How to estimate parameters?How to calculate standard errors of the estimated parameters?How to calculate standard errors of the pricing errors?How to test the model?

Chapter12 Regression-based Tests of linear Factor Models

Time-Series RegressionsCross-Sectional RegressionsFama-MacBeth Procedure

Data structureN Assets , T moments

,1 ,1 ,1 ,11 2 1

,2 ,2 ,2 ,21 2 1

, 1 , 1 , 1 , 11 2 1

, , , 1 ,1 2 1

e e e eT T

e e e eT T

e N e N e N e NT T

e N e N e N e NT T

R R R RR R R R

R R R RR R R R

−

−

− − − −−

−−

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

eitR

12.1 Time-Series Regressions

Express the asset pricing model as:

The Betas are defined by regression coefficients:

(12.1)In the model, the factor is an excess return, the test assets are all excess returns

e i it i i t tR fα β ε= + +

( )e iiE R β λ=

The model states:(12.2)

since the factor is also an excess return,

Comparing (12.2)(12.1),are equal to the pricing errors

( ) ( )eiiE R E fβ=

( ) 1E f λ= ×

iα

( ) ( )eit i i tE R E fα β= +

Black,Jensen and Scholes(1972) suggested: Run time-series regression

1.Estimate the factor risk premium,

2. Run time-series regression for each test asset

3. Use standard OLS formulas for a distribution theory of the parameters, t test

( )1

1 T

T tt

E f fT

λ∧

=

= = ∑

, 1, 2 , ,e i it i i t tR f i Nα β ε= + + =

4. Jointly test the pricing errorsAssuming , no autocorrelation, homoskedastic ,

test:

(12.3)

is the residual covariance matrix , I.e., the sample estimate of

( ) 0i jt tE ε ε ≠

( ) ( )2, 0, 0i it t i t t jE E jε ε σ ε ε −= = ≠

2χ

( )( )

121

T 2ET 1 + ' N

f

fα α χ

σ

−

−∧ ∧ ∧

∧

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟ ∑⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∼

∧

∑

( )'t tE ε ε = ∑

F test for finite-sample: ( are normal)

(12.4)

This is Gibbons, Ross and Shanken (1989) or “GRS” test statistic. This distribution is exact in a finite sample

( )( )

12' 1

T, 1

ET-N-1 1+N N T N

fF

fα α

σ

−

−∧ ∧ ∧

− −∧

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟ ∑⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∼ε

The test can also be interpreted as a test whether f is ex ante mean-variance efficient after accounting for sampling error.

( ) ( ) ( )

( ) ( )

22

2

/ /1

1 /

q q T

T

E f fT N

NE f f

μ σ σ

σ

∧

∧

⎛ ⎞− ⎜ ⎟− − ⎝ ⎠⎛ ⎞+ ⎜ ⎟⎝ ⎠

Multi-factors:The regression equation is :

The asset pricing model is :

Assuming normal I.I.d. errors , the test is :

'ei ii i t tR fα β ε= + +

( ) ( )'eiiE R E fβ=

( ) ( )11 ' 1

',1 T T N T N K

T N K E f E f FN

α α−− −∧ ∧ ∧ ∧

− −

⎛ ⎞− −+ Ω ∑⎜ ⎟

⎝ ⎠∼

Derivation of The Chi Statistic and Distributions with General Errors

Derive (12.3) as an instant of GMMWrite the equations for all N assets together in vector from:

Use the usual OLS moments:

et t tR fα β ε= + +

( )( )( )

0[ ]

eT t t t

T Tet tT t t t

E R fg b E

fE R f f

α β εεα β

⎡ ⎤− − ⎛ ⎞⎡ ⎤⎢ ⎥= = =⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥− − ⎣ ⎦⎝ ⎠⎣ ⎦

Exactly identify, so the matrix inis identity matrix, GMM estimate is : (OLS estimate)

a 0Ta g b∧⎛ ⎞⋅ =⎜ ⎟⎝ ⎠

a I=

( ) ( )

( )( )( )( )

( )( )

,

cov ,

var

eT t T t

e e eT t T t t T t t

T tT t T t t

E R E f

E R E R f R f

fE f E f f

α β

β

∧ ∧

∧

= −

⎡ ⎤−⎣ ⎦= =⎡ ⎤−⎣ ⎦

The matrix is :

The S matrix is :

d

( ) ( )( ) ( )

( )( ) ( )2 2

1

'N N t tT

NN t N t t t

I I E f E fg bd I

b I E f I E f E f E f

⎡ ⎤ ⎡ ⎤∂= =− =− ⊗⎢ ⎥ ⎢ ⎥

∂ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

( ) ( )( ) ( )

' '

' '

t t j t t j t j

j t t t j t t t j t j

E E fS

E f E f f

ε ε ε ε

ε ε ε ε

∞ − − −

=−∞ − − −

⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦

∑

Using the GMM variance formula with a=I:

(11.4)So:

(12.7)1 1 '1v a r d S dT

α

β

∧

− −∧

⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥ =⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

( ) ( )1 11var ' 'b ad aSa adT

∧ − −⎛ ⎞ =⎜ ⎟⎝ ⎠

Get the standard formulas by assuming:

1. The errors are uncorrelated over time and homoskedastic

2. The factor and error are independent as well as orthogonal (multi-factors)

[ ] [ ] [ ][ ]2 2

' '

' '

E f E f E

E f E f E

ε ε ε ε

ε ε ε ε

=

⎡ ⎤ ⎡ ⎤=⎣ ⎦ ⎣ ⎦

Then the S matrix simplifies to :

Plug into (12.7), we obtain:

( ) ( ) ( )( ) ( ) ( ) ( )

( )( ) ( )2 2

' ' 1

' 't t t t t t

t t t t t t t t

E E E f E fS

E f E E E f E f E f

ε ε ε ε

ε ε ε ε

⎡ ⎤ ⎡ ⎤= = ⊗∑⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

( )( ) ( )

( )( ) ( )( )

1

2

2

11v a r

1 1v a rv a r 1

t

t t

t t

t

E f

T E f E f

E f E fT f E f

α

β

α

β

∧ −

∧

∧

∧

⎛ ⎞⎡ ⎤ ⎛ ⎞⎡ ⎤⎜ ⎟ ⎜ ⎟⎢ ⎥ = ⊗ ∑⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎣ ⎦⎢ ⎥ ⎝ ⎠⎣ ⎦⎝ ⎠⎛ ⎞⎡ ⎤ ⎛ ⎞⎡ ⎤−⎜ ⎟⎢ ⎥ ⎜ ⎟= ⎢ ⎥ ⊗ ∑⎜ ⎟⎢ ⎥ ⎜ ⎟−⎢ ⎥⎜ ⎟ ⎣ ⎦⎝ ⎠⎢ ⎥⎣ ⎦⎝ ⎠

The variance of is :

This is the tradition formula (12.3)By simply calculating (12.7), we can easily construct standard errors and test statistic that do not requires these assumptions.

α∧

( )( )

21var 1

varE f

T fα∧ ⎛ ⎞⎛ ⎞ ⎜ ⎟= + ∑⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

12.2 Cross-Sectional Regressions

Start with the K factor model,

The center economics question is why average returns vary across assetsThe model says that average returns should be proportional to betas. ( Figure 12.1)

( ) ' , 1, 2, ,eiiE R i Nβ λ= =

Run a two-pass regression

First, find from time-series regressions

Then, estimate from cross-sectional regression.

pricing errors are

β∧

' , 1,2, ,ei it i i t tR a f t Tβ ε= + + =

( ) ' , 1, 2, ,eiT i iE R i Nβ λ α= + =

λ∧

iα∧

OLS Cross-Sectional Regression

The OLS cross-sectional estimates are:

(12.11)Assuming the true errors are I.I.d. over time and independent of the factors.

(12.12)(12.13)

( ) ( )

( )

1' ' eT

eT

E R

E R

λ β β β

α λ β

∧ −

∧∧

=

= −

( ) ( ) ( )

( ) ( ) ( )

1 12

1 1

1' ' '

1cov ( ' ') ( ' ')

TI I

T

σ λ β β β β β β

α β β β β β β β β

∧ − −

∧ − −

= ∑

= − ∑ −

Since,( ) ( )

( )

1

1 11

1 21 1

11

1

1' ( (

1 1(

T

tt T T

NN t t

t tT

N Nt

t

t

Nt t

Nt

E E ET

ET T

εα

αα α α ε εα

ε

ε

ε ε

ε

=

= =

=

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎛ ⎞ ⎟⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎛ ⎞⎟ ⎜ ⎟⎜ ⎟⎟ ⎜⎜ ⎟⎜= = ⎟⎟ ⎟⎜⎜⎜ ⎟⎟ ⎟ ⎟⎜⎜ ⎜⎜ ⎝ ⎠⎟ ⎟⎜⎜ ⎟ ⎟⎜⎟⎜ ⎟⎟⎜ ⎜⎝ ⎠ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜⎝ ⎠⎟⎜⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜= = ∑⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎟⎜⎜⎝ ⎠⎟

∑∑ ∑

∑

( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( )

1 12

1 1

1 1

' ' cov[ ] '

' ' cov[ ' ] '' ' ' '

eTE R

E

σ λ β β β β β β

β β β β λ α β β ββ β β αα β β β

∧ − −

− −

− −

=

= +=

The test statistic is :(12.14)

The degree of freedom is N-1 not N, and N-K for K-factors model

A test of residuals is unusual in OLS regressions.

1 21' cov( ) Nα α α χ

∧ ∧ ∧−−∼

GLS Cross-Sectional Regression

Since as the error covariance matrix ,GLS estimates are:

The variance of these estimates are:(12.15)

(12.16)

( ) ( )

( )

11 1' ' ,eT

eT

E R

E R

λ β β β

α λ β

∧ −− −

∧∧

= ∑ ∑

= −

( ) ( )

( ) ( )

12 1

1 1

1' ,

1cov ( ' ) ' .

T

T

σ λ β β

α β β β β

∧ −−

∧ − −

= ∑

= ∑− ∑

[ ] 1'ET

αα = Σ

A GLS regression can be understand as a transformation of the space of returnsThe test statistic for the pricing errors

(12.17)1 21' NT α α χ

∧ ∧−−∑ ∼

Correction for the fact that are estimated

Since the betas are estimated, the asymptotic standards errors should be corrected (Shanken 1992)

(12.18)Compare to (12.12)(12.15)

β

( ) ( ) ( )

( ) ( )

1 12 1

12 1 1

1[ ' ' ' (1 ' ) ]

1[ ' (1 ' ) ]

OLS f f

GLS f f

T

T

σ λ β β β β β β λ λ

σ λ β β λ λ

∧ − − −

∧ −− −

= ∑ + ∑ +∑

= ∑ + ∑ +∑

The asymptotic covariance matrix of the pricing errors is :

(12.19)

( 12.20)The test statistic in GLS is :

( ) ( ) ( )

( )( ) ( )( )( )

1 1

1

11 1

1cov ( ' ') ( ' ')

1 '1

cov ' ' 1 '

OLS N N

f

GLS f

I IT

T

α β β β β β β β β

λ λ

α β β β β λ λ

∧ − −

−

−∧ − −

= − ∑ −

× + ∑

= ∑− ∑ + ∑

'1 1 2(1 ' ) GLS GLSf N KT λ λ α α χ∧ ∧− −

−+ ∑ ∑ ∼

How important the corrections ? in CAPM , , so in annual data

It is too large to ignore,but in monthly interval

It makes little difference.

( )emE Rλ =

( ) ( )22 2/ 0.08 / 0.16 0.25emRλ σ ≈ =

( )2 2/ 0.25 /12 0.02emRλ σ ≈ ≈

The additive term can be very important, given some assumption ,write (12.19)as follow

Even with N=1,most factor model have fairly high ,so

Typical assets numbers N=10 to 50even if the the Shanken correction can be

ignored , it should be included in

( ) ( ) ( )2 2 21 1[ ]fT N

σ λ σ ε σ∧

= +

2R( ) ( )2 2 fσ ε σ<

( )σ λ∧

Derivation and formulas that do not require I.I.d. errors

Derivation by GMM, map all effects into GMM framework. The moments are

It is overidentified, since N extra moment

( )( )( )( )

0[ ] 0

0

et te

T t t te

E R a fg b E R a f f

E R

ββ

βλ

⎡ ⎤ ⎡ ⎤− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= − − =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎣ ⎦⎣ ⎦

The ingredients of the recipe:the parameter vector is :

The a matrix is :The d matrix is

[ ]' ', ',b a β λ=2

'

NIa

γ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

( )( )

( ) ( )20

0'

0

N N

TN N

N

I I E fg b

d I E f I E fb

Iλ β

⎡ ⎤− −⎢ ⎥∂ ⎢ ⎥

= = − −⎢ ⎥∂ ⎢ ⎥

⎢ ⎥− −⎢ ⎥⎣ ⎦

The S matrix is:

Calculate the standard error formula (11.4)(11.5) (pricing errors is last N moment)

( ) ( )

( ) ( )

'

'

eet j t jt t

e et t t t j t j t je ejt t j

t t j

t t t j t j

t t t j t j

R a fR a fS E R a f f R a f f

R R

E f ff Ef f Ef

βββ β

βλ βλ

ε εε εβ ε β ε

− −∞

− − −=−∞

−

−

− −

− −

⎛ ⎞⎡ ⎤ ⎡ ⎤− − ⎟− −⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥⎜= − − − − ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎜ ⎟− −⎢ ⎥ ⎢ ⎥ ⎟⎟⎜⎜ ⎢ ⎥ ⎢ ⎥⎝ ⎠⎣ ⎦ ⎣ ⎦⎛ ⎡ ⎤⎡ ⎤⎜ ⎢ ⎥⎜ ⎢ ⎥

⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥− + − +⎢ ⎥⎢ ⎥⎣ ⎦⎝ ⎣ ⎦

∑

j

∞

=−∞

⎞⎟⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎟⎜⎜ ⎠⎟

∑

Recover classic formula (12.18) (12.19) (12.20) by adding assumption:

1.the errors are I.I.d. and independent of the factors

2.the factors are uncorrelated over timeThus:

( )

( ) ( ) ( )

( ) ( )

2

2'

E f

S E f E f E f

E f fββ σ

⎡ ⎤∑ ∑ ∑⎢ ⎥⎢ ⎥⎢ ⎥= ∑ ∑ ∑⎢ ⎥⎢ ⎥∑ ∑ +∑⎢ ⎥⎣ ⎦

Plug into (11.4) get the (3,3) element

(11.4)Plug the items into (11.5) to get the sample distribution of the pricing errors

(11.5)

The result is the same as (12.20)

( ) ( ) ( )1 11var ' 'b ad aSa ad

T

∧ − −=

( ) ( )( ) ( )( )1 1[0, ']TTg b N I d ad a S I d ad a∧ − −→ − −

Once again, it is no need to make assumptions.It is quite easy to estimate an S matrix that does not impose the conditionsIf one is really interesting in efficiency, The GLS cross-sectional estimate should use the spectral density matrix as weighting matrix rather 1−∑

Time series vs. Cross Section

The time series requires factors that are also returns. So that you can estimate factor risk premia by The asset pricing model does predict a restrict in the time-series.If imposed , you can write time-series regression as

( )TE fλ∧=

( ) 'eiiE R β λ=

( )( )' ' , 1,2, ,ei it i i t tR f E f t Tβλ β ε= + − + =

The intercept restriction is

The restriction makes sense. You can see how result in a zero interceptwithout an estimate of ,you can not

check this intercept restriction

( )( )'i ia E fβ λ= −

( )E fλ =λ

When the factor is a return, so that we can compare the two methods:

1. The time-series regression describes the cross section of expected returns by drawing a line as in figure 12.1 that runs through the origin and through the factor

2. The OLS cross-sectional regression picks the slope and intercept to best fit the points

( ) 'eiT i iE R β λ α= +

In including the factor as a test factor The GLS cross-sectional regression =the time-series regression

The time-series regression for is :

The residual covariance matrix of the returns is:

f0 1 0t tf f= + +

[ ]0

'0 1 0 0

eR a fE f f

β ⎡ ⎤∑⎛ ⎞⎡ ⎤− − ⎟⎜ ⎢ ⎥⎢ ⎥ ⎟ =⎜ ⎟ ⎢ ⎥− −⎜⎢ ⎥ ⎟⎜ ⎟⎝ ⎠⎣ ⎦ ⎢ ⎥⎣ ⎦i

Since have 0 residual, GLS puts all its weight on that asset .

Therefore ,just as time-series regression.

The degree of freedom in test is N!Why ignore the pricing errors of the other

asset in estimating the factor risk premium?

f

( )TE fλ∧=

2χ

et t tR a fβ ε= + +

12.3 Fama-MacBeth Procedure

Fama-MacBeth (1973) procedure:1. find beta estimates with a time-series

regression.2. Run a cross-sectional regression at

each time period:

for each time t

' , 1, 2, ,eit i t itR i Nβ λ α= + =

iβ∧

, ittλ α∧ ∧

3. Estimate by:

4. Generate the sampling errors for these estimates:

5.when the time series is autocorrelated

, iλ α

1 1

1 1,

T T

i itt

t tT Tλ λ α α∧ ∧ ∧ ∧

= =

= =∑ ∑

( ) ( ) ( ) ( )2 2

2 22 2

1 1

1 1,

T T

i it it

t tT Tσ λ λ λ σ α α α

∧ ∧ ∧ ∧ ∧ ∧

= =

= − = −∑ ∑

( ) ( )2 1cov ,t t jT

jTσ λ λ λ

∞∧ ∧ ∧

−

=−∞

= ∑

6. Testing whether all the pricing errors are jointly zero:

Write the parameters in vector from:

The test statistic is :

( )1

21

1

1cov ( )( ) '

T

t

tT

t t

t

T

T

α α

α α α α α

∧ ∧

=∧ ∧ ∧ ∧ ∧

=

=

= − −

∑

∑

( )1' 2

1cov Nα α α χ−∧ ∧ ∧

−∼

Fama-MacBeth in depth

Consider a regression:

Pooled time-series cross-section estimate: stack the i and t observations together and

estimate by OLS(contemporaneous correlation)In an expected return-beta asset pricing model,

the xit is the betai and beta is the lamda

' , 1,2, , 1,2, ,it it ity x i N t Nβ ε= + = =

β

Take time-series averages and run a pure cross-sectional regression:

Fama-MacBeth procedure: run a cross-sectional regression at each point in time. Then get the estimates.

( ) ( )' , 1,2, ,T it T it iE y E x u i Nβ= + =

( )1

21

1

1c o v ( ) ( ) '

T

tt

T

t tt

T

T

β β

β β β β β

∧ ∧

=∧ ∧ ∧ ∧ ∧

=

=

= − −

∑

∑

proposition

If the variables do not vary over time,and if the errors are cross-sectionally correlated but not correlated over time, Then:

The Fama-MacBeth estimate =The pure cross-sectional OLS estimate =The pooled time-series cross-sectional OLS

estimateSo to the standard errors ,corrected for residual

correlation.None of them holds if the vary through time

itx

itx

Proof:Having assuming that the variable do

not vary over time, the regression is :

stack up the regressions in vector from:

The error assumptions mean

x'

it i ity x β ε= +

t ty xβ ε= +

( )'t tE ε ε = ∑

Pooled OLS: to run pooled OLS, stack up the time series and cross sections:

and then:

with

1 1

2 2

, ,

T T

y x

y xY X

y x

εε

ε

ε

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Y Xβ ε= +

( )'E εε

⎡ ⎤∑⎢ ⎥⎢ ⎥

=Ω= ⎢ ⎥⎢ ⎥⎢ ⎥∑⎢ ⎥⎣ ⎦

The estimate and its standard error are:

Simplified:

Estimate this sampling variance with

( )

( ) ( ) ( )

1

1 1

' '

c o v ' ' '

O L S

O L S

X X X Y

X X X X X X

β

β

∧ −

∧ − −

=

= Ω

( ) ( )

( ) ( ) ( )

1

1 1

' '1

cov ' ' '

OLS T t

OLS

x x x E y

x x x x x xT

β

β

∧ −

∧ − −

=

= ∑

( )' ,t t t OLST tE y xε ε ε β∧ ∧∧ ∧ ∧∑ = ≡ −

Pure cross-section: take the time-series averages,

Having assumed I.I.d. errors ,so

the cross-sectional estimate and standard errors are:

( ) ( )( ) 1'T t T tE E E

Tε ε = ∑

( ) ( )

( ) ( ) ( )

1

1 12

' '1' ' '

XS T t

XS

x x x E y

x x x x x xT

β

σ β

∧ −

∧ − −

=

= ∑

( ) ( )T y T tE y x Eβ ε= +

Fama-MacBeth: run cross-sectional regression at each moment in time

then the estimate is the average of the cross-sectional regression estimates,

the standard error is :

( ) 1' 't tx x x yβ∧ −=

( ) ( ) ( )1' 'FM tT T tE x x x E yβ β∧ ∧ −= =

( ) ( ) ( ) ( )1 11 1cov( ) cov ' ' cov 'FM tT T tx x x y x x x

T Tβ β∧ ∧ − −= =

with

we have

and finally

End proof

tt FMy xβ ε∧

= +

( ) ( )cov , 't tT t Ty E ε ε∧∧ ∧

= = ∑

( ) ( ) ( )1 11cov ' ' 'FM x x x x x x

Tβ

∧∧ − −= ∑

Varying

none of the three procedures are equal anymore, since it ignore the time-series variance in the an extreme example:

the grand OLS regression is :

x

itx

, 1,2, , , 1,2, ,it t ity x i N t Tα β ε= + + = =

( )2 2 2

1/t it t it t itit t i t iOLS

t t tit t t

x y x y x N y

x N x xβ∧

= = =∑ ∑ ∑ ∑ ∑∑ ∑ ∑

( )Tx x E x= −

Advantage of Fama-MacBeth Method

Allows changing betas.The standard errors are easily to compute.Could be easily modified to consider the estimated beta, Shanken (1992).