of prices of MSFT ticker analyzed in [1] the FARIMA model is used
for description of MSFT time series of prices. The prediction from this
model is constructed.
2000 Mathematics Subject Classification. Primary 60G52, 62M20, 93E10, 93E12.
Key words and phrases. Self-similarity, long-range dependence, MSFT time series,
prediction from long-memory model, Toeplitz normal equations, FARIMA.
114
ANALYSIS AND FORECASTING OF MSFT TIME SERIES 115
2. Self-similarity. Long-range dependence.
Let {xt, t = 0, 1, . . .} be a wide-sense stationary time series with the au-
tocorrelation function r(k), k ≥ 0. For each m ∈ N let x(m) = {x(m)
k , k =
0, 1, . . .} denote a new aggregated time series obtained by averaging of the
original series {xt, t = 0, 1, . . .} over non-overlapping blocks of size m, re-
placing each block by its sample mean. That is x
(m)
k = 1
m
(xkm−m+1 + · · ·+
xkm), k ≥ 1. This aggregated time series is also wide-sense stationary.
The corresponding autocorrelation function is denoted as r(m)(k), k ≥ 0.
A time series {xt, t = 0, 1, . . .} is called asymptotically second-order self-
similar (further simply self-similar) if for large enough k:
(1) r(m)(k) → r(k), m → ∞.
A time series {xt, t = 0, 1, 2, . . .} is said to be long-range dependent if:
(2) r(k) ∼ k−β , k → ∞ and 0 < β < 1.
A long-range dependent time series is self-similar (see, for example, [4]) and
the Hurst parameter is defined as H = 1 − β/2. For long-range dependent
time series 1/2 < H < 1. For H = 1/2 samples (observations) are uncor-
related. For 0 < H < 1/2 time series is said to be short-range dependent
[4].
The predominant way to determine self-similarity is to establish the long-
range dependence, that is, quantify the value of the Hurst parameter. There
are many methods of estimation of the Hurst parameter. The most well-
known of them are: method of standard deviation of aggregates, rescaled
adjusted range statistic (R/S) method, method of autocorrelation function,
method of periodograms, Robinson’s method (see [1], [3], [9], [10]).
If the self-similarity is detected, then it is possible to select the appropri-
ate model for description and prediction of the considered time series.
3. Linear prediction for long-range dependent time series
Let {xt, t = 0, 1, . . .} be a wide-sense stationary long-range dependent
time series. Suppose, that we know values of this time series until some
fixed moment of time T . The prediction problem is to estimate the value
xT , using the information for the past observations x0, x1, . . . , xT−1.
Consider the FARIMA process which is a model for financial long-range
dependent time series (see [2], [5], [7]). This process is a solution of the
equation:
(3) Φp(L)(1 − L)d(yt − μ) = Θq(L) εt, t ∈ Z,
where L denotes the lag operator, d is difference parameter, Φp and Θq are
polynomials:
Φp(z) = 1 − φ1z − φ2z
2 − ... − φpz
p,
Θq(z) = 1 + θ1z + θ2z
2 + ... + θqz
q,
with roots outside the unit circle, εt is assumed to be i.i.d. Gaussian random
variables with zero mean and variance σ2, E[yt] = μ. This process is usually
116 MIKHAIL MOKLYACHUK AND ALEKSEY ZRAZHEVSKY
referred to as the ARIMA(p, d, q) model. By allowing d to be real number
instead of a positive integer, ARIMA model becomes the autoregressive
fractional integrated moving average (FARIMA) model. FARIMA(p, d, q)
model can be equivalently expressed as an AR(∞) model [5].
The prediction from FARIMA model can be considered as prediction from
the AR(p) representation with very large p, where the AR(p) coefficients are
the first p coefficients of the AR(∞) representation of the FARIMA(p, d, q)
model. This method is referred to as the truncation method (see [5]). The
prediction, obtained by this method, is of the form:
(4) x̂T = a1xT−1 + ... + apxT−p,
where ai, i = 1, . . . , p are weighting coefficients selected yield the best linear
predictor, p denotes the model order. To estimate the vector of coefficients
a = [a1, a2, . . . , ap]
� let us form a vector process {x (t), t ≥ L − 1} from
the time series {xt, t = 0, 1, . . .} in such manner: x(t) = [xt, . . . , xt−L+1]
T ,
where 0 ≤ L ≤ T + 1. The vector x(T ) includes the value xT to be
predicted. The problem of prediction of the time series {xt, t = 0, 1, . . .}
at the moment T can be reformulated to the problem of prediction of the
process {x (t), t ≥ L − 1} at the moment T , using the past observations
x (T − 1), . . . ,x (L − 1). This process is stationary and its autocorrelation
function has the same asymptotic behavior as the autocorrelation function
for the initial time series. Construct the prediction for the process {x (t), t ≥
0} from FARIMA model. Denote the forward predicted process vector at
the moment T as a weighted linear combination of previous measurements
similarly to (4) with the vector of coefficients a:
(5) xf(T, a) =
p∑
m=1
am x(T − m),
where p = T − L + 1 denotes the model order.
Evaluation of the “closeness” of the predicted process xf (T, a) to the
true observation x(T ) makes it necessary to define a prediction error vector
ε(T, a) = x(T ) − xf (T, a). We will determine the estimate of a as a vector
â that minimizes the prediction error energy E(T, a):
(6) min
a
E(T, a) = min
a
ε�(T, a) ε(T, a) = E(T, â).
A solution of the least-squares problem (6) results in the normal equations
of linear prediction:
(7) X�(T )X(T ) â = X�(T )x(T ).
where X(T ) = [x(T − 1),x(T − 2), . . . ,x(T − p)]� is the matrix of past
observations.
Let us define the covariance matrix Φ(T ) = [Φi, j(T )], Φi, j(T ) = x(T −
i)x(T − j), 0 ≤ i, j ≤ p for the vector process x(t). Then the normal
ANALYSIS AND FORECASTING OF MSFT TIME SERIES 117
equations (7) can alternatively be expressed as (see [7]):
(8) Φ(T )
⎡
⎣
−1
â
⎤
⎦ =
⎡
⎢⎢⎣
x�(T )X(T ) â− x�(T )x(T )
0
...
0
⎤
⎥⎥⎦ .
The observed vector process is stationary: Φ0, j(T−k) = Φ0, j(T ), 0 ≤ k ≤ p
and the covariance matrix can be replaced with the Toeplitz matrix - matrix
of autocorrelation coefficients of the form ci = Φ0, i/Φ0, 0, 0 ≤ i ≤ p. The
system (8) can be replaced by the Toeplitz Normal Equations:
(9)
⎡
⎢⎢⎢⎣
1 c1 . . . cp−1
c1 1
...
...
. . . c1
cp−1 . . . c1 1
⎤
⎥⎥⎥⎦
⎡
⎢⎢⎣
â1
...
âp−1
âp
⎤
⎥⎥⎦ =
⎡
⎢⎢⎣
c1
...
cp−1
cp
⎤
⎥⎥⎦ .
The system of linear equations (9) is sometimes called the Yule-Walker
equations. Note, that the Toeplitz matrix and matrix from the Normal
Equations A = X�‘(T )X(T ) are symmetric. The equation (9) (or, if the
process is not stationary, (7)) can be solved using classical algorithms for
symmetric linear systems. Most important of them that are used in linear
prediction theory are Cholevsky decomposition, QR decomposition, singular
value decomposition [7].
A solution of the equation (9) provides the least-squares estimation of
a and the forward predicted process vector xf(T, a) can be obtained by
(5). The first element of this vector is the prediction xf
T of the time series
{xt, t = 0, 1, . . .} at the moment T .
Equations (9) may be solved only if we know the vector of autocorre-
lation coefficients c = [c1, c2, . . . , cp]
�. Coefficients c1, c2, . . . , cp−1 can be
computed using the past observations x (T ), . . . ,x (L − 1). The value of
cp is unknown. We propose the linear regression model based on the long
range dependence of the observed time series for predicting unknown auto-
correlation coefficients.
We consider the long-range dependent time series {xt, t = 0, 1, . . .} with
the autocorrelation function r(k) that satisfies [4]:
(10)
r(k)
H(2H − 1)k2H−2
→ 1, k → ∞.
Note, that (10) is the extended version of (2) and is correct for any long-
range dependent time series.
The vector process {x (t), t ≥ 0}, constructed above, has the same as-
ymptotic behavior of the autocorrelation function as the origin time series.
So we will suppose that:
(11) ck ∼ H(2H − 1)k2H−2, k → ∞ and 1/2 < H < 1,
118 MIKHAIL MOKLYACHUK AND ALEKSEY ZRAZHEVSKY
where ck denotes the autocorrelation coefficients for the vector process
{x(t), t ≥ 0}, H is the Hurst parameter.
We will use the linear regression model for fitting values cm, cm+1, . . . , cp−1
on the basis of (11):
(12) ck = α1H(2H − 1) k2H−2 + α2 + ξk, m ≤ k ≤ p − 1,
where α1, α2 are coefficients, ξk, m ≤ k ≤ p, are random errors. The
standard assumptions of linear regression model are: M [ξk] = 0, M [ξk] =
σ2, M [ξk ξl] = 0 for k �= l. Estimates of the coefficients α1, α2 are calculated
to minimize the residual sum of squares
∑
k
ξ2
k [5]:
(13) (α̂1, α̂2) = arg min
(α1, α2)
∑
k
(ck − α1 H (2H − 1) k2H−2 − α2)
2
Estimates of autocorrelation coefficients ck are:
(14) ĉk = α̂1H(2H − 1) k2H−2 + α̂2, m ≤ k ≤ p.
Estimated residuals for the fitted model are ξ̂k = ck − ĉk. Several residuals
diagnostics are used to evaluate the validity of the underlying assumptions
of the model. The most common diagnostic for serial correlation is Durbin-
Watson statistics
(15) DW =
p∑
k=m+1
(ξ̂k − ξ̂k−1)
2
p∑
k=m
ξ̂2
k
.
Values of DW around 2 indicate no serial correlation in the errors: M [ξk ξl] ≈
0 for k �= l ([5]).
The linear regression (12) is not used for first m values because (11)
gives the asymptotic property for autocorrelation function. We obtain the
estimate ĉp from (14) necessary for prediction.
4. Practical results
The practical use of the described above method is shown on the example
of MSFT time series - time series of prices on ticker of firm Microsoft.
This time series is long-range dependent (and self-similar) with the Hurst
parameter close to 1, that implies the possibility for “good” prediction.
This is shown in paper [1] on the basis of four methods. The corresponding
estimates of the Hurst parameter are shown in Table 1.
Table 1. Estimate of the Hurst parameter (see [1]).
Period. Robins. Autocor. R/S
MSFT prices
estim. 0.962 0.977 0.953 0.993
C.I. 0.94-0.98 0.93-1.01 0.94-0.96 0.96-1.02
ANALYSIS AND FORECASTING OF MSFT TIME SERIES 119
Figure 1. Predicted and observed autocorrelation coeffi-
cients with confidence intervals
According to results of paper [1] we may use the prediction from FARIMA
model for MSFT time series of prices. The prediction is computed in two
steps. On the first step the autocorrelation coefficients are fitted and pre-
dicted. On the second step, based on predicted autocorrelation coefficient,
the Toeplitz Normal Equations are solved and prediction for the vector
process is constructed.
We construct a linear regression model (12) for autocorrelation coeffi-
cients cm, cm+1, . . . , cp−1 changing the Hurst parameter with its estimate
shown in the Table 1. The observed autocorrelation coefficients are evalu-
ated using vectors x(t) with length L = 3970. A linear regression is used for
p − m = 128 − 32 = 96 values of observed autocorrelation coefficients with
Ĥ = 0.971 (mean value of estimates computed using different methods). Es-
timates of the coefficients α1, α2 are calculated using (13). Estimates of co-
efficients are α̂1 = 1.0195, α̂2 = 0.1979 with the standard errors 0.0059 and
0.004 correspondingly. The value of Durbin-Watson statistic is 2.0139 that
indicates no serial correlation in the errors. These results show that the cho-
sen regression model is suitable. Estimates of autocorrelation coefficients ĉk
together with the observed values ck and 95%-confidence intervals evaluated
from the linear regression model (12) using the the Student-t distribution
with p − m − 2 degrees of freedom are shown on Fig.1. The prediction for
the next autocorrelation coefficient was found from (14): ĉ128 = 0.8747 with
the standard error 0.013312. All calculations were made in Mathematica 5.0
with the help of function “Regress” [8].
An estimate for the parameter vector a = [a1, a2, . . . , ap]
T is found as a
solution of the Toeplitz Normal Equations (9), where the autocorrelation
coefficients cp are replaced with estimates ĉp. This system of equations was
120 MIKHAIL MOKLYACHUK AND ALEKSEY ZRAZHEVSKY
Figure 2. Predicted and observed vectors for MSFT time series
Table 2. Analysis of errors for prediction
Mean Median Variance Stand.Dev. Stand.Error
ε1 0.00021 0.00027 0.00023 0.015 0.00024
ε2 -0.00056 -0.00012 0.00012 0.01096 0.00028
solved using function “Solve” in Mathematica 5.0 [8]. The forward predicted
vector xf(T ) at the moment T = 4097 is found from (11). This is shown on
Fig.2 along with the corresponding observed vector. The predicted vector
includes all trends that are observed for the initial one. That implies the
goodness of our method for predicting this time series.
Consider the prediction error vector ε1 = x(T ) − xf (T ). The basic
statistic characteristics for the data of errors are shown in Table 2. Note,
ANALYSIS AND FORECASTING OF MSFT TIME SERIES 121
Figure 3. Predicted and observed MSFT time series
that predicted vector have no practical interest except one element x4097
that have to be predicted.
We repeat the described procedure for 256 times. Every time we were
interested in the last element of the predicted vector that was unknown
before prediction. The corresponding data along with the true values of time
series are shown on Fig.3. The errors ε2 = xt−xf
t for t = 4097, . . . , 4353 are
obtained and basic statistic characteristics for them are shown in Table 2.
Note, that values p, T, L, m where chosen from the practical reasons.
5. Conclusions
Time series of prices of MSFT (a ticker of firm Microsoft) is analized
in this paper. Main results from paper [1] on self-similarity of these time
series are reviewed. On the basis of these results a regression model for the
122 MIKHAIL MOKLYACHUK AND ALEKSEY ZRAZHEVSKY
autocorrelation function of the vector process constructed from MSFT time
series of prices is proposed and used for prediction. A prediction problem
from FARIMA long-memory model is considered. A described method of
prediction is applied to MSFT time series of prices. The corresponding
results along with the errors are presented. The predicted time series has
all trends observed at the origin one. Errors are insignificant.
References
1. Moklyachuk, Mikhail P., Zrazhevsky, Aleksey G., Long-range dependence of time
series for MSFT data of shares and returns, Random Operators and Stochastic
Equations, Vol. 14, No. 4, (2006), 393–403.
2. Zhurbenko, I. G.; Kozhevnikova, I. A., Stochastic modelling of processes, Moskva:
Izdatel’stvo Moskovskogo Universiteta. 148 p. (1990).
3. Robinson, P. M., Gaussian semi-parametric estimation of long range dependence,
The Annals of Statistics, 23, (1995), 1630-1661.
4. Beran, J., Statistics for Long-Memory Processes, Monographs on Statistics and
Applied Probability, Chapman and Hall, New York, (1994).
5. Zivot, Eric, Wang, Jiahui, Modeling Financial Time Series with S-PLUS, New
York: Springer, (2003).
6. Strobach, Peter, Linear Prediction Theory - A Mathematical Basis for Adaptive
Systems, Springer-Verlag, (1990).
7. Stoev, S., Taqqu, M.S. Simulation methods for Linear fractional stable motion
and FARIMA using the fast Fourier transform, Fractals, 12(1), (2004), 95-121.
8. Wolfram Research, Inc., Mathematica, Version 5.0, Champaign, IL (2003).
9. Deane, J. H. B., Smythe, C., Jefferies, D. J., Self-similarity in a deterministic
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10. Teverovsky, V., Taqqu, M., Testing for long-range dependence in the presence of
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Department of Probability Theory and Mathematical Statistics, Kyiv
National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: mmp@univ.kiev.ua
Department of Probability Theory and Mathematical Statistics, Kyiv
National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: zalex@univ.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4461 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-11-30T10:26:02Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Moklyachuk, M. Zrazhevsky, A. 2009-11-11T15:23:06Z 2009-11-11T15:23:06Z 2006 Analysis and forecasting of self-similar financial time series / M. Moklyachuk, A. Zrazhevsky // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 114–122. — Бібліогр.: 10 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4461 Time series of prices of MSFT ticker are considered. Results on selfsimilarity of this time series are presented. A method of prediction from FARIMA model for long-range dependent time series is described. This method is used for prediction of MSFT time series of prices that exhibits long-range dependence with the Hurst parameter close to 1. en Інститут математики НАН України Analysis and forecasting of self-similar financial time series Article published earlier |
| spellingShingle | Analysis and forecasting of self-similar financial time series Moklyachuk, M. Zrazhevsky, A. |
| title | Analysis and forecasting of self-similar financial time series |
| title_full | Analysis and forecasting of self-similar financial time series |
| title_fullStr | Analysis and forecasting of self-similar financial time series |
| title_full_unstemmed | Analysis and forecasting of self-similar financial time series |
| title_short | Analysis and forecasting of self-similar financial time series |
| title_sort | analysis and forecasting of self-similar financial time series |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4461 |
| work_keys_str_mv | AT moklyachukm analysisandforecastingofselfsimilarfinancialtimeseries AT zrazhevskya analysisandforecastingofselfsimilarfinancialtimeseries |