M. Schönhofer J. of Economic Behavior Org. 44 2001 71–83 75
if ˆ
γ
k
= 1
T
T −k
X
t =1
ǫ
t
− ˆ µǫ
t +k
− ˆ µ,
1 ≤ k ≤ k
max
. An alternative test for zero autocorrelation is the Box-Pierce-test.
1
3. Consistent adaptive learning in the OLG-model
In this section an adaptive learning process is specified for the OLG-model. In Schönhofer 1996 it was shown that adaptive learning can generate irregular behavior but forecast errors
have not been investigated. In this paper, forecast errors are analyzed in parameter regions where the dynamical system behaves chaotic.
3.1. The overlapping generations model We consider a standard version of the overlapping generations model. There is only
one non-storable good. At the beginning of each period every agent has an endowment of this good. Fiat money is the only store of value between periods. The set of agents in the
economy consists of a government and of consumers. The government is infinitely lived. The set of consumers has the overlapping generations structure.
• Government:
The government purchases a quantity G
t
0 of the commodity in each period at the market price p
t
. At time t = 1 the government owns M
1
0 units of currency. Since fiat money is the only store of value, agents can save by holding currency. The
government finances its consumption by creating additional currency each period, so that the government’s budget constraint is given by
M
t
− M
t −1
= p
t
G
t
. The process for currency creation is given by
M
t
= γ M
t −1
, 9
where γ 1 is the gross rate of currency growth, a policy rule chosen by the monetary authorities. Thus, the government expenditures G
t
are endogenous. • Consumers:
Young agents are endowed with e
1
, old agents are endowed with e
2
. Preferences of young consumers over consumption plans c
t
, c
t +1
are given by an intertemporal utility function u : R
2 +
→ R, which satisfies the following assumptions:
Assumption 1. 1. u is strictly quasi-concave;
2. u is strictly monotone;
1
Compare Box and Pierce 1970 and Greene 1993, p. 557.
76 M. Schönhofer J. of Economic Behavior Org. 44 2001 71–83
3. for any c
t
, c
t +1
∂u ∂c
t
c
t
, c
t +1
→ +∞, if, c
t
→ 0, ∂u
∂c
t +1
c
t
, c
t +1
→ +∞, if, c
t +1
→ 0; 4. c
t
, c
t +1
are normal goods. The young consumer solves the problem
max uc
t
, c
t +1
s.t. c
t
+ s
t
≤ e
1
p
e t +1
c
t +1
≤ p
e t +1
e
2
+ p
t
s
t
, where s
t
is the savings of the young agents, which is also aggregate savings in this context and p
e t +1
is the forecast of the time t + 1 price at time t . This implies that ∂u∂c
t +1
∂u∂c
t
= p
e t +1
p
t
. Therefore, s
t
is a function of the expected gross inflation rate at tθ
e t +1
:= p
e t +1
p
t
s
t
= Sθ
e t +1
. Market clearing on the commodity market requires that aggregate savings be equal to real
money balances M
t
p
t
= Sθ
e t +1
. Together with Eq. 9 we find
p
t
= γ Sθ
e t
Sθ
e t +1
p
t −1
, which is known in the literature as the “actual law of motion”. If we define p
t
p
t −1
= θ
t
, this can also be written as
θ
t
= γ Sθ
e t
Sθ
e t +1
. 10
We assume a Cobb–Douglas utility function uc
t
, c
t +1
= ln c
t
+ ln c
t +1
, and endowments
e
1
= 2, e
2
= 2κ, 0 κ 1,
of the young and old agents. Then, utility maximization leads to a linear savings function
M. Schönhofer J. of Economic Behavior Org. 44 2001 71–83 77
s
t
= Sθ
e t +1
= 1 − κθ
e t +1
. We assume κ ∈ 0, 1 and γ ∈ 1, κ
−1
. Then, a monetary steady state exists. The savings function decreases monotonically in θ
e t +1
. The linear specification is sufficient to rule out cycles and chaos under perfect foresight.
3.2. Specification of the adaptive learning process For a complete description of the economy it is necessary to specify, how agents form
expectations. We assume that agents use the adaptive learning rule β
t
= arg min
β t −1
X
i=1
p
i
− βp
i−1 2
. This yields
β
t
=
t −1
X
s=1
p
2 s−1
−1 t −1
X
s=1
p
s−1
p
s
. 11
The prediction p
e t +1
for the price in period t + 1 is given by p
e t +1
= β
t
p
t
, which means in inflation factors
θ
e t +1
= p
e t +1
p
t
= β
t
. In the following we show that θ
e t +1
can be written as a sequence of predictors {ψ
t
} with θ
e t +1
= ψ
t −2
θ
t −1
. ψ is indexed by t − 2, because the time-dependent parameters in the functional form of ψ
contain the information of past prices up to t − 2. Define
C
1 t −2
:=
t −1
X
s=1
p
2 s−1
−1
, C
2 t −2
:=
t −2
X
s=1
p
s−1
p
s
, C
3 t −2
:= p
2 t −2
. C
1
, C
2
, C
3
is indexed by t − 2, since they only contain information of past prices up to t − 2. Because of
t −1
X
s=1
p
s−1
p
s
=
t −2
X
s=1
p
s−1
p
s
+ p
2 t −2
θ
t −1
,
78 M. Schönhofer J. of Economic Behavior Org. 44 2001 71–83
it follows from 11 θ
e t +1
= β
t
= C
1 t −2
[C
2 t −2
+ C
3 t −2
θ
t −1
] = ψ
t −2
θ
t −1
. The sequences {C
i t
}i = 1, 2, 3 induce an adaptive learning process as a sequence of pre- dictors
ψ
t −2
θ
t −1
= C
1 t −2
[C
2 t −2
+ C
3 t −2
θ
t −1
]. 12
Thus, predictions for the inflation factor in period t + 1 depend on the data up to t − 1. The time-dependent parameter of the predictor include data up to time t − 2. Consider the
economic law
θ
t +1
= f θ
e t +1
, θ
e t +2
= γ Sθ
e t +1
Sθ
e t +2
. Predictions are formed according to
θ
e t +1
= ψ
t −2
θ
t −1
. Thus, we get
θ
t +1
= f ψ
t −2
θ
t −1
, ψ
t −1
θ
t
= ˜ f
t −1
θ
t
, θ
t −1
. In the following proposition it will be shown that the OLG-model with the above described
adaptive learning process can be written as an autonomous dynamical system.
Proposition 1. Consider the economic law of motion
θ
t +1
= f θ
e t +1
, θ
e t +2
= γ Sθ
e t +1
Sθ
e t +2
, and the adaptive learning process {ψ
t
} specified in Eq. 12 with θ
e t +1
= ψ
t −2
θ
t −1
. Then, the trajectories of the non-autonomous dynamical system
θ
t +1
= f ψ
t −2
θ
t −1
, ψ
t −1
θ
t
= ˜ f
t −1
θ
t
, θ
t −1
can be generated by the autonomous dynamical system β
t
, α
t
, g
t
= Gβ
t −1
, α
t −1
, g
t −1
, with
β
t
= β
t −1
+ g
t −1
γ Sα
t −1
Sβ
t −1
− β
t −1
, α
t
= β
t −1
, g
t
= g
−1 t −1
γ Sα
t −1
Sβ
t −1 −2
+ 1
−1
. 13
M. Schönhofer J. of Economic Behavior Org. 44 2001 71–83 79
Proof. Consider
β
t
=
t −1
X
s=1
p
2 s−1
−1 t −1
X
s=1
p
s−1
p
s
, 14
11 can also be written recursively. Define R
t −1
:=
t −1
X
s=1
p
2 s−1
. Eq. 14 implies
t −1
X
s=1
p
s−1
p
s
= R
t −1
β
t
. Furthermore holds
R
t −2
= R
t −1
− p
2 t −2
, ⇒
β
t
= R
−1 t −1
t −2
X
s=1
p
s−1
p
s
+ p
t −2
p
t −1
= R
−1 t −1
R
t −2
β
t −1
+ p
t −2
p
t −1
= R
−1 t −1
h R
t −1
− p
2 t −2
β
t −1
+ p
t −2
p
t −1
i = β
t −1
+ R
−1 t −1
p
t −2
[p
t −1
− β
t −1
p
t −2
] Defining
g
t −1
:= p
2 t −2
t −1
X
s=1
p
2 s−1
−1
, yields
β
t
= β
t −1
+ g
t −1
p
t −1
p
t −2
− β
t −1
= β
t −1
+ g
t −1
θ
t −1
− β
t −1
. 15
For g
t
holds g
t
= p
2 t −1
t
X
s=1
p
2 s−1
−1
= p
2 t −1
t −1
X
s=1
p
2 s−1
+ p
2 t −1
−1
= p
−2 t −1
t −1
X
s=1
p
2 s−1
+ 1
−1
= p
2 t −2
p
2 t −1
p
−2 t −2
t −1
X
s=1
p
2 s−1
+ 1
−1
= h
θ
−2 t −1
g
−1 t −1
+ 1 i
. 16
With 10, 15 and 16 the autonomous dynamical system follows β
t
, α
t
, g
t
= Gβ
t −1
, α
t −1
, g
t −1
,
80 M. Schönhofer J. of Economic Behavior Org. 44 2001 71–83
with β
t
= β
t −1
+ g
t −1
γ Sα
t −1
Sβ
t −1
− β
t −1
, α
t
= β
t −1
g
t
= g
−1 t −1
γ Sα
t −1
Sβ
t −1 −2
+ 1
−1
. With k
t
= β
t
, α
t
, g
t
the dynamical system 13 can also be written as k
t
= Gk
t −1
, with a map G
G : U → R
3
, and a subspace U of R
3
is. For this subspace U holds β ∈ 0, ¯
β], with
S ¯ β = 0,
α ∈ 0, ¯ β],
g ∈ 0,1. 3.3. Consistency of the adaptive learning process
Simulation of the OLG-model with adaptive learning showed for large parameter regions chaotic behavior of the deterministic dynamical system Schönhofer, 1996.
2
With a sav- ings function derived from a Cobb–Douglas utility function the dynamical system possesses
two parameters κ, γ . For an increasing gross rate of currency growth the dynamical system shows cycles and irregular behavior. However, forecast errors have not been investigated. In
this section it will be shown that for certain parameter combinations the adaptive learning process is a-consistent. Note that although the parameter β
t
will fluctuate and prices will diverge to infinity, agents will consider their learning process ex-post as a-consistent.
We assume a usual confidence level 1−α = 0.95 and time–series {ǫ
t
}
T t =1
with T = 100.
3
Consider the following parameter combination γ = 5.75,
κ = 0.1. The trajectory is irregular. Forecast errors are shown in Fig. 1. First the Null hypothesis
H : µ = µ
= 0, is tested against the alternative hypothesis
H
1
: µ = µ 6= 0.
2
Simulations have been performed with MACRODYN see Böhm and Schenk-Hopp´e 1998.
3
For this test with a fixed confidence level agents assume that {ǫ
t
} is normally distributed.
M. Schönhofer J. of Economic Behavior Org. 44 2001 71–83 81
Fig. 1. Forecast errors of a simulated time series γ = 5.75, κ = 0.1.
We get ˆ
µ = −8.336, and
r ˆ
γ T
= 11.044. Thus, the confidence interval is
[−29.982,13.310]. Since µ
= 0 is contained in the confidence interval, the Null hypothesis µ = 0 cannot
be rejected. Now the Null hypothesis H
: ρ
k
= 0, will be tested against the alternative hypothesis
H
1
: ρ
k
6= 0, for k = 1, . . . , 10. In Fig. 2 the autocorrelation coefficients ˆ
ρ
k
are shown for k = 1, . . . , 10. For this trajectory none of the autocorrelation coefficients are significantly different from
0. Agents do not reject the hypothesis H
: ρ
k
= 0, 1 k 10.
82 M. Schönhofer J. of Economic Behavior Org. 44 2001 71–83
Fig. 2. Autocorrelation coefficients with significance level γ = 5.75, κ = 0.1.
The learning process is a-consistent for a confidence level 1 − α = 0.95. Also the Box–Pierce test indicates no autocorrelation. With 10 degrees of freedom we get
Q = 2.074. For a confidence level of 0.95 the hypothesis cannot be rejected that the time series {ǫ
t
}
100 t =1
is not autocorrelated. The phenomenon discovered in this work is robust, since many parameter combinations
lead to a-consistency, mainly for γ 5.
4. Summary and conclusions
