Type I and Type II Fractional Brownian Motions:A Reconsideration

James Davidson and Nigar HashimzadeUniversity of Exeter

1

Long Memory - Some Background Also call strong dependence.

Ideas developed originally by

Harold Hurst (hydrology, weather patterns)

Benoit Mandelbrot (commodity and asset prices).

Consider a stationary zero-mean time series xt, t 1, . . . ,T, variance 2.

1. Short memory. Autocorrelations are summable:

∑t1

Txt OpT1/2, and T−1/2∑

t1

Trxt Br, 0 ≤ r ≤ 1

where B is Brownian motion (BM).

2. Long memory: Autocorrelations are nonsummable:

∑t1

Txt OpT1/2d, 0 d 1

2 ; and T−1/2−d∑t1

Trxt Bdr, 0 ≤ r ≤ 1

where Bd is fractional Brownian motion (fBM).

These continuous-time ‘limit processes’ are essential modelling tools for large sampleinference in time series models.

2

�300

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0

100

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631 703 775 847 919 991 1063 1136 1208 1280

Figure 7. Annual Nile minima (mean deviations)

27

The problem:

There is a single agreed mathematical model of Brownian motion – “Wiener measure”.

However, there are several alternative models of fractional Brownian motion.

There is no consensus among statisticians/ econometricians about which model to use.

Also, not a little misunderstanding of the issues!

4

Fractional Brownian motion:Type I:

Xr 1Γd 1 0

rr − sddBs 1

Γd 1 −0r − sd − −sddBs (1)

Type II:X∗r 1

Γd 1 0

rr − sddBs (2)

where − 12 d 1

2 and B denotes regular Brownian motion.

(Mandelbrot and van Ness 1968, Marinucci and Robinson 1999).

WriteX X∗ X∗∗

where X∗∗r is defined as the second of the two terms in (1).

The processes X∗ and X∗∗ are Gaussian, and independent of each other.

Variance of (1) exceeds that of (2).

Increments of (1) are stationary, whereas those of (2) are not.

5

Motivation for fBM processes1. Postulate a realization, of size n, of discrete long memory processes.2. considering the weak limit of the normalized partial sum, as n → .

Thus, definext 1 − L−dut (3)

where ut− is an i.i.d.0,2, and

1 − L−d ∑j1

bjLj, bj Γd j

ΓdΓ1 j

Define the partial sum process

Xnr 1n1/2d ∑

t1

nr

xt, 0 ≤ r ≤ 1 (4)

It is known (e.g. Davidson and de Jong 2000) that

Xnd→ X.

6

On the other hand, definext∗ 1 − L−dut

∗ (5)where

ut∗ 1t 0ut.

Defining Xn∗ like (4) with xt

∗ replacing xt, it is known (Marinucci and Robinson 2000) that

Xn∗ d→ X∗.

Model (5) used in simulation exercises to generate fractionally integrated processes, asan alternative to setting a fixed, finite truncation of the lag distribution, common to everyt.

Model (5) problematic. Nothing about the date when we start to observe a series tosuggest that we ought to set all shocks preceding it to 0.

Such truncation common in time series modelling, although usually justified byassumption that the effect is asymptotically negligible.

However, if (5) is used to generate the artificial data, the limiting distribution simulatedwill be Type II.

If the observed data ought to be treated as drawn from (3), then estimated critical valuesincorrect even in large samples.

7

Properties of Fractional Brownian MotionsSince

Xr1Xr2 12 Xr12 Xr22 − Xr2 − Xr12

a formula for the variance of an increment Xr2 − Xr1 is sufficient to determine thecomplete covariance structure.

To motivate our discussion, consider cases r1 0 and r2 r ∈ 0,1.

From Mandelbrot and Van Ness (1968), Davidson and Hashimzade (2007),

EXr2 Vdr2d1

where

Vd Γ1 − 2d2d 1Γ1 dΓ1 − d .

By contrast,EX∗r2 V∗dr2d1

whereV∗d 1

2d 1Γd 12 .

8

0.250-0.25

5

3.75

2.5

1.25

0

Plots of V (solid line) and V∗ (dashed line) over −0.5, 0.5

Easy to see how the distributions of functionals such as 0

1Xdr and

0

1X2dr will differ

correspondingly for these two models.

9

Stochastic Integrals

For type I processes X1 (integrand) and X2 (integrator), Davidson and Hashimzade give(2007, Proposition 4.1)

E 0

1X1dX2 12

Γ1 − d1 − d2 sind2

d1 d21 d1 d2

where 12 EX11X21.

On the other hand, for type II processes X1∗ and X2

∗, where 12 is defined analogously:

Proposition 2.1

E 0

1X1∗dX2

∗ 12d21 d1 d2d1 d2Γ1 d1Γ1 d2

10

0.3750.250.1250

2

1.5

1

0.5

E 0

1X1dX2 (solid line) and E

0

1X1∗dX2

∗ (dashed line) as functions of d2, with 12 1,d1 0. 4

11

CommentThese large discrepancies pose important issue - which of these models is the moreappropriate for use in econometric inference? Marinucci and Robinson (1999) remark:

“It is of some interest to note that [type II fBM] is taken for granted as the properdefinition of fractional Brownian motion in the bulk of the econometric time seriesliterature, whereas the probabilistic literature focuses on [type I fBM]. Thisdichotomy mirrors differing definitions of nonstationary fractionally integratedprocesses...”

Two approaches to the fractional model:

1. (“type II”) Continuum of nonstationary models indexed on d ≥ 0. Unified frameworkfor I(1), I(0), I(d).

Must specify finite start date with initial condition given by different mechanism.

2. (“type I”) Integer integration (cumulation from fixed start date) conceptually distinctfrom stationary long memory.

Stationary increments can start at "date" −.

12

Comment, cont’d

The distinction is artificial, but arises because our time series models are highlysimplified representations of (typically) nonlinear/aggregated processes.

What matters, in the choice of linear representation, is how best to represent thejoint distribution of the (finite) observed series.

By representing processes as linear in i.i.d. shocks, stationarity requires "start-up"in the infinite past

Remember, the shocks are unobserved and strictly fictional - just an artificialmodelling device.

Don’t get hung up on what it "means" to remember the infinite past.

13

Working with Type I Processes

Nonstandard asymptotic inference involves distributions that are unavailable in closedform.

Theory becomes operational by simulating the statistic on the computer for large, finiten, using i.i.d. pseudo-Gaussian increments.

Appeal to an invariance principle allows us to use resulting tables of critical values forobserved data.

A simulation strategy is therefore an essential step of the inference procedure.

However, simulating type I processes in some regions of the parameter space isproblematic...

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Simulation Strategies For Type I Processes1. Using Presample LagsApply the formula

xt ∑j0

mt−1

bjut−j

Choosing m “large enough” should approximate type I process to any desired degree ofaccuracy.

Table shows the standard deviations in 10,000 replications of the terminal points

Xn1 1n1/2d ∑

t1

n

xt

where d 0.4, and n 1000.

m 0 1000 3000 6000 9000SD 0.843 0.996 1.036 1.108 1.137

Comment: for comparison, note V∗0.4 0.8401 and V0.4 1. 389 .

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2. Frequency Domain Simulation:

When ut is i.i.d. Gaussian, xt has the harmonic representation

xt 2−

eitgWd (5)

where g 1 − e−i−d,W−d Wd

EWd 0

EWdWd d,

0, otherwise

Process is stationary by construction.

Davidson and Hashimzade (2006, Theorem 2.2) show the weak limit of the partial sumprocess is type I fBM.

16

In principle, could use FFT to compute discrete approximation

xt 2m

∑k1−m

m−1

eiktgkWk, t 0,… ,m − 1

where

Wk Uk iVk, k ≥ 0Uk − iVk, k 0

, Uk,Vk NI0,1

and, for |k| 0,

gk 1 − e−ik −d 2sin k2

−dcos − kd

2 − i sin − kd2

Problem: singularity of g at 0.

Natural solution is to use the series expansion g ∑j0 bje−ij truncated at m terms.

Case: d 0.4, 2 1 and n 1000, SD of Xn1 in 10,000 replications.

m 1000 5000 10,000 20,000SD 1.106 1.128 1.166 1.200

True type I SD 1.389. Comment: not a feasible procedure for routine application.

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3. Choleski Method, Circulant Embedding and Wavelets

Methods based directly on the autocovariance function

k 2 Γ1 − 2dΓk dΓk 1 − d

sind .

do generate a type I process:

Choleski, and CE methods are basically equivalent, but CE much more numericallyefficient.

In practice, wavelet methods use a discrete time ARFIMA process to supply thelow-frequency persistence properties of fBM. Wavelets then fill in the high-frequency"details".

If the ARFIMA is simulated by CE, then the wavelet representation also corresponds totype I.

However, these methods not very suitable to econometric applications.

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4. Simulation by Aggregation

Granger (1981) aggregation scheme.

Sum a large number of independently generated stable AR(1) processes.

Coefficients randomly generated in the interval 0,1 as where is a drawing fromthe Betaa,b distribution.

Granger showed that the resulting aggregate series xt would possess the attributes of afractional sequence with d 1 − b; for example, with d 1

2 the autocovariancesExtxt−k will decrease at the rate k2d−1.

The ‘long memory’ attribute can be identified with the incidence, in a suitableproportion of the aggregate, of AR roots close to 1.

This procedure will generate type II process if micro-series initialized at date t 0.

To get type I process requires ‘remote’ start dates for micro-series.

Comment: Same problem as before - this method also infeasible in practice.

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An Alternative Simulation Strategy

1. Univariate Case If xt defined by (3), write xt xt

∗ xt∗∗ where

xt∗ ∑

j1

t−1

bjut−j, xt∗∗ ∑

jt

bjut−j

X∗ and X∗∗ are:

1. weak limits of the partial sum processes Xn∗ and Xn

∗∗ derived from xt∗ and xt

∗∗

respectively.

2. Gaussian, independent of each other.

Suppose ut process is i.i.d. Gaussian. Then, xt∗ and xt

∗∗ are also independent and thevector x∗∗ x1

∗∗,… ,xn∗∗′ is Gaussian with a known covariance matrix.

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A convenient fact: Formula Ex0x−k k (see above) has alternative representation

Ex0x−k 2∑j0

bjbjk.

Therefore, for any t, s 0,

Ext∗∗xs

∗∗ 2∑j0

bjtbjs Ex0x1−mint,s − 2 ∑j0

mint,s−1

bjbj|t−s|.

The sequence bj is easily constructed by the recursion

bj bj−1j d − 1

jfor j 0 with b0 1.

The n n covariance matrixCn Ex∗∗x∗∗′

can therefore be constructed with minimal computational effort.

21

Idea:Simulate the distribution of x∗∗, by simply making an appropriate collection of Gaussiandrawings, independent of x∗ and Gaussian with covariance Cn.

Theorem 4.1 If

Xn∗∗r 1

n1/2d ∑t1

nr

xt∗∗.

then Xn∗∗ d→ X∗∗.

Let x∗ x1∗,… ,xn

∗ ′ be computed by the usual moving truncation method so that, bystandard arguments,

Xn∗ d→ X∗.

It follows by the continuous mapping theorem that

Xn Xn∗ Xn

∗∗ d→ X Type I fBM!

If ut is either not Gaussian, or is weakly dependent but not i.i.d., this simulation strategywill be inexact in small samples.

However, it will still be asymptotically valid under the usual conditions for theinvariance principle.

22

Points Cn tends rapidly to singularity as n increases.

Hence, only a few Gaussian drawings are needed to generate the complete sequence.

Suppose n is small enough that Cn can be diagonalized numerically (say, n ≤ 150):

1. Obtain the decompositionCn VnVn

′

where Vn is a n s matrix where s rank of smallest positive eigenvalue

(for d 0.4, s ≈ 6!).

2. Draw an independent standard Gaussian vector z s 1, and compute x∗∗ Vnz.

NB: In a Monte Carlo experiment, Vn only has to be computed once.

Computational cost therefore comparable to type II.

23

Large Samples:Where n is too large to perform diagonalization (n 150, say) resort to approximationby interpolation:

tth row of Vn matrix has length Ext∗∗2 (known)

Columns of Vn are orthogonal, and accordingly have a characteristic structure.

Combine these pieces of information as follows:

1. Construct and diagonalize Cp, where p is chosen as the largest whole divisor ofn not exceeding 150.

2. Given Vp, construct Vn as follows:

a) For t 1, n/p, 2n/p,… ,pn/p set the tth row of Vn by taking the pt/nthrow of Vp, renormalized to have squared norm equal to Ext

∗∗2.

b) Fill in missing rows by linear interpolation.

c) Renormalize rows to satisfy vnt′ vnt Ext

∗∗2.

24

The figure plots for the case d 0.4 and n 150, the first 4 columns of Vn by exactcalculation (solid lines) and also by interpolation from p 50 (dashed lines).

-0.2

0

0.2

0.4

0.6

0.8

1

0 50 100

Columns of Vn, n 150: Actual (solid line); interpolated from p 50 (dashed line).

25

Properties of the Simulation

Type I Type IId Theoretical Monte Carlo Theoretical Monte Carlo0.4 1.389 1.383 0.840 0.8420.2 0.997 0.993 0.920 0.9170 1 1.0085 1 1.0085−0.2 1. 176 1.167 1.109 1.104−0.4 1.877 1.76 1.501 1.41

Theoretical standard deviations of the random variables X1 and X∗1, with the samequantities estimated by Monte Carlo from samples of size n 1000 for comparison.

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2.The Multivariate Case

If x t ∑j0

Bjut−j m 1, where Eutu t′ , Bj m m, then

Γk Ex0x0−k′ ∑

j0

BjBjk′ .

By preceding arguments,

Ext∗∗xs

∗∗′ Γs − t −∑j0

t−1

BjBjs−t′ t ≤ s.

Stacking x1∗∗,… ,xm

∗∗ into a vector x∗∗ (mn 1)

Ex∗∗x∗∗′ Cn

C11,n C1m,n

Cm1,n Cmm,n

Letting bk,j′ represent the kth row of Bj, note that

Ckh,n ts Exkt∗∗xhs

∗∗ khs − t −∑j0

t−1

bk,j′ bh,js−t, s ≥ t.

27

Fractional noise example:

Theorem 4.2 For xht and xkt defined by (3) with respect to i.i.d. shock processes uht and uktwith covariance 12,

khs − t hksindh

Γ1 − dh − dkΓdh sΓ1 − dk s

.

Procedure for Multivariate Models: Decompose Cn as

Cn VnVn′

where Vn V1n′ ,… ,Vmn

′ ′.

Use formulaxj∗∗ Vjnz

to generate replications of jth process.

If mn 150, modify the method by the extrapolation step described above.

28

Distributions of Fractional Brownian Functionals

Autoregression: - DF statistic n̂

Without intercept:∑t1

n−1 Stxt1

∑t1n−1 St

2

0

1XdX

0

1X2ds

With intercept: n∑t1

n−1St − S̄x1

∑t1n−1St − S̄2

0

1XdX − X1

0

1Xds

0

1X2ds −

0

1Xds

2

P≤ 0.01 0.05 0.1 0.9 0.95 0. 99no intercept Type I −0.12 0.04 0.28 2.39 2.88 4. 17

Type II −0.24 −0.04 0.16 2.71 3.30 4. 68with intercept Type I −4.51 −2.75 −2. 05 1.97 2.67 4. 25

Type II −4.02 −2.45 −1. 78 2.47 3.15 4. 94

Quantiles of the "Dickey-Fuller" statistics

29

Figure 4: Simulation of unit root autoregression: d = 0.41000 observations, 1000,000 replications

Type I

Type II

Unit root autoregression, no intercept

0

0.01

0.02

0.03

0.04

0.05

0.06

-3 -2 -1 0 1 2 3 4 5 6 7 8

Type I

Type II

Unit root autoregression with intercept

0

0.01

0.02

0.03

0.04

0.05

0.06

-15 -10 -5 0 5 10 15

Bivariate Case - Stochastic Integrals

∑t1n S1tx2t

n1d1d2

0

1X1dX2,

∑t1n S1t − S̄1x2t

n1d1d2

0

1X1dX2 − X21

0

1X1ds,

Cointegration t statistics

n1/2−d2∑t1n S1tx2t

∑t1n S1t

2 ∑t1n x2t

2 − ∑t1n S1tx2t

2

0

1X1dX2

2 01X1

2ds

n1/2−d2∑t1n S1t − S̄1x2t

∑t1n S1t − S̄12∑t1

n x2t2 − ∑t1

n S1t − S̄1x2t2

0

1X1dX2 − X21 0

1X1ds

2 01X1

2 − 0

1X1

2.

31

Figure 5: Simulations of a bivariate distribution with correlation 0.5.Integrand has parameter d1, integrator has parameter d2.

1000 observations, 100,000 replications.

Stochastic Integral, d1 = 0, d2 = 0.4

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-6 -4 -2 0 2 4 6 8

Type I

Type II

Stochastic Integral, d1 = d2 = 0.4

Type I

Type II

0

0.02

0.04

0.06

0.08

0.1

0.12

-6 -4 -2 0 2 4 6

Stochastic Integral (demeaned integrand)d1 = 0, d2 = 0.4

Type I

Type II

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

-1.5 -1 -0.5 0 0.5 1 1.5 2

Stochastic Integral (demeaned integrand)d1 = d2 = 0.4

Type I

Type II

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-1.5 -1 -0.5 0 0.5 1 1.5

Figure 6: Simulations of regression t-value.Processes as for Figure 5.1000 observations, 100,000 replications.

Regression without intercept, d1 = 0, d2 = 0.4

Type I

Type II

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

-3 -2 -1 0 1 2 3 4

Regression without intercept, d1 = d2 = 0.4

Type I

Type II

0

0.01

0.02

0.03

0.04

0.05

0.06

-4 -3 -2 -1 0 1 2 3 4

Regression with intercept, d1 = 0, d2 = 0.4

Type I

Type II

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Regression with intercept, d1 = d2 = 0.4

Type I

Type II

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

P≤ 0.01 0.05 0.1 0.9 0.95 0.99no intercept d1 0 Type I −1. 711 −1.135 −0.879 0.977 1.297 1.810

Type II −0. 672 −0.322 −0.172 1.125 1.375 1.774d1 0.4 Type I −1. 913 −1.353 −1.033 1.206 1.526 2.086

Type II −0. 986 −0.643 −0.446 0.977 1.173 1.566with intercept d1 0 Type I −0. 868 −0.570 −0.437 0.523 0.689 0.954

Type II −0. 381 −0.175 −0.056 0.770 0.888 1.124d1 0.4 Type I −0. 885 −0.623 −0.460 0.487 0.650 0.912

Type II −0. 778 −0.550 −0.387 0.525 0.655 0.916

Quantiles of the cointegrating regression "t statistics"

34

Estimating Type I ARFIMA Processes

Compare the fractional noise model1 − LdYt ut, t 1,… ,n (1)

where ut− is i.i.d.0,2 with feasible counterpart1 − LdYt

∗ ut∗, t 1,… ,n (2)

where ut∗ ut1t ≥ 1, and Yt

∗ defined by the equation.

In other words, if aj repesents coefficients in the expansion of 1 − Ld.Y1∗ u1

Y2∗ u2 − a1Y1

∗

Yn∗ uT − a1Yt−1

∗ − − an−1Y1∗

The asymptotics relevant to models (1) and (2) are those of type I and type II fractionalBrownian motion.

35

Write

tL;d ∑j0

t−1

ajLj

to represent the truncation of the expansion of 1 − L−d at the tth term, note that

tL;−d tL;d−1

With this notation, write the solution of (2) as

Yt∗ 1 − Ldut

∗ tL;−dut.

However, solution of (1) has the approximate form

Yt 1 − Ldut

≈ tL;−dut vtd, ′z

where vtd, ′ is row t of the n s matrix Vn, and z s 1 is a standard normal vector.

36

Consider approximate form of (1)

tL;dYt tL;dvtd, ′z ut

vt∗d,′z ut

Vectors vt∗d can be computed given values for d and .

s is finite, so z can be treated as s additional unknown parameters.

Model (1) can be estimated by inserting a set of s regressors into the equation, andfitting their coefficients.

Asymptotically equivalent to estimating d by fitting (1) as an infinite orderautoregression.

37

Straightforward...... to extend same technique to estimating the ARFIMA(p,d,q) model, with the form

L1 − LdYt − Lut, t 1 maxp,q,… ,n

where EYt.

The approximate model in this case is

L tL;dYt − vt∗d,|1| ′z Lut, t 1 maxp,q,… ,n

Variance of the presample shocks must be calculated as 212, so vt∗ depends

additionally on the moving average parameters.

38

Application: Nile minima series:

622–1284AD 784–1284ADs 0 s 1 s 2 MLE s 0 s 1 s 2 MLE

ARFIMA d0.03160.4182

0.03150.4187

0.03100.4185

0.02990.3932

0.03830.4504

0.03770.4398

0.03150.4289

0.03360.4374

type I Frac.,Z1 −0.516−0.465

0.679−0.908 − −

0.672−0.9841

0.554−0.5301 −

type I Frac., Z2 − −1.7711.894 − − −

1.842−3.485 −

Shock SD2.946

70.5473.004

70.6653.075

70.865 69.903.757

66.9813.891

66.9583.825

66.542 65.37

Student t DF0.2452.345

0.2392.314

0.2342.273 −

0.2142.1248

0.2062.088

0.2142.1248 −

Log-likelihood −3738 −3737 −3737 −3757 −2786 −2783 −2782 −2806Residual Q12 7.6426 7.524 7.027 − 5.250 5.686 5.897 −

39

�300

�200

�100

0

100

200

300

400

631 703 775 847 919 991 1063 1136 1208 1280

Figure 7. Annual Nile minima (mean deviations)

27