A Simple Asymmetric Herding Model Home Page to Distinguish Title Page Contents Among Different Types ◭◭ ◮◮ of Financial Markets ◭ ◮ Page 1 of 32 Go Back Simone Alfarano Full Screen Reiner Franke Close University of Kiel Quit Prepared for CM–Meeting, Marseille, Oct. 2006
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 Home Page 2.4 ML estimation approach . . . . . . . . . . . . . . . . . . . . . . . . . 7 Title Page 3 Stock market estimations 8 3.1 982 estimates from TSE . . . . . . . . . . . . . . . . . . . . . . . . . 8 Contents 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 Page 2 of 32 4 Comparison with foreign exchange markets 20 Go Back 4.1 A critical question . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 FX estimates and class S . . . . . . . . . . . . . . . . . . . . . . . . 21 Full Screen 4.3 A structural interpretation . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4 Specifying class F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Close 4.5 Comparing class F and S . . . . . . . . . . . . . . . . . . . . . . . . 24 4.6 Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Quit 5 On the fatness of the tails 27 5.1 A source of mistrust against estimation results . . . . . . . . . . . . 27
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 Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 3 of 32 Go Back Full Screen Close Quit
1. Architecture Agents: Home Page fundamentalists — noise traders ordinary random walk Title Page demand Contents ◭◭ ◮◮ ◭ ◮ Page 1 of 32 Go Back Full Screen Close Quit 1
Price mechanism Contagion dynamics ( Walrasian ) ( Kirman -like) characterized by eps1 , eps2 daily returns Home Page Title Page Contents tail Equilibrium pdf indices (closed-form solution) ◭◭ ◮◮ ◭ ◮ TSE data (982) ML estimation Page 2 of 32 artificial data of eps1 , eps2 Go Back Full Screen Distinguish S and F markets Close Quit 2
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 : Home Page Title Page π [ (FT) → (NT) | n, ∆ τ ] = ∆ τ [ a 1 + nb ] Contents π [ (NT) → (FT) | n, ∆ τ ] = ∆ τ [ a 2 + ( N − n ) b ] ◭◭ ◮◮ ∆ τ microscopic adjustment period (∆ τ → 0 as N →∞ ) ◭ ◮ a 1 , a 2 autonomous switching (asymmetric) b herding component Page 3 of 32 Note: [ . . . ] ↑ as total N ↑ (“global coupling”) Go Back ⇒ avoids gradual fading out of Full Screen ‘interesting dynamics” as N → ∞ Close Quit 3
2.2. The Langevin equation for the population shares Elementary reasoning (no FPE): n t +∆ τ governed by difference between two binomial distributions Home Page between two normal distributions, for large N Title Page Contents Fix macroscopic time unit = 1 (one day). ◭◭ ◮◮ z := n/N share of noise traders (0 ≤ z ≤ 1.) ◭ ◮ Derive Langevin equation directly: Page 4 of 32 z t +1 = z t − ( a 1 + a 2 ) ( z t − ¯ z ) Go Back � + 2 b z t (1 − z t ) ξ t ξ t ∼ NID(0 , 1) Full Screen z = a 1 / ( a 1 + a 2 ) = mean value of NT ¯ Close Quit 4
2.3. Price dynamics p t market price of the asset (in logs) p f fundamental value (constant) Home Page ED f,t = − ( N − n t ) α f ( p t − p f ) Title Page ED n,t = n t α n λ t , λ t − λ t − 1 =: η t ∼ UID( − 1 , +1) Contents Market clearing, ED f,t + ED n,t = 0, − → ◭◭ ◮◮ z t p t = p f + λ t ( ρ = α f /α n ) ◭ ◮ 1 − z t Page 5 of 32 Returns r t = p t − p t − 1 , z t z t − 1 Go Back r t = ρ [ λ t − λ t − 1 ] 1 − z t 1 − z t − 1 Full Screen Close Quit 5
“Adiabatic approximation” (NT mood λ t changes faster than composition z t ): z t r t = ρ η t , η t ∼ UID( − 1 , +1) 1 − z t Home Page Title Page • Properties of z t -process carry over to r t . Contents • This equation ( r t together with z t ) used for ◭◭ ◮◮ simulations. ◭ ◮ • Problem for estimation: z t is unobserved. Page 6 of 32 • Prob. distribution of r t is analytically tractable − → admits straightforward estimation approach. Go Back Full Screen Close Quit 6
2.4. ML estimation approach Closed-form expression (via FPE) of equilibrium probability density function of returns: Home Page p e = p e ( r ; ε 1 , ε 2 , ρ ) (involves incomplete beta function) Title Page where ε 1 := a 1 /b , ε 2 := a 2 /b Contents ρ = 2( ε 2 − 1) /ε 1 ⇒ E ( | r | ) = 1 ◭◭ ◮◮ ⇒ equil. pdf p e = p e ( r ; ε 1 , ε 2 ) ◭ ◮ Typically exhibits fat tails ! Page 7 of 32 Log-likelihood function (approx. of ‘true’ likelihood), Go Back for normalized return series { r emp } T t =1 : t Full Screen T � ℓ ( ε 1 , ε 2 ; { r emp ln[ p e ( r emp Close } ) := ; ε 1 , ε 2 ) ] = max ε 1 ,ε 2 ! t t t =1 Quit 7
3. Stock market estimations 3.1. 982 estimates from TSE First test of the model: estimations are meaningful: Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 8 of 32 Go Back Full Screen Close Quite dispersed !? Quit 8
3.2. Two sources of variability • Differences in the “true” parameters across shares. • Finite-sample properties. Home Page First quantitative assessment: LR-test (must be based on a given return series) Title Page Choose “representative” reference series { r ref Contents } , t with estimated coefficients (very) close to ε m 1 , ε m 2 . ◭◭ ◮◮ Then, a pair ε 1 , ε 2 is not statistically different from ε m 1 , ε m ◭ ◮ 2 if Page 9 of 32 2 ; { r ref } ) − ℓ ( ε 1 , ε 2 ; { r ref LR( ε 1 , ε 2 ) := 2 [ ℓ ( ε m 1 , ε m } ) ] Go Back t t χ 2 ≤ 2;0 . 95 = 5 . 99 Full Screen Close Quit 9
Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 10 of 32 Shaded area identifies a class of stocks at TSE Go Back (comprises roughly one-third) Full Screen Close But note: based on asymptotic theory ( T = ∞ ), while T emp ≈ 5768. Quit 10
3.3. The bootstrap re-estimation approach Idea: To study small-sample properties, generate artificial data of returns and re-estimate these series. Home Page • Fix ε s 1 = 11 . 0, ε s 2 = 4 . 5 ( ≈ median of ˆ ε 1 , ˆ ε 2 ), T = 5768. Title Page Contents • Generate series of random terms { ξ t , η t } and thus, on the basis of ε s 1 , ε s 2 , returns { r t } . ◭◭ ◮◮ • Re-estimate ε 1 , ε 2 from this sample (generally different ◭ ◮ from true parameters ε s 1 , ε s 2 since T < ∞ ). Page 11 of 32 • Do this 5000 times, getting 5000 bootstrap estimates Go Back ε b ε b ˆ 1 , ˆ 2 ( b = 1 , . . . , 5000). Full Screen ε b ε b • By construction, set of these { ˆ 1 , ˆ 2 } constitutes Close one class of stocks (95% of them). Quit 11
3.4. Results from the bootstrap estimations Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 12 of 32 Go Back Scatter plot and marginal distributions Full Screen from 5000 bootstrap estimates. Close Note : Blue lines in lower panels depict the frequency dis- tributions from the bootstrap estimations, black lines those Quit from the 982 empirical estimations. 12
ε b ε b • Area covered by pairs ˆ 1 , ˆ 2 • High similarity of marginal distributions Home Page suggest that most of the 982 empirical estimates, and not Title Page only one-third, could be assigned to one class Contents (the class constituted by ε s 1 , ε s 2 ). ◭◭ ◮◮ That is, variability of TSE estimates is, “to a large extent”, ◭ ◮ due to finite-sample properties (and not to differences in Page 13 of 32 the “true” parameters). Go Back Full Screen Close Quit 13
3.5. A structural interpretation Observation before going on: Confidence band Home Page for average %-share of noise traders: 48 . 2 ≤ ¯ z ≤ 97 . 7 Title Page (median = 74.2%) Contents Message: on most share markets at TSE, fundamentalists ◭◭ ◮◮ tend to be in a minority position. ◭ ◮ Page 14 of 32 Go Back Full Screen Close Quit 14
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