# CS计算机代考程序代写 finance Observed Factor Models

Observed Factor Models
Chris Hansman
Empirical Finance: Methods and Applications Imperial College Business School
Topic 5
February 4th
1/76

Today
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
2/76

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
3/76

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= . 
. xm,t
4/76

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 =  .  .
fK ,t
􏰒 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
5/76

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= .  .
βK ,i
􏰒 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
6/76

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
7/76

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
8/76

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…
9/76

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,1
··· βm,K 􏰏􏰎
m×K
fK,t εm,t 􏰑􏰐􏰏􏰎􏰑 􏰐􏰏􏰎􏰑
K×1 m×1
m×1 m×1
􏰒 α and B are constant for all t!
9/76

Linear Factor Model
xt =α+Bft+εt
􏰒 Assumptions:
􏰒 {ft} is k-variate covariance stationary: for all t:
E[ft]= μf 􏰐􏰏􏰎􏰑
K×1
Cov[ft] = Ωf = E[(ft −μf )(ft −μf )′]
􏰒 Menti…
􏰒 E[εit|fkt] = 0 for all i,k,t
10/76

Linear Factor Model
xt =α+Bft+εt
􏰒 Assumptions:
􏰒 εit has the following properties:
E[εt]= 0 􏰐􏰏􏰎􏰑
m×1
Cov[εt] = E[εtεt′] = Ψ
􏰐􏰏􏰎􏰑
m×m Cov[εt,εt′]=E[εtεt′′]= 0 fort̸=t′
􏰐􏰏􏰎􏰑
m×m
σ12 0 ··· 0
0 σ2 ··· 0
Cov[εt]=Ψ= . . .  . .
.  .. . 
0 0 ··· σm2
11/76

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 =?
12/76

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] = α +Bft
Cov[xt|ft] = Cov(εt) = Ψ E[xt]=μx =α+Bμf Cov[xt]=Σx =BΩfB′+Ψ
13/76

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
􏰐􏰏􏰎􏰑
T×1
.  .
.
.
1
􏰐􏰏􏰎􏰑 􏰐
T×1
f1,T
··· T×K
fK,T
f2,T
βK,i εi,T 􏰑􏰐􏰏􏰎􏰑 􏰐􏰏􏰎􏰑
K×1 T×1
􏰏􏰎
14/76

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
15/76

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
􏰐 􏰏􏰎 􏰑
T×T
􏰒 This follows from our assumption: Cov[εt,εt′]=E[εtεt′′]= 0 fort̸=t′
􏰐􏰏􏰎􏰑
m×m
16/76

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
17/76

Applications: The Index Model/Testing CAPM
􏰒 Much empirical work testing the CAPM/multifactor models applies this general framework
􏰒 Flashback:
18/76

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?
􏰒 Menti
19/76

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
20/76

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!
21/76

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 ′
1 1
Z=. .
1
Rm1
Rm2 .  .
Rmt
βˆ =(ZZ) (Zxi) i
22/76

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 Jonathan Regenstein for the example (check his great R for finance tutorials)
23/76

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
24/76

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):
(T − 2)
􏰒 Write Ψˆ as a diagonal matrix of all these variances:
σˆ12 0 ··· 0 0 σˆ2 ··· 0
ˆˆ
εˆ = x − αˆ 1 − R βˆ iiiTmi
εˆ ′ εˆ σˆi2= i i
2 Cov[εt]=Ψ= . . … .  
0 0 ··· σˆm2
25/76

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 =
T −1
26/76

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
27/76

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:
28/76

Estimating Covariance Matrix–Flashback
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
29/76

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
30/76

Estimating Covariance Matrix
􏰒 Significantly fewer parameters in Σˆx
Σˆ x = Bˆ Ωˆ f Bˆ ′ + Ψˆ
􏰐􏰏􏰎􏰑 􏰐􏰏􏰎􏰑 􏰐􏰏􏰎􏰑
m1m
􏰒 Number of parameters grows exponentially faster with m in general
Assets Parameters in Sample Σx Parameters in Model Σˆx 235
5 15 11
10 55 21 100 5050 201 1000 500500 2001
31/76

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…
32/76

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)
33/76

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
34/76

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
35/76

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?
36/76

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: Ken French’s website (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/f-f_portfolios.html)
36/76

Fama-French Three Factor Model
Source: Ken French’s website (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/bench_m_buy.html)
36/76

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
1 3
3
􏰒 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 2
− 1 (Small Growth+Big Growth) 2
(Small Value+Small Neutral+Small Growth)
SMB =
− 1 (Big Value+Big Neutral+Big Growth)
HML =
(Small Value+Big Value)
37/76

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
38/76

Estimating The Fama French 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
39/76

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
40/76

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
41/76

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
σˆR ,HML
􏰒 Each entry is just a sample variance or covariance e.g.:
T ̄ ̄ σˆSMB,HML = ∑t=1(SMBt −SMB)(HMLt −HML)
T −1
σˆSMB,R σˆ2
σˆHML,R  σˆHML,SMB
SMB σˆSMB ,HML
σˆ 2 HML
42/76

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  
σˆ12
  0
0 ···
σˆ 2 2 · · ·
0 
0  
σˆ2
σˆSMB,R σˆHML,R
βˆSMB,1 βˆSMB,2 · · · βˆSMB,m 
Σˆx=. . .σˆ σˆ2 σˆ . . .    R,SMB SMB HML,SMB  … 2… . . .σˆR,HMLσˆSMB,HML σˆHML. . . 
βˆR ,m
βˆSMB ,m βˆHML,m
βˆHML,1 βˆHML,2
· · ·
βˆHML,m
+…
 .  …. ….
0 0 ··· σˆm2
43/76

Fama- French—Covariance Matrix
􏰒 More parameters in Σˆx in 3 factor vs. 1 factor model: 􏰒 3m βs
􏰒 mσis
􏰒 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
5 15 26
10 55 46 100 5050 406 1000 500500 4006
44/76

Fama French—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
45/76

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
46/76

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
47/76

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
48/76

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
49/76

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
50/76

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
􏰐 􏰏􏰎 􏰑
m×K
􏰒 A matrix of (fixed) asset specific attributes
􏰒 Market-cap, industry classification, style classification, etc
51/76

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 􏰐 􏰏􏰎 􏰑
K×1
􏰒 ̃xt = [x1,t x2,t · · · xm,t ]′ is a vector of de-meaned returns
􏰐 􏰏􏰎 􏰑
M×1
􏰒 Var (εit ) = σi2 is different for each asset i 􏰒 Different assets have different variances
52/76

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
53/76

BARRA approach—Issue with OLS
􏰒 One small issue here:
̃x t = B f t + ε t ˆfOLS = (B′B)−1B′ ̃x
tt
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
54/76

BARRA approach—Solution: GLS
􏰒 Flashback (he uses u instead of εt )
55/76

BARRA approach—Solution: GLS
􏰒 Flashback (he uses Ω instead of Ψ)
56/76

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
57/76

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
t
􏰒 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
58/76

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
T −1
σˆ12 0 ··· 0 0 σˆ2 ··· 0
2 Cov(εt)=Ψ= . . … .  
􏱋ˆ
0 0 ··· σˆm2
59/76

BARRA approach—Implementing GLS: Step 3
̃x t = B f t + ε t
􏰒 Use Ψˆ to compute ˆfFGLS separately for each time t
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
60/76

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
􏰒 Where
 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 )
t=1
61/76

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)
62/76

Application: BARRA Industry Model
􏰒 Three industries: Financial Services, Tech, and Other
Source: Tsay, R.S. (2010) Analysis of Financial Time Series. 3rd Edition, John Wiley & Sons, Hoboken.
62/76

Application: BARRA Industry Model
x ̃i,t = βfin,iffin,t +βtech,iftech,t +βo,ifo,t +εit
􏰒 For each asset i, define the factor loading (βi,k) for industry k as:
􏰍1 if asset i is in industry k βi,k = 0 otherwise
􏰒 For example βfin,i = 1 for Citigroup while other elements of βi are 0: βi =(1,0,0)
􏰒 Note that these are known for each i and fixed over time 􏰒 Dell Inc. is always a tech company
63/76

Application: BARRA Industry Model
x ̃i,t = βfin,iffin,t +βtech,iftech,t +βo,ifo,t +εit 􏰒 Lets estimate the industry model
􏰒 First, load data on 10 stocks
􏰒 10 stocks in three categories
􏰒 Monthly data from 1990-2003 (168 months)
􏰒 Here m=10, k =3, T =168
Source: Tsay, R.S. (2010) Analysis of Financial Time Series. 3rd Edition, John Wiley & Sons, Hoboken.
64/76

Application: BARRA Industry Model
x ̃i,t = βfin,iffin,t +βtech,iftech,t +βo,ifo,t +εit
􏰒 Next, demean the data (so we have x ̃i,t instead of xit)
􏰒 And lets calculate the sample covariance/correlation of returns
cov return <- var(X) corr return <- cor(X) 􏰒 Because T >> m here, calculating these directly is no issue
65/76

Application: BARRA Industry Model
̃x t = B f t + ε t
􏰒 Finally, generate B the matrix of loadings: 􏰒 4 Financials, 3 Tech, 3, other
􏰒 This is just a matrix of dummy variables
66/76

Application: BARRA Industry Model
̃x t = B f t + ε t 􏰒 Now, step 1 of the GLS procedure: OLS
ˆfOLS = (B′B)−1B′ ̃x tt
F hat<-solve(t(B)%*%B)%*%t(B)%*%t(X) 􏰒 NotethatXisT×m 􏰒 Each cross section ̃xt is a row 􏰒 So we transpose: X′ 􏰒 And calculate ˆfOLS for all T periods in one line t 67/76 Application: BARRA Industry Model ̃x t = B f t + ε t 􏰒 Step 2 of the GLS procedure : Calculate residuals: εˆOLS =x ̃ −BˆfOLS ttt 􏰒 Use these residuals to calculate σˆ 12 = ∑ εˆ 12 t t T−1 68/76 Application: BARRA Industry Model ̃x t = B f t + ε t 􏰒 Step 2 of the GLS procedure Continued: Calculate Ψˆ: σˆ12 0 ··· 0 0 σˆ2 ··· 0 2 Cov(εt)=Ψ= . . ... .   􏱋ˆ 0 0 ··· σˆm2 Psi hat <- diag(apply(e hat gls,1,var)) 􏰒 Note that we take the transpose of e hat ols because we want a time series variance within asset 69/76 Application: BARRA Industry Model ̃x t = B f t + ε t 􏰒 Step 3 of the GLS procedure: estimate ˆfFGLS : t ˆfFGLS =(B′Ψˆ−1B)−1B′Ψˆ−1 ̃x tt F hat gls<-solve(t(B)%*%Psi inv%*%B)%*%t(B)%*%Psi inv%*%t(X) 􏰒 Once again each cross section ̃xt is a row 􏰒 So we transpose: X′ 􏰒 And calculate ˆfFGLS for all T periods in one line t 70/76 Application: BARRA Industry Model ̃x t = B f t + ε t Σˆ x = B Ωˆ f B ′ + Ψˆ 􏰒 Finally, lets compute the covariance of asset returns 􏰒 First: Ωˆf =Cov(ft) 􏰒Then,usingBandourestimatedΨˆ,wecancalculateσˆ x 71/76 BARRA Models and Factor Mimicking Portfolios 􏰒 Consider a single factor model with m assets ̃x t = β 1 f t + ε t 􏰒 Imagine we want to choose weights ωi for each asset to find the minimum (residual) variance portfolio such that: ∑ωiβ1 =1 i 􏰒 Interpretation: the minimum (residual) variance portfolio whose return moves exactly with the factor 72/76 BARRA Models and Factor Mimicking Portfolios 􏰒 If ω =(ω1 ω2 ··· ωm), we can restate this as argmin 1ω′Ψω ω′β1 =1 􏰒 This problem has solution: ω′ = (β1′ Ψ−1β1)−1β1′ Ψ−1 􏰒 These are the weights on the minimum variance portfolio ω2 Subject to 73/76 BARRA Models and Factor Mimicking Portfolios 􏰒 So our portfolio weights are: ω′ =(β1′Ψ−1β1)β1′Ψ−1 􏰒 And the return on this portfolio is just: ω′x ̃ = (B′Ψ−1B)B′Ψ−1 ̃x = ˆfGLS it tt 􏰒 This return is just the estimated factor realization 􏰒 When scaled so that ∑t ωi = 1, it is called the Factor Mimicking Portfolio 74/76 Factor Mimicking Portfolios–Application 􏰒 Our portfolio weights are: ω′ =(β1′Ψ−1β1)β1′Ψ−1 omega=(inv(B’*inv(Psi hat)*B)*B’*inv(Psi hat))’; 75/76 Today 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 76/76