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Pairs Trading Ch6. Pairs Selection in Equity Markets 郭孟鑫. Stationary and Nonstationary. Stationary. For all t-s, and t-s-j. Stationary and Nonstationary. Spurious regression. Downward bias Using unit root test to verify. Introduction. In chapter 5, we need three steps

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Pairs trading ch6

Pairs TradingCh6

PairsSelectioninEquityMarkets

郭孟鑫


Stationary and nonstationary
StationaryandNonstationary

  • Stationary

Forallt-s,andt-s-j


Stationary and nonstationary1
StationaryandNonstationary

  • Spuriousregression.

    • Downwardbias

  • Usingunitroottesttoverify.


Introduction
Introduction

Inchapter5,weneedthreesteps

  • Identificationofthestockpairs

  • Cointegrationtesting

  • Tradingruleformulation

    Thischapterwillfocusoncointegrationtestingandfurtheranalysis.


Introduction1
Introduction

  • WedrawparallelsbetweenthecommontrendsmodelforcointegrationandtheideaofAPT.


Common trends cointegration model
CommonTrendsCointegrationModel

  • Inference1:Inacointegrationsystemwithtwotimeseries,theinnovationssequencesderivedfromthecommontrendcomponentsmustperfectlycorrelated.(correlationmustbe+1or-1)


Common trends cointegration model1
CommonTrendsCointegrationModel


Common trends cointegration model2
CommonTrendsCointegrationModel

  • Inference2: Thecointegrationcoefficientmaybeobtainedbyaregressionoftheinnovationsequencesofthecommontrendsagainsteachother.


Discussion
Discussion

  • First,theinnovationsequencesderivedfromthecommontrendsofthetwoseriesmustbeidenticaluptoascalar.

  • Next,thespecificcomponentsofthetwoseriesmustbestationary.


Common trends model and apt
CommonTrendsmodelandAPT

Isthereturnduetothenonstationarytrendcomponent.

Isthereturnduetothestationarycomponent.


Common trends model and apt1
CommonTrendsmodelandAPT

  • RecallingtheAPT,thestockreturnsmaybeseparatedintocommonfactorreturnsandspecificreturns.

  • Ifthetwostockssharethesameriskfactorexposureprofile,thenthecommonfactorreturnsforboththestocksmustbethesame.


Common trends model and apt2
CommonTrendsmodelandAPT

  • Observation1: A pairs of stocks with the same risk factor exposure profile satisfies the necessary conditions for cointegration.

    • Condition 1: The factor exposure vectors in this case are identical up to scalar.

      Stock A:

      Stock B:


Common trends model and apt3
CommonTrendsmodelandAPT

  • If is the factor returns vector, and and are the specific returns for the stocks A and B.

    then,

    and,


Common trends model and apt4
CommonTrendsmodelandAPT

  • Condition 2: Consider the linear combination of the returns.

    Where,

    If A and B are cointegrated, then the common factor return becomes zero.


Common trends model and apt5
CommonTrendsmodelandAPT

  • Summary

    • The common factor return of a long-short portfolio of the two stocks is zero and the integration of the specific returns of the stocks is stationary, then the two stocks are cointegration.


The distance measure
The Distance Measure

  • The closer the absolute value of this measure to unity, the greater will be the degree of co-movement.


Interpreting the distance measure
Interpreting The Distance Measure

  • Calculating the Cosine of the Angle between Two Vectors.


Interpreting the distance measure1
Interpreting The Distance Measure

  • Appendix:EigenvalueDecomposition

Let

Disadiagonalmatrixwiththeeigenvalues

Uisamatrixwitheachcolumncorrespondingto

aneigenvectors.


Interpreting the distance measure2
Interpreting The Distance Measure

  • Geometric interpretation

    • Key to doing the geometric interpretation is the idea of eigenvalue decomposition.



Reconciling theory and practice
Reconciling Theory And Practice

  • Deviations from Ideal Conditions

    • When two stocks are not have their factor exposures perfectly aligned.

      Signal-to-Noise(weneedbigSNR)


Example
Example

  • ConsiderthreestocksA,B,andCwithfactorexposuresinatwofactorasfollows:


Example1
Example

  • Step1:Calculatethecommonfactorvarianceandcovariance.


Example2
Example

  • Step2:Calculatethecorrelation(absolutevalueofcorrelationisthedistancemeasure)

    • Ifwehavetochooseonepairforpurposetrading,ourchoicewouldthereforebethepair(A,B)


Example3
Example

  • Step3:Calculatethecointegrationcoefficient

    • Wewilldiscussonthenextchapter


Example4
Example

  • Step4:Calculatetheresidualcommonfactorexposeinpairedportfolio.Thisistheexposurethatcausesmeandrift.


Example5
Example

  • Step5:Calculatethecommonfactorportfoliovariance/varianceofresidualexposure.


Example6
Example

  • Step6:Calculatethespecificvarianceoftheportfolio.(assumethespecificvarianceforallofthestockstobe0.0016)


Example7
Example

  • Step7:CalculatetheSNRratiowithwhitenoiseassumptionsforresidualstockreturn.


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