A diagnostic criterion for approximate factor structure

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1 A diagnostic criterion for approximate factor structure Patrick Gagliardini a, Elisa Ossola b and Olivier Scaillet c a Università della Svizzera Italiana (USI Lugano) and Swiss Finance Institute, b European Commission, Joint Research Centre, c University of Genève and Swiss Finance Institute. February, 2017

2 Introduction Goal of the paper Dene a simple diagnostic criterion: (i) for approximate factor structure in large cross-sectional equity datasets; (ii) to determine the number of omitted common factors. Main idea: If the set of observable factors is correctly specied, the errors are only weakly cross-sectionally correlated. Relevant because the no-arbitrage restrictions from APT do not hold when factors are omitted.

3 Introduction Building blocks of the paper 1. A new simple diagnostic criterion Conditional linear factor model in a large economy, with an approximate factor structure for excess returns. Gagliardini-Ossola-Scaillet (GOS, 2016). Large unbalanced panel of returns. Cross-section and time-series dimensions: n, T s.t. either T /n = o (1) or T /n c, where c > 0. Observable factors. Link with criteria for unobservable factors. Large balanced panels: Connor-Korajczyk (1993), Stock-Watson (2002); Procedure to estimate the number of factors: Bai-Ng (2002); Onatski (2010), Ahn-Horenstein (2013), Caner-Han (2014); Procedure for hypotheses on the number of factors: Onatski (2009);

4 Introduction 2. Empirical analysis results with CRSP individual stock returns Use of individual stocks with monthly and quarterly returns: n >> T. Several linear multi-factor models built by using a large number of empirical factors. Monthly data: Time invariant and time-varying specications of the nancial factor models. We conclude for no omitted factors in the errors for the time-invariant models with at least four nancial factors. We conclude for no omitted factor structure in the errors for a scaled three factor Fama-French specication. Quarterly data: We cannot select macroeconomic models without the market factor.

5 Introduction Outline of the presentation Introduction! Model setting Conditional factor model Rival models Diagnostic criterion Determining the number of omitted factors Empirical results Conclusions

6 Model setting Model setting: Conditional linear factor model Excess returns generation The excess return R t (γ) of asset γ [0, 1] at date t = 1, 2,..., satises where: R t (γ) = β t (γ) x t + ε t (γ), (1) x t = (1, f t ) and f t is the K 1 random vector of observable factors; β t (γ) = ( a t (γ), b t (γ) ) contains time-varying coecients; ε t (γ) is a random vector of error terms s.t. E [ε t (γ) F t 1 ] = 0 and Cov [ε t (γ), f t F t 1 ] = 0 for any γ [0, 1]. (Hansen-Richard (1987))

7 Model setting Approximate factor structure: (Chamberlain-Rothschild (1983)) nondiagonal conditional error var-cov matrix Σ ε,t,n = [Cov [ε t (γ i ), ε t (γ j ) F t 1 ]] i,j=1,...,n with bounded largest eigenvalue if model is correct. No asymptotic arbitrage opportunities: there are no portfolios that approximate arbitrage opportunities when the number of assets increases. Asset pricing restriction: a t (γ) = b t (γ) ν t holds a.s. in probability (GOS (2016)). E [R t (γ) F t 1 ] = b t (γ) λ t, where λ t = ν t + E [f t F t 1 ] is the vector of risk premia. Large economy with a continuum of assets: robustness of factor structures to asset repackaging (Al-Najjar (1995, 1998, 1999), GOS (2016)). Unbalanced nature of the panel: I t (γ) admits value 1 if the return of asset γ is observable at date t, and 0 otherwise (Connor-Korajczyk (1987)).

8 Model setting Functional specication of time-varying coecients Information set F t 1 contains lagged observations of: Z t R p, vector of common instruments: the constant and observable factors f t, additional observable variables Z t. Z t (γ) R q, vector of asset-specic instruments: rm characteristics, stocks returns. Factor loadings: b t (γ) = B (γ) Z t 1 + C (γ) Z t 1 (γ), where B (γ) R K p and C (γ) R K q, for any γ [0, 1] and t = 1, 2,...; Risk premia: λ t = ΛZ t 1, where Λ R K p, for any t; Factors: E [f t F t 1 ] = FZ t 1, where F R K p, for any t.

9 Model setting The sampling scheme: (Andrews (2005)) A sample of n assets is obtained by drawing i.i.d. indices γ i according to a probability distribution G on [0, 1]. cross-sectional limits exist and are invariant to reordering of assets. sample of n assets and T observations of excess returns R i,t = R t (γ i ), I i,t = I t (γ i ), ε i,t = ε t (γ i ), Z i,t = Z t (γ) for i = 1,..., n and t = 1,..., T. random coecient panel model with β i,t = β t (γ i ).

10 Model setting The conditional factor model (1), for the sample observations, becomes R i,t = x i,tβ i + ε i,t, (2) where regressor x i,t involves cross-terms of instruments Z t 1, Z i,t 1 and f t ; time-invariant parameters β i = ( β 1,i, β 2,i) are (unconditional) transformations of matrices B i, C i, Λ and F. In matrix notation, for any asset i, we have R i = X i β i + ε i, where R i and ε i are T 1 vectors.

11 Diagnostic criterion Rival models M 1 : the linear regression model (2), where the errors (ε i,t ) are weakly cross-sectionally correlated (approximate factor structure); M 2 : the linear regression model (2), where the errors (ε i,t ) satisfy a factor structure: ε i,t = θ ih t + u i,t, (3) where vector h t includes m unobservable common factors. In matrix notation, for any asset i, under M 2, we have ε i = Hθ i + u i, where H is the T m matrix of unobservable factor values, and u i is a T 1 vector.

12 Diagnostic criterion Assumption 1: Presence of some common factors in the errors Under model M 2, 1 (i) plim h t h t = Σ h, where Σ h is a positive denite matrix; T T ( t ) 1 (ii) µ 1 θ i θ i C, with probability approaching 1, for a constant n i C > 0, where µ 1 ( ) denotes the largest eigenvalue of a symmetric matrix. Assumption 1(i) is a standard condition on the latent factor (equivalent to the Assumption A in Bai-Ng (2002)); Assumption 1(ii) requires that at least one factor in the error terms is strong (equivalent to the Assumption B in Bai-Ng (2002)).

13 Diagnostic criterion Assumption 2: Granger non causality assumption The best linear prediction of the unobservable factor EL (h t {x i,t, i = 1, 2,...}) is independent of { x i,t, i = 1, 2,...} ( where the regressor x i,t = x t, x i,t) is decomposed in the common component x t = ( vech [X t ], f t Z t 1) and the stock-specic component (. x i,t = Z Z, f t 1 i,t 1 t Z i,t 1) Assumption 2 implies orthogonality between latent factors and observable factors for all stocks, i.e. E [x i,t h t] = 0, i. If x i,t = x t, we get E [x t h t] = 0 by a transformation of the latent factors: identication restriction. Assumption 2 is mantained under model M 2. Assumption 3: The cross-sectional dimension n and time series dimension T are such that n = O(T γ ), γ > 0, and T = O(n γ ), γ (0, 1].

14 Diagnostic criterion Diagnostic criterion The diagnostic criterion is ( 1 ξ = µ 1 nt ) 1 χ i ε i ε i g (n, T ), i where the vector ε i R T contains ε i,t = I i,t ˆε i,t with ˆε i,t = R i,t x i,t ˆβ i, and ˆβ i are estimated by OLS regression on (2) as in GOS (2016); the penalty g (n, T ) is such that g(n, T ) 0 and Cn,T 2 g(n, T ), when n, T, for Cn,T 2 = min{n, T }.

15 Diagnostic criterion Proposition 1: Model selection rule Under Assumptions 1 and 2, when n, T, (a) we select M 1 if ξ < 0, (b) we select M 2 if ξ > 0, since Pr (M 1 ξ < 0) 1 and Pr (M 2 ξ > 0) 1. In the balanced case: ξ = SS 0 SS 1 g (n, T ), where SS 0 is the sum of squared residuals, 1 SS 1 = min (ˆε i,t θ i h t ) 2, subject to the H R T, Θ R n nt i t normalization constraint H H T = 1. Penalized criteria for zero- and one-factor model in Bai-Ng (2002): ξ = PC (0) PC (1), where PC (0) = SS 0, and PC (1) = SS 1 + g (n, T ).

16 Diagnostic criterion The diagnostic criterion based on a logarithmic transform: ( ) 1 ˇξ = ln 1 χ i ε 2 i,t nt i t ( ( 1 ln 1 χ i ε 2 1 i,t µ 1 nt nt i t )) 1 χ i ε i ε i g(n, T ). i In the balanced case: ˇξ = ln(ss 0 /SS 1 ) g(n, T ). Information criteria for zero- and one-factor model in Bai-Ng (2002): ˇξ = IC (0) IC (1). The selection rule is the same as in Proposition 1 with ˇξ substituted for ξ.

17 Determining the number of omitted factors Do we have one, two, or more omitted factors? Rival models M 1 (k): the linear regression model (2), where the errors (ε i,t ) satisfy a factor structure with exactly k unobservable factors; M 2 (k): the linear regression model (2), where the errors (ε i,t ) satisfy a factor structure with at least k + 1 unobservable factors. Assumption 3: Identication of unobservable factors in the errors ( ) 1 Under model M 2 (k), we have µ k+1 θ i θ i C, with probability nt approaching 1, for a constant C > 0, where µ k+1 ( ) denotes the (k + 1)-largest eigenvalues of a symmetric matrix. Assumption 3 requires that there are at least k + 1 strong factors under M 2 (k). i

18 Determining the number of omitted factors Diagnostic criterion The diagnostic criterion is ( ) 1 ξ(k) = µ k+1 1 χ i ε i ε i g(n, T ). nt i Proposition 2: Model selection rule Under Assumptions 1(i), 2 and 3, when n, T : (a) we select M 1 (k) if ξ (k) < 0, (b) we select M 2 (k) if ξ (k) > 0, since Pr (M 1 (k) ξ (k) < 0) 1 and Pr (M 2 (k) ξ (k) > 0) 1. The number of omitted factors is ˆk = min {k = 0, 1,..., T 1 : ξ(k) < 0}.

19 Determining the number of omitted factors Data description Base assets: 10,442 stocks with monthly and quarterly returns from Jan1968 to Dec2011 after merging CRSP and Compustat databases. Linear factor specications: involving nancial factors: CAPM and Fama-French models, among others; involving nancial and macro factors: CCAPM and Epstein-Zin model, among others. Conditional specications: (i) common variables: Z t 1 = (1, divy t 1 ) ; (ii) common variables: Z t 1 = (1, divy t 1 ) and rm characteristics Z i,t 1 = bm i,t 1.

20 Determining the number of omitted factors Implementation: the rescaled diagnostic criterion ξ(k) = µ ( 1 k+1 nt i 1χ i ε i ε ) i g(n, T ) ˆσ 2 measures the contribution (in %) of the (k + 1)th eigenvalue to the variance ˆσ 2 = 1 nt 1 χ i ε 2 i,t = ( ) T 1 µ j 1 χ i ε i ε i. nt i t j=1 i Rescaled eigenvalues are easier to interpret! In practice, we standardize each time series of residuals ε i to have unit ε i,t variance, i.e. ε i,t = (see Bai-Ng (2002)) and we compute 1 T t ε2 i,t ( 1 ξ(k) = µ k+1 n χ T i 1 χ i ε i ε i ) g(n χ, T ), where n χ = i 1χ i and the eigenvalues are interpreted as percentage of the variance of the normalised residuals.

21 Determining the number of omitted factors We also investigate the ratio µ 2 j ( 1 nt i 1 χ i ε i ε i ) / T l=1 µ 2 l ( 1 nt i 1 χ i ε i ε i ). This quantity is a measure the contributions of the omitted factors in terms of the o-diagonal terms (correlation part) in addition to the diagonal terms (residual variance).

22 Determining the number of omitted factors Time-invariant CAPM 3 Panel A 10 Panel B Eigenvalues in % Cumulated eigenvalues in % Number of latent factors Eigenvalues

23 Determining the number of omitted factors Time-invariant CAPM 20 Panel A 40 Panel B Ratio of squared eigenvalues in % Cumulated ratio of squared eigenvalues in % Number of latent factors Eigenvalues

24 Determining the number of omitted factors Time-invariant three-factor Fama-French model 3 Panel A 10 Panel B Eigenvalues in % Cumulated eigenvalues in % Number of latent factors Eigenvalues

25 Determining the number of omitted factors Time-invariant three-factor Fama-French model Ratio of squared eigenvalues in % Panel A Number of latent factors Cumulated ratio of squared eigenvalues in % Panel B Eigenvalues

26 Determining the number of omitted factors Time-invariant ve-factor Fama-French model 3 Panel A 10 Panel B Eigenvalues in % Cumulated eigenvalues in % Number of latent factors Eigenvalues

27 Determining the number of omitted factors Time-varying CAPM with Z t 1 = (1, divy t 1 ) 3 Panel A 10 Panel B Eigenvalues in % Cumulated eigenvalues in % Number of latent factors Eigenvalues

28 Determining the number of omitted factors Time-varying three-factor Fama-French model with Z t 1 = (1, divy t 1 ) 3 Panel A 10 Panel B Eigenvalues in % Cumulated eigenvalues in % Number of latent factors Eigenvalues

29 Determining the number of omitted factors Time-invariant CCAPM 10 Panel A 25 Panel B 9 Eigenvalues in % Cumulated eigenvalues in % Number of latent factors Eigenvalues

30 Determining the number of omitted factors Time-invariant Epstein-Zin model 4 Panel A 15 Panel B Eigenvalues in % Cumulated eigenvalues in % Number of latent factors Eigenvalues

31 Conclusions Conclusions A new diagnostic criterion for approximate factor structure in large cross-sectional datasets The simple criterion is based on three steps: (i) compute the largest eigenvalue of a variance-covariance matrix, (ii) substract a penalty, (iii) conclude on the validity of the approximate factor structure if criterion value is negative. The theoretical results are obtained on residuals, instead of true errors, allowing for unbalanced panel and considering an asymptotics with n >> T.

32 Conclusions Empirical results: interpretation of the diagnostic criterion as the percentage of the residuals' variance explained by omitted latent factors; interpretation of the squared eigenvalues as the percentage of the correlation part explained by omitted latent factors (see Fiorentini-Sentana, 2015); we can choose either among time-invariant specications with at least four nancial factors, or a scaled Fama-French model; the latent factors are more representative of the correlation part than the variance part of the residuals; we cannot select macroeconomic models without the market factor.