Factor Models for Asset Returns

Prof. Daniel P. Palomar

The Hong Kong University of Science and Technology (HKUST)

MAFS6010R - Portfolio Optimization with R

MSc in Financial Mathematics
Fall 2019-20, HKUST, Hong Kong
Outline

1 Introduction

2 Linear Factor Models

3 Macroeconomic Factor Models

4 Fundamental Factor Models

5 Statistical Factor Models

Outline

1 Introduction

2 Linear Factor Models

3 Macroeconomic Factor Models

4 Fundamental Factor Models

5 Statistical Factor Models

Why Factor Models?

To decompose risk and return into explainable and unexplainable

components.
Generate estimates of abnormal returns.
Describe the covariance structure of returns
without factors: the covariance matrix for N stocks requires
N(N + 1)/2 parameters (e.g., 500(500 + 1)/2 = 125, 250)
with K factors: the covariance matrix for N stocks requires N (K + 1)
parameters with K  N (e.g., 500(3 + 1) = 2, 000)
Predict returns in specified stress scenarios.
Provide a framework for portfolio risk analysis.

D. Palomar Factor Models 4 / 45

Types of Factor Models

Factor models decompose the asset returns into two parts:

low-dimensional factors and idiosynchratic residual noise.1
Three types:
1 Macroeconomic factor models
factors are observable economic and financial time series
no systematic approach to choose factors
2 Fundamental factor models
factors are created from observable asset characteristics
no systematic approach to define the characteristics
3 Statistical factor models
factors are unobservable and extracted from asset returns
more systematic, but factors have no clear interpretation

1
R. S. Tsay, Analysis of Financial Time Series. John Wiley & Sons, 2005.
D. Palomar Factor Models 5 / 45
Outline

1 Introduction

2 Linear Factor Models

3 Macroeconomic Factor Models

4 Fundamental Factor Models

5 Statistical Factor Models

Linear Factor Model

Data:
N assets/instruments/indexes: i = 1, . . . , N
T time periods: t = 1, . . . , T
N-variate random vector for returns at t: xt = (x1,t , . . . , xN,t )T .

Factor model for asset i:

xi,t = αi + β1,i f1,t + · · · + βK ,i fK ,t + i,t , t = 1, . . . , T .

K : the number of factors

αi : intercept of asset i
ft = (f1,t , . . . , fK ,t )T : common factors (same for all assets i)
βi = (β1,i , . . . , βK ,i )T : factor loading of asset i (independent of t)
i,t : residual idiosyncratic term for asset i at time t

D. Palomar Factor Models 7 / 45

Cross-Sectional Factor Model
Factor model:
xt = α + Bft + t , t = 1, . . . , T
   T   
α1 β1 1,t
where α =  ...
 ..  . 
 (N × 1), B =  .  (N × K ), t =  .. 
  

αN βNT N,t
(N × 1).
α and B are independent of t
the factors {ft } (K -variate) are stationary with
E [ft ] = µf
Cov [ft ] = E [(ft − µf ) (ft − µf )T ] = Σf
the residuals {t } (N-variate) are white noise with
E [t ] = 0
2 2
Cov [t , s ] = E [t T


s ] = Ψδts , Ψ = diag σ1 , . . . , σN
the two processes {ft } and {t } are uncorrelated
D. Palomar Factor Models 8 / 45
Linear Factor Model
Summary of parameters:
α: (N × 1) intercept for N assets
B: (N × K ) factor loading matrix
µf : (K × 1) mean vector of K common factors
Σf : (K × K ) covariance
matrix of K common factors
Ψ = diag σ12 , . . . , σN
2 : N asset-specific variances

Properties of linear factor model:

the stochastic process {xt } is a stationary multivariate time series with
conditional moments
E [xt | ft ] = α + Bft
Cov [xt | ft ] = Ψ
unconditional moments
E [xt ] = α + Bµf
Cov [xt ] = BΣf BT + Ψ.
D. Palomar Factor Models 9 / 45
Time-Series Regression

Factor model:

xi = 1T αi + Fβi + i , i = 1, . . . , N

f1T
     
xi,1 1
 ..  .  .. 
where xi =  .  (T × 1), 1T =  ..  (T × 1), F = 
 
. 
xi,T 1 fTT
 
i,1
(T × K ), i = 
 ..  (T × 1).
. 
i,T
i is the (T -variate) vector of white noise with

E [i ] = 0
Cov [i ] = σi2 IT .

D. Palomar Factor Models 10 / 45

Multivariate Regression

Factor model (using compact matrix notation):

XT = α1T + BFT + ET
 T   T 
T
 
x1 f1 1
where X =  ...  (T × N), F =  ...  (T × K ), E =  .. 
    
. 
xTT fTT TT
(T × N).

D. Palomar Factor Models 11 / 45

Expected Return (α–β) Decomposition

Recall the factor model for asset i:

xi,t = αi + βiT ft + i,t , t = 1, . . . , T .

Take the expectation:

E [xi,t ] = αi + βiT µf
where
βiT µf is the explained expected return due to systematic risk factors
αi = E [xi,t ] − βiT µf is the unexplained exptected return (abnormal
return).

According to the CAPM model, the alpha should be zero, but is it?

D. Palomar Factor Models 12 / 45

Portfolio Analysis
Let w = (w1 , . . . , wN )T be a vector of portfolio weights (wi is the fraction
of wealth in asset i). The portfolio return is
N
X
rp,t = wT rt = wi ri,t , t = 1, . . . , T .
i=1
Portfolio factor model:
rt = α + Bft + t ⇒
rp,t = w α + w Bft + wT t = αp + βpT ft + p,t
T T

where
αp = w T α
βpT = wT B
p,t = wT t
and  
var (rp,t ) = wT BΣf BT + Ψ w.

D. Palomar Factor Models 13 / 45

Outline

1 Introduction

2 Linear Factor Models

3 Macroeconomic Factor Models

4 Fundamental Factor Models

5 Statistical Factor Models

Macroeconomic Factor Models

Recall the factor model for asset i:

xi,t = αi + βiT ft + i,t , t = 1, . . . , T .

In this model, the factors {ft } are observed economic/financial time series.

Econometric problems:
choice of factors
estimation of mean vector and covariance matrix of factors µf and Σf
from observed history of factors
estimation of factor betas βi ’s and residual variances σi2 ’s using time
series regression techniques

D. Palomar Factor Models 15 / 45

Macroeconomic Factor Models
Single factor model of Sharpe (1964) (aka CAPM):
xi,t = αi + βi RM,t + i,t , t = 1, . . . , T
where
RM,t is the return of the market (in excess of the risk-free asset rate):
market risk factor (typically a value weighted index like the S&P
500)
xi,t is the return of asset i (in excess of the risk-free rate)
K = 1 and the single factor is f1,t = RM,t
the unconditional cross-sectional covariance matrix of the assets is
2
Cov [xt ] = Σ = σM ββ T + Ψ
where
2
σM = var (RM,t )
T
β = (β1 , . . . , βN )
Ψ = diag σ1 , . . . , σN2
2

D. Palomar Factor Models 16 / 45

Macroeconomic Factor Models

Estimation of single factor model:

xi = 1T α̂i + β̂i rM + ˆi , i = 1, . . . , N

where rM = (RM,1 , . . . , RM,T ) with estimates:

β̂i = cd
ov (xi,t , RM,t ) /var
c (RM,t )
α̂i = x̄i − β̂i r¯M
1
σ̂i2 = ˆT Ψ̂ = diag σ̂12 , . . . , σ̂N
2


T −2  ˆi ,
i 

The estimated single factor model covariance matrix is

2
Σ̂ = σ̂M β̂ β̂ T + Ψ̂

D. Palomar Factor Models 17 / 45

Macroeconomic Factor Models: Market Neutrality

Recall the single factor model:

xt = α + βRM,t + t , t = 1, . . . , T

When designing a portfolio w, it is common to have a market-neutral

constraint:
β T w = 0.
This is to avoid exposure to the market risk. The resulting risk is given by

wT Ψw.

D. Palomar Factor Models 18 / 45

Capital Asset Pricing Model: AAPL vs SP500
AAPL regressed against the SP500 index (using risk-free rate
rf = 2%/252):
0.1
Scatter
CAMP model

0.05

0
AAPL log−return

−0.05

−0.1

−0.15
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06
Market log−return

D. Palomar Factor Models 19 / 45

R2
The R 2 from the time series regression is a measure of the proportion of
“market” risk and 1 − R 2 is a measure of asset specific risk (σi2 is a
measure of the typical size of asset specific risk).

Given the variance decomposition:

var (xi,t ) = βi2 var (RM,t ) + var (i,t ) = βi2 σM
2
+ σi2
R 2 can be estimated as
β̂i2 σ̂M
2
R2 =
var
c (xi,t )
Extensions:
robust regression techniques to estimate βi , σi2 , and σM
2
2 ,
factor model not constant over time, obtaining time-varying βi,t , σi,t
2
and σM,t
βi,t can be estimated via rolling regression or Kalman filter techniques
2 2
σi,t and σM,t (i.e., conditional heteroskedasticity) can be captured via
GARCH models or exponential weights
D. Palomar Factor Models 20 / 45
Macroeconomic Factor Models

General multifactor model:

xi,t = αi + βiT ft + i,t , t = 1, . . . , T .

where the factors {ft } represent macro-economic variables such as2

market risk
price indices (CPI, PPI, commodities) / inflation
industrial production (GDP)
money growth
interest rates
housing starts
unemployment
In practice, there are many factors and in most cases they are very
expensive to obtain. Typically, investment funds have to pay substantial
subscription fees to have access to them (not available to small investors).

2
Chen, Ross, Roll (1986). “Economic Forces and the Stock Market”
D. Palomar Factor Models 21 / 45
Macroeconomic Factor Models
Estimation of multifactor model (K > 1):

xi = 1T αi + Fβi + i , i = 1, . . . , N
= F̃γi + i
 
  αi
where F̃ = 1T F and γi = .
βi
Estimates:
T
γ̂i = (F̃T F̃)−1 F̃T xi , B̂ = β̂1 · · · β̂N


ˆi = xi − F̃γ̂i
1
σ̂i2 = T −K ˆT ˆi , Ψ̂ = diag σ̂12 , . . . , σ̂N
2


−1  i 

T
Σ̂f = T 1−1 T ft − f̄ , f̄ = T1 T
P
P
t=1 ft − f̄ t=1 ft

The estimated multifactor model covariance matrix is

Σ̂ = B̂Σ̂f B̂T + Ψ̂
D. Palomar Factor Models 22 / 45
Interlude: LS Regression
Consider the LS regression:

minimize kx − F̃γk2
γ

We set the gradient to zero

2F̃T F̃γ − 2F̃T x = 0

to find the estimator as the optimal solution

γ̂ = (F̃T F̃)−1 F̃T x.

Now we can compute the residual of the LS regression as ˆ = x − F̃γ̂ and

its variance with the sample estimator
1 1
σ̂ 2 = k2 = kx − F̃γ̂k2 .
kˆ
T T
D. Palomar Factor Models 23 / 45
Interlude: Maximum Likelihood Estimation
The pdf for the residual under a Gaussian distribution is
1 1 2
p (t ) = √ e − 2σ2 t
2πσ 2
and the log-likelihood of the parameters γ, σ 2 given the T uncorrelated

observations in  is
T 1
L γ, σ 2 ;  = − log 2πσ 2 − 2 kx − F̃γk2 .



2 2σ
We can then formulate the MLE as the optimization problem
minimize T2 log σ 2 + 2σ1 2 kx − F̃γk2 .


γ,σ 2

Setting the gradient with respect to γ and 1/σ 2 , respectively, we get

1 T 1 T 1
2
F̃ F̃γ̂ − 2 F̃T x = 0 and − σ̂ 2 + kx − F̃γ̂k2 = 0
σ̂ σ̂ 2 2
leading to the estimators
1
γ̂ = (F̃T F̃)−1 F̃T x and σ̂ 2 = kx − F̃γ̂k2 .
T
D. Palomar Factor Models 24 / 45
Interlude: MLE is Biased
The MLE is consistent and asymptotically efficient and unbiased. However,
it is biased for finite number of observations:
the estimation of γ is unbiased
h i
E [γ̂] = E (F̃T F̃)−1 F̃T x = (F̃T F̃)−1 F̃T E [x] = (F̃T F̃)−1 F̃T F̃γ = γ

the estimation of σ 2 is, however, biased. First, rewrite it as

1 1 1
σ̂ 2 = kx − F̃γ̂k2 = kx − F̃(F̃T F̃)−1 F̃T xk2 = kP⊥ xk2
T T T F̃
where P⊥F̃
= I − F̃(F̃T F̃)−1 F̃T is the projection onto the subspace
orthogonal to the one spanned by F̃ (contains T − (K + 1)
eigenvalues equal to one and K + 1 zero eigenvalues). Now, the bias is
1 h i 1  ⊥ h T i
E σ̂ 2 = E xT P⊥
 
F̃
x = Tr PF̃ E xx =
T T
1    σ 2 T − (K + 1)
= Tr P⊥ F̃
σ 2
I + F̃γγ T T
F̃ = Tr(P⊥F̃
) = σ2 .
T T T
D. Palomar Factor Models 25 / 45
Interlude: MLE is Biased

Since we now know there is a bias in the MLE of the variance, we could
consider correcting this bias to obtain an unbiased estimator (Bessel’t
correction):

2 T 1
σ̂unbiased = × σ̂ 2 = kx − F̃γ̂k2 .
T − (K + 1) T − (K + 1)

The interpretation is that T − (K + 1) represent the degrees of freedom.

(Note that if γ was known perfectly, then the degrees of freedom would be
T .)

D. Palomar Factor Models 26 / 45

Outline

1 Introduction

2 Linear Factor Models

3 Macroeconomic Factor Models

4 Fundamental Factor Models

5 Statistical Factor Models

Fundamental Factor Models

Fundamental factor models use observable asset specific characteristics

(fundamentals) like industry classification, market capitalization, style
classification (value, growth), etc., to determine the common risk factors
{ft }.
factor loading betas are constructed from observable asset
characteristics (i.e., B is known)
factor realizations {ft } are then estimated/constructed for each t
given B
note that in macroeconomic factor models the process is the opposite,
i.e., the factors {ft } are given and B is estimated
in practice, fundamental factor models are estimated in two ways:
BARRA approach and Fama-French approach.

D. Palomar Factor Models 28 / 45

BARRA Approach

The BARRA approach was pioneered by Bar Rosenberg, founder of

BARRA Inc. (cf. Grinold and Kahn (2000), Conner et al. (2010), Cariño et
al. (2010)).

Econometric problems:
choice of betas
estimation of factor realizations {ft } for each t given B (i.e., by
running T cross-sectional regressions)

D. Palomar Factor Models 29 / 45

Fama-French Approach

This approach was introduced by Eugene Fama and Kenneth French

(1992):
For a given observed asset specific characteristics, e.g. size, determine
factor realizations for each t with the following two steps:
1 sort the cross-section of assets based on that attribute
2 form a hedge portfolio by longing in the top quintile and shorting in the
bottom quintile of the sorted assets
Define the common factor realizations with the return of K of such
hedge portfolios corresponding to the K fundamental asset attributes.
Then estimate the factor loadings using time series regressions (like in
macroeconomic factor models).

D. Palomar Factor Models 30 / 45

BARRA Industry Factor Model
Consider a stylized BARRA-type industry factor model with K mutually
exclusive industries. Define the K factor loadings as
(
1 if asset i is in industry k
βi,k =
0 otherwise
The industry factor model is (note that α = 0)
xt = Bft + t , t = 1, . . . , T
LS estimation (inefficient due to heteroskedasticity in Ψ):
f̂t = (BT B)−1 BT xt , t = 1, . . . , T
but since BTP B = diag (N1 , . . . , NK ), where Nk is the count of assets in
industry k ( K k=1 Nk = N), then f̂t is a vector of industry averages!!
The residual covariance
matrix unbiased estimator is
Ψ̂ = diag σ̂12 , . . . , σ̂N
2 where  ˆt = xt − Bf̂t and
T T
2 1 X ¯


¯

T
¯ 1 X
σ̂i = ˆi,t − ˆi ˆi,t − ˆi and ˆi = ˆi,t .
T −1 T
t=1 t=1
D. Palomar Factor Models 31 / 45
Outline

1 Introduction

2 Linear Factor Models

3 Macroeconomic Factor Models

4 Fundamental Factor Models

5 Statistical Factor Models

Statistical Factor Models: Factor Analysis

In statistical factor models, both the common-factors {ft } and the factor
loadings B are unknown. The primary methods for estimation of factor
structure are
Factor Analysis (via maximum likelihood EM algorithm)
Principal Component Analysis (PCA)
Both methods model the covariance matrix Σ by focusing on the sample
covariance matrix Σ̂SCM computed as follows:

XT = x1 · · · xT
 
(N × T )
 
1
X̄T = XT I − 1T 1T T (demeaned by row)
T
1
Σ̂SCM = X̄T X̄
T −1

D. Palomar Factor Models 33 / 45

Factor Analysis Model
Linear factor model as cross-sectional regression:
xt = α + Bft + t , t = 1, . . . , T
with E [ft ] = µf and Cov [ft ] = Σf .

Invariance to linear transformations of ft :

The solution we seek for B and {ft } is not unique (this problem was
not there when only B or {ft } had to be estimated).
For any K × K invertible matrix H we can define f̃t = Hft and
Be = BH−1 .
We can write the factor model as
xt = α + Bft + t = α + B̃f̃t + t
with
E [f̃t ] = E [Hft ] = Hµf
Cov [f̃t ] = Cov [Hft ] = HΣf HT .
D. Palomar Factor Models 34 / 45
Factor Analysis Model

Standard formulation of factor analysis:

Orthogonal factors: Σf = IK
This is achieved by choosing H = Λ−1/2 ΓT , where Σf = ΓΛΓT is
the spectral/eigen decomposition with orthogonal K × K matrix Γ
and diagonal matrix Λ = diag (λ1 , . . . , λK ) with λ1 ≥ λ2 ≥ · · · ≥ λK .
Zero-mean factors: µf = 0
This is achieved by adjusting α to incorporate the mean contribution
from the factors: α̃ = α + Bµf .

Under these assumptions, the unconditional covariance matrix of the

observations is

Σ = BBT + Ψ.

D. Palomar Factor Models 35 / 45

Factor Analysis: Variance Decomposition

Recall the covariance matrix for the returns

Σ = BBT + Ψ.

For a given asset i, the return variance may then be expressed as

K
X
2
var (xi,t ) = βik + σi2
k=1

PK 2
variance portion due to common factors, k=1 βik , is called the
communality
variance portion due to specific factors, σi2 , is called the uniqueness
another quantity of interest is the factor’s marginal contribution to
active risk (FMCAR).

D. Palomar Factor Models 36 / 45

Factor Analysis: MLE
Maximum likelihood estimation:
Consider the model
xt = α + Bft + t , t = 1, . . . , T
where
α and B are vector/matrix constants
all random variables are Gaussian/Normal:
ft i.i.d. N (0, I)
σ12 , . . . , σN2


t i.i.d. N (0, Ψ) with Ψ = diag


xt i.i.d. N α, Σ = BBT + Ψ
Probability density function (pdf):
T
Y
p (x1 , . . . , xT | α, Σ) = p (xt | α, Σ)
t=1
T  
Y − N2 − 21 1 T −1
= (2π) |Σ| exp − (xt − α) Σ (xt − α)
2
t=1

D. Palomar Factor Models 37 / 45

Factor Analysis: MLE
Likelihood of the factor model:
The log-likelihood of the parameters (α, Σ) given the T i.i.d. observations
is
L (α, Σ) = log p (x1 , . . . , xT | α, Σ)
T
TN T 1X
=− log (2π) − log |Σ| − (xt − α)T Σ−1 (xt − α)
2 2 2
t=1

Maximum likelihood estimation (MLE):

PT
minimize T
2 log |Σ| + 1
2 t=1 (xt − α)T Σ−1 (xt − α)
α,Σ,B,Ψ
subject to Σ = BBT + Ψ
Note that without the constraint, the solution would be
T T
1 X 1 X
α̂ = xt and Σ̂ = (xt − α̂) (xt − α̂)T .
T T
t=1 t=1

D. Palomar Factor Models 38 / 45

Factor Analysis: MLE
MLE solution:
PT
minimize T
2 log |Σ| + 1
2 t=1 (xt − α)T Σ−1 (xt − α)
α,Σ,B,Ψ
subject to Σ = BBT + Ψ

Employ the Expectation-Maximization (EM) algorithm to compute α̂,

B̂, and Ψ̂.
Estimate factor realization {ft } using, for example, the GLS estimator:
f̂t = (B̂T Ψ̂−1 B̂)−1 B̂T Ψ̂−1 xt , t = 1, . . . , T .
The number of factors K can be estimated with a variety of methods such
as the likelihood ratio (LR) test, Akaike information criterion (AIC), etc.
For example:
1 2  
LR(K ) = −(T − 1 − (2N + 5) − K ) log |Σ̂SCM | − log |B̂B̂T + Ψ̂|
6 3
which is asymptotically chi-square with 12 ((N − K )2 − N − K ) degrees of
freedom.
D. Palomar Factor Models 39 / 45
Principal Component Analysis
PCA: is a dimension reduction technique used to explain the majority of
the information in the covariance matrix.
N-variate random variable x with E [x] = α and Cov [x] = Σ.
Spectral/eigen decomposition Σ = ΓΛΓT where
Λ = diag (λ1 , . . . , λK ) with λ1 ≥ λ2 ≥ · · · ≥ λK
Γ orthogonal K × K matrix: ΓT Γ = IN
Principal component variables: p = ΓT (x − α) with
h i
E [p] = E Γ (x − α) = ΓT (E [x] − α) = 0
T

h i
Cov [p] = Cov ΓT (x − α) = ΓT Cov [x] Γ = ΓT ΣΓ = Λ.

p is a vector of zero-mean, uncorrelated random variables, in order of

importance (i.e., the first components explain the largest portion of
the sample covariance matrix)
In terms of multifactor model, the K most important principal
components are the factor realizations.
D. Palomar Factor Models 40 / 45
Factor Models: Principal Component Analysis
Factor model from PCA:
From PCA, we can write the random vector x as
x = α + Γp
where E [p] = 0 and Cov[p] = Λ = diag (Λ1 , Λ2 ).
Partition Γ = Γ1 Γ2 where Γ1 corresponds to the K largest
eigenvalues of Σ. 
p1
Partition p = where p1 contains the first K elements
p2
Then we can write
x = α + Γ1 p1 + Γ2 p2 = α + Bf + 
where B = Γ1 , f = p1 and  = Γ2 p2 .
This is like a factor model except that Cov [] = Γ2 Λ2 ΓT
2 , where Λ2
is a diagonal matrix of last N − K eigenvalues but Cov [] is not
diagonal...
D. Palomar Factor Models 41 / 45
Factor Models: Why PCA?
But why is PCA the desired solution to the statistical factor model?
The idea is to obtain the factors from the returns themselves:

xt = α + Bft + t , t = 1, . . . , T
ft = CT xt + d

or, more compactly,

 
xt = α + B CT xt + d + t , t = 1, . . . , T

The problem formulation is

minimize T1 T
P T

2
α,B,C,d t=1 kxt − α + B C xt + d k

Solution is involved to derive and is given by: ft = ΓT

1 (xt − α) or, if
−1
normalized factors are desired, ft = Λ1 2 ΓT
1 (xt − α).
D. Palomar Factor Models 42 / 45
Factor Models via PCA
1 Compute sample estimates
T T
1 X 1 X
α̂ = xt and Σ̂ = (xt − α̂) (xt − α̂)T .
T T
t=1 t=1
2 Compute spectral decomposition:
Σ̂ = Γ̂Λ̂Γ̂T ,
 
Γ̂ = Γ̂1 Γ̂2
3 Form the factor loadings, factor realizations, and residuals (so that
xt = α̂ + B̂f̂t + ˆt ):
1
−1
B̂ = Γ̂1 Λ̂12 , f̂t = Λ̂1 2 Γ̂T
1 (xt − α̂) , ˆt = xt − α̂ − B̂f̂t
4 Covariance matrices:
T T
1 X T 1 X
Σ̂f = f̂t f̂t = IK , Ψ̂ = ˆt ˆT
t = Γ̂2 Λ̂2 Γ̂2 (not diagonal!)
T T
t=1 t=1

Σ̂ = Γ̂1 Λ̂1 Γ̂1 + Γ̂2 Λ̂2 Γ̂2 = B̂Σ̂f B̂T + Ψ̂.

D. Palomar Factor Models 43 / 45
Principal Factor Method
Since the previous method does not lead to a diagonal covariance matrix for
the residual, let’s refine the method with the following iterative approach
1 PCA:

sample mean: α̂ = x̄ = T1 XT 1T
demeaned matrix: X̄T = XT − x̄1T T
1
sample covariance matrix: Σ̂ = T −1 X̄T X̄
eigen-decomposition: Σ̂ = Γ̂0 Λ̂0 Γ̂T
0
set index s = 0
2 Estimates: 1
2
B̂(s) = Γ̂(s−1) Λ̂(s−1)
Ψ̂(s) = diag(Σ̂ − B̂(s) B̂T
(s) )
Σ̂(s) = B̂(s) B̂T
(s) + Ψ̂(s)

3 Update the eigen-decomposition as Σ̂ − Ψ̂(s) = Γ̂(s) Λ̂(s) Γ̂T

(s)
4 Update s ← s + 1 and repeat Steps 2-3 generating a sequence of
estimates (B̂(s) , Ψ̂(s) , Σ̂(s) ) until convergence.
D. Palomar Factor Models 44 / 45
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