8. Intertemporal Equilibrium Models
8.1 The Stochastic Discount Factor
Random variable Mt is a stochastic discount
factor if
(1)
1 = Et (1 + Rt+1)Mt+1
This equation can be derived merely from the
arbitrage theory, without assuming that investors maximize a well behaved utility function.
In the discrete-state setting with states s =
1, . . . , S and assets i = 1, . . . , N this goes as
follows:
Let q = (q1, . . . , qN ) be the asset price vector, and Xsi the payo® of asset i in the state
s. The S × N payo® matrix is then
X
⎛
X11 · · ·
...
⎞
X1N
⎟
..
⎠.
XS1 · · · XSN
⎜
= ⎝ ..
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We call p = (p1, . . . , pS ) the state price vector if it satis¯es
X p = q,
i.e.,
S
(2)
qi =
psXsi.
s=1
Thus ps gives the price of one dollar to be
paid in state s.
The gross return of the ith asset in state s
is
(3)
gsi = Xsi/qi = 1 + Rsi.
Dividing both sides of (2) by qi gives
S
ps(1 + Rsi).
1=
s=1
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An important result is that there exists a positive state price vector if and only if there are
no arbitrage opportunities.
Let πs > 0 denote the probability of state s,
and de¯ne Ms = ps/πs, which is positive in
the no arbitrage case. Then
1 =
(4)
=
S
s=1 ps(1 + Rsi)
S
s=1 πsMs(1 + Rsi)
= E [(1 + Ri)M ] ,
which is the static discrete equivalent of (1).
Note. Expectation (1) holds also unconditionally, because by taking expectations from both sides yields
(after reindexing)
(5)
1 = E [(1 + Rit )Mt].
Furthermore, using the de¯nition of covariance, we
have
E[(1 + Rit)Mt ] = E[1 + Rit]E[Mt ] + Cov[Rit , Mt ],
from which we obtain an expression for the cross return
1
(6)
E[1 + Rit ] =
(1 + Cov[Rit , Mt ]).
E[Mt ]
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8.1.1 Volatility Bounds
Any model of expected returns may be viewed
as a model of the stochastic discount factor.
Question: What asset return data may tell
about the behavior of the stochastic discount
factor.
• Lower bound for the stochastic discount
factor∗
Rede¯ne
(7)
1 = E [(1 + Rt)Mt],
where 1 = (1, . . . , 1) is an N -vector of ones,
and Rt = (R1t, . . . , RNt) is the return vector.
∗ Hansen,
L. and J. Jagannathan (1991). Econometric evaluation of asset pricing models. Journal of
Political Economy 99, 225{262.
85
¹ ) be any stochastic discount facLet Mt∗(M
¹ . Hansen and Jagannathan
tor with mean M
show that using asset pricing theory, there
exist a coe±cient βM
¹ such that
(8)
¹) = M
¹ + (Rt − E[Rt]) βM
Mt∗(M
¹.
¹ ) is a stochastic discount facThen if Mt∗(M
tor, it must satisfy (7),
¹) .
1 = E (1 + Rt)Mt∗(M
Expanding this yields
(9)
¹ E[1 + Rt] + −βM
1=M
¹,
where − = Cov(Rt), assumed nonsingular.
Then
(10)
−1 1 − M
¹ E[1 + Rt] .
βM
¹ =−
86
¹ ) becomes then
The variance of Mt∗(M
¹
Var[Mt∗(M)]
= βM¹ −βM¹
=
(11)
¹ 1 + Rt] − 1 − ME[
¹ 1 + Rt ] .
1 − ME[
The right hand side of (11) is the lower
¹.
bound of any discount factor with mean M
The Benchmark Portfolio
Let R̃t = (R0, Rt) be an N + 1 vector, where
¹ − 1 is the return of an arti¯cial
R0t = 1/M
riskless asset.
De¯ne then the benchmark portfolio return
as
¹)
Mt∗(M
Rbt(m) =
(12)
− 1.
∗
2
¹
E[Mt (M ) ]
87
Exercise: Show that this return can be obtained by forming a portfolio of the risky assets and the arti¯cial riskless asset, and that
it satis¯es the condition (7) on returns.
Exercise:
(P1) Show that Rbt is mean-variance e±cient.
¹)
(P2) Any stochastic discount factor Mt(M
has a greater correlation with Rbt than with
any other portfolio.
(P3) All asset returns obey a beta-pricing relation with the benchmark portfolio. That is,
(13)
E Rit −
1
−1
¹
M
= βib E[Rbt ] −
1
−1
¹
M
.
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Two further properties are:
(14)
¹ − E[1 + Rbt]
1/M
σ[Rbt]
=
.
E[1 + Rbt]
σ[Rbt]
(15)
¹ )]
σ[Mt(M
σ[1 + Rbt]
≤
,
¹
E[1 + Rbt]
E[Mt(M )]
i.e., the left hand side is the lower bound for
the ratio of the standard deviation to mean
of a stochastic discount factor.
The above analysis applies to returns themselves. Let Zit = Rit − Rkt denote the excess return on asset i over asset k, and let
Zt = (Z1t, . . . , ZNt) . Then (7) implies
(16)
0 = E[ZtMt].
89
¹) = M
¹ +
Proceeding as before, from Mt∗(M
~=−
~ −1(−M
¹ E[Zt]),
(Zt − E[Zt]) β~M
¹ , we get β
~ = Cov[Zt].
where −
The lower bound for the stochastic discount
factor is then
~ −1E[Z].
¹ )] = M
¹ 2E[Z] −
(17) Var[Mt∗(M
Specially in the case of a single asset this
simpli¯es to
(18)
¹ )]
E[Zt]
σ[Mt∗(M
=
.
¹
M
σ[Zt]
Equity Premium
Mehra and Prescott (1985) observe that SPindex annual standard deviation is about 18%,
and mean over a commercial paper of 6%.
Then the right hand side of (17) would be
¹ )) ≥ 0.33,
6/18 = 0.33, meaning that σ(Mt∗(M
¹ = 1 (which is close to what
i.e. 33%, if M
empirically is a plausible value).
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Nevertheless, empirical evidence indicates that
annual standard deviation of the stochastic
discount factor is much less than 33%. An
implication of this would be that the equity
premium cannot be explained easily with these
kinds of asset pricing models.
8.2 Consumption-Based Asset Pricing with
Power Utility Function
Let
1−γ
Ct
−1
(19)
,
U (Ct) =
1−γ
where γ is the relative risk aversion. Not that
limγ→1 = log Ct.∗
∗ It
is assumed here that Ct represents all individuals
in the economy, and is the aggregate consumption.
The associate asset pricing model is then Consumption CAPM or CCAPM.
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Now
U (Ct) = C −γ ,
and
−γ
U (Ct+1)
Ct+1
=δ
Mt+1 = δ
U (Ct)
Ct
Implying that 1 = Et[(1 + Ri,t+1)Mt+1] becomes
⎡
(20) 1 = Et ⎣(1 + Rt+1)δ
Ct+1
Ct
−γ
⎤
⎦.
To test the restrictions imposed by this model
we need to consider the distributional assumptions of the random variables. Assume
that the joint conditional distribution of the
asset returns and consumption homoscedastic and lognormal.
92
For a lognormally distributed random variable
X holds
(21)
1
log Et [X] = Et [log X] − Vart [log X].
2
Furthermore, the conditional homoscedasticity implies that
Vart[log X] = Var[log X − Et[log X]].
Using these, and taking logs on both sides of
(20) yields
(22)
0 = Et[rt+1 + log δ − γEt[¢ct+1]
2
2 2
+1
2 [σi + γ σc − 2γσic],
where the lowercase letter denote logarithms
of the corresponding variables, and
σic = Cov rit − Et[ri,t+1], ¢ct − Et[¢ct+1] .
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The riskless (real) interest rate return is then
(23)
1
rf,t+1 = − log δ − γ 2σc2 + γEt[¢ct+1].
2
Note that we can write also
(24)
Et[¢ct+1] =
1 2
γσ + ψ(rf,t+1 + log δ),
2 c
where ψ = 1/γ.
Furthermore, the homoscedasticity assumption makes the log risk premium on any asset
over the riskless real rate constant, so that
1 2
(25) Et[ri,t+1 − rf,t+1] + σi = γσic,
2
or remembering that e.g. rit = log(1 + Rit)
and using (21) we can write
(26)
log Et (1 + Ri,t+1)/(1 + Rf,t+1) = γσic.
This shows that the risk premia are determined by the coe±cient of relative risk
aversion times covariance with consumption
growth.
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Table 8.1 Sample statistics of US consumption growth and asset returns
================================================================
Correl. with Covar with
Standard
consumption
consumption
Variable
Mean
deviation
growth
growth
---------------------------------------------------------------Consumption growth 0.0172
0.0328
1.0000
0.0011
Stock return
0.0601
0.1674
0.4902
0.0027
CP return
0.0183
0.0544
-0.1157
-0.0002
Stock-CP return
0.0418
0.1774
0.4979
0.0029
================================================================
Using these ¯gures in (25) we obtain γ = 19
to ¯t the equity premium, which is much
greater than 10, the maximum value considered plausible by Mehra and Prescott!
95
Time Varying Expected Returns and Consumption Growth
Equation (22) gives a relation between rational expectations of asset returns and rational
expectations of consumption growth.
De¯ne an error term
ηi,t+1 = ri,t+1 − Et[ri,t+1] − γ(¢ct+1 − Et[¢ct+1]),
so that we can rewrite (22)
(27)
ri,t+1 = µi + γ¢ct+1 + ηi,t+1.
In general the error term ηi,t+1 are correlated
with realized consumption growth. So OLS
is not an appropriate estimation method. However, ηi,t+1 is not correlated with any information variables at time t. Hence any lagged
variables can be used as instruments in IV
regression.
96
8.3 GMM Estimation in Discount Factor Models
The Generalized Method of Moments (GMM)
is natural estimation method in discount factor models (DFM). The asset pricing model
predicts
(28)
E [Pt] = E M (datat+1, b) Pt+1 ,
where b denote the parameters of the model.
Note that Pt is usually a vector of prices.
For example with the power utility function
(29)
Ct+1 −γ
M (datat+1, b) = δ
,
Ct
so that b = (δ, γ) .
97
Natural estimates of the left and right hand
sides are the sample averages
1 T
Pt,
T t=1
(30)
and
(31)
1 T
M (datat+1, parms)Pt+1 .
T t=1
GMM estimates the parameters by making the sample averages (30) and (31) as close to each other as
possible.
98
Estimation Procedure
De¯ne errors
(32)
ut+1(b) = Mt+1(b)Pt+1 − Pt,
pricing error
(33)
1 T
gT (b) =
ut(b),
T t=1
GMM estimate:
(34)
^ = argminb gT (b) S
^−1gT (b),
b
^−1 is an estimate of
where S
(35)
S=
∞
E ut(b)ut−j (b) ,
j=−∞
An estimate of S is
(36)
1 T
^)ut(b
^)
^=
ut(b
S
T t=1
In practice the estimation is worked out numerically with suitable starting values. In the
^ = I, the identity matrix.
¯rst round S
99
The essential point here is that error, ut+1(b)
should be unpredictable.∗ That is
(37) E ut+1(b)|It = Et ut+1(b) = 0,
where It is the available information at time
point t. We say that the prediction errors are
orthogonal (uncorrelated) to It.
Note. Because prices are nonstationary, returns are
preferred in practice such that (32) becomes
(38)
ut+1(b) = Mt+1(b)(1 + Rt+1) − 1,
where Rt+1 can be a vector of returns of di®erent
assets.
∗ We
say that a stochastic process, Yt , is unpredictable
if its conditional mean is the same as its unconditional mean. That is Et [Yt+1] = E [Yt ] = µY .
100
Usually the information set It consist of some
predictior variables, say zt. These are also
called as instrumental variables. The conditions that the instrument variables are not
correlated with the prediction errors are called
orthogonality conditions, and are mathematically de¯ned as
(39)
E ut+1(b)zt = 0,
i.e., ut+1(b) is uncorrelated with each component of zt.
Note. The expectation (37) is of the form (39) with
zt = 1.
If the number of equations (moments) in (39)
is less than the number of parameters in b
then we say that the estimation problem is
underidenti¯ed, and the estimates of the parameters cannot be solved uniquely.
101
If there are equally many moments as parameters then the problem is exactly identi¯ed
and the estimation problem with given sample can be directly solved by selecting b such
^)
that (denote the solution as b
(40)
1 T
^)zt = 0,
ut+1(b
T t=1
where T is the number of observations.
Example. Sample mean and variance a GMM estimator: Let Y1, . . . YT be a sample from a random variable
with E[Y ] = µ, and Var[Y ] = σ 2. Then b = (µ, σ 2) .
Now σ 2 = Var[Y ] = E (Y − µ)2 = E Y 2 − µ2. So
that E Y 2 = σ 2 + µ2. Then we can de¯ne
ut(b) =
Yt − µ
Yt2
−
(σ 2
+
µ2)
.
We have an exactly identi¯ed case, and using (40)
with zt = 1, we get
⎛ 1 T
⎞
^
t=1 Yt − µ
T
⎝
⎠ = 0.
T
1
2
σ2 + µ
^2)
t=1 Yt − (^
T
102
Thus the GMM estimators of µ and σ 2 are
1
¹ =
µ
^=Y
T
T
Yt,
t=1
the sample mean, and
1
σ
^2 = s2 =
T
T
1
¹2 =
Yt2 − Y
T
t=1
T
t=1
¹ )2,
(Yt − Y
the sample variance.
Example. OLS estimator as a GMM estimator.
yt = β0 + β1x1t + · · · + βpxpt + et
= xt b + et,
where xt = (1, x1t, . . . , xpt ) and b = (β0, β1, . . . , βp) .
Given observations, we can write
y = Xb + e,
where y = (y1, . . . , yT ) , X = (x1, . . . , xT ) : T × (p + 1)
matrix of x-observations, and e = (e1, . . . , eT ) .
103
In addition to
E [y − Xb] = E [e] = 0,
an essential condition in regression is that E [X e] = 0.
That is the residuals are uncorrelated with the explanatory variables. Thus we can use X as instruments, so that (39) becomes
E (yt − xt b)xt = 0.
Again we have an exactly identi¯ed case, and can write
(40) as
^) X = 0,
(y − Xb
from which we get
^ = X y,
(X X)b
or
^ = (X X)−1X y,
b
the OLS estimator.
104
Finally if there are more equations (moments)
in (39) than parameters in b the estimation
problem is over identi¯ed. The GMM estimator b is then the one which satis¯es (40)
"as closely as possible". This is achieved by
rede¯ning (33) as
(41)
1 T
gT (b) =
ut(b)zt,
T t=1
and applying (34).
De¯ne a statistic, called the J-statistic as
^) S−1gT (b
^).
JT = gT (b
If the moment conditions are satis¯ed then
asymptotically
(42)
(43)
T JT ∼ χ2(df),
where df = #(moments) − #(parameters),
i.e., number of overidentifying restrictions.
This can be used to test the null hypothesis
that the moment conditions (39) are satis¯ed, and is called a test for the overidenti¯cation restrictions.
Rejection of the null hypothesis indicates that
the model does not ¯t the data.
105
Note. If in (39) ut+1(b) is a vector, then (39) must
be expressed in a bit more general form
(44)
E [ut+1(b) ⊗ zt ] = 0,
where ⊗ is the Kronecker product, which means that
each element in z are multiplied by u (b).∗
t
t+1
Example. Let Xt = (Xta, Xtb) = (1 + Rta, 1 + Rtb) and
instrument zt = (1, zt ) then
⎡
⎤
a
Mt+1(b) Xt+1 − 1
⎢ Mt+1(b) X b − 1 ⎥
t+1
⎥
E [(Mt+1(b)Xt+1 − 1) ⊗ zt ] = E ⎢
a z − z ⎦ = 0.
⎣ Mt+1(b) Xt+1
t
t
a z −z
Mt+1(b) Xt+1
t
t
∗ If
a = (a1, a2) and c = (c1, c2) then
⎛
⎞
⎛
⎞
a1
a
c
1 1
c1 ⎟
⎜
a
2
a c1
⎜
⎟ ⎜ a2c1 ⎟
a⊗c=
=⎜
⎟=⎝ a c ⎠
a c2
1 2
⎝
⎠
a1
c2
a2c2
a2
106
GMM in EViews
Application of GMM requires speci¯cation of
the moment condition and instrument variables. Estimation can be applied both in single equation as well as system estimation. In
EViews we simply de¯ne the moment conditions
(45)
Mt+1(b)(1 + Rt) − 1
and specify the instruments.
Example. Consider the consumption CAPM.
107