Lecture 3: Euler-Equation Estimation
Simon Gilchrist
Boston Univerity and NBER
EC 745
Fall, 2013
Euler equation tests
This lecture considers some further implications and tests of the
“standard” asset pricing model: for any gross return Rt+1 , we
have the equation:
0
βu (Ct+1 )
Rt+1 = 1,
Et
u0 (Ct )
Hence for any gross return Rt+1 , we have
0
βu (Ct+1 ) f
Et
Rt+1 − Rt+1
= 0.
u0 (Ct )
These equations hold for any asset which you can buy or sell (at
the margin): stocks, bonds, options, commodities, etc.
Key intuition: assets are priced according to their covariance with
consumption growth. Assets which pay off when consumption
growth is high (in good times) are risky, hence they will need to
have high expected returns for investors to want to hold them.
Euler equation methods
Earlier lecture studies the implication of the Euler equation, plus
an assumption that consumption growth is iid and dividends =
consumption.
Euler equation methods requires very few assumptions - no need
to specify the shocks, the technology (as you would in a GE
model), or the endowment process.
The logic of Euler equation methods applies to models others
than the CRRA representative agent model.
Example: Suppose utility depends on ct and other observable xt ,
say U (ct , xt ), then we have
Mt+1 =
βU1 (ct+1 , xt+1 )
,
U1 (ct , xt )
and Et (Mt+1 Rt+1 ) = 1 can similarly be tested.
Euler equation tests
Specify u(C) =
have:
C 1−γ
1−γ .
For any variable Zt known at time t, we
!
Ct+1 −γ
Rt+1 − 1 = 0,
β
Ct
Zt Et
Since Zt known at t:
Et
Zt
β
Ct+1
Ct
!!
−γ
Rt+1 − 1
=0
Take unconditional expectations:
E
Zt
β
Ct+1
Ct
−γ
!!
Rt+1 − 1
= 0.
This is a set of restrictions that the theory imposes on the joint
distribution of consumption growth, returns and Zt .
Alternative interpretation
Express the Euler equation as:
0
βu (Ct+1 )
Rt+1 − 1 = εt+1 ,
u0 (Ct )
where
εt+1 =
0
βu0 (Ct+1 )
βu (Ct+1 )
R
−
1
−
E
R
−
1
t+1
t
t+1
u0 (Ct )
u0 (Ct )
By construction εt+1 is a forecast error – the difference between
realized and expected values. Therefore it is uncorrelated with
any variable known at time t:
E (Zt εt+1 ) = 0
No need for distributional assumption, except that returns and
consumption growth must be stationary.
Instrument Relevance
Any stationary variable Zt realized at time t is a valid instrument.
if Zt is uncorrelated with future returns and consumption, then it
does not add any restriction since
!!
Ct+1 −γ
0 = E Zt β
Rt+1 − 1
Ct
!
Ct+1 −γ
= E(Zt )E β
Rt+1 − 1 .
Ct
Instrument relevance: Zt must forecast future returns or future
consumption growth.
Empirical Implementation
This equation holds for any returns Rt+1 , Zt , so we have N K
equations, where N = # of returns and K = # of “instruments”
Zt , and we have 2 parameters (β, γ).
Intuitively, we simply pick β and γ to set the sample version of
these equations equal to zero, i.e.
!
T
1X
Ct+1 −γ
Rt+1 − 1 = 0.
Zt β
T
Ct
t=1
Exact identification:
Let Zt = 1 and apply to excess aggregate stock returns: choose γ
such that
T 1 X Ct+1 −γ e
f
Rt+1 − Rt+1
= 0.
T
Ct
t=1
Note: this may not be possible! i.e. with the data there may not
be a γ s.t. this holds.
Let Zt = 1 find β and γ to match both stock returns and the
risk-free rate:
!
T
Ct+1 −γ e
1X
Rt+1 − 1 = 0,
β
T
Ct
t=1
!
T
1X
Ct+1 −γ f
β
Rt+1 − 1 = 0.
T
Ct
t=1
Two eqns in two unknowns. Again, this may not be possible to
find both β and γ in sample.
Over-identification
If more than equations, then of course (generically) cannot set all
the equations equal to zero. In practice minimize the weighted
errors:
min gT0 WT gT ,
β,γ
where gT =
1
T
−γ
PT
Ct+1
Rt+1 − 1 and WT is a
t=1 Zt β
Ct
weighting matrix.
Can pick WT = I, or WT so that moments have the same scale,
or choose W in a statistically optimal way. Formula for WT is
the inverse of the asymptotic variance of the moment vector:
"
W = V ar
!!#−1
T
Ct+1 −γ
1X
β
Rt+1 − 1
T
Ct
t=1
Over-identifying tests
Overidentification: if N K > # of parameters the system is
over-identified.
−γ
P
Test that T1 Tt=1 Zt β CCt+1
R
−
1
is close to zero0.
t+1
t
GMM general formulas
Express a model as
E[f (xt , b)] = 0
The general GMM estimate b̂ is given by:
aT gT (b̂) = 0,
where
gT (b) ≡
T
1X
f (xt , b),
T
t=1
and aT is a matrix that defines which linear combination of
gT (b) will be set to zero.
Asymptotics
The asymptotic distribution of the GMM estimate is:
√
T (b̂ − b) → N [0, (ad)−1 aSa0 (ad)−10 ],
where
d ≡ E
∂f
∂gT (b)
(x
,
b)
=
,
t
∂b0
∂b0
a ≡ aT ,
∞
X
E[f (xt , b)f (xt−j , b)0 ].
S ≡
j=−∞
Hansen (1982) gives the sampling distribution of the moments
gT (b):
√
T gT (b̂) → N [0, (I − d(ad)−1 a)S(I − d(ad)−1 a)0 ].
Efficieny
The efficient estimate is obtained by setting:
a = d0 S −1
In this case,
√
T (b̂ − b) → N [0, (d0 S −1 d)−1 ]
The quadratic objective function provides a test of
over-identifying restrictions:
T JT = T gT (b̂)0 S −1 gT (b̂) = χ2 (N K − x)
Testing parameter restrictions
Suppose we estimate a model that imposes R restrictions on b.
We can test these restrictions using:
T (JT,R − JT ) = χ2 (R).
where JT,R is the value of objective estimated under the
restriction.
Comment: when imposing the restrictions we should use the
unrestricted weighting matrix. Otherwise the test statistic has a
non-central χ2 distribution.
The test statistic is constructed as a quadratic function of the
difference between the two moment vectors:
(gT,R − gT )0WT (gT,R − gT )
If we used alternative weighting matrices the difference in the J
statistics is not guaranteed to be positive.
Ordinary Least Squares
β solves:
M inβ ET [(yt − β 0 xt )2 ]
In this case:
f (xt , β) = xt (yt − β 0 xt ) = xt εt ,
gT (β) = ET [xt (yt − β 0 xt )],
aT
= I,
d = −ET (xt x0t ),
β̂ = ET [xt x0−1
t ET (xt yt )
∞
X
1
0−1
ET [xt xt [
E(εt xt x0t−j ε0t−j )]ET [xt x0−1
var(β̂) =
t
T
j=−∞
An implication of Euler equations
With Euler equations we have
gT (b) =
T
1X
Zt εt+1 (b)
T
t=1
We then choose b to minimize the quadratic
gT (b)WT gT
This is equivalent to choosing b so that
d(gT (b))0
WT gT = 0
db
In this case, the optimal weighting matrix is WT =V−1
gT where
VgT = E gT (b)gT (b)0
Variance-covariance matrix
Note that in Euler equation case:
0
E (f (xt , b)f (xt−s , b)) = E Zt εt+1 ε0t−s+1 Zt−s
If we confine attention to one period returns then εt is serially
uncorrelated and this exprssion is zero for s 6= 0. In this case, the
variance-covariance matrix can be constructed as
T
1X
Zt εt+1 ε0t+1 Zt0
E gT (b)gT (b)0 =
T
t=1
If we use multi-period returns then we induce serial correlation
and we need a Newey-West style estimator of the variance of the
moment matrix.