Fundamental properties of linear factor models
Damir Filipovic Paul Schneider


 









Fundamental properties of linear factor models


    Peter M. Kraus
    3 September 2024
================================================

§ ABSTRACT 
We study conditional linear factor models in the context of asset pricing panels. Our analysis focuses on conditional means and covariances to characterize the cross-sectional and inter-temporal properties of returns and factors as well as their interrelationships. We also review the conditions outlined in <cit.> and show how the conditional mean-variance efficient portfolio of an unbalanced panel can be spanned by low-dimensional factor portfolios, even without assuming invertibility of the conditional covariance matrices. Our analysis provides a comprehensive foundation for the specification and estimation of conditional linear factor models.





Keywords: asset pricing, factor models, characteristics, covariances, mean-variance efficient portfolio, stochastic discount factor, covariance estimation 

JEL classification: G11, G12, C38



 

§ INTRODUCTION


Since the capital asset pricing model <cit.> and the arbitrage pricing theory of <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>,  academia and industry alike have shown a strong desire to compress vast asset pricing panels into low-dimensional linear factor representations. Accordingly, an extensive econometric literature has simultaneously developed that exploits statistical arbitrage relations and asymptotic results in the time series and cross section to develop estimators for factor loadings and return covariances <cit.>. 
Recent literature devises models for conditional means of panels  <cit.>, or conditional covariances <cit.>.
To date, however, there is no comprehensive collection of properties that any conditional linear factor model should satisfy, how they relate to first and second conditional moments, and how these properties change when the factors are tradable. 


In this short note, we fill the gap in this literature and discuss exhaustively and from first principles the fundamental properties obeyed by (un)conditional linear factor models. Our analysis focuses on conditional means and covariances to characterize the cross-sectional and inter-temporal properties of returns and factors as well as their interrelationships. 

The focus is on a concise, rigorous discussion with minimal assumptions and the identification of pitfalls. For example, we do not assume invertibility of the covariance matrices, which is a strong assumption given the large cross-sectional dimensions in modern unbalanced asset pricing panels.  

In particular, we investigate under which conditions

  
  *  factors and residuals are conditionally uncorrelated,
  
  *  tradable factors (i.e. factor portfolios) span the conditional mean-variance efficient portfolio,
  
  *  a generative risk factor model can be represented by tradable factors that load on the same coefficient matrix and fulfill the first two properties,

and how these three questions are interrelated.[Question <ref> is also studied in <cit.>, but under more restrictive assumptions. We further extend their paper by relating <ref> to aspects of <ref> and <ref>.] For example, we find that a linear factor model with tradable factors and conditionally uncorrelated residuals, as in <ref>, cannot possibly have an invertible residual covariance matrix. We also review the conditions outlined in <cit.> and show how the conditional mean-variance efficient portfolio of an unbalanced panel can be spanned by low-dimensional factor portfolios, relating to <ref>. We find that  property <ref> holds almost universally, which is remarkable and suggests a general structure for consistent estimators of conditional means and covariance matrices for unbalanced asset pricing panels. 

Our analysis is relevant in particular also for the broader context of models that are linear in nonlinear functions of factors, as they are commonly used in financial machine learning based upon neural nets, or kernel-based methods.
It is complemented by extensive examples and provides a comprehensive foundation for the specification and estimation of conditional linear factor models.




The structure of this short note is as follows. In Section <ref> we introduce the formal setup for conditional linear factor models and give an overview of our main results. In Section <ref> we answer question <ref> and derive related results based on the covariances. We also give an example that serves as counterexample in some of our proofs. In Section <ref> we discuss risk premia and show that characteristics are covariances under certain conditions. In Section <ref> we address question <ref>. Using the counterexample, we also show some pitfalls with false implications. In Section <ref> we address question <ref>. In Section <ref> we conclude. The appendix collects all the proofs.  

 




 


§ CONDITIONAL LINEAR FACTOR MODELS


We study conditional linear factor models of the form

    x_t+1 = Φ_t  f_t+1 + ϵ_t+1,

where x_t+1 denotes the vector of excess returns of n_t assets over period [t,t+1], for a n_t× m matrix Φ_t of asset characteristics that are measurable with respect to the information set (i.e.  observable) at t, loading on a vector of m common risk factors f_t+1, and residuals ϵ_t+1.[Formally, all random variables are modeled on a probability space (Ω,,) along with a sequence of information sets (σ-algebras) _t⊆_t+1⊆⋯⊆. We write _t[·] and _t[·] for the conditional mean and covariance given _t. We assume a regular conditional probability _t such that _t[1_A]=_t[A] for any event A∈. Conditional expectations given _t hence amount to expectations under _t, which we always assume to exist and be finite. Equality between random variables means _t-almost sure equality. Assuming a trivial information set, _t={∅,Ω}, this setup includes also unconditional moments.] 



The above panel specification inherently combines two notions, one within generative statistical models, and one within the study of optimal portfolios. The first interprets the right-hand side of (<ref>) as a data-generating process, where abstract risk factors f_t+1 and idiosyncratic risk ϵ_t+1 generate returns x_t+1, given Φ_t. The second assumes that the factors are tradable portfolios of the form

    f_t+1 =  W_t^⊤ x_t+1,

for a n_t× m matrix W_t of weights that are measurable with respect to the information set at t. Such tradable factors should serve as an accurate low-rank representation of the full cross section x_t+1, with implied residuals ϵ_t+1 = x_t+1 - Φ_t  f_t+1, given Φ_t. In this portfolio context an eminent question concerns the conditional Sharpe ratios that can be attained from trading in the full cross section x_t+1 or only in the factor portfolios f_t+1, respectively.[Our setup is the same as in <cit.> and in fact more general, as they assume that matrices have full rank. In view of Footnote <ref>, our analysis even applies for m≥ n_t.] 


Our analysis of the introductory questions <ref>–<ref> is based exclusively on the first two conditional moments. We therefore denote by μ_t_t[ x_t+1] and Σ_t_t[ x_t+1] the conditional mean and covariance matrix of the excess returns, and analogously we write μ_ f, t_t[ f_t+1], Σ_ f, t_t[ f_t+1] and Σ_ϵ,t_t[ϵ_t+1] for those of the factors and residuals, respectively. We do not assume that these matrices have full rank. Instead of the regular matrix inverses we use the pseudoinverse, denoted by A^+ for any matrix A.[Also called Moore–Penrose generalized inverse. Let A= V  D  W^⊤ be the singular value decomposition of a matrix A, with orthogonal matrices V and W. Then the pseudoinverse of A is given by A^+ =  W D^+  V^⊤, where D^+ is the transpose of D in which the positive singular values are replaced by their reciprocals, see <cit.> or <cit.>. Consequently, A^+ A and A A^+ are the orthogonal projections on the images of A^⊤ and A, respectively. If A^⊤ A is invertible then A^+ = ( A^⊤ A)^-1 A^⊤.]



To simplify the exposition, we will omit the qualifier “conditional” from probabilities, expectations, covariances, and correlations in the following. We will also refer to excess returns simply as “returns”. Figure <ref> gives an overview of some of our main findings. 







 



§ FACTOR AND RESIDUAL CORRELATION


   

Our first result gives necessary and sufficient conditions that factors and residuals are  uncorrelated, which answers the introductory question <ref>, and relates this property to the covariance matrix of the returns. We refer to Φ_t  f_t+1 as the factor-spanned component of the returns whose covariance is given by _t[Φ_t  f_t+1]=Φ_t Σ_ f,tΦ_t^⊤.

The factor-spanned component and the residuals are  uncorrelated,

    _t[Φ_t  f_t+1, ϵ_t+1] = 0,

if and only if the covariance matrix of the returns is given by the sum

    Σ_t = Φ_t Σ_ f,tΦ_t^⊤  + Σ_ϵ,t.

Moreover, if 

    (Φ_t)=m

then any of (<ref>) or (<ref>) is equivalent to 

    _t[ f_t+1, ϵ_t+1] = 0.






We obtain stronger results under the assumption that the factors are tradable. 





Assume that (<ref>) holds. Then any of (<ref>) or (<ref>) is equivalent to

    Σ_t   W_t Φ_t^⊤   = Φ_t  W_t^⊤Σ_t W_t Φ_t^⊤ = Φ_t  W_t^⊤Σ_t.

Moreover, if 

    W_t^⊤∩Φ_t={ 0},

then any of (<ref>), (<ref>) or (<ref>) is equivalent to (<ref>).










Factor-spanned components that are uncorrelated with residuals imply a zero matrix-product relation between covariance matrices, but not conversely, as the following proposition shows.


Assume that (<ref>) holds. Then any of (<ref>), (<ref>) or (<ref>) implies the zero matrix-product relation

    Φ_t  W_t^⊤Σ_ϵ,t = 0.

However, the converse is not true, as (<ref>) does not imply any of (<ref>), (<ref>) or (<ref>).




As an important corollary of Proposition <ref>, we furthermore obtain an impossibility result.

Assume that (<ref>) holds and the factor model (<ref>) is non-degenerate, Φ_t  W_t^⊤≠ 0. Then the residual covariance matrix cannot have full rank,

    Σ_ϵ,t<n_t,

under any of the equivalent conditions (<ref>), (<ref>) or (<ref>).



Corollary <ref> implies in particular that there exists no linear factor model (<ref>) with tradable factors (<ref>), which are portfolios in x _t+1, and uncorrelated idiosyncratic risk components ϵ_t+1 with nonsingular covariance matrix Σ_ϵ,t. Corollary <ref> thus has ramifications for the literature on shrinkage of covariance matrices with factor structure <cit.>, where oftentimes the spectrum of the residual covariance is lifted in order to obtain invertibility.


 

Tradable factors with weight matrix inducing a projection imply cross-sectional orthogonality, which is stronger than (<ref>), in the following sense.




Assume that (<ref>) holds for a weight matrix such that the matrix product

    Φ_t W_t^⊤ =(Φ_t W_t^⊤)^2

is a (generally non-orthogonal) projection. Then the rows of Φ_t W_t^⊤ and the residuals are cross-sectionally orthogonal, Φ_t W_t^⊤ϵ_t+1= 0, and therefore (<ref>) holds. If in addition the matrix product

    Φ_t W_t^⊤ = W_tΦ_t^⊤

is self-adjoint, and thus an orthogonal projection, then the factor-spanned component and the residuals are cross-sectionally orthogonal, 

    (Φ_t f_t+1)^⊤ϵ_t+1= 0.

Even in this case, uncorrelatedness (<ref>) generally does not follow.
 




We are able to  further qualify the above results under more specific assumptions on the weight matrix, as the following lemma shows.


The weight matrix W_t satisfies  

    W_t^⊤Φ_t W_t^⊤ =  W_t^⊤

if and only if (<ref>) and (<ref>) hold. In this case, we also have (Φ_t W_t^⊤)= W_t^⊤. Property (<ref>) holds in particular for weight matrices of the form

    W_t^⊤ =  R_t ( S_tΦ_t  R_t)^+  S_t,
 
 for some m× m-matrix R_t and n_t× n_t-matrix S_t. If in addition we have
 
    ( Φ_t R_t)∩ S_t ={ 0},

 then (Φ_t W_t^⊤) = (Φ_t R_t).







Weight matrices of the form (<ref>) include the following two examples.

For R_t= I_m and S_t= I_n_t in (<ref>), we obtain the OLS factors 

    f^ OLS_t+1:=Φ_t^+ x_t+1.

Here, Φ_t W_t^⊤=Φ_tΦ_t^+ is the orthogonal projection in ^n_t onto the image of Φ_t. The OLS factors f^ OLS_t+1 minimize the cross-sectional least-squares problem x_t+1 - Φ_t f_t+1_2 over all factors f_t+1 in ^m.[ If (Φ_t)=m then the OLS problem has a unique solution. In general, all factors of the form f_t+1= f^ OLS_t+1 + ( I_m - Φ_t^+Φ_t) w_t+1, for any w_t+1∈^m, are OLS solutions. Hence f^ OLS_t+1 is the unique OLS solution f_t+1 with minimal norm f_t+1_2.]




For R_t= I_m and S_t in (<ref>) such that S_t^⊤ S_t= Σ_ϵ,t^+, we obtain the GLS factors 

    f^ GLS_t+1:=( S_tΦ_t)^+ S_t x_t+1.

The GLS factors f^ GLS_t+1 minimize the squared Mahalanobis length ( x_t+1 - Φ_t f_t+1)^⊤Σ_ϵ,t^+( x_t+1 - Φ_t f_t+1) over all factors f_t+1 in ^m.[Subject to similar aspects as discussed in Footnote <ref>.] More generally, we see that (<ref>) allows for rotated GLS factors by an appropriate choice of R_t.







The following example illustrates the dichotomy between cross-sectional orthogonality as defined in Lemma <ref>, and orthogonality in the time series, as it shows OLS factors that are correlated with residuals. It serves as counterexample in the proofs of Proposition <ref> and Lemmas <ref> and <ref>, and prepares and complements Proposition <ref> below.
[label=exspanning2]
  Let n_t=3, m=2, and let ξ_1, ξ_2, ξ_3  be  uncorrelated random variables with  variances _t[ξ_i]=a_i>0 and  means b_i=_t[ξ_i]∈, for i=1,2,3. Assume returns and their characteristics are given by
 
    [ x_t+1,1; x_t+1,2; x_t+1,3 ][                        ξ_1;                        ξ_2; ρ/a_1ξ_1 + ρ/a_2ξ_2  + ξ_3 ] ,   Φ_t [ 1 0; 1 0; 0 1 ],

for some ρ∈. This gives the following mean and covariance matrix of x_t+1,
  
    μ_t = [                         b_1;                         b_2; ρ/a_1 b_1 + ρ/a_2 b_2 + b_3 ],  Σ_t =[                 a_1                   0                   ρ;                   0                 a_2                   ρ;                   ρ                   ρ ρ/a_1 + ρ/a_2 + a_3 ].

Matrix Φ_t has full column rank m=2 and its pseudoinverse equals
   
    Φ_t^+ = [ 1/2 1/2   0;   0   0   1 ].

We set the portfolio weight matrix W_t^⊤Φ_t^+, so that we obtain the OLS factors (cf. Example <ref>),

    f_t+1 =  f^ OLS_t+1 = Φ_t^+  x_t+1 = [ 1/2 x_t+1,1 + 1/2 x_t+1,2;                   x_t+1,3 ],

and 
    Φ_t  W_t^⊤ =Φ_t Φ_t ^+ =[ 1/2 1/2   0; 1/2 1/2   0;   0   0   1 ]
 is the orthogonal projection onto the image of Φ_t. As shown in (the first part of) Lemma <ref>, the factor-spanned component and residuals, given by

    Φ_t f_t+1= [ 1/2 x_t+1,1 + 1/2 x_t+1,2; 1/2 x_t+1,1 + 1/2 x_t+1,2;                   x_t+1,3 ]  and  ϵ_t+1 =   [ 1/2 x_t+1,1 -1/2 x_t+1,2; 1/2 x_t+1,2 -1/2 x_t+1,1;                        0 ],

are cross-sectionally orthogonal, (<ref>), and the zero matrix-product condition (<ref>) holds. However, their  covariance,

    _t[ Φ_t f_t+1,ϵ_t+1] = [ 1/2(a_1-a_2) 1/2(a_2-a_1)            0; 1/2(a_1-a_2) 1/2(a_2-a_1)            0;            0            0            0 ] ,

is nonzero if a_1≠ a_2.                                                                                                   


















































§ RISK PREMIA

The preceding   properties of linear factor models have not yet directly used the asset risk premia μ_t so far.  In this section, we investigate the role of  the  mean in particular with respect to the time series properties of linear factor models. 
The following elementary lemma gives an algebraic condition in terms of μ_t such that residual risk is unpriced for tradable factors.

Assume that (<ref>) holds. Then residual risk is unpriced,

    _t[ϵ_t+1]= 0,

if and only if

    μ_t =Φ_t W_t^⊤μ_t.






The following proposition further extends our knowledge of the properties of linear factor models, by showing that characteristics are covariances under the assumption of uncorrelated factors and residuals and unpriced residual risk. This property in turn concerns optimality of the characteristics matrix Φ_t in the time series.  It extends the discussion of OLS factors in <cit.>, and GLS factors in <cit.> with respect to their times series, cross-sectional, and asset pricing properties <cit.>.



Assume that (<ref>) and (<ref>) hold. Then the characteristics Φ_t are also covariances for the factors, in the sense that

    Φ _t∈β _t _t  x_t+1 - β_t  f_t+1_2^2

where the minimum is taken over all n_t× m-matrices β_t that are measurable with respect to the information set at t. The minimizer in (<ref>) is unique if and only if the Gram matrix of the factors, Σ_ f,t + μ_ f,tμ_ f,t^⊤, has full rank. Sufficient conditions for (<ref>) to hold for tradable factors are given in Proposition <ref> and Lemma <ref>.


 





§ SPANNING FACTORS

We henceforth assume absence of arbitrage, in the weak sense that[An arbitrage is a strategy w_t that yields a nonnegative return, w_t^⊤ x_t+1≥ 0, and such that _t[ w_t^⊤ x_t+1> 0]>0. Condition (<ref>) is necessary but not sufficient for the absence of such arbitrage in the strict sense, which is outside the scope of this short note. Indeed, the orthogonal decomposition μ_t = μ_t,0 + μ_t,1 according to Σ_t⊕Σ_t shows that μ_t,0^⊤ x_t+1 = μ_t,0_2^2 is risk-free and strictly positive, and thus μ_t,0 is an arbitrage, unless μ_t,0= 0. ]

    μ_t ∈Σ_t .

This is equivalent to the existence of the mean-variance efficient (MVE) portfolio maximizing the objective w_t^⊤μ_t - 1/2 w_t^⊤Σ_t  w_t over w_t∈^n_t.[The MVE portfolio is unique up to scaling by the risk aversion parameter, which cancels out in the Sharpe ratio and which we therefore set equal to one.] Its weights are given by w_t = Σ_t^+μ_t, and it attains the maximum squared Sharpe ratio, SR_t^2=μ_t^⊤Σ_t^+ μ_t.

Absence of arbitrage (<ref>) is also equivalent to the existence of the minimum-variance stochastic discount factor (SDF), which solves the problem

    M_t+1∈ L^2_ _tminimize    _t[ M_t+1^2 ] 
    subject to    _t[ M_t+1 x _t+1 ]= 0  and _t[ M_t+1 ]=1,

and is given in terms of the MVE portfolio by M_t+1= 1 - μ_t^⊤Σ_t^+ ( x_t+1 - μ_t).


We henceforth assume that factors are tradable,  and consider m factor portfolios f_t+1 with returns given by (<ref>). We denote by μ_ f,t_t[ f_t+1] =   W_t^⊤μ_t their  mean and, as above, by Σ_ f,t = W_t^⊤Σ_t  W_t their  covariance. Then the MVE portfolio spanned by f_t+1 with weights given by Σ_ f,t^+ μ_ f,t is equally well defined and yields the squared Sharpe ratio SR_ f,t^2=μ_ f,t^⊤Σ_ f,t^+μ_ f,t.[In fact, we have μ_ f,t∈Σ_ f,t, which follows from (<ref>) and using that A = A A^⊤ for any matrix A.] There exists also a minimum-variance SDF that prices only factors. 

In general, we have SR_ f,t^2≤ SR_t^2. The following proposition gives sufficient and necessary conditions for equality and answers the introductory question <ref>. It generalizes <cit.> to the case where the matrices Σ_t, Σ_ f,t and Φ_t do not have full rank.[<cit.> derive the proof of their Lemma 1 from <cit.>, which assumes that matrices are invertible.]



The following are equivalent:

  
  *  the factor MVE portfolio attains the maximum Sharpe ratio, 
  
    SR_ f,t^2= SR_t^2;

 
  *  asset risk premia μ_t are given as covariances with the factors, _t[ x_t+1, f_t+1]=Σ_t W_t, as
 
    μ_t ∈( Σ_t  W_t);
  
  
  *  factors f _t+1 span the full MVE portfolio,
  
    μ_ f,t^⊤Σ_ f,t^+  f_t+1 = μ_t^⊤Σ_t^+  x_t+1;


  *  factors f _t+1 span the minimum-variance SDF, M_t+1= 1 - μ_ f,t^⊤Σ_ f,t^+ ( f_t+1 - μ_ f,t).
  





For a large class of tradable factors, and under the assumption that factors and residuals are uncorrelated, the spanning condition (<ref>) holds if and only if the residual risk is unpriced. This is the content of the following proposition.


  Assume that any of (<ref>), (<ref>) or (<ref>) holds. Then any of (<ref>) or (<ref>) implies the spanning condition (<ref>). Conversely, if the weight matrix W_t satisfies (<ref>), then the spanning condition (<ref>) implies any of (<ref>) or (<ref>).






From Proposition <ref> we deduce that any of (<ref>), (<ref>) or (<ref>) and any of (<ref>) or (<ref>) together imply the spanning condition (<ref>). However, the converse is not true, as the following continuation of Example <ref> shows.[Proposition 2 in <cit.> is similar to our Proposition <ref>, but differs in some important points. First, their proposition assumes that the weight matrix is of the form W_t =Φ_t  R_t, for some invertible m× m matrix R_t. However, this excludes GLS factors (Example <ref>) unless they are OLS factors (Example <ref>) and Φ_t={ 0} holds, in which case R_t=(Φ_t^⊤Φ_t)^-1. Second, they replace the spanning condition (<ref>) by the stronger assumption that Φ_t⊆(Σ_t W_t) for the converse implication in their proposition. Third, they replace our decomposition (<ref>) by the weaker property that Σ_t =Φ_tΨ_tΦ_t^⊤ +  U_tΩ_t U_t^⊤ for some conformable matrices Ψ_t, Ω_t, and a n_t× (n_t-m) matrix U_t for which U_t^⊤Φ_t= 0. The latter holds automatically in our case, see (<ref>). Hence their proposition makes no inference on the correlation between factors and residuals as we do.] 

[continues=exspanning2]
Assume additionally b_1=b_2 and ρ≠ 0. Then (<ref>) follows by inspection. Moreover, some algebra shows that

    Σ_t  W_t [     0; b_1/ρ ]=Σ_t (Φ_t^+)^⊤[     0; b_1/ρ ] = μ_t,

if b_3 = (1-ρ)b_1/a_1 + (1-ρ)b_1/a_2 + b_1/ρa_3, which shows (<ref>). Hence (<ref>), (<ref>) and the spanning condition (<ref>) hold, while (<ref>), (<ref>) and (<ref>) do not.


The following lemma puts Example <ref> into a larger perspective and shows what goes wrong with the failed implication.[In fact, in Example <ref> we have b_t= W_t c_t for c_t= [     0; b_1/ρ ].]


Assume that the weight matrix W_t satisfies either (<ref>) or Φ_t⊆ W_t. Then any of (<ref>) or (<ref>) and the spanning condition (<ref>) imply that there exists a vector b_t∈^n_t such that the vector equality Σ_t  W_tΦ_t^⊤ b_t=Φ_t W_t^⊤Σ_t  W_tΦ_t^⊤ b_t holds. However, not the matrix equality (<ref>), in general.   





§ GENERATIVE MODELS HAVE SPANNING FACTORS


In the following we answer the introductory question <ref>. Concretely, we establish sufficient conditions on a data-generating linear factor model (<ref>), which we write here as  

    x_t+1 =Φ_t  g_t+1 +  z_t+1,

such that GLS-type factors are spanning. Here g_t+1∈^m denote some abstract (generally non-tradable) systematic risk factors and z_t+1 the idiosyncratic risk components of the returns x_t+1. We denote their mean and covariance matrices by μ_ g,t_t[ g_t+1], Σ_ g,t_t[ g_t+1], and Σ_ z,t_t[ z_t+1].
We first prove two technical lemmas.


Assume that systematic risk factors g_t+1 and idiosyncratic risk z_t+1 are  uncorrelated, _t[ g_t+1, z_t+1]= 0,  covariance matrix Σ_ g,t is invertible, and idiosyncratic risk is unpriced, _t[ z_t+1]= 0. Then the implied  return covariance matrix decomposes as

    Σ_t = Φ_t Σ_ g,tΦ_t^⊤  + Σ_ z,t,

and absence of arbitrage (<ref>) holds.




The following auxiliary lemma gives a matrix extension in the case where the idiosyncratic covariance matrix does not have full rank.


    Let U_t be a n_t× n_t-matrix such that U_t^⊤ U_t= Σ_ z,t^+. Then Σ_ z,t= U_t^⊤ and Σ_ z,t= U_t. Moreover, there exists an invertible n_t× n_t-matrix  S_t extending U_t in the sense that S_t  =  U_t on U_t^⊤ and S_t^⊤= U_t^⊤ on U_t, and S_t: U_t→ U_t^⊤ is a bijection. In particular, S_t= U_t if Σ_ z,t is invertible, and a possible choice is S_t = I_n_t in the degenerate case where Σ_ z,t= 0.




We can now prove our announced result.




Under the assumptions of Lemmas <ref> and <ref>, the GLS-type factors 

    f_t+1( S_tΦ_t)^+ S_t x_t+1

satisfy  spanning condition (<ref>). The implied residuals ϵ_t+1 x_t+1 - Φ_t f_t+1 have zero  mean (<ref>) and are  uncorrelated with the factors (<ref>). The implied matrix

    Φ_t W_t^⊤ = Φ_t ( S_tΦ_t)^+ S_t is a (generally non-orthogonal) projection onto Φ_t,

and the residuals are equal to the projection of idiosyncratic risk,

    ϵ_t+1 = ( I_n_t -Φ_t W_t^⊤) z_t+1.

Mean and covariance matrices of f_t+1 and ϵ_t+1 are given by the expressions

    μ_ f,t    = ( S_tΦ_t)^+ S_tΦ_tμ_ g,t,
    Σ_ f,t   = ( S_tΦ_t)^+  S_tΦ_t Σ_ g,tΦ_t^⊤ S_t^⊤ (Φ_t^⊤ S_t^⊤)^+   +  Q_t ,
    Σ_ϵ,t   =     Σ_ z,t - Φ_t W_t^⊤Σ_ z,t W_tΦ_t^⊤ = Σ_ z,t -Φ_t  Q_t Φ_t^⊤,

where Q_t  ( S_tΦ_t)^+  S_t Σ_ z,t S_t^⊤ (Φ_t^⊤ S_t^⊤)^+ = ( S_tΦ_t)^+ U_t  U_t^+   (Φ_t^⊤ S_t^⊤)^+.




Note  that S_t S_t^⊤≠Σ_ϵ,t in general in the above statement. Hence, the factors f_t+1 in Proposition <ref> are only of “GLS-type”, rather than the  GLS factors as described in Example <ref>. Nevertheless, Proposition <ref> remarkably promises that any return panel linearly generated by abstract risk factors, which are not necessarily tradable, also has a linear representation in terms of tradable factors f_t+1 that load on the same characteristics Φ_t. However, the same statement also reveals that the properties of these tradable factors crucially rely on Φ _t, as well as on Σ _g,t and Σ _z,t in their population formulation. Any empirical application thus necessitates consistent estimators of Φ _t and Σ _t, or, alternatively Σ_ g,t and Σ_ z,t.  


 



§ CONCLUSION

This short note provides a collection of elementary and fundamental properties of conditional linear factor models for unbalanced panels, in particular in the context of asset pricing. Our  results are derived for a finite cross section, they are comprehensive, and based upon  elementary linear algebra.

We exhaustively describe the relation between the covariance matrix of returns, the covariance matrix of the residuals, risk premia, and Sharpe ratios. Our results range from the question when a covariance matrix of a panel generated by a linear factor structure can be decomposed into  a factor, and a residual part, to the question when the maximum Sharpe ratio attained by the full return panel can also be attained a linear factor portfolio. If the factors are themselves portfolios of the returns, a number of additional powerful properties and qualifications arise that are useful in a portfolio context. We show that  any return panel whose data-generating process is linear in abstract, non-tradable factors, has a linear factor representation in terms of tradable factors that load on the same characteristics. The same result also shows that consistent estimators for factor, residual, and return covariances are indispensable for econometric factor analysis.

Our analysis provides a simple framework together with simple guidelines for the estimation and specification of linear factor models, comprising also modern formulations based upon neural nets or kernels. 
Future work could tackle the connection of the results in this short note, to the asymptotic notions introduced in <cit.> and <cit.>.








§ PROOFS

The appendix collects all proofs.



 §.§ Proof of Lemma <ref>


Equivalence of (<ref>) and (<ref>) follows from the elementary decomposition

    Σ_t = Φ_t Σ_ f,tΦ_t^⊤ + _t[Φ_t  f_t+1, ϵ_t+1] + _t[ϵ_t+1,Φ_t  f_t+1] + Σ_ϵ,t.

Assuming (<ref>), equivalence of (<ref>) and (<ref>) follows because _t[Φ_t  f_t+1, ϵ_t+1]  = Φ_t _t[ f_t+1, ϵ_t+1].




 §.§ Proof of Proposition <ref>


Given (<ref>), it follows that

    _t[Φ_t  f_t+1,ϵ_t+1] = _t[Φ_t  f_t+1, x_t+1 - Φ_t  f_t+1] = Φ_t  W_t^⊤Σ_t - Φ_t  W_t^⊤Σ_t W_t Φ_t^⊤.

Hence (<ref>) is equivalent to (<ref>). 
Assuming (<ref>), equivalence of (<ref>) and (<ref>) follows because _t[Φ_t   f_t+1, ϵ_t+1]  =  Φ_t _t[ f_t+1, ϵ_t+1] and _t[ f_t+1, ϵ_t+1]=    W_t^⊤_t[ x_t+1, ϵ_t+1].





 §.§ Proof of Proposition <ref>


Given (<ref>), we have

    Σ_ϵ,t = _t[ x_t+1 - Φ_t  f_t+1] = Σ_t - Σ_t  W_t Φ_t^⊤- Φ_t  W_t^⊤Σ_t  + Φ_t  W_t^⊤Σ_t  W_t Φ_t^⊤ .

Hence (<ref>) implies Σ_ϵ,t =Σ_t   - Σ_t  W_t Φ_t^⊤ and therefore Φ_t  W_t^⊤Σ_ϵ,t =Φ_t  W_t^⊤Σ_t   - Φ_t  W_t^⊤Σ_t  W_t Φ_t^⊤ = 0, which yields (<ref>). That the converse implication is not true is proved by means of the counterexample given in Example <ref>.






 §.§ Proof of Corollary <ref>


This follows from (<ref>).





 §.§ Proof of Lemma <ref>


Given (<ref>), the residuals ϵ_t+1=( I_n_t-Φ_t W_t^⊤) x_t+1 satisfy Φ_t W_t^⊤ϵ_t+1= 0, and therefore (<ref>). The second statement follows by elementary linear algebra. The last statement follows by means of the counterexample given in Example <ref>.






 §.§ Proof of Lemma <ref>


Property (<ref>) clearly implies (<ref>). Now let v∈ W_t^⊤∩Φ_t. This means that v= W_t^⊤ c for some c∈^n_t such that Φ_t W_t^⊤ c= 0. Given (<ref>), this implies that v= W_t^⊤ c= W_t^⊤Φ_t W_t^⊤ c= 0, which proves (<ref>). This also implies that (Φ_t W_t^⊤)= W_t^⊤.[By elementary linear algebra, this again implies that the adjoint W_tΦ_t^⊤ is a projection with ( W_tΦ_t^⊤)= W_t^⊤. On the other hand, we only have (Φ_t W_t^⊤)⊆Φ_t, without equality in general.] Conversely, (<ref>) reads Φ_t W_t^⊤Φ_t W_t^⊤=Φ_t W_t^⊤, and given (<ref>) this again implies (<ref>), which proves the first part of the lemma.

For the second part of the lemma, given (<ref>), matrix algebra shows that W_t^⊤Φ_t W_t^⊤ =  R_t ( S_tΦ_t  R_t)^+  S_t Φ_t  R_t ( S_tΦ_t  R_t)^+  S_t= R_t ( S_tΦ_t  R_t)^+  S_t= W_t^⊤, where we used that A^+ A A^+= A^+ for any matrix A, which implies (<ref>). 

For the last statement, we first note that (Φ_t W_t^⊤) ⊆(Φ_t R_t) and consider the identity S_tΦ_t W_t^⊤ (Φ_t R_t) = S_tΦ_t R_t ( S_tΦ_t  R_t)^+  S_t Φ_t R_t   = S_t (Φ_t R_t). Given (<ref>), we infer that Φ_t W_t^⊤ (Φ_t R_t)=Φ_t R_t, and therefore (Φ_t W_t^⊤) = (Φ_t R_t).





 §.§ Proof of Lemma <ref>


This follows from the identity x_t+1=Φ_t W_t^⊤ x_t+1 +  ϵ_t+1. 





 §.§ Proof of Proposition <ref>


    The first claim follows from the  L_ _t^2-orthogonality of the factors and residuals, _t[  f_t+1ϵ_t+1^⊤]= 0, which holds due to (<ref>) and (<ref>). The second claim about the uniqueness of the minimizer follows elementary. 






 §.§ Proof of Proposition <ref>


<ref>⇔<ref>: Given (<ref>), we can write μ_t =Σ_t^1/2ν_t, and thus μ_ f,t =   W_t^⊤Σ_t^1/2ν_t, for some ν_t∈^n_t. We obtain SR_t^2 = ν_t^⊤ν_t and, using the matrix identity A^⊤ ( A A^⊤)^+ =  A^+ for any matrix A,  
  
    SR_ f,t^2   =ν_t^⊤ ( W_t^⊤Σ_t^1/2)^⊤(( W_t^⊤Σ_t^1/2) ( W_t^⊤Σ_t^1/2)^⊤)^+ ( W_t^⊤Σ_t^1/2)   ν_t 
       = ν_t^⊤ ( W_t^⊤Σ_t^1/2)^+( W_t^⊤Σ_t^1/2)   ν_t = ν_t^⊤ P_Σ_t^1/2 W_t ν_t ,

where P_Σ_t^1/2 W_t denotes the orthogonal projection onto the image of Σ_t^1/2 W_t. It follows that (<ref>) holds if and only if ν_t =  P_Σ_t^1/2 W_t ν_t lies in the image of Σ_t^1/2 W_t, which again is equivalent to (<ref>).  

<ref>⇒<ref>: Given (<ref>), and because (Σ_t^1/2 W_t)⊆(Σ_t W_t), there exists a vector b_t∈( W_t^⊤Σ_t^1/2) such that μ_t=Σ_t  W_t   b_t and therefore Σ_ f,t^+μ_ f,t=( W_t^⊤Σ_t  W_t)^+  W_t^⊤Σ_t W_t  b_t = b_t. On the other hand, we have x_t+1=Σ_tΣ_t^+  x_t+1 with probability one. We obtain μ_ f,t^⊤Σ_ f,t^+  f_t+1 =  b_t^⊤ W_t^⊤ x_t+1 =  b_t^⊤ W_t^⊤Σ_tΣ_t^+  x_t+1=μ_t^⊤Σ_t^+  x_t+1, with probability one, which is (<ref>).

<ref>⇒<ref>: This direction is trivial. 

<ref>⇔<ref>: This follows from (<ref>) and the aforementioned expression of M_t in terms of the full MVE portfolio.





 §.§ Proof of Proposition <ref>


Using (<ref>) and (<ref>), property (<ref>) implies μ_t = Φ_t W_t^⊤μ_t =Φ_t W_t^⊤Σ_t  b_t = Σ_t W_tΦ_t^⊤ b_t= Σ_t W_t c_t, for c_tΦ_t^⊤ b_t, for some b_t∈^n_t, which proves (<ref>)⇒(<ref>).

Conversely, using (<ref>) and (<ref>), the spanning condition (<ref>) implies Φ_t W_t^⊤μ_t =Φ_t W_t^⊤Σ_t  W_t c_t =  Σ_t  W_tΦ_t^⊤ W_t c_t =Σ_t   W_t c_t=μ_t, for some c_t∈^m, which proves (<ref>)⇒(<ref>). 





 §.§ Proof of Lemma <ref>


Under either assumption, we have that (<ref>) implies that there exists a vector b_t∈^n_t such that μ_t =Σ_t W_tΦ_t^⊤ b_t. The claimed vector equality now follows from (<ref>). The last statement follows by means of the counterexample given in Example <ref>.




 §.§ Proof of Lemma <ref>


    The decomposition (<ref>) follows from Lemma <ref>. This clearly implies that Σ_t = (Φ_t Σ_ g,tΦ_t^⊤)  + Σ_ z,t, and as μ_t=Φ_tμ_ g,t∈Φ_t =(Φ_t Σ_ g,tΦ_t^⊤) by assumption, absence of arbitrage (<ref>) follows.




 §.§ Proof of Lemma <ref>


We have Σ_ z,t= ( U_t^+ ( U_t^+)^⊤) =  U_t^+ = U_t^⊤, and Σ_ z,t= U_t, where we used that ( A^⊤ A)^+ = A^+ ( A^+)^⊤ for any matrix A. This proves the first statement.

We now construct S_t. By elementary linear algebra, we have ^n_t= U_t⊕ U_t^⊤ =  U_t^⊤⊕ U_t, and U_t:  U_t^⊤→ U_t is a bijection, and U_t= U_t^⊤ k≥ 0. Let {ξ_1,…,ξ_k} and {ζ_1,…,ζ_k} be any orthonormal bases of U_t and U_t^⊤, respectively. Setting S_t U_t on U_t^⊤ and S_tξ_iζ_i on U_t gives the desired extension. 




 §.§ Proof of Proposition <ref>


Lemma <ref> implies that S_t Σ_ z,t S_t^⊤ = U_t ( U_t^⊤ U_t)^+ U_t^⊤=  U_t U_t^+ is the orthogonal projection on U_t, where we used that ( A^⊤ A)^+  A^⊤ =  A^+ for any matrix A. We obtain ( S_t Σ_ z,t S_t^⊤)  S_t  =  U_t U_t^+ U_t =  U_t. Given that ( U_tΦ)⊆ ( S_tΦ), by definition of S_t, we thus have proved that
 
    S_tΣ_ z,t S_t^⊤ =   U_t U_t^+ maps ( S_tΦ_t) into itself.


As S_t is invertible, condition (<ref>) for R_t= I_m holds, and Lemma <ref> implies property (<ref>). This again implies (<ref>) and (<ref>) and, by Lemma <ref>, thus (<ref>).

By the same token and given (<ref>), condition (<ref>) is equivalent to Σ_ z,t W_tΦ_t^⊤   =  Φ_t W_t^⊤Σ_ z,t W_tΦ_t^⊤, which again is equivalent to

    S_tΣ_ z,t W_tΦ_t^⊤   =  S_tΦ_t W_t^⊤Σ_ z,t W_tΦ_t^⊤.

Plugging in (<ref>) for Φ_t W_t^⊤, the right hand side of (<ref>) can be written as S_tΦ_t ( S_tΦ_t)^+ S_t Σ_ z,t S_t^⊤ (Φ_t^⊤ S_t^⊤)^+ Φ_t^⊤. Given (<ref>), and as ((Φ_t^⊤ S_t^⊤)^+ Φ_t^⊤)⊆ ((Φ_t^⊤ S_t^⊤)^+)= ( S_tΦ_t) and S_tΦ_t ( S_tΦ_t)^+ is the orthogonal projection on ( S_tΦ_t), we infer that the right hand side of (<ref>) is equal to S_t Σ_ z,t S_t^⊤ (Φ_t^⊤ S_t^⊤)^+ Φ_t^⊤. This again is equal to the left hand side of (<ref>), which implies (<ref>). Proposition <ref> and Lemma <ref> prove (<ref>).

Proposition <ref> now implies that the spanning condition (<ref>) holds for f_t+1.

Expression (<ref>) and the first summand in expression (<ref>) follow directly from (<ref>). For the second summand in (<ref>), we have the identity ( S_tΦ_t)^+ S_t Σ_ z,t S_t^⊤ (Φ_t^⊤ S_t^⊤)^+ = ( S_tΦ_t)^+ U_t  U_t^+   (Φ_t^⊤ S_t^⊤)^+, using (<ref>). For (<ref>), as in the proof of Proposition <ref>, we first derive that (<ref>) implies Σ_ϵ,t =Σ_t   - Φ_t  W_t^⊤Σ_t  W_t Φ_t^⊤ =Σ_ z,t   -Φ_t  W_t^⊤Σ_ z,t W_t Φ_t^⊤, where we used (<ref>). By similar calculations as above, we derive Φ_t  W_t^⊤Σ_ z,t W_t Φ_t^⊤ = Φ_t ( S_tΦ_t)^+ U_t   U_t^+ (Φ_t^⊤ S_t^⊤)^+ Φ_t^⊤, which completes the proof.






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