A Bayesian Approach to Stein-Optimal Covariance Matrix Estimation

Transcription

1 A Bayesan Approach to Sten-Optmal Covarance Matrx Estmaton Ben Gllen Calforna Insttute of Technology Unversty of Calforna Irvne Econometrcs Semnar November 5 013

2 Estmatng Large Covarance Matrces Central parameter n portfolo mean-varance optmzaton and hgh-dmensonal IV problem

3 Estmatng Large Covarance Matrces Central parameter n portfolo mean-varance optmzaton and hgh-dmensonal IV problem Dffcult to Estmate due to Curse of Dmensonalty N assets requre estmatng N(N+1)/ parameters Number of equtes under consderaton commonly exceeds number of observatons (T)

4 Estmatng Large Covarance Matrces Central parameter n portfolo mean-varance optmzaton and hgh-dmensonal IV problem Dffcult to Estmate due to Curse of Dmensonalty N assets requre estmatng N(N+1)/ parameters Number of equtes under consderaton commonly exceeds number of observatons (T) Sample Covarance Matrx Estmator Problems Estmator s effectvely undentfed when N > T Frequently non-nvertble n fnte samples Performs poorly n out of sample applcatons

5 Sparse Approach to Covarance Estmaton Lnear k-factor structure for the covarance matrx r F F F t 1 1 t k kt t t t cov F t Fjt j cov Ft jt 0 cov t jt 0 j j Jacobs Levy & Markowtz (013) Sharpe (1963) Markowtz & Perold (1985)

6 Sparse Approach to Covarance Estmaton Lnear k-factor structure for the covarance matrx r F F F t 1 1 t k kt t t t cov F t Fjt j cov Ft jt 0 cov t jt 0 j j Impled covarance matrx structure & plug-n estmator BB Bˆˆ Bˆˆ F S F Jacobs Levy & Markowtz (013) Sharpe (1963) Markowtz & Perold (1985)

7 Sparse Approach to Covarance Estmaton Lnear k-factor structure for the covarance matrx r F F F t 1 1 t k kt t t t cov F t Fjt j cov Ft jt 0 cov t jt 0 j j Impled covarance matrx structure & plug-n estmator BB Bˆˆ Bˆˆ F S F Factor structure facltates varance optmzaton Jacobs Levy & Markowtz (013) Sharpe (1963) Markowtz & Perold (1985)

8 Sparse Approach to Covarance Estmaton Lnear k-factor structure for the covarance matrx r F F F t 1 1 t k kt t t t cov F t Fjt j cov Ft jt 0 cov t jt 0 j j Impled covarance matrx structure & plug-n estmator BB Bˆˆ Bˆˆ Factor structure facltates varance optmzaton F S F Number of factors and factor models can be selected usng lkelhood-rato type approaches Connor & Koracyzk (1996) Ba & Ng (00) Jacobs Levy & Markowtz (013) Ng (008) Sharpe (1963) Markowtz & Perold (1985)

9 Sparse Approach to Covarance Estmaton Lnear k-factor structure for the covarance matrx r F F F t 1 1 t k kt t t t cov F t Fjt j cov Ft jt 0 cov t jt 0 j j Impled covarance matrx structure & plug-n estmator BB Bˆˆ Bˆˆ Factor structure facltates varance optmzaton F S F Number of factors and factor models can be selected usng lkelhood-rato type approaches Recent approaches apply LASSO model-selecton devces to select factor model Connor & Koracyzk (1996) Ba & Ng (00) Lam & Fan (009) Ng (008) Fan Lao & Mncheva (011) Fan Lv & Q (011)

10 Ledot & Wolf Shrnkage Approach Model structured covarance matrx as a pror belef Estmator shrnks unbased covarance matrx estmator toward a target covarance matrx: ˆ * S (1 ) S F Ledot & Wolf ( ) Ledot & Wolf (01 013)

11 Ledot & Wolf Shrnkage Approach Model structured covarance matrx as a pror belef Estmator shrnks unbased covarance matrx estmator toward a target covarance matrx: * ˆ S (1 ) F S Optmzes Shrnkage Intensty () to mnmze expected loss under Frobenus Norm: * arg mn E SF 1S 01 Ledot & Wolf ( ) Ledot & Wolf (01 013)

12 Ledot & Wolf Shrnkage Approach Model structured covarance matrx as a pror belef Estmator shrnks unbased covarance matrx estmator toward a target covarance matrx: * ˆ S (1 ) F S Optmzes Shrnkage Intensty () to mnmze expected loss under Frobenus Norm: * arg mn E SF 1S 01 Shrnkage towards dentty matrx corresponds to standard regularzaton technques Tkhonov regularzaton for covarance matrx nverse Quadratc constrant on optmal portfolo weghts Ledot & Wolf ( ) Ledot & Wolf (01 013) Smon & Smon (011) Garlapp Uppal & Wang (010) Fan Zhang & Yu (01)

13 Bayesan Regresson Model of Covarance Objectves of Analyss Provde flexble devce for nterpretng structured covarance matrx estmators Characterze pror belefs consstent wth standard regularzaton technques Derve optmal regularzaton wthn ths framework

14 Bayesan Regresson Model of Covarance Objectves of Analyss Provde flexble devce for nterpretng structured covarance matrx estmators Characterze pror belefs consstent wth standard regularzaton technques Derve optmal regularzaton wthn ths framework Represent unrestrcted return generatng process as factor model wth N mean zero orthogonal factors r F F F t 1 1 t k Nt t t t Central Idea: Use pror belefs to mpose structure on posteror expected factor loadngs ( 1 N )

15 Bayesan Structure for Covarance Matrx Estmaton Bayesan Lnear Regresson Model as a devce for mposng structure on the Posteror Expected Covarance Matrx

16 Bayesan Regresson Model of Covarance Represent unrestrcted return generatng process as factor model wth N mean zero orthogonal factors r F F F t 1 1 t k Nt t t t Assumpton: Means and factor covarance matrx known

17 Bayesan Regresson Model of Covarance Represent unrestrcted return generatng process as factor model wth N mean zero orthogonal factors r F F F t 1 1 t k Nt t t t Assumpton: Means and factor covarance matrx known Example: Prncpal component factors and covarance matrx egenstructure Factor covarance matrx corresponds to egenvalues The factor loadngs ( 1 N ) correspond to the th row of the matrx of egenvectors The cross secton of factor loadngs ( 1k Nk ) correspond to the k th egenvector Ledot & Wolf (01 013)

18 Bayesan Regresson Model of Covarance Represent unrestrcted return generatng process as factor model wth N mean zero orthogonal factors r F F F Structure: Informatve prors shrnk factor loadngs t 1 1 t k Nt t t t Sngle Factor Pror: f k 1 k ~ pror N 0 Vk Vk c otherwse Cross-Sectonal Mean-Revertng Pror: 1 N ~ N V ˆ V c k pror k k k N 1 k k

19 Bayesan Regresson Model of Covarance Represent unrestrcted return generatng process as factor model wth N mean zero orthogonal factors r F F F t 1 1 t k Nt t t t Problem: Zero resdual varance mples degenerate lkelhood for data Smlar to overft n nonparametrc regresson

20 Bayesan Regresson Model of Covarance Represent unrestrcted return generatng process as factor model wth N mean zero orthogonal factors r F F F Problem: Zero resdual varance mples degenerate lkelhood for data Smlar to overft n nonparametrc regresson Requres ntroducng addtonal dosyncratc nose wth a bandwdth parameter h such that: Var h t T t 1 1 t k Nt t t t The bandwdth affects optmal mplementaton but not propertes of models wth pre-specfed prors

21 Normal-Gamma Conjugate Model of Covarance Lkelhood of data for seres n terms of ˆ 1 p r 1 1 ;0 h N FF p G T

22 Normal-Gamma Conjugate Model of Covarance Lkelhood of data for seres n terms of ˆ 1 p r 1 1 ;0 h N FF p G T Conjugate Pror for Factor Loadngs 1 s ~ - v pror NG

23 Normal-Gamma Conjugate Model of Covarance 1 s ~ - v pror NG 1 F lkelhood N G FF 1 ˆ ˆ h 1 L ; ~ - T

24 Normal-Gamma Conjugate Model of Covarance 1 s ~ - v pror NG 1 F lkelhood N G FF 1 ˆ ˆ h 1 L ; ~ - T Happly Natural Conjugate Posteror T 1 s ~ - 1 v r t Ft N G t 1 1 FF 1 FF ˆ 1 1 FF v v T s vs T N s FF 1 1 ˆ ˆ v T 1

25 Normal-Gamma Conjugate Model of Covarance 1 s ~ - v pror NG 1 F lkelhood N G FF 1 ˆ ˆ h 1 L ; ~ - T Happly Natural Conjugate Posteror T 1 s ~ - 1 v r t Ft N G t 1 1 FF 1 FF ˆ 1 1 FF v v T 1 1 ˆ s ˆ v T vs T N s FF 1

26 Posteror Covarance Matrx Expectaton The posteror covarance matrx expectaton s: B T r F N G s v ~ - 1 t t t1 1 1 FF 1 FF ˆ s vs T N s FF 1 1 ˆ ˆ v T 1 B B * k s k 0 s s N k1 k kk N

27 Posteror Factor Loadngs for Orthogonal Model Lkelhood and Data 1 ˆ p r 1 1 ;0 h N FF p G T dag... ; FF T * 11 NN

28 Posteror Factor Loadngs for Orthogonal Model Lkelhood and Data 1 ˆ p r 1 1 ;0 h N FF p G T Conjugate Pror 1 s ~ - v pror NG dag... ; FF T * 11 dag NN... ; 1 N

29 Posteror Factor Loadngs for Orthogonal Model Lkelhood and Data 1 ˆ p r 1 1 ;0 h N FF p G T Conjugate Pror 1 s ~ - v pror NG dag... ; FF T * 11 dag... ; 1 N Then T 1 s ~ - 1 v r t Ft N G t where ˆ 1 ˆ k Tkk k k Tkk k 1 k k k k k NN

30 Bayesan Representatons for Shrnkage Estmators Emprcally-specfed prors mpose structure consstent wth shrnkage-based estmators

31 Shrnkage Representaton for Orthogonal Model Denote T 1 s ~ - 1 v r t Ft N G t 1 ˆ k k k k k B B 1 Bˆ k 1 k k N k k k k k

32 Shrnkage Representaton for Orthogonal Model Denote T 1 s ~ - 1 v r t Ft N G t 1 ˆ Here the posteror covarance matrx expectaton has a shrnkage representaton N kk k k kk k k k k k k k1 k1 N k 1 BB BB 1 BB ˆ ˆ BB 1 BB ˆ ˆ k k k k k B B 1 Bˆ k 1 k k N k k k k k N k kk k k k kk k k

33 Pror Representaton for Ledot & Wolf Shrnkage Specfy orthogonal (emprcal) pror belefs such that: 1 s ~ v pror N G dag N ˆ 1 k 1 1 k ; k Tkk 0 k 0 1 vs 1 v T ˆ ˆ FF ˆ

34 Pror Representaton for Ledot & Wolf Shrnkage Specfy orthogonal (emprcal) pror belefs such that: 1 s ~ v pror N G dag N ˆ 1 k 1 1 k ; k Tkk 0 k 0 1 vs 1 v T ˆ ˆ FF ˆ Then the posteror covarance matrx s equvalent to the Ledot & Wolf Sngle-Factor Shrnkage Model 1 ˆ ˆ 1 BB BB S SF SF

35 Pror Representaton for General Shrnkage Suppose a covarance matrx estmator takes the form: * ˆ 1 S 0

36 Pror Representaton for General Shrnkage Suppose a covarance matrx estmator takes the form: * ˆ 1 S Consder the egenvalue decomposton: B B ˆ

37 Pror Representaton for General Shrnkage Suppose a covarance matrx estmator takes the form: * ˆ 1 S Consder the egenvalue decomposton: B B ˆ Then s the posteror expected covarance matrx n the orthogonal model wth pror belefs: s ~ - v pror NG dag... 1 N 1 B T k 0 k k kk

38 Pror Belefs and Non-negatve GMVP Weghts Non-negatvty of GMVP s equvalent to a style of shrnkage of the sample covarance matrx Jagannathan & Ma (003)

39 Pror Belefs and Non-negatve GMVP Weghts Non-negatvty of GMVP s equvalent to a style of shrnkage of the sample covarance matrx Consder the problem: * w arg mn wsw w N 1 st.. w 0 Jagannathan & Ma (003)

40 Pror Belefs and Non-negatve GMVP Weghts Non-negatvty of GMVP s equvalent to a style of shrnkage of the sample covarance matrx Consder the problem: * w arg mn wsw w N 1 st.. w 0 Let be the vector of shadow costs for the nonnegatvty constrant then the soluton s equvalent to: * * w arg mn w w w N 1 * N 0.5S 0.5 S 1 1 N Jagannathan & Ma (003)

41 Pror Belefs and Non-negatve GMVP Weghts Non-negatvty of GMVP s equvalent to a style of shrnkage of the sample covarance matrx Consder the problem: * w arg mn wsw w N 1 st.. w 0 Let be the vector of shadow costs for the nonnegatvty constrant then the soluton s equvalent to: * * w arg mn w w w N 1 * N 0.5S 0.5 S 1 1 As such weght constrants can be ratonalzed by pror belefs but belefs are not a pror ntutve Jagannathan & Ma (003) N

42 Admssble Bayesan Covarance Matrx Estmators Dervng optmal pror belefs n the Bayesan lnear regresson settng for covarance matrx estmaton

43 Posteror Mean Square Error for Covarance Matrx Advantage of shrnkage formulaton les n ts facltaton of dervng optmal shrnkage ntenstes * arg mn E SF 1S 01 Ths optmzaton mmedately mples admssblty wthn the class of shrnkage estmators wth a fxed target

44 Posteror Mean Square Error for Covarance Matrx Advantage of shrnkage formulaton les n ts facltaton of dervng optmal shrnkage ntenstes * arg mn E SF 1S 01 Ths optmzaton mmedately mples admssblty wthn the class of shrnkage estmators wth a fxed target For orthogonal model we can consder the problem of specfyng admssble pror belefs: N k 1 1 ˆ ˆ k kkbb k k k kkbb k k * arg mn 01 N E

45 Admssble Posteror for Sten Loss Admssble posteror shrnkage solves a set of lnear equatons balancng estmator bas and varance

46 Admssble Posteror for Sten Loss Admssble posteror shrnkage solves a set of lnear equatons balancng estmator bas and varance RHS = varance-weghted expected posteror bas N k l 1 F lk l

47 Admssble Posteror for Sten Loss Admssble posteror shrnkage solves a set of lnear equatons balancng estmator bas and varance RHS = varance-weghted expected posteror bas N k l 1 F lk l The ndvdual bas and varance terms are defned as: jk l jl l jl N N E N N cov ˆ ˆ ˆ ˆ kl 1 j1 k jk k kl 1 j1 k jk l jl

48 Admssble Posteror for Sten Loss Admssble posteror shrnkage solves a set of lnear equatons balancng estmator bas and varance RHS = varance-weghted expected posteror bas LHS = varance-weghted expected posteror MSE The ndvdual bas and varance terms are defned as: N N E N N cov ˆ ˆ ˆ ˆ kl 1 j1 k jk k jk l jl l jl N k l 1 F lk l Fl kl kl kl kl 1 j1 k jk l jl

49 Admssble Pror Belefs for Sten Loss We can reverse-engneer the pror belefs that are consstent wth admssble posteror shrnkage

50 Admssble Pror Belefs for Sten Loss We can reverse-engneer the pror belefs that are consstent wth admssble posteror shrnkage In the orthogonal model wth prncpal component factors and zero pror expectaton the optmal pror belefs are: 1 s ~ - v pror NG dag... 1 N T kk k 0 k 1 N 1 ˆ ˆ ˆ N h vs v T FF

51 Proof by Puddng Smulatons compare fnte-sample performance of estmators

52 Monte Carlo Smulatons Standard battery of tests: Mean Square Error Mnmum Varance portfolo (Constraned and Unconstraned)

53 Monte Carlo Smulatons Standard battery of tests: Mean Square Error Mnmum Varance portfolo (Constraned and Unconstraned) Asset Unverses: Country portfolos Industry portfolos Style portfolos US Stocks European Stocks European Mutual Funds

54 Monte Carlo Smulatons Standard battery of tests: Mean Square Error Mnmum Varance portfolo (Constraned and Unconstraned) Asset Unverses: Country portfolos Industry portfolos Style portfolos US Stocks European Stocks European Mutual Funds Usual suspects n the lneup: Sample covarance matrx Ledot & Wolf Factor Model (Post-LASSO and absolute) 1/N 1/V Pror structures Sngle Factor Pror (V chosen by cross-valdaton) Mean-Revertng Pror (V chosen by cross-valdaton) Sten-Optmal Pror (h chosen by cross-valdaton)

55 Horse Race Summary: Portfolos T = 5 Best Estmator Sten Optmal Underperformance Reference Portfolos N MSE Volatlty MSE Volatlty Country 0 MR SOP 11% Sze & Book to Market 5 LWCC LW1P 4% 0.5 Industry 49 MR SOP 9% Sze & Momentum 5 LW1P LW1P 5% 0.54 Sze & Reversal 5 LW1P LW1P 9% 0.4 Sze & Long Term Rev 5 LW1P LW1P 4% Industry 30 MR SOP 10% Gbl Sze & Book Market 5 LW1P LW1P 4% 0.49 Gbl Sze & Momentum 5 LW1P LW1P 3% 0.51 Sze & Book Market 100 LWCC SOP 8%

56 Horse Race Summary: Portfolos T = 5 Best Estmator Sten Optmal Underperformance Reference Portfolos N MSE Volatlty MSE Volatlty Country 0 MR SOP 11% Sze & Book to Market 5 LWCC LW1P 4% 0.5 Industry 49 MR SOP 9% Sze & Momentum 5 LW1P LW1P 5% 0.54 Sze & Reversal 5 LW1P LW1P 9% 0.4 Sze & Long Term Rev 5 LW1P LW1P 4% Industry 30 MR SOP 10% Gbl Sze & Book Market 5 LW1P LW1P 4% 0.49 Gbl Sze & Momentum 5 LW1P LW1P 3% 0.51 Sze & Book Market 100 LWCC SOP 8%

57 Horse Race Summary: Portfolos T = 5 Best Estmator Sten Optmal Underperformance Reference Portfolos N MSE Volatlty MSE Volatlty Country 0 MR SOP 11% Sze & Book to Market 5 LWCC LW1P 4% 0.5 Industry 49 MR SOP 9% Sze & Momentum 5 LW1P LW1P 5% 0.54 Sze & Reversal 5 LW1P LW1P 9% 0.4 Sze & Long Term Rev 5 LW1P LW1P 4% Industry 30 MR SOP 10% Gbl Sze & Book Market 5 LW1P LW1P 4% 0.49 Gbl Sze & Momentum 5 LW1P LW1P 3% 0.51 Sze & Book Market 100 LWCC SOP 8%

58 Horse Race Summary: Securtes T = 5 Best Estmator Sten Optmal Underperformance Asset Unverse N MSE Volatlty MSE Volatlty US Stocks 5 LWCC SOP 34% (CRSP) 50 LWCC SOP 37% 100 LWCC LWSF 35% 0.09 European Stocks 5 LWCC SOP 47% (DataStream) 50 LWCC SOP 53% 100 LWCC SOP 58% European Mutual Funds 5 LWCC LW1P 3% 0.98 (Lpper) 50 LWCC LW1P 5% LW1P LWSF 6% 0.47

59 Horse Race Summary: Securtes T = 5 Best Estmator Sten Optmal Underperformance Asset Unverse N MSE Volatlty MSE Volatlty US Stocks 5 LWCC SOP 34% (CRSP) 50 LWCC SOP 37% 100 LWCC LWSF 35% 0.09 European Stocks 5 LWCC SOP 47% (DataStream) 50 LWCC SOP 53% 100 LWCC SOP 58% European Mutual Funds 5 LWCC LW1P 3% 0.98 (Lpper) 50 LWCC LW1P 5% LW1P LWSF 6% 0.47

60 Key Contrbutons Develop an ntutve Bayesan model for analyzng and estmatng covarance matrces

61 Key Contrbutons Develop an ntutve Bayesan model for analyzng and estmatng covarance matrces Derve a closed-form soluton for the posteror expectaton of the covarance matrx n a factor model Reverse-engneer pror belefs consstent wth shrnkage covarance matrx estmators Optmze belefs for fxed expected pror factor loadngs

62 Key Contrbutons Develop an ntutve Bayesan model for analyzng and estmatng covarance matrces Derve a closed-form soluton for the posteror expectaton of the covarance matrx n a factor model Reverse-engneer pror belefs consstent wth shrnkage covarance matrx estmators Optmze belefs for fxed expected pror factor loadngs Introduce new covarance matrx estmator that does qute well n fnte sample MSE

63 A Bayesan Approach to Sten-Optmal Covarance Matrx Estmaton Ben Gllen Calforna Insttute of Technology Thanks for your Tme!