Generalized IV Estimation of
Nonlinear Rational Expectations
Models
Hansen and Singleton (1982)
What is a Rational Expectations (RE)
Model?
• An equilibrium of a dynamic model can typically be
described by a probability distribution over sequences of
data
• RE à Each agent’s subjective belief about the data is a
conditional of this equilibrium probability distribution,
where the conditioning is on the agent’s information set
• Expectations are thus consistent with outcomes
generated by the model
• This means we can use the moment conditions derived
from agent’s Euler equations to estimate the model’s
parameters using GMM
2 Ways to estimate RE models
Estimation methods for RE models can
be categorized by the amount of info
they require
1. Full-Information: MLE
2. Limited-Information: GMM
MLE: Full Information
• Goal: Estimate the entire model
• This estimation method is efficient and produces
estimates for all the parameters in the model
• Requires econometrician to specify the entire
structure of the model, including the distribution
of shocks.
• VERY HARD outside of linear models
GMM: Limited Information
• Goal: Estimate only some of the model
parameters
• Trade-off: Generally less efficient than MLE, but
avoids contaminating the estimation results by
model misspecification
• E.g. it avoids having to specify the shock
distributions, production technology etc.
• Good for non-linear models
Asset-Pricing Example
• Simple version of Lucas (1978)
– One consumption good and one asset*
• Representative Agent
– Infinite horizon, CRRA utility
– Each period chooses between consuming
good or investing in asset
The Consumers Problem
max Et
P1
°
j Ct+j
j=0 ¯
°
s.t. Ct+j + It+j · rt+j It+j¡1 + Wt+j
j = 0; 1; : : : ; 1
Ct : Consumption
It : Investment in (one-period) asset
Wt : Labor Income
rt : Asset Return
¯ : Discount Factor
° : Preference parameter
Euler Equation
• The agents EE:
· ³
¸
´°¡1
Et ¯ CCt+1
rt+1 ¡ 1 = 0
t
• 1 orthogonality condition for 2 unknowns: (°; ¯)
• Adding instruments zt, and conditioning
down the EE becomes:
· ³
¸
´°¡1
N
Ct+1
E ¯ Ct
rt+1 ¡ 1
zt = 0
GMM
• Let yt+1 =
Ct+1
; rt+1
Ct
and µ = [°; ¯]
" µ
#
¶°¡1
Ct+1
rt+1 ¡ 1 - zt
• Define f (yt+1 ; zt ; µ) = ¯ C
t
T
1 X
gT (µ) =
f (yt+1 ; zt ; µ)
T t=1
• GMM: µ^ = arg minµ gT (µ)WT gT (µ)
where WT is the weighting matrix
Optimal Weighting Matrix
• Define the jth sample autocovariance:
T
1 X
T
Rt (j) =
f (yt+1 ; zt ; µT ) f (yt+1 ; zt ; µT )
T t=1+j
• Optimal Weighting Matrix:
n
o¡1
£
¤
P
n¡1
WT¤ = RT (0) + j=1 RT (j) + RT (j)T
where µT is obtained from the first-step
GMM using a suboptimal weighting matrix
MLE
• Assume that consumption growth and the
asset return are conditionally log-normal,
·³
¸
so
´°¡1
h
i
2
Et
Ct+1
Ct
• Recall the EE:
rt+1 = exp ¹t +
Et
·³
Ct+1
Ct
´°¡1
¸
rt+1 =
• Combining these equations
¾
2
1
¯
¹t = ¡ log(¯) ¡
¾2
2
MLE
³
Ct+1
Ct
´
• Let
• Define the error term Vt+1 :
Xt = log
Vt+1 ´ log
·³
Ct+1
Ct
• Assume:
• System:
´°¡1
;
Rt = log(rt );
Yt = [Xt ; Rt ]T
¸
rt+1 ¡ ¹t = ®Xt+1 + Rt+1 + log ¯ +
Et (Xt+1 ) = a(L)T Yt + ¹x
·
¸·
¸ ·
¸· ¸ ·
T
¹x
1 0 Xt+1
a(L)
Xt
=
+
® 1 Rt+1
0
Rt
log ¯ +
| {z } | {z } | {z } | {z } |
{z
A0
Yt+1
¾2
2
A1 (L)
Yt
¹
¸
·
¸
Wt+1
¾2 + V
t+1
2
} | {z }
´ t+1
MLE
• We have ´t+1 = A0 Yt+1 ¡ A1 (L)Yt ¡ ¹;
´t+1 » N (¹; §)
• Define µ = ° ¯ ¹x a(L) parameters §
• Given T observations of Yt we can
estimate µ via:
µ^ = arg maxµ ¡ T2 log j§j ¡
1
2
T
t=0
T
´t+1
§¡1 ´t+1
Empirical Results
• GMM:
• MLE:
Empirical Results
• All of the GMM estimates of gamma are
between .03 and .32
• All of the estimates of beta are greater than .99,
but less than 1
• Much smaller S.E.’s on beta than gamma à
beta estimated more precisely than gamma
• High chi-square p-vals à All estimates reject the
stock pricing model with CRRA returns
Hall, Chapter 3
• Iterated GMM estimates:
°^ = ¡0:34
¯^ = 0:99
• Continuous updating GMM estimates:
°^ = 0:52
¯^ = 0:99