Chapter 18 GMM in explicit
discount factor models
Fan Longzhen
Our task
How to estimate and test discount factor model.
1. Bring an asset pricing model to data to estimate free
parameters. For example, parameter in
? Or the b in
2. Evaluate the model, is it a good model or not? Is another
model better?
?
γβ,
γ
β
?
+
= )/(
1 tt
ccm
fbm '=
)),((
111 +++
=
tttt
xparameterdatamEEp
GMM in explicit discount factor model
? Asset pricing model predicts that
? The most natural way to check this prediction is to
examine sample average, i,.e., to calculate
? and
]),([)(
11 ++
=
ttt
xparametersdatamEpE
∑
=
T
t
t
p
T
1
1
∑
=
++
T
t
tt
xparametersdatam
T
1
11
]),([
1
GMM and asset pricing model
? Any asset pricing model implies
? Equivalently or
? Where x and p are typically vectors; we typically check
whether a model for m can price a number of assets
simultaneously. So the equation is often called moment
conditions.
? Define errors as
? The sample mean is
? The first stage estimate of b minimizes a quadratic form of
the sample mean of the errors,
?
? For some arbitrary matrix W (often W=I)
])([)(
11 ++
=
ttt
xbmEpE
0])([
11
=?
++ ttt
xbmpE
tttt
pxbmbu ?=
++ 11
)()(
∑
=
==
T
t
tTtT
buEbu
T
bg
1
)]([)(
1
)(
{}
)
?
()'
?
(minarg
?
?1
bWgbgb
TT
b
=
0]1)([
11
=?
++ tt
RbmE
GMM and asset pricing model---
continued
? Using , form an estimate of
?
? Second-stage estimate
? is a consistent, asymptotically normal, and
asymptotically efficient estimate of the parameter vector b.
? The variance-covariance matrix of is
?
?Where
1
?
b
S
?
{}
)
?
(
?
)'
?
(minarg
?
1
?2
bgSbgb
TT
b
?
=
∑
∞
?∞=
?
=
j
jtt
bubuES ])'()([
2
?
b
2
?
b
11
2
)'(
1
)
?
var(
??
= dSd
T
b
b
bg
d
T
?
?
=
)(
Test of parameters
? This variance-covariance matrix can be used to test
whether a parameter or a group of parameters is equal to
zero, via
? Finally, the test of overidentifying restriction is a test of the
overall fit of the model,
)1,0(~
)
?
var(
?
N
b
b
ii
i
)(#~
?
])
?
[var(
?
21
jjjjj
ofbbbb χ
?
{}
)#(#~)]()'([
21
min
parametersofmomentsofbgSbgTTJ
TT
b
T
?=
?
χ
Interpreting the GMM procedure—pricing
errors
? In the language of expected returns, is proportional to the
difference between actual and predicted returns: Jensen’s alphas.
? Because
? So we can write
? =(actual mean return-predicted mean return)/Rf
? If we express the model in expected return-beta language
then the GMM object is proportional to the Jensen’s alpha measure of mispricing
][])([)(
11 tTttTT
pExbmEbg ?=
++
)(bg
T
)(/),cov()( mERmRE
ee
?=
)))(/),cov(()()(()()( mERmREmEmREbg
eee
??==
λβα ')(
ii
ei
RE +=
f
i
Rbg /)( α=
Why
? This fact suggests that a good weighting matrix
might be inverse of S. Hansen (1982) shows
formally that the choice is statistically
optimal weighting matrix, meaning that it
produces estimates with lowest asymptotic
variance.
1?
S
∑∑
∞
?∞=
?
=
+
=→=
j
jtt
T
t
tT
S
T
uuE
T
u
T
g
1
)'(
1
)
1
var()var(
1
1
1?
= SW
Standard errors
? The formula for the standard error of the estimate,
? Where it come from? –”Delta method”
? Delta method: the asymptotic variance of f(x) is
? S/T is the variance matrix of the moment .
?is
? Then the delta method gives
11
2
)'(
1
)
?
var(
??
= dSd
T
b
)var()('
2
xxf
T
g
1?
d
T
T
g
b
bg
?
?
=??
?1
]/[
11
2
1
)var(
1
)
?
var(
??
=
?
?
?
?
= Sdd
Tg
b
g
g
b
T
b
T
T
T
test
? You have estimated parameters that make a model “fit
best”. The natural question is how does it fit?
? It is natural to look at pricing errors and see if they are
“big”.
? The asks whether they are big by statistical method. If
it is big, the model is “rejected”.
? The test is
T
J
T
J
{}
)#(#~)]()'([
21
min
parametersofmomentsofbgSbgTTJ
TT
b
T
?=
?
χ
Applying GMM
? Simply mapping a given problem into the very general
notation:
? Using price
? Using returns (1)
? It is common to add instruments as well, you multiply both
sides of
? By any variable observed at time t, and take
unconditional expectation, resulting in
?Or
0])([
11
=?
++ ttt
pxbmE
0]1)([
11
=?
++ tt
RbmE
1])([
11
=
++ ttt
RbmE
t
z
)(])([
11 ttttt
zEzRbmE =
++
{})2(0]1)([
11
=?
++ tttt
zRbmE
(1) And (2) are both important for asset pricing test
For example
? Using returns
? Start with two returns and one instrument z,the
model that we will test is
?
? Denote and using the Kronecker product
],[
ba
RR
?
?
?
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=
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++
++
++
++
0
0
0
0
1
1
)(
)(
)(
)(
11
11
11
11
t
t
t
b
tt
t
a
tt
b
tt
a
tt
z
z
zRbm
zRbm
Rbm
Rbm
E
?
0)])'1,1(())(([
11
=???
++ tttt
zzRbmE
0]1)([
11
=?
++ tt
RbmE
)',1(
tt
zz =