程序代写 Observed Factor Models

Observed Factor Models
Chris Hansman
Empirical Finance: Methods and Applications Imperial College Business School
February 7th and 8th

1. General Framing of Linear Factor Models
2. Single Index Model and the CAPM
3. Multi-Factor Models 􏰀 Fama-French
􏰀 Macroeconomic Factors
4. Barra approach

Part 1: Linear Factor Models
1. Clarifying the Assumptions Behind the Linear Factor Model 2. Time-Series and Cross-Sectional Notation
3. Conditional and Unconditional Covariances of Factor Returns

Linear Factor Models
􏰀 Suppose we observe the returns on m assets (i = 1,2,··· ,m) 􏰀 Often excess returns (rit − rf )
􏰀 And often in logs: rit = log Pt Pt −1
􏰀 Over T time periods (t = 1,2,··· ,T)
􏰀 Can think of this as a panel of returns
􏰀 Denote each return by xi,t, so every t we see a vector of length m: x1,t 
x2,t  xt= . 

Linear Factor Models
􏰀 Assume that returns xit are driven by K common factors:
xi,t = αi +β1,if1,t +β2,if2,t +···+βK,ifK,t +εit 􏰀 The set of common factors is:
f1,t  f2,t 
ft =  .  .
􏰀 These are the same for all assets (constant over i) 􏰀 But change over time (different for t, t+1)
􏰀 Each ft has dimension (K × 1)
􏰀 T different versions of this vector in sample
􏰀 One for each time period
􏰀 In some applications we will assume we know ft —in others we will estimate it

Linear Factor Models
xi,t = αi +β1,if1,t +β2,if2,t +···+βK,ifK,t +εit 􏰀 The set of factor loadings is
β1,i  β2,i 
βi= .  .
􏰀 An asset has a fixed relationship with each factor
􏰀 Do not change over time
􏰀 This means K different parameters for each asset i
􏰀 Each βi has dimension (K × 1)
􏰀 m different versions of this vector in sample:
􏰀 One for each asset
􏰀 In some applications we will assume we know βi , in most we will estimate it

Linear Factor Models
􏰀 Assume that returns xit are driven by K common factors:
xi,t = αi +β1,if1,t +β2,if2,t +···+βK,ifK,t +εit 􏰀 αi is the intercept for each asset
􏰀 εit is the error or asset specific factor

Linear Factor Models: Cross Sectional
􏰀 Suppose we focus on a single cross section of the data: time t 􏰀 For each individual asset i we have:
xi,t =αi +βi,1f1,t +βi,2f2,t +···+βi,KfK,t +εit 􏰀 Or written for all m assets at once in matrix notation:
xt =α+Bft +εt 􏰀 Looks similar to OLS—but not quite

Linear Factor Models: Cross Sectional
xt =α+Bft +εt 􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏
m×1 m×1 K×1 m×1 􏰀 Looks similar to OLS—but not quite
􏰀 Menti…

Linear Factor Models: Cross Sectional
􏰀 Assume that returns xt are driven by K unobserved common factors:
xi,t = αi +β1,if1,t +β2,if2,t +···+βK,ifK,t +εit
􏰀 For all m assets can be written more concisely for each period t as:
xt =α+Bft+εt
x1,t  α1  β1,1 ··· x2,t  α2  β2,1 ···
β1,K f1,t  ε1,t 
β2,K f2,t  ε2,t  .  . + .  ... . ...
 . = . + . ..
xm,t αm 􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏 􏰎
··· βm,K 􏰍􏰌
fK,t εm,t 􏰏􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏
􏰀 α and B are constant for all t!

Linear Factor Model
xt =α+Bft+εt
􏰀 Assumptions:
􏰀 {ft} is k-variate covariance stationary: for all t:
E[ft]= μf 􏰎􏰍􏰌􏰏
Cov[ft] = Ωf = E[(ft −μf )(ft −μf )′]
􏰀 Menti…
􏰀 E[εit|fkt] = 0 for all i,k,t

Linear Factor Model
xt =α+Bft+εt
􏰀 Assumptions:
􏰀 εit has the following properties:
E[εt]= 0 􏰎􏰍􏰌􏰏
Cov[εt] = E[εtεt′] = Ψ
m×m Cov[εt,εt′]=E[εtεt′′]= 0 fort̸=t′
σ12 0 ··· 0
0 σ2 ··· 0
Cov[εt]=Ψ= . . .  . .
.  .. . 
0 0 ··· σm2

Linear Factor Model
xt =α+Bft+εt 􏰀 Summary of Parameters
􏰀 α: (m×1) intercepts for m assets
􏰀 B:(m×K)loadings(βik)onK factorsformassets 􏰀 μf : (K × 1) vector of means for K factors
􏰀 Ωf : (K × K ) variance covariance matrix of factors
􏰀 Ψ: (m×m) diagonal matrix of asset specific variances
􏰀 Given our assumptions xt is m-variate covariance stationary with: E[xt|ft] =?
Cov[xt|ft]=? E[xt]=μx =? Cov[xt] = Σx =?

Linear Factor Model: Time Series
􏰀 Can also write the linear factor model for each asset i’s time series: xi = αi 1T + Fβi + εi
xi,1  1 f1,1 xi,2  1 f1,2
 . =αi.+ .
f2,1 f2,2 .
fK,1 β1,i  εi,1 
fK,2 β2,i  εi,2  .  . + .  .  .   . 
 .  xi,T
βK,i εi,T 􏰏􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏

Linear Factor Model: Time Series
􏰀 Can also write the linear factor model for each asset i’s time series: xi = αi 1T + Fβi + εi
􏰀 Much closer to the OLS specifications we have seen in the past: 􏰀 We are used to Y=Xβ+v
􏰀 xi is analogous to Y
􏰀 Our factor realizations F are analogous to X
􏰀 αi 1T is just an explicit way of specifying the constant term

Linear Factor Model: Time Series
􏰀 Can also write the linear factor model for each asset i’s time series: xi = αi 1T + Fβi + εi
􏰀 What about the covariance of εi ?
σi2 0 ··· 0
0 σ2 ··· 0 i
Cov(εi) =  . . … .  
0 0 ··· σi2
􏰀 This follows from our assumption: Cov[εt,εt′]=E[εtεt′′]= 0 fort̸=t′

Part 2: The Index Model/CAPM
1. The Index Model as a Special Case of the General Framework
2. Review of Two-Pass Approach and Testing CAPM
3. Estimating Covariances of Factor Returns 􏰀 Why are factor models useful

Applications: The Index Model/Testing CAPM
􏰀 Much empirical work testing the CAPM/multifactor models applies this general framework
􏰀 Flashback:

Applications: The Index Model/Testing CAPM
xi,t = αi +β1,if1,t +εit
􏰀 Test of the CAPM consider models of this form with a single factor 􏰀 What do we use for f1,t?

Applications: The Index Model/Testing CAPM
xi,t = αi +β1,iRm,t +εit
􏰀 Test of the CAPM consider models of this form with a single factor
􏰀 We often perform “two pass” strategies:
􏰀 First pass estimates β1,i , αi for each asset
􏰀 Second pass uses these estimated βs to test the CAPM 􏰀 These two passes use different aspects of the data
(cross-section/time series)
􏰀 Does first pass use cross-sectional or time series approach? 􏰀 Menti

The Index Model/Testing CAPM: Cross-Section
xt =α+BRmt+εt 􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏
m×1 m×1 m×1 1×1 m×1
􏰀 Suppose we only had one cross-section of data (one period)
􏰀 Need to estimate m different αs and m different βs 􏰀 But only have m observations (of xit )!
􏰀 And only one Rmt
􏰀 Can’t estimate more than m parameters with m data points!

The Index Model/Testing CAPM: First Pass
xi =αi1T +Rmβi +εi
􏰀 Estimate OLS regression on time-series version of our factor specification
􏰀 One regression for each asset i
􏰀 Recover two parameters αˆi and βˆi for each asset i 􏰀 OLS estimates are like always, if we define
􏰀 Then our estimates are just:
􏰅αˆi􏰆 ′ −1 ′
Rm2 .  .
βˆ =(ZZ) (Zxi) i

Estimating The Index Model
xi =αi1T +Rmβi +εi
􏰀 We will estimate this model on log montly returns for a set of 5 assets
1. SPY: S&P 500 ETF
2. EFA: A non US equities ETF
3. IJS: A small-cap value ETF
4. EEM: an emerging markets ETF 5. AGG: A bond fund
􏰀 Monthly from January 2013-December 2017
􏰀 Thanks to for the example (check his great R for finance tutorials)

Estimating The Index Model
xi =αi1T +Rmβi +εi
􏰀 As a proxy for the market portfolio, we use the S&P
􏰀 What does this imply about β1?
􏰀 Data available on insendi: returns.csv

The Index Model/Testing CAPM: First Pass
xi =αi1T +Rmβi +εi 􏰀 With αˆi and βˆi, then for each i we can:
􏰀 Estimate residuals
􏰀 Use these to estimate asset specific variances (for each i):
􏰀 Write Ψˆ as a diagonal matrix of all these variances:
σˆ12 0 ··· 0 0 σˆ2 ··· 0
εˆ = x − αˆ 1 − R βˆ iiiTmi
εˆ ′ εˆ σˆi2= i i
2 Cov[εt]=Ψ= . . … .  
0 0 ··· σˆm2

Estimating Covariance Matrix
xi =αi1T +Rmβi +εi 􏰀 A major benefit of having βˆi, and Ψˆ?
􏰀 We can now estimate the covariance of asset returns! Cov[Xt]=Σx =BΩfB′+Ψ
􏰀 In this case B is just the vector of βi s for all m assets: Bˆ = [βˆ1,βˆ2,··· ,βˆm]′
􏰀 The only missing piece is Ωf
􏰀 Because we have only one factor Rmt , it’s easy to estimate:
ˆ 2 ∑Tt=1(RMt −R ̄m)2
Ωf =Var(Rmt)=σˆR =

Estimating Covariance Matrix
􏰀 So we may estimate: 􏰀 Or, written out fully:
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
xi =αi1T +Rmβi +εi Cov[Xt]=Σx =BΩfB′+Ψ
βˆ σˆ2 0 ··· 0 11
ˆ2 β22ˆˆ ˆ 0σˆ2···0
Σˆx =  . ·σˆR ·(β1,β2,··· ,βm)+ . .
. .   .  . . . .
βˆm 0 0 ··· σˆm2

Estimating Covariance Matrix
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
􏰀 Why do we want to estimate the covariance matrix?
􏰀 Natural to want to understand relationship between returns of
different assets
􏰀 In general, hopeless without some structure:

Estimating Covariance Matrix–Flashback
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ

Estimating Covariance Matrix
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
􏰀 Σx is an m×m symmetric matrix
􏰀 Without any structure on Σx how many different parameters are
there to estimate?
􏰀 How many parameters are included in our Σˆx? 􏰀 Menti

Estimating Covariance Matrix
􏰀 Significantly fewer parameters in Σˆx
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏 􏰎􏰍􏰌􏰏
􏰀 Number of parameters grows exponentially faster with m in general
Assets Parameters in Sample Σx Parameters in Model Σˆx 235
10 55 21 100 5050 201 1000 500500 2001

Estimating Covariance Matrix
􏰀 More technically, Σx is m×m
􏰀 We have m×T data points (one for each asset in each period)
􏰀 If we estimate the sample analogue of Σx directly (e.g. by computing each sample variance and covariance directly):
􏰀 Our estimated matrix can’t be more than rank T
􏰀 Notinvertibleifm>T!
􏰀 See if you can work this out…

The Index Model/Testing CAPM: Second Pass
xit = αi +βiRmt +εit
􏰀 Take expectations of both sides:
E[xi] = E[αi]+βiE[Rm]
􏰀 CAPM predicts E [αi ] = 0 so should be the case that: E[xi]=βiE[Rm] (1)
􏰀 Now we have αˆi, βˆi, σˆi in hand for each i
􏰀 Second pass uses these parameters to test (1)

The Index Model/Testing CAPM: Second Pass
x ̄ i = γ 0 + γ 1 βˆ i + γ 2 σˆ i + η i
􏰀 CAPM tests: expected excess return should be determined only by systemic risk (β)
1. γ0=0oraverageαis0
2. γ2 = 0 (idiosyncratic risk shouldn’t be priced) 3. γ1=R ̄m

Part 3: Multi-Factor Models
1. The Fama-French Three Factor Model 􏰀 Details of Construction
􏰀 Calculating the Covariance of Asset Returns 2. Extensions of the Three Factor Model
3. Macroeconomic Factors

Fama-French Three Factor Model
􏰀 Recall our general linear factor model:
xi,t = αi +β1,if1,t +β2,if2,t +···+βK,ifK,t +εit
􏰀 Fama-French is just another version of this with three particular factors:
xi,t = αi +β1,if1,t +β2,if2,t +β3,if3,t +εit
􏰀 The factors are:
1. f1,i = Rmt : proxy for excess market return—same as before 2. f2,i = SMBt : size factor
3. f2,i = HMLt : value factor
􏰀 How do we get these last two?

How to get the Fama-French Factors
􏰀 First, rank assets according to two variables:
1. Market equity (ME)
2. Book-to-market (book equity over market equity)
􏰀 Cut them into buckets:
Source: ’s website (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_portfolios.html)

Fama-French Three Factor Model
Source: ’s website (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/bench_m_buy.html)

Fama-French Three Factor Model
􏰀 To construct the size factor SMB (small minus big):
􏰀 Average return on the three small portfloios minus average return on
the three big portfolios
􏰀 To construct the value factor HML (high minus low):
􏰀 Average return on the two value portfolios minus average return on
the two growth portfolios
− 1 (Small Growth+Big Growth) 2
(Small Value+Small Neutral+Small Growth)
− 1 (Big Value+Big Neutral+Big Growth)
(Small Value+Big Value)

Fama-French Three Factor Model
􏰀 So the general form:
xi,t = αi +βR,iRm,t +βSMB,iSMBt +βHML,iHMLt +εit
􏰀 Just like before, (first pass) time series regression to estimate αi, βs:
xi =αi1T +βR,iRm+βSMB,iSMB+βHML,iHML+εt 􏰀 For each i, can collect αˆi, βˆR,i βˆSMB,i and βˆHML,i

Estimating The Model
xi,t = αi +βR,iRm,t +βSMB,iSMBt +βHML,iHMLt +εit 􏰀 Data available on the hub: ff returns.csv
􏰀 Use R excess this time… 􏰀 What is the SMB β for EFA? 􏰀 Menti

Fama-French Three Factor Model
xi =αi1T +βR,iRm+βSMB,iSMB+βHML,iHML+εt
􏰀 With αˆi, βˆR,i, βˆSMB,i, and βˆHML,i (for each i) can do two things
􏰀 Second pass regression to assess model
􏰀 Construct the covariance matrix of asset returns: Σˆx
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ

Fama-French Three Factor Model
xi =αi1T +βR,iRm+βSMB,iSMB+βHML,iHML+εt Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
􏰀 What are the dimensions of these objects now? 􏰀 Menti

Fama-French Three Factor Model
xi =αi1T +βR,iRm+βSMB,iSMB+βHML,iHML+εt Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
􏰀 Very similar to single factor model, but Ωˆf takes a bit more work:
 σˆR2 Ωˆf =σˆR,SMB
􏰀 Each entry is just a sample variance or covariance e.g.:
T ̄ ̄ σˆSMB,HML = ∑t=1(SMBt −SMB)(HMLt −HML)
σˆSMB,R σˆ2
σˆHML,R  σˆHML,SMB
SMB σˆSMB ,HML

Fama-French Three Factor Model
xi =αi1T +βR,iRm+βSMB,iSMB+βHML,iHML+εt Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
βˆ βˆ ··· βˆ  R,1 R,2 R,m
􏰀 Written out fully:
βˆ βˆ βˆ 
R,1 SMB,1 HML,1   βˆR,2 βˆSMB,2 βˆHML,2 
 R  
σˆ 2 2 · · ·
σˆSMB,R σˆHML,R
βˆSMB,1 βˆSMB,2 · · · βˆSMB,m 
Σˆx=. . .σˆ σˆ2 σˆ . . .    R,SMB SMB HML,SMB  … 2… . . .σˆR,HMLσˆSMB,HML σˆHML. . . 
βˆSMB ,m βˆHML,m
βˆHML,1 βˆHML,2
 .  …. ….
0 0 ··· σˆm2

Fama- French—Covariance Matrix
􏰀 More parameters in Σˆx in 3 factor vs. 1 factor model: 􏰀 3m βs
􏰀 6 factor variance/covariance parameters
􏰀 Still way fewer than the general form of Σx
Assets Parameters in Sample Σx Parameters in Model Σˆx 2 3 14
10 55 46 100 5050 406 1000 500500 4006

—Extensions
􏰀 Numerous other factors have been proposed
􏰀 Fama and French have suggested a two additional factors:
1. Profitability (robust minus weak operating profitablity) 2. Investment (conservative minus aggressive asset growth)
􏰀 Another, momentum, has been one of the most popular
􏰀 Tendency of good or bad performance to persist over several months
􏰀 Usually defined as the average returns of winners minus average returns of losers in the last x months
􏰀 Actual implementation is the same with more factors—just a few more parameters
􏰀 Can also let xit represent excess returns on portfolios of assets, rather than assets themselves

Macroeconomic Factors
􏰀 An alternative approach uses key macro variables as factors 􏰀 For example, Chen, Roll, and Ross use:
􏰀 IP: Growth rate in industrial production
􏰀 EI: Changes in expected inflation
􏰀 UI: Unexpected inflation
􏰀 CG: Unexpected changes in risk premiums 􏰀 GB: Unexpected changes in term premia

Macroeconomic Factors
􏰀 In this case, our general model becomes:
xi,t =αi +βR,iRm,t +βIp,iIPt +βEI,iEIt +βUI,iUIt +βCG,iCGt +βGB,iGCt +εit
􏰀 Can use two-pass procedure to estimate βˆs, evaluate the model
􏰀 Like before, can use estimated βˆs, asset specific variances, and factor covariances to estimate asset covariance matrix

Part 4: BARRA approach
1. Details of the BARRA model 􏰀 Flipped roles for β, f
2. Estimation details 􏰀 A review of GLS
3. An application in R
􏰀 BARRA industry model
4. Factor mimicking portfolios

BARRA approach
􏰀 Developed by Bar Rosenburg for BARRA (now owned by MSCI) 􏰀 Flipped approach to our linear factor model:
x ̃i,t = β1,if1,t +β2,if2,t +···+βK,ifK,t +εit 􏰀 Instead of knowing fk,t, suppose we know all the β’s
􏰀 We know asset i′s exposure to underlying factor fk,t 􏰀 Do not know the value of fk,t in period t
􏰀 The goal is then to estimate fk,t in each period 􏰀 Rather than to estimate some β

BARRA approach
x ̃i,t = β1,if1,t +β2,if2,t +···+βK,ifK,t +εit 􏰀 Silly example
􏰀 Suppose the only thing that matters for returns is a “tech factor” f1,t 􏰀 Define βi = 1 if asset i is a tech stock, 0 otherwise
􏰀 Note: I’ve written x ̃it instead of xit
􏰀 This is just the demeaned excess return for each xit
x ̃it = xit − ∑Tt=1 xit T
􏰀 Lets us drop αi, interpret fi,t as mean 0

BARRA approach
x ̃i,t = β1,if1,t +β2,if2,t +···+βK,ifK,t +εit
􏰀 For each cross section, we can write this in matrix notation: ̃x t = B f t + ε t
􏰀 The difference is that here B is our “data” —
β1,1 ··· βK,1
β1,2 ··· βK,2 B=. .. .
… β1,m ··· βK,m
􏰀 A matrix of (fixed) asset specific attributes
􏰀 Market-cap, industry classification, style classification, etc

BARRA approach
x ̃i,t = β1,if1,t +β2,if2,t +···+βK,ifK,t +εit
􏰀 For each cross section, we can write this in matrix notation: ̃x t = B f t + ε t
􏰀 ft = [f1,t f2,t · · · fK ,t ]′ parameters to be estimated in each t 􏰎 􏰍􏰌 􏰏
􏰀 ̃xt = [x1,t x2,t · · · xm,t ]′ is a vector of de-meaned returns
􏰀 Var (εit ) = σi2 is different for each asset i 􏰀 Different assets have different variances

BARRA approach
̃x t = B f t + ε t
􏰀 This looks just like the standard OLS matrix notation 􏰀 And we can estimate our ft like always:
ˆfOLS = (B′B)−1B′ ̃x tt
􏰀 A bit weird conceptually because the role of the βs flips 􏰀 But no technical difference

BARRA approach—Issue with OLS
􏰀 One small issue here:
̃x t = B f t + ε t ˆfOLS = (B′B)−1B′ ̃x
σ12 0 ··· 0 0 σ2 ··· 0
2 Cov(εt)=Ψ= . . … .  
0 0 ··· σm2
􏰀 Heteroskedasticity!
􏰀 The classic assumptions for OLS to be efficient require
σ 12 = σ 2 2 · · · = σ m2

BARRA approach—Solution: GLS
􏰀 Flashback (he uses u instead of εt )

BARRA approach—Solution: GLS
􏰀 Flashback (he uses Ω instead of Ψ)

BARRA approach—Implementing GLS
̃x t = B f t + ε t 􏰀 Our GLS estimator is just:
ˆfGLS = (B′Ψ−1B)−1B′Ψ−1 ̃x tt
􏰀 Issue: we don’t know Ψ
􏰀 Solution: three step procedure (special case of feasible GLS)
(1) Estimate ˆfOLS using regular OLS for each t t
􏰀 Compute residuals ˆεit for each t and m
(2) Estimate Ψˆ using the time series of residuals
(3) EstimateˆfFGLS usingΨˆ t

BARRA approach—Implementing GLS: Step 1
̃x t = B f t + ε t
􏰀 Estimate ˆfOLS for each of the t cross-sections:
ˆfOLS = (B′B)−1B′ ̃x tt
􏰀 This gives T different versions of the vector ˆfOLS t
􏰀 For each i and t, compute:
εˆ =x ̃ −β fˆOLS−β fˆOLS−···−β fˆOLS
it i,t 1,i 1,t 2,i 2,t K,i K,t
􏰀 This gives m×T different versions of the scalar εˆit 􏰀 Oneforeachi andt

BARRA approach—Implementing GLS: Step 2
̃x t = B f t + ε t
􏰀 Use the times series of εˆit for each i to compute σˆi2
􏰀 Create the Ψˆ matrix:
∑T εˆ2 σˆi2= t=1 it
σˆ12 0 ··· 0 0 σˆ2 ··· 0
2 Cov(εt)=Ψ= . . … .  
0 0 ··· σˆm2

BARRA approach—Implementing GLS: Step 3
̃x t = B f t + ε t
􏰀 Use Ψˆ to compute ˆfFGLS separately for each time t
ˆfFGLS =(B′Ψˆ−1B)−1B′Ψˆ−1 ̃x
tt 􏰀 This gives a vector of length K for each period t
ˆfFGLS =[fˆGLS fˆGLS ··· fˆGLS]′ t 1,t 2,t K,t

BARRA approach—Covariance of Asset Returns
̃x t = B f t + ε t
Σˆ x = B Ωˆ f B ′ + Ψˆ
􏰀 Almost the same as before–but this time we know B
􏰀 Have to estimate Ωf = Cov (ft ) using our GLS estimates
 σˆ2 σˆf1,f2 ··· σˆf1,fK f1
σˆf 2,f 1 σˆ2 ··· σˆf 2,fK  ˆf2
Ωf = . . … .  
σˆfK,f 1 σˆfK,f 2 ··· σˆ2 fK
1 T ˆFGLS ̄FGLS σˆfk,fl = T −1 ∑(fk,t −fk
ˆFGLS ̄FGLS )(fl,t −fl )

Application: BARRA Industry Model
̃x t = B f t + ε t
􏰀 Let’s take the silly example seriously: suppose we believe there are industry specific factors driving asset returns
􏰀 But we don’t know what the factors are in any given month…
􏰀 Lets suppose we have 10 stocks (m=10) in three industries (K=3)

程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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