156 B
. Lucke, C. Gaggermeier Economics Letters 71 2001 155 –163
In this paper, we argue that a fundamental problem of Euler equation estimation lies in the fact that if the model is true, then the data used to estimate the Euler equation are the solution to the complete
set of first order conditions including, in particular, the transversality condition. The Euler equation to be estimated represents only a subset of the restrictions the optimal trajectory is required to meet.
Specifically, Euler equations in saddlepath stable systems contain an explosive root, while the observed data which fulfil the transversality condition do not. Hence the explosive root cannot be
estimated from the data and the identifiability of parameters related to this root is not ensured.
We also show that the problem is absent in the textbook time separable case, where the change in consumption is unrelated to lagged consumption. This is Hall’s 1978 random walk scenario. While
the unit root is unstable, it is not explosive and does hence not violate the transversality condition. Empirically, however, the change in consumption is often correlated to lagged consumption, so that
momentary utility cannot be viewed as time separable, possibly due to habit persistence, durable goods, or liquidity constraints. In this case, Euler equation estimates of some of the deep parameters
will be nonsensical. As an illustration we show that estimates of the representative agents’s discount rate which should be between 0 and 1 will tend to infinity due to this problem.
The body of this paper is organized as follows: in Section 2 we propose a very general representative consumer model and derive log linear approximations to selected first order conditions.
We work with log linear approximations here since the solutions to linear systems of difference equations are well known. For nonlinear Euler equations the problem is probably at least as severe,
and certainly less well analytically tractable. Further, we derive simple properties of solutions to the Euler equation and compare these to the analogous properties of solutions to the complete set of first
order conditions. We argue that saddlepath stability induces an identifiability problem that renders Euler equation estimates grossly biased or causes estimation algorithms to break down. We illustrate
our point in an empirical application with consumption data of selected OECD countries in Section 3. Section 4 concludes.
2. The model
Consider, for simplicity, a non-growing economy, in which a representative household maximizes expected utility,
` t
E
O
b u ? 1
F G
t 50
subject to the budget constraint C 1 A
5 w L 1 1 1 r A . 2
t t 11
t t
t t
Here, C is consumption, L is the household’s labor supply, A is real wealth, and r and w are the
t t
t t
t
interest rate and the real wage in period t. b [ 0, 1 is a discount factor. We assume that factor prices
are stochastic, for instance due to productivity shocks as the source of uncertainty. We assume standard properties for momentary utility u ? , time invariance in particular. Typically,
u ? will depend on C , but it may also depend on lagged consumption or any other sensible variables.
t
B . Lucke, C. Gaggermeier Economics Letters 71 2001 155 –163
157
In the following, we will study the time non-separable case u 5 uC , C to allow for habit
t t 21
1
persistence or durable consumption goods. Differentiating the Lagrangean with respect to C and A
yields the first order conditions
t t 11
t t 11
l 5 b u C , C 1
b E [u C
, C ] 3
t 1
t t 21
t 2
t 11 t
and l 5 E [1 1 r
l ]
4
t t
t 11 t 11
where partial derivatives are indicated by indices. The transversality condition is of the usual form lim
l A 5 0.
TVC
t t 11
t →
`
We combine 3 and 4 to get the Euler equation u C , C
1 bE u C
, C 5 bE 1 1 r
u C , C 1
bu C , C
5
f g
f s
dg
1 t
t 21 t
2 t 11
t t
t 11 1
t 11 t
2 t 12
t 11 2
which is a nonlinear stochastic difference equation of odd order. If the model is true, then the observed consumption time series will obey Eq. 5 — and this fact is precisely the idea of Euler
equation estimation. Unfortunately, as we will show, the parameters are not identified unless momentary utility is time separable, i.e. u
; 0.
2
To illustrate this claim, we would need the solution of 5. Since this is not known, we will follow
3
the standard approach in the literature and study a linearized version of 5. We will argue that the transversality condition eliminates the unstable root of the difference equation, so that the observed
data, which also obey the transversality condition, are uninformative with respect to the unstable root. Not knowing the unstable root, however, it is impossible to retrieve all structural parameters. We
further argue that nothing of this line of reasoning is special to linear models, except for the fact that the general solution of the linear equation is known. The key point in the argument is not the linearity
4
but the saddle path property — which should be identical for nonlinear Euler equations. We linearize the Euler equation about the steady state values C and 1 1 r. As the procedure is
standard cf. King et al. 1988, we do not give the details here. After some calculus, we obtain u
u u
u 1
1
11 22
11 22
]] ]
]] ]
] ]
ln C ¯ 1 2
2 ln C
1 1
2 ln C 1
ln C
S D
S D
t 12 t 11
t t 21
bu u
bu u
b b
21 21
21 21
u 1 bu
1 2
]]]] 2
r 1
´
t 11 t 12
bu C
21
\ ¯ a ln C 1
a ln C 1 a ln C 1
dr 1
´ 6
1 t 11
2 t
3 t 21
t 11 t 12
1
It is not difficult to show that all of our arguments remain valid if higher consumption lags up to C , say or other
t 2k
contemporaneous variables affect momentary utility. The textbook time separable case is, of course, contained as a special case in the non-separable set-up we study.
2
This odd order property holds for arbitrary k in the general utility function uC , . . . , C .
t t 2k
3
Of course, if momentary utility is quadratic then the linearized equation and the true Euler equation coincide. Thus in this case our argument is immune against criticisms of the quality of log-linear approximation as advanced e.g. by Carroll
1997.
4
If it were not, then most of the RBC literature would be besides the point.
158 B
. Lucke, C. Gaggermeier Economics Letters 71 2001 155 –163
5
where ´
collects the expectational errors.
t 12
The following Euler equation properties EEP of the linearized Eq. 6 are noteworthy: EEP 1: the lag polynomial in log consumption has a unit root,
oa 5 1.
i
EEP 2: the order of the lag polynomial in log consumption is odd. EEP 3: the coefficient of the highest log consumption lag,
a , is larger than one.
3
While the first property dates back to Hall’s 1978 seminal paper and comes as no surprise, the
6
very robust properties EEP 2 and EEP 3 have not received similar attention. Let us study the solution of the homogenous part of 6. The lag polynomial is of degree three, i.e.
there are three roots m , m , and m which govern the dynamic properties of the general solution. One
1 2
3
root, by EEP 1, is equal to one, and at least one more root is explosive i.e. smaller than one in absolute value by EEP 3 and by virtue of the following.
k j
Lemma. Let p z 5 1 2
o a z be a polynomial defined on the complex numbers with
ua u . 1. Then
k j 51
j k
there is at least one root of the polynomial with absolute value smaller than one.
k k
j
Proof. Any polynomial p z can be factorized as p z 5 a
P
z 2 m . Since p z 5 1 2
o a z , we
k k
k j
k j
j 51 j 51
k k
21
have a
P
2 m 5 1. Hence
P
um u 5 ua u , 1 and consequently at least one
m must have absolute
k j
j k
j j 51
j 51
value less than one. h
The general solution to the homogenous part of 6 is given by
t t
1 1
] ]
ln C 5 cnst 1 cnst 1 cnst
7
S D S D
t 1
2 3
m m
2 3
where cnst , cnst , cnst are arbitrary constants and m is assumed to be the unit root. Without loss of
1 2
3 1
generality, let us further assume that m be inside the unit circle. Thus 7 is generally explosive,
3
unless cnst equals zero.
3
It is well known that every explosive trajectory given by 7 violates the transversality condition
7
TVC. Hence it is necessary for utility maximization to set cnst 50, which immediately determines
3
the initial value of the optimal consumption trajectory. Thus the optimal and observed consumption stream satisfies
5
Note that I have used ln 1 1 r ¯ r
, 1 1 r b 5 1, and u
5 u in deriving 6.
t 11 t 11
21 12
6
These properties refer to the homogenous part of 6 only. This is why we could allow momentary utility to depend on other contemporaneous variables without affecting EEP 1–EEP 3. Note that the properties continue to hold if momentary
utility depends on further lagged consumption terms: u 5 uC , . . . , C . The coefficient of the highest consumption lag is
t t 2k
2k
then given by a
5 b
, which is larger than one for k .0. Only in the time separable case of k 50 do we get
112k
a 5 a 5 1; i.e. consumption is a random walk.
1 112k
7
Note that the unit root is not explosive and does hence not violate the transversality condition. This is easy to see: TVC states that the product of wealth and the present value shadow price converge to zero. Standard analysis of such models
reveals that the product of wealth and the current value shadow price converges to a constant if the homogenous part of the central difference equation features a unit root, but no explosive roots. Since the present value multiplier equals the
t
discounted current value multiplier, TVC is fulfilled as b converges to zero.
B . Lucke, C. Gaggermeier Economics Letters 71 2001 155 –163
159
t
1 ]
ln C 5 cnst 1 cnst 8
S D
t 1
2
m
2
which happens to be the general solution of a second-order homogenous difference equation with a unit root, i.e. an equation of the form
ln C 5 a ln C
1 1 2 aln C
. 9
t t 21
t 22
If m is stable the usual case, then 12a is smaller than one in absolute value, i.e. 9 is stable. Note
2
that any time series generated from 9 also obeys the homogenous part of 6, but asymptotically the estimate of
a from this time series would be zero and hence inconsistent.
3
Thus, the dynamical properties of trajectories which solve the full system of first order conditions the Euler equation and the transversality condition are very different from the dynamical properties
of general solution trajectories of the Euler equation alone. In the saddlepath case in which just one root of 6 is explosive, we can sum up the full system properties FSP as:
FSP 1: the lag polynomial in log consumption has a unit root. FSP 2: the order of the lag polynomial in log consumption is even.
FSP 3: the coefficient of the highest log consumption lag is smaller than one.
If, in rare circumstances, more than one root of the lag polynomial in the linearized Euler equation lies inside the unit circle, then properties FSP 1 and FSP 3 are unchanged. Property FSP 2 does not
necessarily hold, however: all that can be said instead is that the order of the lag polynomial in log consumption is definitely smaller than the order suggested by the Euler equation.
The jeopardies of Euler equation estimation should now be clear: so long as the transversality condition and its implications for initial conditions are neglected, efforts to estimate 6 will fall prey
to an identifiability problem: by writing the homogenous part of 6 in growth rates, c [ln C 2 ln
t t
C , we have
t 21
c 5 a 2 1 c
1 a 1 a 2 1 c
. 10
s d
s d
t 1
t 21 1
2 t 22
After some algebra, we can write 10 in terms of its roots m and m as
1 2
m 1 m 1
2 3
]]] ]]
c 5 c
2 c
. 11
t t 21
t 22
m m m m
2 3
2 3
Apparently, estimating a and a in 10 from data generated by 8 is tantamount to estimating m
1 2
2
and m in 11 from data generated by 8. But the latter task is obviously impossible, since 8 does
3
not contain any information about the instable root m . Hence m is not identified and any attempt to
3 3
estimate it is doomed to failure. Not knowing m , however, it is impossible to retrieve a and a .
3 1
2
To see how this phenomenon works out in the estimation of the deep structural parameters, imagine that a researcher uses a nonlinear estimation method e.g. GMM or maximum likelihood to
retrieve b and possibly some other structural parameters directly from 6. Note that the data originate
from a law of motion 9 in which the analogue of a is zero, while in 6 a 5 1 b. Hence the
3 3
estimation routine will tend to drive b up to plus infinity Consequently, either the procedure will
160 B
. Lucke, C. Gaggermeier Economics Letters 71 2001 155 –163
crash when the Hessian becomes singular, or it will stop with a biased estimate of b when the cost in
terms of the implied coefficients a and a which also depend on b becomes too large.
1 2
3. Empirical findings
