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A Simple Asymmetric Herding Model to Distinguish Among Different Types

• f Financial Markets

Simone Alfarano Reiner Franke

University of Kiel

Prepared for CM–Meeting, Marseille, Oct. 2006

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Contents

1 Architecture 1 2 The model 3 2.1 Transition probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The Langevin equation for the population shares . . . . . . . . . . . 4 2.3 Price dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 ML estimation approach . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Stock market estimations 8 3.1 982 estimates from TSE . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Two sources of variability . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 The bootstrap re-estimation approach . . . . . . . . . . . . . . . . . 11 3.4 Results from the bootstrap estimations . . . . . . . . . . . . . . . . . 12 3.5 A structural interpretation . . . . . . . . . . . . . . . . . . . . . . . . 14 3.6 Constructing a 2D confidence region . . . . . . . . . . . . . . . . . . 15 4 Comparison with foreign exchange markets 20 4.1 A critical question . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 FX estimates and class S . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 A structural interpretation . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4 Specifying class F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5 Comparing class F and S . . . . . . . . . . . . . . . . . . . . . . . . 24 4.6 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5 On the fatness of the tails 27 5.1 A source of mistrust against estimation results . . . . . . . . . . . . 27

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5.2 The Hill tail index of a distribution . . . . . . . . . . . . . . . . . . . 28 5.3 Reconciliation of Hill and theoretical tail index . . . . . . . . . . . . 29 5.4 Isoclines of the Hill tail index . . . . . . . . . . . . . . . . . . . . . . 30 5.5 Hill estimators of the artificial data . . . . . . . . . . . . . . . . . . . 31 5.6 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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1. Architecture Agents: fundamentalists — noise traders

• rdinary

random walk demand

1

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Contagion dynamics

(Kirman-like)

characterized by eps1, eps2 Price mechanism

(Walrasian)

daily returns

Equilibrium pdf (closed-form solution) ML estimation

• f eps1, eps2

Distinguish S and F markets tail indices TSE data (982) artificial data

2

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2. The model

2.1. Transition probabilities

Individual probabilities to switch between FT and NT within time span ∆τ, given n NT’s and population size N: π[ (FT) → (NT) | n, ∆τ ] = ∆τ [a1 + nb] π[ (NT) → (FT) | n, ∆τ ] = ∆τ [a2 + (N −n)b] ∆τ microscopic adjustment period (∆τ →0 as N →∞) a1, a2 autonomous switching (asymmetric) b herding component Note: [ . . . ] ↑ as total N ↑ (“global coupling”) ⇒ avoids gradual fading out of ‘interesting dynamics” as N → ∞

3

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2.2. The Langevin equation for the population shares

Elementary reasoning (no FPE): nt+∆τ governed by difference between two binomial distributions between two normal distributions, for large N Fix macroscopic time unit = 1 (one day). z := n/N share of noise traders (0 ≤ z ≤ 1.) Derive Langevin equation directly: zt+1 = zt − (a1 + a2) (zt − ¯ z) +

• 2b zt(1 − zt) ξt

ξt ∼ NID(0, 1) ¯ z = a1 / (a1 + a2) = mean value of NT

4

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2.3. Price dynamics

pt market price of the asset (in logs) pf fundamental value (constant) EDf,t = −(N − nt) αf (pt − pf) EDn,t = nt αn λt , λt − λt−1 =: ηt ∼ UID(−1, +1) Market clearing, EDf,t + EDn,t = 0, − → pt = pf + zt 1 − zt λt (ρ = αf/αn) Returns rt = pt − pt−1, rt = ρ [ zt 1 − zt λt − zt−1 1 − zt−1 λt−1 ]

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“Adiabatic approximation” (NT mood λt changes faster than composition zt): rt = ρ zt 1 − zt ηt , ηt ∼ UID(−1, +1)

• Properties of zt-process carry over to rt.
• This equation (rt together with zt) used for

simulations.

• Problem for estimation: zt is unobserved.
• Prob. distribution of rt is analytically tractable

− → admits straightforward estimation approach.

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2.4. ML estimation approach

Closed-form expression (via FPE) of equilibrium probability density function of returns: pe = pe(r; ε1, ε2, ρ)

(involves incomplete beta function)

where ε1 := a1/b, ε2 := a2/b ρ = 2(ε2 − 1)/ε1 ⇒ E(|r|) = 1 ⇒

• equil. pdf

pe = pe(r; ε1, ε2) Typically exhibits fat tails ! Log-likelihood function (approx. of ‘true’ likelihood), for normalized return series {remp

t

}T

t=1 :

ℓ(ε1, ε2; {remp

t

}) :=

T

• t=1

ln[ pe(remp

t

; ε1, ε2) ] = max

ε1,ε2 ! 7

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3. Stock market estimations

3.1. 982 estimates from TSE

First test of the model: estimations are meaningful: Quite dispersed !?

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3.2. Two sources of variability

• Differences in the “true” parameters across shares.
• Finite-sample properties.

First quantitative assessment: LR-test (must be based on a given return series) Choose “representative” reference series {rref

t

}, with estimated coefficients (very) close to εm

1 , εm 2 .

Then, a pair ε1, ε2 is not statistically different from εm

1 , εm 2

if LR(ε1, ε2) := 2 [ ℓ(εm

1 , εm 2 ; {rref t

}) − ℓ(ε1, ε2; {rref

t

}) ] ≤ χ2

2;0.95 = 5.99 9

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Shaded area identifies a class of stocks at TSE (comprises roughly one-third) But note: based on asymptotic theory (T =∞), while T emp ≈ 5768.

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3.3. The bootstrap re-estimation approach

Idea: To study small-sample properties, generate artificial data of returns and re-estimate these series.

• Fix εs

1 = 11.0, εs 2 = 4.5 ( ≈ median of ˆ

ε1, ˆ ε2 ), T = 5768.

• Generate series of random terms {ξt, ηt} and thus, on

the basis of εs

1, εs 2, returns {rt}.

• Re-estimate ε1, ε2 from this sample (generally different

from true parameters εs

1, εs 2 since T <∞).

• Do this 5000 times, getting 5000 bootstrap estimates

ˆ εb

1, ˆ

εb

2 (b = 1, . . . , 5000).

• By construction, set of these {ˆ

εb

1, ˆ

εb

2} constitutes

• ne class of stocks (95% of them).

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3.4. Results from the bootstrap estimations

Scatter plot and marginal distributions from 5000 bootstrap estimates.

Note: Blue lines in lower panels depict the frequency dis- tributions from the bootstrap estimations, black lines those from the 982 empirical estimations.

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• Area covered by pairs ˆ

εb

1, ˆ

εb

2

• High similarity of marginal distributions

suggest that most of the 982 empirical estimates, and not

• nly one-third, could be assigned to one class

(the class constituted by εs

1, εs 2).

That is, variability of TSE estimates is, “to a large extent”, due to finite-sample properties (and not to differences in the “true” parameters).

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3.5. A structural interpretation

Observation before going on: Confidence band for average %-share of noise traders: 48.2 ≤ ¯ z ≤ 97.7 (median = 74.2%) Message: on most share markets at TSE, fundamentalists tend to be in a minority position.

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3.6. Constructing a 2D confidence region

Task: quantify “to a large extent”. Set up a confidence region of pairs (ε1, ε2) from 95% of the scatter plot of boot- strap estimates. According to what criterion:

• 1. discard “extreme” pairs (ˆ

εb

1, ˆ

εb

2);

• 2. discard pairs in rectangles with low frequencies ;
• 3. adopt LR statistic.

Options 1 and 2 require judgement (which may not be shared). Hence Option 3.

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Problem 1: Option 3 is anchored to a definite series {rt}. So take one of the artificial {rb

t}. But isn’t any such series

purely arbitrary? Solution: take a consistent series {rb⋆

t },

whose estimation yields precisely the parameters from which it was generated: (ˆ εb⋆

1 , ˆ

εb⋆

2 ) = (εs 1, εs 2) = (11.0, 4.5)

Five such series do exist (precision at least 5 digits). Problem 2: χ2

2;0.95 = 5.99 on RHS of LR-inequality is too

low: there would be less than 95% of the (ˆ εb

1, ˆ

εb

2) inside the

thus defined set. Solution: increase number on RHS until inequality is sat- isfied by exactly 95% of the (ˆ εb

1, ˆ

εb

2). 16

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Doing this, the confidence region is given by set of all (ε1, ε2) satisfying LR(ε1, ε2) := 2 [ ℓ(ˆ εb⋆

1 , ˆ

εb⋆

2 ; {rb⋆ t }) − ℓ(ε1, ε2; {rb⋆ t }) ]

= 2 [ ℓ(εs

1, εs 2; {rb⋆ t }) − ℓ(ε1, ε2; {rb⋆ t }) ]

≤ ˜ χ2

2;0.95 := 34.4 17

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Class S of parameter pairs ε1, ε2 (the shaded area).

Note: The dots are the empirical TSE estimates ˆ ε1, ˆ ε2. 11.6% of them are outside set S.

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Conclusion on a firm methodological basis: The large range of the empirical estimates is not exclu- sively, but mainly or even predominantly, due to the finite sample size of the time series.

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4. Comparison with foreign exchange markets

4.1. A critical question

Isn’t the set S much too large, and thus rather uninformative? To check, estimate FX markets.

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4.2. FX estimates and class S

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4.3. A structural interpretation

Points above the zz-line imply ¯ z ≤ 48.2%. Hence, FX markets tend to have, on average, a significantly larger share of fundamentalist traders.

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4.4. Specifying class F

Generate 5000 artificial return series on the basis of (εf

1, εf 2) = (5.0, 10.0).

Construct 2D confidence region in the same way as before.

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4.5. Comparing class F and S

Sets S and F are largely disjunct, but not completely.

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Degree of overlapping:

• 20.3% of the bootstrap estimates ˆ

εb

1, ˆ

εb

2

• riginating with εf

1, εf 2 fall into class S.

• 16.3% of the bootstrap estimates ˆ

εb

1, ˆ

εb

2

• riginating with εs

1, εs 2 fall into class F.

• 20.4% of the TSE estimates ˆ

ε1, ˆ ε2 fall into class F.

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4.6. Predictions

• (ε1, ε2)-estimates for stock markets tend to fall into

class S.

• (ε1, ε2)-estimates for FX markets tend to fall into class

F.

• 16 – 20 percent of them contained in inconclusive re-

gion (find reasons why these estimates are not so clear- cut??). To be tested in further work (NYSE).

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5. On the fatness of the tails

5.1. A source of mistrust against estimation results

Model-implied fatness contradicts empirical measures of fatness.

• Tail index := highest finite moment of prob. distribu-

tion = ε2 (analytical result).

• Model estimations yield ε2 >

> 5 for FX markets.

• Hill estimators of tail index ≤ 5 for FX markets.

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5.2. The Hill tail index of a distribution

With respect to absolute returns v =|r|; specifying the tail by a threshold vo, referring to empirical series sampling entries vk in the tail (k ∈ index set I), Hill estimator ˆ αH = [(1/m)

k∈I(ln vk − ln vo)]−1

(fatter tails associated with lower ˆ αH). Concept of Hill index αH refers to known pdf, mimics the sampling process, and advances a formula for this. Putting vo = 3, compute: εs

1=11, εs 2=4.5

εf

1 =5, εf 2 =10

αH(vo; ε1, ε2) : 3.10 4.07

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5.3. Reconciliation of Hill and theoretical tail index

αH(vo; ε1, ε2) quite distinct from theoretical tail index ε2, for moderate thresholds vo. αH(vo; ε1, ε2) → ε2 as vo → large (25%, 50%). Power law takes effect in too extreme a range of returns, ε2 as “the” tail index is too academic. For practical reasons, αH or ˆ αH with vo ≈ 3 (say) are more appropriate concepts.

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5.4. Isoclines of the Hill tail index

Shape qualitatively similar to classes S and F. Hill tail index as a complementary, and atheoretical, characterization of S and F?

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5.5. Hill estimators of the artificial data

95% : 2.67 − 4.13 empS 95% : 2.42 − 3.70 S 19.5% : 3.26 − 3.70 17% : 3.26 − 3.70 F 95% : 3.26 − 5.11 Quantile intervals of the Hill estimator (vo = 3).

Note: empS refers to the frequency distribution of the Hill estimators of the 982 Japanese stocks; S and F to the dis- tributions of ˆ αH from 5000 simulations of εs

1, εs 2 and εf 1, εf 2,

respectively.

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5.6. Predictions

• ˆ

αH(vo) for stock markets tend to be lower than for FX markets.

• Stock markets

tend to have fatter tails than FX markets.

• Degree of overlapping: ≈ 20 percent.

Note: Hill estimator is an atheoretical measure, but generation of artificial data to substantiate the statement resorts to a theoretical model.

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