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Applications of the generalized nonparametric method to the analysis of a stock market crash

• N. I. Klemashev

Lomonosov Moscow State University

September 2016

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Plan

• 1. Chinese stock market crash in Summer 2015.
• 2. Generalized nonparametric method.
• 3. Analysis.
• 4. Conclusions.
• 5. References.
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Chinese stock market crash in 2015

Figure 1: Price indices.

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Chinese stock market crash in 2015

Figure 2: Price indices.

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Chinese stock market crash in 2015

Figure 3: Volume indices.

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Chinese stock market crash in 2015

Figure 4: Volume indices.

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Structure of Chinese stock market

Two types of agents; Major investors;

Long-term strategies; Objective: savings and low-risk profit;

Minor investors (speculators);

Short-term strategies; Objective: fast and risky profit;

Two types of preferences; Two types of utility functions. How to reveal minor investors?

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Rationalizability of trade statistics

Trade statistics: TS = {(Pt, X t)}T

t=1, Pt ∈ Rm ++, X t ∈ Rm +.

ΦH – set of all functions defined on Rm

+ which are positively

homogeneous of degree 1, continuous, concave, non-satiated, and taking nonzero values for all points in Rm

+.

TS is ratinalizable (in ΦH) if there exists F ∈ ΦH such that for all t ∈ {1, . . . , T} X t ∈ Arg max{F(X) | X ∈ Rm

+,

• Pt, X
• Pt, X t

}. (1)

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Afriat-Varian theorem on rationalizability in ΦH

The following statements are equivalent: 1) trade statistics {(Pt, X t)}T

t=1 is rationalizable in ΦH;

2) there exist numbers λt > 0 (t = 1, . . . , T), such that λt Pt, X τ λτ Pτ, X τ , ∀t, τ = 1, . . . , T; (2) 3) trade statistics {(Pt, X t)}T

t=1 satisfies Homothetic Axiom of

Revealed Preference (HARP), which means that for all subsets of indices {t1, . . . , tk} from {1, . . . , T}

• Pt1, X t2

Pt2, X t3 . . .

• Ptk, X t1
• Pt1, X t1

Pt2, X t2 . . .

• Ptk, X tk

(3) 4) the function u(X) = minτ∈{1,...,T}{λτ Pτ, X}, where {λt}T

t=1 satisfy (2) and λt > 0 for all t ∈ {1, . . . , T},

rationalizes trade statistics {(Pt, X t)}T

t=1.

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Kon¨ us-Divisia indices (nonparametric method)

trade statistics {(Pt, X t)}T

t=1 satisfies HARP;

λt > 0 (t = 1, . . . , T) satisfy λt Pt, X τ λτ Pτ, X τ , ∀t, τ = 1, . . . , T; Kon¨ us-Divisia consumption index: F t = λt Pt, X t; Kon¨ us-Divisia price index: Qt = 1

λt .

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Generalized nonparametric method

trade statistics TS = {(Pt, X t)}T

t=1 does not satisfy HARP;

irrationality index – minimum ω such that TS satisfies HARP(ω): for all subsets of indices {t1, . . . , tk} from {1, . . . , T}

• Pt1, X t2

Pt2, X t3 . . .

• Ptk, X t1
• 1

ωk

• Pt1, X t1

Pt2, X t2 . . .

• Ptk, X tk

Afriat-Varian theorem: TS satisfies HARP(ω) iff there exist λt > 0 (t = 1, T) such that λt Pt, X t ωλτ Pτ, X t . ∀t, τ = 1, T

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Forecasting

Forecasting set: K(P; TS, ω) = {X ∈ Rm

+ | TS ∪ {(P, X)} satisfies HARP(ω)}

Theorem [Grebennikov, Shananin, 2008] Assume that the trade statistics TS = {(Pt, X t)}T

t=1 satisfies

HARP(ω) with ω 1 and P is not equal to one of Pts. Then K(P; TS, ω) =

• X ∈ Rm

+ | γs (P, ω) Ps, X P, X ∀s ∈ {1, . . . , T}

• ,
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Forecasting

where C ∗

ts(ω) = max

• ω−k−1Ctt1Ct1t2 . . . Ctk−1tkCtks
• {t1, . . . , tk} ⊂ {1, . . . , T}, k 0
• ,

γs (P, ω) = min

t∈{1,...,T}

• ω2

C ∗

ts(ω)

P, X t Pt, X t

• ,

and Ctτ = Pτ, X τ Pt, X τ . (4)

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Analysis – irrationality indices MW

Figure 5: 12-months moving window for irrationality indices.

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Analysis – acceptable level of irrationality

Figure 6: 31-days moving window for log-irrationality index.

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Analysis – Trade statistics

100 stocks; Daily data aggregated to monthly data. January 2014 – August 2015. Acceptable level of log-irrationality: 0.035. Actual log-irrationality: 0.049. Objective: Find stocks most responsible for the increased irrationality.

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Analysis – irrationality index

log-Paasche indices: ctτ = log

• Pτ, X τ

Pt, X τ

• .

log(ω) → min

ω,λt,

(5) log(ω) + log(λt) − log(λτ) ctτ, (t, τ = 1, T, t = τ) (6) log(ω) 0 (7)

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Analysis – irrationality indices for pairs of periods

T

• t=1

T

• τ=1

τ=t

log(ωtτ) → min

ωtτ,λt,

(8) log(ωtτ) + log(λt) − log(λτ) ctτ, (t, τ = 1, T, t = τ) (9) log(ωtτ) 0 (t, τ = 1, T, t = τ) (10)

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Analysis – selecting periods

t τ log(ωtτ) 2015–01 2014–01 0.076 2015–01 2014–02 0.073 2015–01 2014–03 0.065 2015–04 2014–02 0.063 2015–04 2014–03 0.058 2015–04 2014–01 0.056 2014–12 2014–01 0.051 2015–03 2014–03 0.051 2014–12 2014–03 0.051 2015–03 2014–02 0.049 2015–03 2014–01 0.049 2014–12 2014–02 0.041 2015–02 2014–02 0.040 2015–02 2014–01 0.039 2014–02 2014–12 0.038 2015–02 2014–03 0.037

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Analysis – selecting periods

Selected periods: 2014–12; 2015–01; 2015–02; 2015–03; 2015–04.

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Analysis – volume projection

Given Trade statistics satisfying HARP(ω), An observation (P, X), find the projection of X on K(P; TS, ω): X − Y 2 → min

Y ∈Rm,

(11) Y ∈ K(P; TS, ω) (12)

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Selecting stocks

Let us split the set {1, . . . , T} into two subsets with empty intersection: {1, . . . , T} = V ∗ ∪ ({1, . . . , T} \ V ∗), (13) where V ∗ is the set of periods selected on the previous step. We

• rder elements of V ∗ in ascending order, project the observed

volumes X t for t ∈ V ∗ with consequent adding the projected volumes to the trade statistics and collect the differences between the observed volumes and the projected ones.

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Checking procedure – partial projection

Fix a set of stocks I. X − Y 2 → min

Y ∈Rm,

(14) Y ∈ K(P; TS, ω) (15) Yi = Xi (i / ∈ I) (16)

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The resulting set of stocks

CITIC Securities Co Ltd (ticker 600030).

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Conclusions

New method for analysis of stock marker crises. It allows an analyst to select only few stocks for further analysis. The method is computationally efficient. We managed to reduce the number of stocks for detailed analysis from one hundred to just one.

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References

Varian, H. (1983). Non-parametric tests of consumer

• behavior. Review of Economic Studies, 50(1):99–110.

Grebennikov V., Shananin A. (2009). Generalized nonparametrical method: Law of demand in problems of

• forecasting. Mathematical Models and Computer Simulations,

1(5):591–604. Shananin A. (2009). Integrability problem and the generalized nonparametric method for the consumer demand analysis (Russian). Proceedings of MIPT, 1(4):84–98. Klemashev N., Shananin A. (2016). Inverse problems of demand analysis and their applications to computation of positively-homogeneous Kon¨ us-Divisia indices and forecasting. Journal of Inverse and Ill-posed Problems, 24(4):367–391.

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