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Estimating Asset Pricing Factors from Large-Dimensional Panel Data Markus Pelger 1 Martin Lettau 2 1 Stanford University 2 UC Berkeley March 24th, 2017 Western Conference in Mathematical Finance University of Washington Intro Model Weak

  1. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Weak Factor Model Large eigenvalues converge either to A biased value characterized by the Stieltjes transform of the bulk spectrum To the bulk of the spectrum if the true eigenvalue is below some critical threshold ⇒ Phase transition phenomena: estimated eigenvectors orthogonal to true eigenvectors if eigenvalues too small Onatski (2012): Weak factor model with phase transition phenomena Problem: All models in the literature assume that random processes have mean zero ⇒ RP-PCA implicitly uses non-zero means of random variables ⇒ New tools necessary! 15

  2. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Assumption 1: Weak Factor Model Residual matrix can be represented as e = ǫ Σ with ǫ t , i ∼ N (0 , 1). The 1 empirical eigenvalue distribution function of Σ converges to a non-random spectral distribution function with compact support. The supremum of the support is b . The factors F are uncorrelated among each other and are independent of 2 e and Λ and have bounded first two moments.   σ 2 · · · 0 F 1 � T µ F := 1 := 1   . . p p ... ˆ T F t F ⊤ ˆ → µ F Σ F → Σ F = . . F t   . . t T t =1 σ 2 0 · · · F K The column vectors of the loadings Λ are orthogonally invariant and 3 independent of ǫ and F (e.g. Λ i , k ∼ N (0 , 1 N ) and Λ ⊤ Λ = I K Assume that N T → c with 0 < c < ∞ . 4 16

  3. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Definition: Weak Factor Model Average idiosyncratic noise σ 2 e := trace (Σ) / N T e ⊤ e . The 1 Denote by λ 1 ≥ λ 2 ≥ ... ≥ λ N the ordered eigenvalues of Cauchy transform (also called Stieltjes transform) of the eigenvalues is the almost sure limit: � � − 1 � N 1 1 1 ( zI N − 1 T e ⊤ e ) G ( z ) := a . s . lim z − λ i = a . s . lim N trace N T →∞ T →∞ i =1 B -function � N c λ i B ( z ) := a . s . lim N ( z − λ i ) 2 T →∞ i =1 �� �� � − 2 � 1 c ( zI N − 1 T e ⊤ e ) T e ⊤ e = a . s . lim N trace T →∞ 17

  4. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Estimator Risk-premium PCA (RP-PCA): Apply PCA estimation to T X ⊤ � � I T + γ ✶✶ ⊤ S γ := 1 X T PCA : Apply PCA to estimated covariance matrix T X ⊤ � � I T − ✶✶ ⊤ S − 1 := 1 X , i.e. γ = − 1. T ⇒ PCA special case of RP-PCA “Signal” Matrix for Covariance PCA   σ 2 F 1 + c σ 2 0 · · · e  . .  ... M Var = Σ F + c σ 2 . . e I K =   . . σ 2 F K + c σ 2 0 · · · e ⇒ Intuition: Largest K “true” eigenvalues of S − 1 . 18

  5. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Lemma: Covariance PCA Assumption 1 holds. Define the critical value σ 2 1 crit = lim z ↓ b G ( z ) . The first K largest eigenvalues ˆ λ i of S − 1 satisfy for i = 1 , ..., K  � �  G − 1 if σ 2 F i + c σ 2 e > σ 2 1 ˆ p crit σ 2 Fi + c σ 2 λ i → e  otherwise b The correlation between the estimated and true factors converges to   ̺ 1 · · · 0   . . ... Corr ( F , ˆ � p . . F ) →   . . 0 · · · ̺ K with � 1 if σ 2 F i + c σ 2 e > σ 2 p 1+( σ 2 Fi + c σ 2 e ) B (ˆ crit ̺ 2 λ i )) → i 0 otherwise 19

  6. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Corollary: Covariance PCA for i.i.d. errors Assumption 1 holds, c ≥ 1 and e t , i i.i.d. N (0 , σ 2 e ). The largest K eigenvalues of S − 1 have the following limiting values: � F > √ c σ 2 σ 2 σ 2 ( c + 1 + σ 2 if σ 2 F i + c σ 2 e > σ 2 crit ⇔ σ 2 F i + e e ) p ˆ σ 2 e λ i → e (1 + √ c ) 2 Fi σ 2 otherwise The correlation between the estimated and true factors converges to   ̺ 1 0 · · ·  . .  ... � p Corr ( F , ˆ . . F ) →   . . 0 ̺ K · · · with  1 − c σ 4  e  σ 4  Fi if σ 2 F i + c σ 2 e > σ 2 p ̺ 2 crit 1+ c σ 2 + σ 4 → e e ( c 2 − c ) i  σ 2 σ 4   Fi Fi 0 otherwise 20

  7. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model “Signal” Matrix for RP-PCA “Signal” Matrix for RP-PCA � � Σ 1 / 2 Σ F + c σ 2 F µ F (1 + ˜ γ ) e M RP = F Σ 1 / 2 F µ F + c σ 2 µ ⊤ (1 + γ )( µ ⊤ F (1 + ˜ γ ) 2 ) γ = √ γ + 1 − 1 and note that (1 + ˜ γ ) 2 = 1 + γ . Define ˜ ⇒ Projection on K demeaned factors and on mean operator. Denote by θ 1 ≥ ... ≥ θ K +1 the eigenvalues of the “signal matrix” M RP and by ˜ U the corresponding orthonormal eigenvectors :   θ 1 0 · · ·  . .  ... U ⊤ M RP ˜ ˜ . . U =   . . 0 θ K +1 · · · ⇒ Intuition: θ 1 , ..., θ K +1 largest K + 1 “true” eigenvalues of S γ . 21

  8. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Theorem 1: Risk-Premium PCA under weak factor model Assumption 1 holds. The first K largest eigenvalues ˆ θ i i = 1 , ..., K of S γ satisfy � G − 1 � � 1 if θ i > σ 2 1 crit = lim z ↓ b p ˆ θ i → θ i G ( z ) b otherwise The correlation of the estimated with the true factors converges to   ρ 1 0 · · · 0   0 ρ 2 · · · 0   � � ˜   . p ... Corr ( F , ˆ � D 1 / 2 Σ − 1 / 2 ˆ F ) 0 . → I K U   0 0 . K ˆ   � �� � F � �� �   0 · · · 0 ρ K rotation rotation 0 · · · 0 with � 1 if θ i > σ 2 p ρ 2 1+ θ i B (ˆ crit θ i )) → i 0 otherwise 22

  9. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Theorem 1: continued     ⊤ ρ 1 · · · 0 ρ 1 · · · 0 � � I K �     . . . . ... ... . . 0 . .     ˆ F = D 1 / 2 . . U ⊤ ˜ ˜ . . Σ ˆ   U   K 0 0     0 · · · ρ K 0 · · · ρ K 0 · · · 0 0 · · · 0   1 − ρ 2 � · · · 0 1   . . ... D 1 / 2 + . .   . . K 1 − ρ 2 0 · · · K �� ˆ �� ˆ D K = diag θ 1 · · · θ K 23

  10. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Lemma: Detection of weak factors If γ > − 1 and µ F � = 0, then the first K eigenvalues of M RP are strictly larger than the first K eigenvalues of M Var , i.e. θ i > σ 2 F i + c σ 2 e For θ i > σ 2 crit it holds that ∂ ˆ θ i ∂ρ i ∂θ i > 0 ∂θ i > 0 i = 1 , ..., K Thus, if γ > − 1 and µ F � = 0, then ρ i > ̺ i . ⇒ For µ F � = 0 RP-PCA always better than PCA. 24

  11. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Example: One-factor model Assume that there is only one factor, i.e. K = 1. The “signal matrix” M RP simplifies to � σ 2 � F + c σ 2 σ F µ (1 + ˜ γ ) e M RP = ( µ 2 + c σ 2 µσ F (1 + ˜ γ ) e )(1 + γ ) and has the eigenvalues: θ 1 , 2 =1 e + ( µ 2 + c σ 2 2 σ 2 F + c σ 2 e )(1 + γ ) � ± 1 e + ( µ 2 + c σ 2 F + µ 2 + c σ 2 e )(1 + γ )) 2 − 4(1 + γ ) c σ 2 ( σ 2 F + c σ 2 e ( σ 2 e ) 2 The eigenvector of first eigenvalue θ 1 has the components µσ F (1 + ˜ γ ) ˜ � U 1 , 1 = e )) 2 + µ 2 σ 2 ( θ 1 − ( σ 2 F + c σ 2 F (1 + γ ) θ 1 − σ 2 F + c σ 2 ˜ e U 1 , 2 = � e )) 2 + µ 2 σ 2 ( θ 1 − ( σ 2 F + c σ 2 F (1 + γ ) 25

  12. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model Weak Factor Model Corollary: One-factor model The correlation between the estimated and true factor has the following limit: ρ 1 � p Corr ( F , ˆ F ) → � ( θ 1 − ( σ 2 F + c σ 2 e )) 2 +1 ρ 2 1 + (1 − ρ 2 1 ) µ 2 σ 2 F (1+ γ ) 26

  13. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Strong Factor Model Strong Factor Model Strong Factor Model Strong factors affect most assets: e.g. market factor N Λ ⊤ Λ bounded (after normalizing factor variances) 1 Statistical model: Bai and Ng (2002) and Bai (2003) framework Factors and loadings can be estimated consistently and are asymptotically normal distributed RP-PCA provides a more efficient estimator of the loadings Assumptions essentially identical to Bai (2003) 27

  14. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Strong Factor Model Strong Factor Model Asymptotic Distribution (up to rotation) PCA under assumptions of Bai (2003): Asymptotically ˆ Λ behaves like OLS regression of F on X . Asymptotically ˆ F behaves like OLS regression of Λ on X . RP-PCA under slightly stronger assumptions as in Bai (2003): Asymptotically ˆ Λ behaves like OLS regression of FW on XW � � with W 2 = I T + γ ✶✶ ⊤ . T Asymptotically ˆ F behaves like OLS regression of Λ on X . Asymptotic Expansion Asymptotic expansions (under slightly stronger assumptions as in Bai (2003)): � � � √ � √ � 1 � − 1 H ⊤ ˆ T F ⊤ W 2 F 1 T F ⊤ W 2 e i + O p T T Λ i − Λ i = + o p (1) 1 √ N � � � √ � √ � 1 � − 1 H ⊤− 1 ˆ 1 N Λ ⊤ Λ N Λ ⊤ e ⊤ N N F t − F t = t + O p + o p (1) 2 √ T with known rotation matrix H . 28

  15. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Strong Factor Model Strong Factor Model Assumption 2: Strong Factor Model Assume the same assumptions as in Bai (2003) (Assumption A-G) hold and in addition � � � T � Ω 1 , 1 � 1 t =1 F t e t , i √ Ω 1 , 2 D T � T → N (0 , Ω) Ω = 1 Ω 2 , 1 Ω 2 , 2 t =1 e t , i √ T 29

  16. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Strong Factor Model Strong Factor Model Theorem 2: Strong Factor Model Assumption 2 holds and γ ∈ [ − 1 , ∞ ). Then: For any choice of γ the factors, loadings and common components can be estimated consistently pointwise. � � √ √ H ⊤ ˆ D N If → 0 then Λ i − Λ i → N (0 , Φ) T T � � − 1 � � Σ F + ( γ + 1) µ F µ ⊤ Ω 1 , 1 + γµ F Ω 2 , 1 + γ Ω 1 , 2 µ F + γ 2 µ F Ω 2 , 2 µ F Φ = F � � − 1 Σ F + ( γ + 1) µ F µ ⊤ · F For γ = − 1 this simplifies to the conventional case Σ − 1 F Ω 1 , 1 Σ − 1 F . The asymptotic distribution of the factors is not affected by the choice of γ . The asymptotic distribution of the common component depends on γ if and only if N T does not go to zero. For T N → 0 � � √ � � ˆ D 0 , F ⊤ t Φ F t T C t , i − C t , i → N 30

  17. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Strong Factor Model Strong Factor Model Example 2: Toy model with i.i.d. residuals and K = 1 √ i . i . d . ∼ (0 , σ 2 T Assume K = 1 and e t , i e ). If Assumption 2 holds and → 0, then N � � √ D ˆ T Λ i − Λ i → N (0 , Ω) with � F (1 + γ ) 2 � σ 2 F + µ 2 Ω = σ 2 e F (1 + γ )) 2 ( σ 2 F + µ 2 ⇒ Optimal choice minimizing the asymptotic variance is risk-premium weight γ = 0. ⇒ Choosing γ = − 1, i.e. the covariance matrix for factor estimation, is not efficient. 31

  18. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Extension Extension: Time-varying loadings Model with time-varying loadings Observe panel of excess returns and L covariates Z i , t − 1 , l : ⊤ g X t , i = F t ( Z i , t − 1 , 1 , ..., Z i , t − 1 , L ) + e t , i 1 × K K × 1 Loadings are function of L covariates Z i , t − 1 , l with l = 1 , ..., L e.g. characteristics like size, book-to-market ratio, past returns, ... Factors and loading function are latent Literature (partial list) Projected PCA: Fan, Liao and Wang (2016) Dynamic semiparametric factor model: Park, Mammen, H¨ ardle and Borak (2009) Nonparametric regression model: Connor and Linton (2007) 32

  19. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Extension Extension: Time-varying loadings Projected RP-PCA (work in progress) Assume additive nonparametric loading model: L � g k ( Z i , t − 1 ) = g k , l ( Z i , t − 1 , l ) l =1 Each additive component of g k is estimated by the sieve method. Choose appropriate basis functions φ 1 ( . ) , ..., φ D ( . ) (e.g. splines, polynomial series, kernels, etc.) Define projection P t − 1 as regression on L · D × N matrix φ ( Z t − 1 ) with elements φ d ( Z i , t − 1 , l ), i = 1 , ..., N , l = 1 , ..., L , d = 1 , ..., D . Apply RP-PCA to projected data ˜ X t = X t P t − 1 . Empirical results promising: We recover size, value, momentum and volatility factors from individual stock price data 33

  20. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Illustration Illustration Illustration: Anomaly-sorted portfolios (Size and accrual) Factors PCA: Estimation based on PCA of correlation matrix, K = 3 1 RP-PCA: Estimation based on PCA of X ⊤ � T ✶✶ ⊤ � I + γ X 2 (normalized standard deviation of X ), K = 3 and γ = 100 Fama-French 5 factor model: market, size, value, profitability 3 and investment Specific factors: market, size and accrual 4 Data Double-sorted portfolios according to size and accrual (from Kenneth French’s website) Monthly return data from July 1963 to December 2013 (T = 606) for N = 25 portfolios 34

  21. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Illustration Comparison among estimators Goodness-of-fit-measures: �� � F Σ − 1 µ ⊤ SR: Sharpe ratio of the stochastic discount factor: F µ F . Cross-sectional pricing error α : Time-series estimator: Intercept of regression: X i = α i + F Λ i + e i Cross-sectional estimator: Regression of E [ X ] = E [ F ]Λ ⊤ + α Results the same. This presentation: Time-series regression α . � � N 1 i =1 α i 2 RMS α : Root-mean-squared pricing errors N Out-of-sample estimation: Rolling window of 10 years (T=120) to estimate loadings for next month: � − 1 Λ ⊤ � α t , i = X t , i − ˆ C t , i with ˆ Λ ⊤ ˆ C t = X t (Λ t − 1 t − 1 Λ t − 1 t − 1 ). Fama-MacBeth test-statistic (weighted sum of squared α ′ s , with χ 2 N − K distribution under H 0 ). 35

  22. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Illustration Portfolio Data: In-sample (Size and accrual) SR RMS α Fama-MacBeth RP-PCA 0.305 0.068 44.570 PCA 0.135 0.141 89.946 Fama-French 0.344 0.154 61.979 Specific 0.173 0.155 76.041 Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics. K = 3 statistical factors and risk-premium weight γ = 100. ⇒ RP-PCA significantly better than PCA and quantile-sorted factors. 36

  23. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Illustration Cross-sectional α ’s for sorted portfolios (Size and Accrual) Pricing Errors Size and Accrual 0.4 PCA RP-PCA 0.35 Fama-French 5 Specific 0.3 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 ⇒ RP-PCA avoids large pricing errors due to penalty term. 37

  24. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Illustration Loadings for statistical factors (Size and Accrual) Loadings of 1. PCA factor Loadings of 2. PCA factor Loadings of 3. PCA factor 0.3 0.4 0.4 0.2 0.25 0.2 Loadings Loadings Loadings 0 0.2 0 -0.2 0.15 -0.2 -0.4 0.1 -0.4 -0.6 0 10 20 30 0 10 20 30 0 10 20 30 Portfolio Portfolio Portfolio Loadings of 1. RP-PCA factor Loadings of 2. RP-PCA factor Loadings of 3. RP-PCA factor -0.1 0.4 0.8 0.6 -0.15 0.2 0.4 Loadings Loadings Loadings -0.2 0 0.2 0 -0.25 -0.2 -0.2 -0.3 -0.4 -0.4 0 10 20 30 0 10 20 30 0 10 20 30 Portfolio Portfolio Portfolio ⇒ RP-PCA detects accrual factor while 3rd PCA factor is noise. 38

  25. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Illustration Maximal Incremental Sharpe Ratio PCA RP-PCA 1 Factor 0.134 0.137 2 Factors 0.135 0.139 3 Factors 0.135 0.305 Table: Maximal Sharpe-ratio by adding factors incrementally. For the statistical factor estimators we use K = 3 factors and γ = 100. ⇒ 1st and 2nd PCA and RP-PCA factors the same. ⇒ Better performance of RP-PCA because of third accrual factor. 39

  26. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Illustration Portfolio Data: Objective function (Size and Accrual) PCA TS RP-PCA TS PCA XS RP-PCA XS 1 Factor 3.308 3.617 0.014 0.002 2 Factors 1.937 2.240 0.014 0.002 3 Factors 1.623 1.751 0.014 0.000 Table: Time-series and cross-sectional objective functions. ⇒ RP-PCA and PCA explain the same amount of variation. ⇒ PR-PCA explains cross-sectional pricing much better. ⇒ Motivation for risk-premium weight γ = 100. 40

  27. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Illustration Portfolio Data: Out-of-sample (Size and Accrual) Out-of-sample In-sample RP-PCA 0.097 0.090 PCA 0.128 0.146 Fama-French 5 0.111 0.102 Specific 0.134 0.126 Table: Root-mean-squared pricing errors. Out-of-sample factors are estimated with a rolling window. K = 3 statistical factors and risk-premium weight γ = 100. ⇒ RP-PCA performs better in- and out-of-sample. 41

  28. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Illustration Cross-sectional α ’s out-of-sample (Size and Accrual) Out-of-sample Pricing Errors Size and Accrual 0.35 PCA RP-PCA 0.3 Fama-French 5 Specific 0.25 0.2 Alpha 0.15 0.1 0.05 0 0 5 10 15 20 25 Portfolio ⇒ RP-PCA avoids large pricing errors due to penalty term. 42

  29. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Portfolio Data Portfolio Data Data Monthly return data from July 1963 to December 2013 ( T = 606) 13 double sorted portfolios (consisting of 25 portfolios) from Kenneth French’s website and 49 industry portfolios Factors PCA: K = 3 1 RP-PCA: K = 3 and γ = 100 2 Fama-French 5 factor model: market, size, value, profitability 3 and investment Specific factors: market + two specific anomaly long-short 4 factors 43

  30. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Pricing errors α (in-sample) RP-PCA PCA FF 5 Specific Size and BM 0.13 0.14 0.12 0.20 BM and Investment 0.07 0.12 0.14 0.13 BM and Operating Profits 0.11 0.12 0.14 0.17 Size and Accrual 0.07 0.14 0.15 0.16 Size and Beta 0.06 0.07 0.08 0.17 Size and Investment 0.11 0.13 0.11 0.20 Size and Operating Profits 0.06 0.07 0.08 0.16 Size and Short-Term Reversal 0.15 0.16 0.24 0.33 Size and Long-Term Reversal 0.11 0.13 0.09 0.20 Size and Res. Var. 0.17 0.18 0.21 0.22 Size and Total Var. 0.18 0.19 0.22 0.21 Operating Profits and Investment 0.11 0.14 0.12 0.14 Size and Net Share Iss. 0.14 0.16 0.13 0.17 49 Industries 0.14 0.16 0.13 0.29 44

  31. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Pricing errors α (out-of-sample) RP-PCA PCA FF 5 Specific Size and BM 0.17 0.19 0.14 0.21 BM and Investment 0.12 0.16 0.11 0.14 BM and Operating Profits 0.15 0.18 0.15 0.17 Size and Accrual 0.10 0.13 0.11 0.13 Size and Beta 0.09 0.10 0.07 0.09 Size and Investment 0.14 0.17 0.12 0.19 Size and Operating Profits 0.09 0.12 0.09 0.18 Size and Short-Term Reversal 0.17 0.19 0.09 0.18 Size and Long-Term Reversal 0.13 0.14 0.09 0.14 Size and Res. Var. 0.17 0.20 0.18 0.26 Size and Total Var. 0.17 0.21 0.20 0.26 Operating Profits and Investment 0.13 0.17 0.13 0.16 Size and Net Share Iss. 0.14 0.21 0.16 0.18 49 Industries 0.26 0.24 0.21 0.25 45

  32. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Maximum Sharpe-Ratios RP-PCA PCA Specific Size and BM 0.25 0.22 0.16 BM and Investment 0.26 0.17 0.24 BM and Operating Profits 0.24 0.22 0.25 Size and Accrual 0.30 0.13 0.17 Size and Beta 0.23 0.23 0.17 Size and Investment 0.30 0.26 0.23 Size and Operating Profits 0.22 0.21 0.18 Size and Short-Term Reversal 0.26 0.20 0.25 Size and Long-Term Reversal 0.23 0.18 0.15 Size and Res. Var. 0.33 0.30 0.32 Size and Total Var. 0.32 0.28 0.32 Operating Profits and Investment 0.31 0.24 0.34 Size and Net Share Iss. 0.33 0.25 0.35 49 Industries 0.35 0.25 0.11 46

  33. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Portfolio Data Portfolio Data Monthly return data from July 1963 to December 2013 ( T = 606) for N = 199 portfolios Novy-Marx and Velikov (2014) data: 150 portfolios sorted according to 15 anomalies (same data as in Kozak, Nagel and Santosh (2015)) 49 industry portfolios from Kenneth French’s website Fama-French 5: The five factor model of Fama-French 1 (market, size, value, investment and operating profitability, all from Kenneth French’s website). Specific : Market, value, value-momementum-profitibility and 2 volatility factors. Number of statistical factors K = 4 and γ = 100. 47

  34. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Portfolio Data I: 15 Novy-Marx factors and portfolios Size Gross Profitability Value Value Prof Accruals Net Issuance Asset Growth Investment Piotrotski F-Score ValMomProf ValMom Idiosyncratic Vol Momentum Long Run Reversal Beta Arbitrage. 48

  35. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Portfolio Data: In-sample SR RMS α Fama-MacBeth RP-PCA 729.944 0.417 0.135 PCA 0.155 0.213 820.804 Fama-French 0.344 0.225 801.013 Specific 0.413 0.152 731.392 Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics. K = 4 statistical factors and risk-premium weight γ = 100. RP-PCA strongly dominates PCA and Fama-French 5 factors Specific factors (Market, Value, Value-Momementum-Profitibility and Volatility) perform similar to RP-PCA. 49

  36. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Portfolio Data: Out-of-sample Out-of-sample In-sample RP-PCA 0.178 0.145 PCA 0.202 0.208 Fama-French 5 0.182 0.182 Specific 0.154 0.137 Table: Root-mean-squared pricing errors. Out-of-sample factors are estimated with a rolling window. K = 4 statistical factors and risk-premium weight γ = 100. ⇒ RP-PCA performs well in- and out-of-sample. 50

  37. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Portfolio Data: Interpreting factors PCA RP-PCA 1. Gen. Corr. 0.997 0.997 2. Gen. Corr. 0.898 0.925 3. Gen. Corr. 0.809 0.888 4. Gen. Corr. 0.032 0.741 Table: Generalized Correlations between specific factors and statistical factors. Problem in interpreting factors: Factors only identified up to invertible linear transformations. Generalized correlations close to 1 measure of how many factors two sets have in common. Specific factors: Market, Value, Value-Momementum-Profitability and Volatility factors. ⇒ Specific factors approximate RP-PCA factors. 51

  38. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Maximal Incremental Sharpe Ratio PCA RP-PCA 1 Factor 0.127 0.137 2 Factors 0.149 0.381 3 Factors 0.153 0.412 Table: Maximal Sharpe-ratio by adding factors incrementally. K = 4 statistical factors and risk-premium weight γ = 100. 52

  39. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Empirical Results Portfolio Data: Objective function PCA TS RP-PCA TS PCA XS RP-PCA XS 1 Factor 44.771 51.623 0.298 0.037 2 Factors 39.846 42.326 0.268 0.001 3 Factors 36.112 37.849 0.263 0.000 Table: Time-series and cross-sectional objective functions. ⇒ RP-PCA and PCA explain the same amount of variation. ⇒ PR-PCA explains cross-sectional pricing much better. ⇒ Motivation for risk-premium weight γ = 100. 53

  40. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Conclusion Conclusion Methodology New estimator for estimating priced latent factors from large data sets Combines time-series and cross-sectional criterion function Asymptotic theory under weak and strong factor model assumption Detects weak factors with high Sharpe-ratio More efficient than conventional PCA Empirical Results Strongly dominates estimation based on PCA of the covariance matrix Potential to provide benchmark factors for horse races. Promising empirical results. 54

  41. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Cumulative returns of factors 100 Mkt-Rf 80 Cumulative return RP-PCA 1 RP-PCA 2 60 RP-PCA 3 RP-PCA 4 40 20 0 -20 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005 2008 2011 2014 Year 80 Mkt-Rf 60 Cumulative return PCA 1 PCA 2 PCA 3 40 PCA 4 20 0 -20 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005 2008 2011 2014 Year A 1

  42. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Predicted excess return in-sample (Size and Accrual) Expected excess return In-sample RP-PCA 1 0.8 0.6 0.4 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Predicted excess return Expected excess return In-sample PCA 1 0.8 0.6 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Predicted excess return Expected excess return In-sample 5 Fama-French factors 1 0.8 0.6 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Predicted excess return Expected excess return In-sample Specific factors 1 0.8 0.6 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 A 2 Predicted excess return

  43. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Predicted excess return out-of-sample (Size and Accrual) Expected excess return OLS out-of-sample RP-PCA 1 0.8 0.6 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Predicted excess return Expected excess return OLS out-of-sample PCA 1 0.8 0.6 0.4 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Predicted excess return Expected excess return OLS out-of-sample 5 Fama-French factors 1 0.8 0.6 0.4 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Predicted excess return Expected excess return OLS out-of-sample Specific factors 1 0.8 0.6 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 A 3 Predicted excess return

  44. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Strong Factor Model Asymptotic Expansion Asymptotic expansions (under slightly stronger assumptions as in Bai (2003)): � � � √ � √ � 1 � − 1 H ⊤ ˆ T F ⊤ W 2 F 1 T F ⊤ W 2 e i + O p T Λ i − Λ i = + o p (1) 1 T √ N � � � √ � √ � 1 � − 1 H ⊤− 1 ˆ N Λ ⊤ Λ 1 N Λ ⊤ e ⊤ N = t + O p + o p (1) N F t − F t 2 √ T � � √ ˆ δ C t , i − C t , i = 3 � 1 � − 1 � 1 � − 1 √ √ δ T F ⊤ W 2 F 1 T F ⊤ W 2 e i + δ 1 T F ⊤ N Λ ⊤ N Λ ⊤ Λ N Λ ⊤ e ⊤ t + o p (1) √ √ √ √ t i � � � � 1 T F ⊤ W 2 F N Λ ⊤ ˆ 1 V − 1 with H = Λ TN , δ = min( N , T ) and � � W 2 = I T + γ ✶✶ ⊤ . T A 4

  45. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix The Model Weighted Combined objective function: Straightforward extension to weighted objective function: 1 NT trace ( Q ⊤ ( X − F Λ ⊤ ) ⊤ ( X − F Λ ⊤ ) Q ) min Λ , F � � + γ 1 ✶ ⊤ ( X − F Λ ⊤ ) QQ ⊤ ( X − F Λ ⊤ ) ⊤ ✶ N trace � M Λ Q ⊤ X ⊤ � T ✶✶ ⊤ � � I + γ s.t. F = X (Λ ⊤ Λ) − 1 Λ ⊤ = min Λ trace XQM Λ Cross-sectional weighting matrix Q Factors and loadings can be estimated by applying PCA to Q ⊤ X ⊤ � T ✶✶ ⊤ � I + γ XQ . Today: Only Q equal to inverse of a diagonal matrix of standard deviations. For γ = − 1 corresponds to PCA of a correlation matrix. Optimal choice of Q : GLS type argument A 5

  46. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Simulation Simulation parameters N = 250 and T = 350. Factors: K = 4 1. Factor represent the market with N (1 . 2 , 9): Sharpe-ratio of 0.4 2. Factor represents an industry factors following N (0 . 1 , 1): Sharpe-ratio of 0.1. 3. Factor follows N (0 . 4 , 1): Sharpe-ratio of 0.4. 4. Factor has a small variance but high Sharpe-ratio. It follows N (0 . 4 , 0 . 16): Sharpe-ratio of 1. Loadings normalized such that 1 N Λ ⊤ Λ. Λ i , 1 = 1 and Λ i , k ∼ N (0 , 1) for k = 2 , 3 , 4. Errors: Cross-sectional and time-series correlation and heteroskedasticity in the residuals. Half of the variation due to non-systematic risk. A 6

  47. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Simulation 1. Factor 2. Factor 3. Factor 4. Factor 400 40 180 140 160 350 35 120 140 300 30 100 120 250 25 100 80 200 20 80 60 60 150 15 40 40 100 10 True factor 20 PCA Var PCA Corr 20 50 5 RP-PCA 0 RP-PCA Corr 0 0 -20 0 0 200 400 0 200 400 0 200 400 0 200 400 Figure: Sample path of the first four factors and the estimated factor processes. γ = 50. A 7

  48. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Simulation PCA Var PCA Corr RP-PCA RP-PCA Corr 1. Factor 0.094 0.086 0.042 0.040 2. Factor 0.023 0.022 0.025 0.022 3. Factor 0.100 0.095 0.079 0.074 4. Factor 0.312 0.312 0.183 0.170 Table: Average root-mean-squared (RMS) errors of estimated factors relative to the true factor processes. γ = 50. A 8

  49. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Simulation Statistical Model 1 0.8 0.6 PCA 0.4 RP-PCA ( . =0) RP-PCA ( . =10) 0.2 RP-PCA ( . =50) 0 0 0.5 1 1.5 2 Monte-Carlo Simulation 1 0.8 0.6 PCA 0.4 RP-PCA ( . =0) RP-PCA ( . =10) 0.2 RP-PCA ( . =50) 0 0 0.5 1 1.5 2 Squared correlations between estimated and true factor based on the weak factor model prediction and Monte-Carlo simulations for different variances of the factor. The Sharpe-ratio of the factor is 1, i.e. the mean equals µ F = σ F . The normalized variance of the factors is σ 2 A 9 F · N .

  50. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Weak Factor Model: Dependent residuals 1 0.8 0.6 0.4 0.2 dependent residuals i.i.d residuals 0 0 10 20 30 40 Figure: Values of ρ 2 1 θ i )) if θ i > σ 2 i ( crit and 0 otherwise) for different 1+ θ i B (ˆ signals θ i . The average noise level is normalized in both cases to σ 2 e = 1 and c = 1. For the correlated residuals we assume that Σ 1 / 2 is a Toeplitz matrix with β, β, β, β 2 on the right four off-diagonals with β = 0 . 7. A 10

  51. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Simulation parameters Errors Residuals are modeled as e = σ e D T A T ǫ A N D N : ǫ is a T × N matrix and follows a multivariate standard normal distribution Time-series correlation in errors: A T creates an AR(1) model with parameter ρ = 0 . 1 Cross-sectional correlation in errors: A N is a Toeplitz-matrix with ( β, β, β, β 2 ) on the right four off-diagonals with β = 0 . 7 Cross-sectional heteroskedasticity: D N is a diagonal matrix with independent elements following N (1 , 0 . 2) Time-series heteroskedasticity: D T is a diagonal matrix with independent elements following N (1 , 0 . 2) Signal-to-noise ratio: σ 2 e = 10 Parameters produce eigenvalues that are consistent with the data. A 11

  52. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Simulation True Factors PCA Var PCA Corr RP-PCA PR-PCA Corr SR 1.330 0.515 0.517 0.865 0.883 Table: Maximal Sharpe Ratio with K = 4 factors. γ = 50. True PCA Var PCA Corr RP-PCA RP-PCA Corr 1. Factor 1.20 1.10 1.11 1.16 1.16 2. Factor 0.10 0.11 0.10 0.12 0.11 3. Factor 0.40 0.31 0.31 0.49 0.48 4. Factor 0.40 0.08 0.08 0.21 0.22 Table: Estimated mean of factors. γ = 50. A 12

  53. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Simulation True PCA Var PCA Corr RP-PCA RP-PCA Corr 1. Factor 9.000 8.608 8.615 8.494 8.510 2. Factor 1.000 0.697 0.716 0.683 0.706 3. Factor 1.000 0.801 0.820 0.674 0.690 4. Factor 0.160 0.028 0.028 0.066 0.070 Table: Estimated variance of factors. γ = 50. A 13

  54. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Fama-MacBeth Test-Statistics: χ 2 22 : 34(95 %) RP-PCA PCA FF 5 Specific Size and BM 85.66 94.50 79.99 105.15 BM and Investment 14.52 37.04 26.14 31.61 BM and Operating Profits 19.45 25.95 15.40 21.92 Size and Accrual 44.57 89.95 61.98 76.04 Size and Beta 30.74 32.90 31.76 31.96 Size and Investment 87.89 104.53 93.88 103.60 Size and Operating Profits 29.17 32.98 29.16 42.32 Size and Short-Term Reversal 87.70 103.35 88.86 108.31 Size and Long-Term Reversal 53.92 65.07 44.09 68.69 Size and Res. Var. 134.57 147.18 125.28 163.77 Size and Total Var. 120.14 133.46 120.71 143.01 Operating Profits and Investment 29.21 51.63 34.38 35.89 Size and Net Share Iss. 121.13 149.78 119.91 126.64 49 Industries 140.76 175.77 140.59 206.47 A 14

  55. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Predicted excess return in-sample Expected excess return In-SampleRP-PCA 1.5 1 0.5 0 -0.5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Predicted excess return Expected excess return In-SamplePCA 1.5 1 0.5 0 -0.5 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Predicted excess return Expected excess return In-Sample5 Fama-French factors 1.5 1 0.5 0 -0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Predicted excess return Expected excess return In-SampleSpecific factors 1.5 1 0.5 0 -0.5 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Predicted excess return A 15

  56. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Predicted excess return out-of-sample Expected excess return OLS out-of-sample RP-PCA 1.5 1 0.5 0 -0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Predicted excess return Expected excess return OLS out-of-sample PCA 1.5 1 0.5 0 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Predicted excess return Expected excess return OLS out-of-sample 5 Fama-French factors 1.5 1 0.5 0 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Predicted excess return Expected excess return OLS out-of-sample Specific factors 1.5 1 0.5 0 -0.5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Predicted excess return A 16

  57. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Cumulative returns of optimal portfolios RP-PCA Optimal Portfolios 300 1 Factor 2 Factors Cumulative return 200 3 Factors 4 Factors 100 0 -100 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005 2008 2011 2014 Year PCA Optimal Portfolios 100 1 Factor 80 2 Factors Cumulative return 3 Factors 60 4 Factors 40 20 0 -20 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005 2008 2011 2014 Year A 17

  58. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Portfolio Data: In-sample (BM and Investment) SR RMS α Fama-MacBeth RP-PCA 0.256 0.074 14.520 PCA 0.169 0.123 37.038 Fama-French 0.344 0.140 26.144 Specific 0.236 0.127 31.611 Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics for different set of factors. For the statistical factor estimators we use K = 3 factors and γ = 100. A 18

  59. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Cross-sectional α ’s for sorted portfolios (BM and Investment) Pricing Errors BM and Investment 0.3 PCA RP-PCA Fama-French 5 0.25 Specific 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 A 19

  60. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Loadings for statistical factors (BM and Investment) Loadings of 1. PCA factor Loadings of 2. PCA factor Loadings of 3. PCA factor 0.25 0.6 0.4 0.4 0.2 Loadings Loadings Loadings 0.2 0 0.2 0 -0.2 -0.2 -0.4 0.15 -0.4 -0.6 0 10 20 30 0 10 20 30 0 10 20 30 Portfolio Portfolio Portfolio Loadings of 1. RP-PCA factor Loadings of 2. RP-PCA factor Loadings of 3. RP-PCA factor 0.3 0.6 0.6 0.4 0.4 0.25 Loadings Loadings Loadings 0.2 0.2 0 0 0.2 -0.2 -0.2 0.15 -0.4 -0.4 0 10 20 30 0 10 20 30 0 10 20 30 Portfolio Portfolio Portfolio A 20

  61. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Maximal Incremental Sharpe Ratio (BM and Investment) PCA RP-PCA 1 Factor 0.144 0.149 2 Factors 0.167 0.193 3 Factors 0.169 0.256 Table: Maximal Sharpe-ratio by adding factors incrementally. For the statistical factor estimators we use K = 3 factors and γ = 100. A 21

  62. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Portfolio Data: Objective function (BM and Investment) PCA TS RP-PCA TS PCA XS RP-PCA XS 1 Factor 5.543 5.989 0.021 0.002 2 Factors 4.416 4.647 0.014 0.001 3 Factors 3.944 4.098 0.013 0.000 Table: Time-series and cross-sectional objective functions. A 22

  63. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Portfolio Data: Out-of-sample (BM and Investment) Out-of-sample In-sample RP-PCA 0.123 0.065 PCA 0.157 0.156 Fama-French 5 0.111 0.103 Specific 0.138 0.138 Table: Root-mean-squared pricing errors for different set of factors. Out-of-sample factors are estimated with a rolling window. For the statistical factor estimators we use K = 3 factors and γ = 100. A 23

  64. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Cross-sectional α ’s out-of-sample (BM and Investment) Out-of-sample Pricing Errors BM and Investment 0.4 PCA RP-PCA 0.35 Fama-French 5 Specific 0.3 0.25 Alpha 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 Portfolio A 24

  65. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Predicted excess return in-sample (BM and Investment) Expected excess return In-sample RP-PCA 1.2 1 0.8 0.6 0.4 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Predicted excess return Expected excess return In-sample PCA 1.2 1 0.8 0.6 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Predicted excess return Expected excess return In-sample 5 Fama-French factors 1.2 1 0.8 0.6 0.4 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Predicted excess return Expected excess return In-sample Specific factors 1.2 1 0.8 0.6 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 A 25 Predicted excess return

  66. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Predicted excess return out-of-sample (BM and Invest.) Expected excess return OLS out-of-sample RP-PCA 1.2 1 0.8 0.6 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Predicted excess return Expected excess return OLS out-of-sample PCA 1.2 1 0.8 0.6 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Predicted excess return Expected excess return OLS out-of-sample 5 Fama-French factors 1.2 1 0.8 0.6 0.4 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Predicted excess return Expected excess return OLS out-of-sample Specific factors 1.2 1 0.8 0.6 0.4 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 A 26 Predicted excess return

  67. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Portfolio Data: In-sample (Size and BM) SR RMS α Fama-MacBeth RP-PCA 0.248 0.126 85.664 PCA 0.217 0.137 94.505 Fama-French 0.344 0.116 79.990 Specific 0.163 0.197 105.153 Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics for different set of factors. For the statistical factor estimators we use K = 3 factors and γ = 100. A 27

  68. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Cross-sectional α ’s for sorted portfolios (Size and BM) Pricing Errors Size and BM 0.6 PCA RP-PCA Fama-French 5 0.5 Specific 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 A 28

  69. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Loadings for statistical factors (Size and BM) Loadings of 1. PCA factor Loadings of 2. PCA factor Loadings of 3. PCA factor -0.15 0.6 0.6 0.4 0.4 -0.2 Loadings Loadings Loadings 0.2 0.2 -0.25 0 0 -0.3 -0.2 -0.2 -0.35 -0.4 -0.4 0 10 20 30 0 10 20 30 0 10 20 30 Portfolio Portfolio Portfolio Loadings of 1. RP-PCA factor Loadings of 2. RP-PCA factor Loadings of 3. RP-PCA factor 0.35 0.8 0.4 0.6 0.3 0.2 0.4 Loadings Loadings Loadings 0.25 0.2 0 0.2 0 -0.2 0.15 -0.2 0.1 -0.4 -0.4 0 10 20 30 0 10 20 30 0 10 20 30 Portfolio Portfolio Portfolio A 29

  70. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Maximal Incremental Sharpe Ratio PCA RP-PCA 1 Factor 0.148 0.156 2 Factors 0.155 0.212 3 Factors 0.217 0.248 Table: Maximal Sharpe-ratio by adding factors incrementally. For the statistical factor estimators we use K = 3 factors and γ = 100. A 30

  71. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Portfolio Data: Objective function (Size and BM) PCA TS RP-PCA TS PCA XS RP-PCA XS 1 Factor 4.263 4.981 0.035 0.003 2 Factors 2.663 3.213 0.032 0.001 3 Factors 1.756 1.889 0.011 0.000 Table: Time-series and cross-sectional objective functions. A 31

  72. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Portfolio Data: Out-of-sample (Size and BM) Out-of-sample In-sample RP-PCA 0.171 0.160 PCA 0.187 0.180 Fama-French 5 0.141 0.140 Specific 0.212 0.196 Table: Root-mean-squared pricing errors for different set of factors. Out-of-sample factors are estimated with a rolling window. For the statistical factor estimators we use K = 3 factors and γ = 100. A 32

  73. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Cross-sectional α ’s out-of-sample (Size and BM) Out-of-sample Pricing Errors Size and BM 0.7 PCA RP-PCA 0.6 Fama-French 5 Specific 0.5 0.4 Alpha 0.3 0.2 0.1 0 0 5 10 15 20 25 Portfolio A 33

  74. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Generalized correlations for time-varying loadings (Size and BM) RP-PCA Generalized Correlation 1 1st GC 0.98 2nd GC 3rd GC 0.96 0.94 0.92 0.9 0 50 100 150 200 250 300 350 400 450 Year PCA Generalized Correlation 1 1st GC 0.99 2nd GC 3rd GC 0.98 0.97 0.96 0.95 0 50 100 150 200 250 300 350 400 450 Year A 34

  75. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Generalized correlations for time-varying loadings (Size and BM) RP-PCA Generalized Correlation 1 1st GC 2nd GC 0.99 3rd GC 0.98 0.97 0 50 100 150 200 250 300 350 400 Year PCA Generalized Correlation 1 1st GC 2nd GC 0.995 3rd GC 0.99 0.985 0 50 100 150 200 250 300 350 400 Year A 35

  76. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Time-varying loadings (Size and BM) Loadings of 1. RP-PCA factor Loadings of 2. RP-PCA factor Loadings of 3. RP-PCA factor 1.5 2 2 1 1 1 0 Loadings Loadings Loadings 0 0.5 -1 -1 -2 0 -2 -3 -0.5 -4 -3 0 10 20 30 0 10 20 30 0 10 20 30 Portfolio Portfolio Portfolio Loadings of 1. PCA factor Loadings of 2. PCA factor Loadings of 3. PCA factor 1.4 2 2 1.2 1 1 Loadings Loadings Loadings 1 0 0 0.8 -1 -1 0.6 0.4 -2 -2 0 10 20 30 0 10 20 30 0 10 20 30 Portfolio Portfolio Portfolio A 36

  77. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Predicted excess return in-sample (Size and BM) Expected excess return In-sample RP-PCA 1.5 1 0.5 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Predicted excess return Expected excess return In-sample PCA 1.5 1 0.5 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Predicted excess return Expected excess return In-sample 5 Fama-French factors 1.5 1 0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Predicted excess return Expected excess return In-sample Specific factors 1.5 1 0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 A 37 Predicted excess return

  78. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Predicted excess return out-of-sample (Size and BM) Expected excess return OLS out-of-sample RP-PCA 1.5 1 0.5 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Predicted excess return Expected excess return OLS out-of-sample PCA 1.5 1 0.5 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Predicted excess return Expected excess return OLS out-of-sample 5 Fama-French factors 1.5 1 0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Predicted excess return Expected excess return OLS out-of-sample Specific factors 1.5 1 0.5 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 A 38 Predicted excess return

  79. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Portfolio Data: In-sample (Size and Momentum) SR RMS α Fama-MacBeth RP-PCA 0.255 0.146 87.702 PCA 0.199 0.160 103.350 Fama-French 0.344 0.238 88.855 Specific 0.253 0.329 108.315 Table: Maximal Sharpe-ratios, root-mean-squared pricing errors and Fama-MacBeth test statistics for different set of factors. For the statistical factor estimators we use K = 3 factors and γ = 100. A 39

  80. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Cross-sectional α ’s for sorted portfolios (Size and Momentum) Pricing Errors Size and Short-Term Reversal 0.7 PCA RP-PCA 0.6 Fama-French 5 Specific 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 A 40

  81. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Loadings for statistical factors (Size and Momentum) Loadings of 1. PCA factor Loadings of 2. PCA factor Loadings of 3. PCA factor -0.15 0.4 0.6 0.2 0.4 -0.2 Loadings Loadings Loadings 0 0.2 -0.25 -0.2 0 -0.3 -0.4 -0.2 -0.35 -0.6 -0.4 0 10 20 30 0 10 20 30 0 10 20 30 Portfolio Portfolio Portfolio Loadings of 1. RP-PCA factor Loadings of 2. RP-PCA factor Loadings of 3. RP-PCA factor 0.4 0.4 0.6 0.2 0.4 0.3 0 Loadings Loadings Loadings 0.2 0.2 -0.2 0 -0.4 0.1 -0.2 -0.6 0 -0.8 -0.4 0 10 20 30 0 10 20 30 0 10 20 30 Portfolio Portfolio Portfolio A 41

  82. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Portfolio Data: Out-of-sample (Size and Momentum) Out-of-sample In-sample RP-PCA 0.171 0.148 PCA 0.193 0.187 Fama-French 5 0.090 0.106 Specific 0.181 0.201 Table: Root-mean-squared pricing errors for different set of factors. Out-of-sample factors are estimated with a rolling window. For the statistical factor estimators we use K = 3 factors and γ = 100. A 42

  83. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Cross-sectional α ’s out-of-sample (Size and Momentum) Out-of-sample Pricing Errors Size and Short-Term Reversal 0.45 PCA 0.4 RP-PCA Fama-French 5 Specific 0.35 0.3 0.25 Alpha 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 Portfolio A 43

  84. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Predicted excess return in-sample (Size and Momentum) Expected excess return In-sample RP-PCA 1.5 1 0.5 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Predicted excess return Expected excess return In-sample PCA 1.5 1 0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Predicted excess return Expected excess return In-sample 5 Fama-French factors 1.5 1 0.5 0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Predicted excess return Expected excess return In-sample Specific factors 1.5 1 0.5 0 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 A 44 Predicted excess return

  85. Intro Model Weak Factors Strong F. Extension Illustration Empirical Results Conclusion Appendix Predicted excess return out-of-sample (Size and Moment.) Expected excess return OLS out-of-sample RP-PCA 1.5 1 0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Predicted excess return Expected excess return OLS out-of-sample PCA 1.5 1 0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Predicted excess return Expected excess return OLS out-of-sample 5 Fama-French factors 1.5 1 0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Predicted excess return Expected excess return OLS out-of-sample Specific factors 1.5 1 0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 A 45 Predicted excess return

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