Applications of the generalized nonparametric method to the analysis - PowerPoint PPT Presentation

Applications of the generalized nonparametric method to the analysis of a stock market crash N. I. Klemashev Lomonosov Moscow State University September 2016 N. I. Klemashev 1/26

Plan 1. Chinese stock market crash in Summer 2015. 2. Generalized nonparametric method. 3. Analysis. 4. Conclusions. 5. References. N. I. Klemashev 2/26

Chinese stock market crash in 2015 Figure 1: Price indices. N. I. Klemashev 3/26

Chinese stock market crash in 2015 Figure 2: Price indices. N. I. Klemashev 4/26

Chinese stock market crash in 2015 Figure 3: Volume indices. N. I. Klemashev 5/26

Chinese stock market crash in 2015 Figure 4: Volume indices. N. I. Klemashev 6/26

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? N. I. Klemashev 7/26

Rationalizability of trade statistics t =1 , P t ∈ R m ++ , X t ∈ R m Trade statistics: TS = { ( P t , X t ) } T + . Φ H – set of all functions defined on R m + which are positively homogeneous of degree 1, continuous, concave, non-satiated, and taking nonzero values for all points in R m + . TS is ratinalizable (in Φ H ) if there exists F ∈ Φ H such that for all t ∈ { 1 , . . . , T } X t ∈ Arg max { F ( X ) | X ∈ R m P t , X P t , X t � � � � + , } . (1) � N. I. Klemashev 8/26

Afriat-Varian theorem on rationalizability in Φ H The following statements are equivalent: 1) trade statistics { ( P t , X t ) } T t =1 is rationalizable in Φ H ; 2) there exist numbers λ t > 0 ( t = 1 , . . . , T ), such that � λ τ � P τ , X τ � , λ t � P t , X τ � ∀ t , τ = 1 , . . . , T ; (2) 3) trade statistics { ( P t , X t ) } T t =1 satisfies Homothetic Axiom of Revealed Preference (HARP), which means that for all subsets of indices { t 1 , . . . , t k } from { 1 , . . . , T } P t 1 , X t 2 � � P t 2 , X t 3 � P t k , X t 1 � � � . . . � P t 1 , X t 1 � � P t 2 , X t 2 � P t k , X t k � � � . . . (3) � 4) the function u ( X ) = min τ ∈{ 1 ,..., T } { λ τ � P τ , X �} , where t =1 satisfy (2) and λ t > 0 for all t ∈ { 1 , . . . , T } , { λ t } T rationalizes trade statistics { ( P t , X t ) } T t =1 . N. I. Klemashev 9/26

Kon¨ us-Divisia indices (nonparametric method) trade statistics { ( P t , X t ) } T t =1 satisfies HARP; λ t > 0 ( t = 1 , . . . , T ) satisfy � λ τ � P τ , X τ � , λ t � P t , X τ � ∀ t , τ = 1 , . . . , T ; us-Divisia consumption index: F t = λ t � P t , X t � ; Kon¨ us-Divisia price index: Q t = 1 Kon¨ λ t . N. I. Klemashev 10/26

Generalized nonparametric method trade statistics TS = { ( P t , X t ) } T t =1 does not satisfy HARP; irrationality index – minimum ω such that TS satisfies HARP( ω ): for all subsets of indices { t 1 , . . . , t k } from { 1 , . . . , T } P t 1 , X t 2 � � P t 2 , X t 3 � P t k , X t 1 � � � . . . � � 1 P t 1 , X t 1 � � P t 2 , X t 2 � P t k , X t k � � � . . . ω k Afriat-Varian theorem: TS satisfies HARP( ω ) iff there exist λ t > 0 ( t = 1 , T ) such that λ t � P t , X t � � ωλ τ � P τ , X t � . ∀ t , τ = 1 , T N. I. Klemashev 11/26

Forecasting Forecasting set: K ( P ; TS , ω ) = { X ∈ R m + | TS ∪ { ( P , X ) } satisfies HARP( ω ) } Theorem [Grebennikov, Shananin, 2008] Assume that the trade statistics TS = { ( P t , X t ) } T t =1 satisfies HARP( ω ) with ω � 1 and P is not equal to one of P t s. Then X ∈ R m + | γ s ( P , ω ) � P s , X � � � P , X � ∀ s ∈ { 1 , . . . , T } � � K ( P ; TS , ω ) = , N. I. Klemashev 12/26

Forecasting where � � C ∗ ω − k − 1 C tt 1 C t 1 t 2 . . . C t k − 1 t k C t k s ts ( ω ) = max � � � { t 1 , . . . , t k } ⊂ { 1 , . . . , T } , k � 0 , ω 2 � P , X t � � � γ s ( P , ω ) = min , C ∗ ts ( ω ) � P t , X t � t ∈{ 1 ,..., T } and C t τ = � P τ , X τ � � P t , X τ � . (4) N. I. Klemashev 13/26

Analysis – irrationality indices MW Figure 5: 12-months moving window for irrationality indices. N. I. Klemashev 14/26

Analysis – acceptable level of irrationality Figure 6: 31-days moving window for log-irrationality index. N. I. Klemashev 15/26

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. N. I. Klemashev 16/26

Analysis – irrationality index � � � P τ , X τ � log-Paasche indices: c t τ = log . � P t , X τ � log( ω ) → min ω,λ t , (5) log( ω ) + log( λ t ) − log( λ τ ) � c t τ , ( t , τ = 1 , T , t � = τ ) (6) log( ω ) � 0 (7) N. I. Klemashev 17/26

Analysis – irrationality indices for pairs of periods T T � � log( ω t τ ) → min ω t τ ,λ t , (8) t =1 τ =1 τ � = t log( ω t τ ) + log( λ t ) − log( λ τ ) � c t τ , ( t , τ = 1 , T , t � = τ ) (9) log( ω t τ ) � 0 ( t , τ = 1 , T , t � = τ ) (10) N. I. Klemashev 18/26

Analysis – selecting periods t τ log( ω t τ ) 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 N. I. Klemashev 19/26

Analysis – selecting periods Selected periods: 2014–12; 2015–01; 2015–02; 2015–03; 2015–04. N. I. Klemashev 20/26

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 ∈ R m , (11) Y ∈ K ( P ; TS , ω ) (12) N. I. Klemashev 21/26

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 order 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. N. I. Klemashev 22/26

Checking procedure – partial projection Fix a set of stocks I . � X − Y � 2 → min Y ∈ R m , (14) Y ∈ K ( P ; TS , ω ) (15) Y i = X i ( i / ∈ I ) (16) N. I. Klemashev 23/26

The resulting set of stocks CITIC Securities Co Ltd (ticker 600030). N. I. Klemashev 24/26

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. N. I. Klemashev 25/26

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. N. I. Klemashev 26/26