FAMA-FRENCH 1992 REDUX WITH ROBUST STATISTICS

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1 FAMA-FRENCH 1992 REDUX WITH ROBUST STATISTICS R. Douglas Martin* Professor Emeritus Applied Mathematics and Statistics University of Washington IAQF, NYC December 11, 2018** *Joint work with Christopher G. Green. See Green and Martin (2017), SSRN Abstract ID ** Appendices and some details slides added 12/21/2018 1

2 Main Reference is Chapter 5 of: Robust Statistics: Theory and Methods (2019). 2 nd Ed. Maronna, Martin, Yohai and Salibian-Barrera, Wiley. Companion R package: RobStatTM (2019, beta) Maintainer: Matias Salibian-Barrera To install, load, and view functions and data sets: > devtools::install_github("msalibian/robstattm") > library(robstattm) > ls("package:robstattm") > data(package = "RobStatTM") 2

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4 OUTLINE 1. Robust regression theory overview 2. Fama-French 1992 method & results 3. FF92 Redux with robust regression 4. Two models not studied in FF92 5. Use in fundamental factor models 4

5 1. ROBUST REGRESSION OVERVIEW Data Oriented Viewpoint Not much influenced by outliers A good fit to the bulk of the data Reliable multi-d outlier detection Provides a diagnostic check on classical estimates Important Fact: Normal distribution MLEs are not robust toward outliers! 5

6 Least Squares n i= ( r ) ˆ = min i i ( ) θ x θ θ XX Xr A normal distribution MLE Blessed by Gauss-Markov (BLUE). But not at all robust: r Outliers in and/or have unbounded influence! i x i The culprit is quadratic loss and estimator linearity! 6

7 Robust vs. LS Prediction of EPS INVENSYS EARNINGS EARNINGS PER SHARE ROBUST LS The ROBUST line fits bulk of data well by down weighting the 2-D outliers, The LEAST SQUARES line is a poor fit to the bulk of the data, and is a very poor predictor of EPS YEAR 7

8 The Fathers of Robust Statistics First date below is date of first seminal contribution. John W. Tukey (1961, 1979) Peter J. Huber (1964, 1973) Frank R. Hampel (1971, 1974) 8

9 Tukey (1979) Redacted just which robust methods you use is not important what is important is that you use some. It is perfectly proper to use both classical and robust methods routinely, and only worry when they differ enough to matter. But when they differ, you should think hard. 9

10 Tukey (1979) Non-redacted and updated just which robust methods you use is not important what is important is that you use some. It is perfectly proper to use both classical and robust methods routinely, and only worry when they differ enough to matter. But when they differ, you should think hard. Our 2018 Update: A lot is now known about which robust regression estimators are (much) better than others. 10

11 ROBUSTNESS THEORY BRIEF Large-sample (asymptotic) in nature Tukey-Huber model in regression context: r = x θ + ε i i i r F= (1 γ) Nr (, x) + γ Hr (, x), H i Choices for : Hrx (, ) 1. All with symmetric error distributions: no bias problem Hrx (, ) 2. All : must control maximum bias 11

12 Robust Regression Theory First theoretical result (Huber, 1964, 1973) M-estimators that minimize maximum variance (ignores bias) Better theoretical result (Yohai & Zamar, 1997, MMYS Chap. 5) MM-estimators with high breakdown point (BP = 0.5) Efficient min-max bias estimators: ( θˆ ) MLE ( θˆ ) ROBUST var EFF( θ ˆ ) ROBUST = = 99% var Theory improvement (Maronna & Yohai, 2015 ; Ch MMYS) Distance constrained maximum-likelihood estimator (DCML) See Appendix 2 for details 12

13 Maximum Bias Curves Maximum bias of over all H in F = (1 γ) F +γ H B (;) γ θˆ max ˆθ 25% trimmed mean sample median Sample mean N 2 (, ) θ σ θ ˆθ 1 ˆθ 2 Breakdown points BP 1 =.25 BP 1 =.5 γ 13

14 Regression M-Estimators M = maximum-likelihood type Huber (1964, 1973) ˆ β = argmin β n i=1 ρ r i x iθ sˆ o n i=1 x i r x θˆ = 0, = sˆ o i i ψ ψ ρ MM-estimators : M-estimators with a highly robust initial estimator crucial for non-convex. Yohai (1987) ρ 14

15 Huber Min-Max Variance Optimal ρ and ψ rho k k 2 3 x Good news: It is a convex optimization problem. Bad news: It can result in arbitrarily large bias for the Tukey-Huber model (Martin, Yohai & Zamar, 1989). psi k k None-the-less, very useful as a complement x N.B. Axioma uses this robust regression estimator, see., e.g., Axioma paper 062 (2015), and Axioma AXWWW21-1 (2015). 15

16 Efficient Min-Max Bias Robust Regression Yohai and Zamar (1997) and Svarc, Yohai & Zamar (2002) Solution was found in terms of the ψ function (not the rho!)

17 It was only discovered this year that the analytic form of the optimal rho functions shown below are those of an imaginary error function. This is used in the RobStatTM package. NOTE: the robust and robustbase packages use a polynomial approximation for ψ, and its integral provides ρ. 17

18 We recommend the 99% normal efficiency ρ and ψ shown below A non-convex optimization problem. But a very good MM-estimator method exists well with a deterministic initial estimate. For the latter, see Appendix 1. Smoothed outlier hard rejection rule Reject data for which: r x θˆ > sˆ i i o Formula for psi function in MMYS Section

19 Weighted Least Squares (WLS) Version of M-Estimator n n r ˆ ˆ i i ri i sˆ i i ( ˆ o ψ x θ W xθ x r = i i ) = 0 i=1 sˆ x i=1 ˆ o s xθ o W opt () t ψ = opt t () t We use a good starting point and solve by iterative re-weighting. 19

20 CAPM/Single-Index Model Example > devtools::install_github('kecoli/mpo') > library(mpo) > (names(retvhi)) [1] "VHI" "MKT" "RF" > ret12 = retvhi[,1:2] > tsplot(ret12,cex =.8) 20

21 LS Fit versus Robust Fit with lmrobdetmm() > library(robstattm) > x=(retvhi[,2]-retvhi [,3])*100 > y=(retvhi[,1]-retvhi [,3])*100 > fit.ls = lm(y~x) > ctrl=lmrobdet.control(efficiency = 0.99, family = "optimal ) > fit.rob = lmrobdetmm(y~x,control = ctrl) > coef(fit.ls) (Intercept) x > coef(fit.rob) (Intercept) x

22 LS beta is almost twice the robust beta Robust beta standard error is smaller than that of LS beta 22

23 Important fact: good outliers are not rejected robdetmm() does not reject good outliers that are consistent with the fit to the bulk of the returns 23

24 Bailer, Maravina & Martin (2011). Robust Betas in Asset Management, Handbook of Quant. Mgm t, Oxford Univ. Press SIZE25 SIZE50 Robust Beta SIZE75 SIZE Robust Betas 24

25 OLS Beta - Robust Beta SIZE25 SIZE75 SIZE50 SIZE Paired Differences Between OLS and Robust Betas 25

26 Main ψ Competitor: Tukey bisquare (90% Efficiency) ρ BS ( x ) ψ BS ( x ) WBS ( x) 26

27 Finite-Sample MSE Performance MSE under Tukey-Huber model (for a range of n and p values) Search for best performance Use extensive simulation - Concludes that ψ () opt t is much better than ψ opt () t ψ BS () t - DCML shrinkage of to LS performs even better For details on finite-sample MSE study, see MMYS Section 5.9, Table 5.9 and Table

28 2. FF92 METHOD AND RESULTS Eugene F. Fama and Kenneth R. French (1992). The Cross- Section of Expected Stock Returns, Journal of Finance. Fama: Nobel Prize, 2013 (shared with Hansen and Schiller) For an overview of empirical asset pricing, including brief discussion of research on many pricing anomalies, see: Bali, Engle and Murray (2016). Empirical Asset Pricing: The Cross-Section of Stock Returns, Wiley 28

29 Cross-Section Regression Models r = X θ + ε t t 1 t t, t = 1, 2,, T Factor exposures Regression slopes Least Squares (LS) Fitted Models r = X θˆ + ε θˆ = t t 1 t t t ˆ ˆ ˆ ( θ, θ,, θ ) ˆ 1, t 2, t Kt, t-tests of Significance Sample mean of time-series of slopes, θ ˆ = 1, 2,, T kt, t 29

30 Fama-French 1992 Goal Determine which of the factors below explain the cross-section of expected returns (which price risk ) β ln(me) ln(be/me) E(+)/P E/P Dummy ln(a/me), ln(a/be) CAPM beta (special portfolios to reduce EV) Size (ME is market equity in $M) Book-to-Market (often just B/M) Positive Earnings to Price Negative Earnings to Price Dummy Leverage factors (A = book assets) 30

31 Fama-French (1992) Table III Ignore strange beta does not price risk! returns are negatively related to size returns are positively related to BE/ME 31

32 3. FF92 REDUX WITH ROBUST REGRESSION For the vast majority of the stocks (~ 97%) we found: Different conclusions with robust regression than FF92: - Beta relationship is significant and negative - Equity returns are positively related to firm size - E/P dummy is significantly negative New results for two models not in FF92: - Full E/P prices risk (no negative EP dummy) - Model with beta and size interaction term Use robust t-stats for time series of regression slopes - With autocorrelation correction 32

33 Note: In the following slides, as in Green and Martin (2017) SSRN ID , we did the following: ψ opt Used a polynomial approximation to. ρ opt The reason is that an analytic expression is needed for in order to obtain the robust regression coefficients via numeric optimization, and such an expression was not known until this year. Used 99.9% normal distribution efficiency Under normality the result is totally indistinguishable from LS (but this is also the case for 99% efficiency) 33

34 Returns vs Beta NOTE: There is a January Beta positive effect 34

35 Monthly Returns vs Beta LS & Robust Fits 53 obs. instead of obs. instead of 520 Beta January effect 35

36 Returns vs Size KR97 = Knez & Ready (1997) CCW04 = Chou, Chou & Wang (2004) LTS = least trimmed squares 36

37 Monthly Slopes of Returns Regressed on Size red dots = Januaries, a well-known January effect N.B. Existence of outliers and serial correlation, thus one should use a robust location estimator with HAC: Croux et al. (2003). 37

38 Nov Returns vs Size LS & Robust Fits Full vertical range view 38

39 View with.1% vertical trimming 39

40 Percent Outliers Rejected Returns vs Size 40

41 Monthly Returns vs Size LS & Robust Fits The well-known January effect A not-so-well-known Q4 effect?? 41

42 What about the Common Return vs Size Wisdom? Returns decrease with firm size But: The opposite is true with robust location estimator weights for decile portfolio weights from time series of decile portfolios 42

43 Advice for Evaluating Factor Premia Tom Philips: Attempt to replicate the returns of a factor using publicly available indices, preferably ones that discard the bottom 5%-10% of the market s total capital. Such a replication allows the investigator to determine if a strategy is tradeable, and also real-time permits performance monitoring. 43

44 4. TWO MODELS NOT TREATED IN FF92 Returns vs Earnings-to-Price uncorrected = not corrected with Newey-West (should be done for classic t-test) Croux et al. = using Croux et al. (2003) correction for serial correlation N.B. Typically using a robust mean estimator and corresponding robust t-test will improve the power of the test with auto-correlation correction (AC), or even without AC 44

45 Slopes of Returns Regressed on E/P 45

46 Size-Beta Interaction Model With LS the only significant coefficient is interaction for 2 time periods, but Robust Regression coefficients are all highly significant for all 3 time periods. returns = β.4 SIZE +.51 β SIZE + noise SIZE = ln(me): (5, 6, 7, 8) = ($148M, $403M, $1.1B, $3.0B) 46

47 Conclusions Don t Depend on Choice of ρ and Efficiency 95%, 99% and 99.9% normal distribution efficiency Optimal and bisquare rho t-stats change but conclusions don t change! 47

48 5. USE IN FUNDAMENTAL FACTOR MODELS Axioma has responded to the need for robust regression in fundamental factor models. Outliers abound in returns and in factor exposures, more so in the latter than one may think Price paid for using LS is more volatile factor returns and crosssection correlation in residuals. The former can result in overstating the factor contribution to risks. The following two slides illustrate the last point. 48

49 Robust versus Classical Factor Returns Three factors: size, E/P, B/M, monthly returns Times Series of Factor Returns Classical Robust LOG.MARKET.CAP.MM EARN2PRICE BOOK2MARKET.MM Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q

50 Residuals Cross-Section Correlations Densities of residual correlations Classical Robust Density residual correlations 50

51 6. OPEN QUESTIONS & TAKE-AWAY Connection with low-vol anomaly? - Blitz & van Vliet, 20017, Baker et al., 2011) Outliers and Asness et al. (2015) Quality-Junk factor? Life-time and other properties of positive outliers? What is the full story about the negative beta relationship? Financially meaningful arguments for the size-beta interaction model? Main Take-Away: Empirical asset pricing studies can benefit considerably by using robust regression (and other robust statistics) as a complement to LS (and other classical statistics). Furthermore: Fundamental factor model construction for portfolio optimization and risk management could similarly benefit. 51

52 Appendix 1: lmrobdetmm() in RobStatTM MM use a highly robust initial estimator that consists of a fast first-step initial estimator, followed by secondstep initial S-estimator with lower efficiency but small maximum bias, and then uses a fast IRWLS to find the nearest local minimum to the M-estimator equation. det : Special fast deterministic first-step estimator due to Pena and Yohai (1999). Much better than previous resampling method. See MMYS Section Very important: A 99% normal distribution efficiency optimal bias robust regression has very small maximum bias over Tukey-Huber model distributions. 52

53 M-Estimator for Scale and S-Estimators M-scale estimator n i= 1 ( r ) σ 1 i n ρ = b Regression S-estimator: 1 n n i= 1 ρ ( r ) i x β σ i =.5 ˆ σ( β) βˆ = arg min β ˆ σ ( β) 53

54 Appendix 2. lmrobdetdcml() in RobStatTM For details, see Section of MMYS. Distance constrained maximum-likelihood (DCML) β ˆ 0 : a highly robust estimator, typically an MM - estimator βˆ = arg min L( β r,r,,r ) subject to d( βˆ, β) δ β 1 2 n 0 For linear regression with normal errors, DCML uses the Kullback-Leibler (KL) distance estimate: ˆ ˆ 1 ˆ = β β = β β V β βˆ VX X 2 ˆ σ 0 ( ) ˆ ( ) ˆ d KL,V ( 0, ) 0 X 0 robust estimate

55 The DCML optimization is: n i= 1 2 min ( ) subject to ( ˆ β ri β d β, β) δ Maronna and Yohai (2015) show that: KL,Vˆ 0 x 1. This leads to: βˆ = t βˆ + (1 t) βˆ LS 0 where t = min 1, δ ˆV X δ = δpn, = 0.3 p n 2. The DCML estimator is asymptotically fully efficient, and best finite-sample efficiency and MSE. 55

56 VHI data lmrobdetdcml() fit vs. lmrobdetmm() Modifying the code on slide 21 as shown below gives the results comparative lmrobdetmm() and lmrobdetdcml() results: > x=(retvhi[,2]-retvhi [,3])*100 > y=(retvhi[,1]-retvhi [,3])*100 > ctrl=lmrobdet.control(efficiency=0.99,family="optimal") > fit.dcml = lmrobdetdcml(y~x,control = ctrl) > fit.mm = lmrobdetmm(y~x,control = ctrl) > round(summary(fit.dcml)[9]$coefficients[2,1:2],3) Estimate Std. Error > round(summary(fit.mm)[9]$coefficients[2,1:2],3) Estimate Std. Error