The benefit from using second-order approximations tostochastic dynamic rational expec- tations models is explained. By example of the neoclassical growth model, this note as- sesses the accuracy of the obtained approximation. The implications for optimal policy are discussed.Second-order approximation, accuracy, optimalpolicy

Matthias Paustian

English

Research Papers in Economics

Bonn Econ Discussion Papers
Discussion Paper 25/2003
Gains from second-order approximations
by
Matthias Paustian
November 2003
Bonn Graduate School of Economics
Department of Economics
University of Bonn
Adenauerallee 24 - 42
D-53113 Bonn                                     The Bonn Graduate School of  Economics is
                                                             sponsored by theGains from second-order approximations
Matthias Paustian∗
Center for European Integration Studies, Walter-Flex-Str. 3, D-53113 Bonn, Germany
Abstract
The beneﬁt from using second-order approximations to stochastic dynamic rational expec-
tations models is explained. By example of the neoclassical growth model, this note as-
sesses the accuracy of the obtained approximation. The implications for optimal policy are
discussed.
Keywords: Second-order approximation; accuracy; optimal policy
JEL classiﬁcation: E0; C63
1 Introduction
Various authors have pointed to shortcomings in using ﬁrst order approximations to
dynamic general equilibrium models. Kim and Kim (2003) have shown that such a
method can lead to spurious welfare reversals, i.e. full risk sharing appears inferior
to autarky in a simple two country model. This note assesses the accuracy of the
second-order approximation to the policy function developed by Schmitt-Groh´ e
and Uribe (2004). Furthermore, it suggests a welfare measure usable for policy
rankings that can be easily constructed and takes account of the effect of variances
on means.
2 Model representation and form of the solution
Toﬁxnotation, considerthegenericrepresentationforrationalexpectationsmodels
introduced by Schmitt-Groh´ e and Uribe (2004)
Et f(yt+1,yt,xt+1,xt) = 0. (1)
f is a known function describing the equilibrium conditions of the model economy,
yt is a vector of co-state variables and xt a vector of state variables partitioned as
xt = [x1,t;x2,t]. x1,t is a vector of endogenous state variables and x2,t a vector of
state variables following an exogenous stochastic process
x2,t+1 = Lx2,t + ˜ Nσt. (2)
∗Tel.: +49-228-73-4024, fax: +49-228-73-1809.
E-mail address: matthias.paustian@wiwi.uni-bonn.de (M. Paustian)
1Land ˜ N areknowncoefﬁcientmatrices, t isavectorofinnovationswithbounded
support, independently and identically distributed with mean zero and covariance
matrix I. σ is a parameter scaling the standard deviation of the innovations. The
solution to the model described by (1) is of the form
yt =g(xt,σ), (3)
xt+1 =h(xt,σ) + σNt+1, with: N =

0
˜ N

. (4)
Schmitt-Groh´ e and Uribe (2004) derive the second-order Taylor approximation to
the policy functions g(·) and h(·) and provide MATLAB codes for the numerical
implementation. The approximate model dynamics obtained from their second-
order approximation can be compactly expressed as
yt =Gxt +
1
2
G∗(xt ⊗ xt) +
1
2
gσ2, (5)
xt+1 =Hxt +
1
2
H∗(xt ⊗ xt) +
1
2
hσ2 + σNt+1. (6)
Here, the vectors yt and xt denote deviation or log-deviation from the steady state.
G and H are coefﬁcient matrices representing the linear part of the Taylor ap-
proximation. The matrices G∗ and H∗ form the second-order part jointly with the
vectors g and h.
3 Accuracy
The neoclassical growth model is employed to assess the accuracy of the second-
order approximation . The model consists of the following equilibrium conditions
c
−γ
t =βEtc
−γ
t+1[αAt+1kα−1
t+1 + (1 − δ)], (7)
ct + kt+1 =Atkα
t + (1 − δ)kt, (8)
lnAt+1 =ρlnAt + σt+1. (9)
3.1 den Haan - Marcet χ2 test
The den Haan and Marcet (1994) test exploits that for an exact solution the pre-
diction error, ut = βct+1(αkα−1
t θt+1 + 1 − δ) − c
−γ
t , must be orthogonal to any
function φ(xt) of state variables included in period t information set. The test
is carried out by constructing a simulated time series of length T for the model’s
variables and computing
BT ≡
ΣT
t=1(utφ(xt))
T
. (10)
The choice of φ(xt) in this analysis is the vector valued function of a constant, cur-
rent period state variables and two lags of the states. den Haan and Marcet (1994)
2show how to construct from BT a test statistic that has χ2
qm distribution, where
q is the number of instruments and m the number of Euler equations. Note that
the growth model is solved in log-deviations using the choice of parameters in den
Haan and Marcet (1994), i.e. γ = 0.5and 3.0, α = 0.33,β = 0.99,δ = 0.025,ρ =
0.95,σ = 0.03. The following table shows the percentages of 500 repetitions of
the test statistics for sample size T = 3,000 and T = 10,000 falling into the up-
per or lower 5% percentile of the χ2 distribution. A test statistic belonging to this
region is evidence against the accuracy of the solution. The χ2-test delivers ample
γ = 0.5 γ = 3
T lower 5% upper 5% lower 5% upper 5%
1st order 3,000 0.0% 52% 0.3% 13.6%
2nd order 3,000 5.6% 6.4% 5.6% 4.2%
1st order 10,000 0.0% 98% 0.2% 33.6%
2nd order 10,000 5% 5.2% 4% 4.2%
Table 1: χ2 accuracy test
evidence for inaccuracy of the log-linear solution method, too much probability
mass is in the upper tail. This is very clear for T = 10,000. As the sample size in-
creases, the null will be rejected more often for any approximate solution method.
However, even for this large sample size, there is little evidence for inaccuracy of
the second-order approximation.
A more comprehensive way of illustrating the test results is the percentile plot
in Figure 1. This is a plot of the analytical χ2 - c.d.f . against the simulated
distribution function of the test statistic. Up to sampling variability, if the test
statistic follows the χ2 distribution, the points should lie on the diagonal, which
is plotted as a reference. For the second-order approximation, this is roughly the
case. In contrast, the test statistic from the ﬁrst order approximation is clearly not
compatible with the χ2 distribution.
However, the relationship between orthogonal prediction errors and deviation
of the approximate policy function from the unknown exact policy function has
not yet been sharply characterized. Therefore it is instructive to consider Euler
equation residuals, which is done in the next subsection.
3.2 Euler equation residuals
Let xt+1 = hs(xt) denote the transition function for the state variables obtained
under solution method s. For notational simplicity, the dependence of this function
on σ and t+1 is suppressed. The residual arising from the Euler equation is
Rs(xt) = 1 −

βEt

c(hs(xt))
−γ 
αAt+1k(xt)α−1 + 1 − δ
		− 1
γ
c(xt)
. (11)
30.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
analytical c.d.f.
s
i
m
u
l
a
t
e
d
 
c
.
d
.
f
.
1st, g = 0.5
2nd, g = 0.5
1st, g = 3
2nd, g = 3
Figure 1: Percentile plot for T = 10,000
The Euler equation residual expresses the error from following the approximated
policy rule as a fraction of current period consumption. Clearly, for an exact so-
lution, the error is zero. Under certain conditions the approximation error of the
policy function is of the same order of magnitude as the Euler equation residual as
pointed out by Santos (2000).
Figures 2 and 3 plot the Euler equation residual as a function of the percentage
deviation of capital and technology from the steady state for the calibration of the
previous subsection with γ = 0.5. The Euler equation residuals for the linear solu-
tion method are of the order of magnitude of 10−3, an error of $ 1 for every 1,000
dollars spent. However the error does not have mean zero, the residual function is
positive everywhere on its domain.
The second-order approximation yields residuals which are of an order of mag-
nitude smaller than those of the linear approximation. It also seems to center the
residuals better around zero.
4 Welfare evaluation
Given the superior accuracy of the second-order approximation, it remains to be
demonstrated in what way this matters for the economic analysis. This section
shows that the effect of variances on means cannot be captured by a ﬁrst order
approximation and suggests a simple way to do undertake welfare ranking of alter-
native policies.
4−0.2
−0.1
0
0.1
0.2
−0.2
−0.1
0
0.1
0.2
0.5
1
1.5
2
2.5
3
x 10
−3
deviation technology deviation capital
E
u
l
e
r
 
r
e
s
i
d
u
a
l
Figure 2: Euler equation error based on linear approximation
4.1 Unconditional welfare
A natural welfare measure for rankings of ﬁscal or monetary policy that can be
easily constructed from the second-order approximation is the unconditional ex-
pectation of period utility.
The second-order approximation to an arbitrary utility function u(yt) of co-
states yt is
u(yt) ≈ u(y) + ∇u(y)yt +
1
2
vec
 
∇2u(y)
0 (yt ⊗ yt), (12)
such that upon taking expectations
E(u(yt)) ≈ u(y) + ∇u(y)µy +
1
2
vec
 
∇2u(y)
0 vec(Σy + µyµ0
y). (13)
Here, µy,Σy denote unconditional mean and covariance matrix of y, respectively.
To construct ﬁrst and second moments of the co-state variables assume covariance
stationarity and take expectation of (5) and (6)
µy =Gµx +
1
2
G∗vec(Σx + µxµ0
x) +
1
2
gσ2, (14)
µx =Hµx +
1
2
H∗vec(Σx + µxµ0
x) +
1
2
hσ2. (15)
Note that while under the linear approximation unconditional means do not differ
from the steady state values, the second-order approximation is able to capture
5−0.2
−0.1
0
0.1
0.2
−0.2
−0.1
0
0.1
0.2
−3
−2
−1
0
1
2
3
x 10
−4
deviation technology deviation capital
E
u
l
e
r
 
r
e
s
i
d
u
a
l
Figure 3: Euler equation error based on quadratic approximation
the effect of variances on means. Since variances can be computed accurately
up to second-order from the linear part of the policy function, it is sufﬁcient to
approximate vec(Σx + µxµ0
x) ≈ vec(Σx) and vec(Σy + µyµ0
y) ≈ vec(Σy). It
is then possible to construct these using the simple formulas
vec(Σy) =(G ⊗ G)vec(Σx), (16)
vec(Σx) =σ2(I − H ⊗ H)−1(N ⊗ N)vec(I). (17)
Given these approximations for the variances, the means can be computed from
(14) and (15). The described welfare measure has the following compact represen-
tation, which can easily be veriﬁed by applying the rules of the partitioned inverse.
E(u(yt)) ≈ u(y) +

∇u(y),
1
2
vec
 
∇2u(y)
0

×
 
G 1
2G∗
0 G ⊗ G

I −

H 1
2H∗
0 H ⊗ H
−1  1
2h
N ⊗ Nvec(I)

σ2 +
1
2g
0

σ2
!
(18)
The task of computing optimal monetary or ﬁscal policy in dynamic general equi-
libriummodelsthenamountstonumericallyoptimizingthiswelfaremeasurethrough
choice of the coefﬁcients in the policy rules.
If the second-order approximation to the welfare function can be re-written so
as to involve quadratic terms only, then linear and quadratic approximations to the
policy functions employed as suggested here will yield the same level of welfare.
6This is the case for some models of optimal monetary policy, where welfare is
equal to the weighted sum of the variances of inﬂation and the output gap. Up to
second-order, there is no bias in welfare calculations based on linear policy rules
in such a case.
4.2 Conditional welfare and transitional dynamics
The welfare measure employed so far is unconditional expectation of period utility.
However this measure neglects the effects of transitional dynamics. The following
paragraphs present a straightforward extension to the work of Kim et al. (2003)
who develop a formula for conditional welfare. These authors assume that utility
depends on state variables directly. Here, their approach is adapted to the case
where utility depends on co-states and the second-order approximation is obtained
using the code of Schmitt-Groh´ e and Uribe (2004).
In line with the approach used so far of evaluating second moments from the
linear part of the policy function only, append to the system recursions for the
variances of the states and co-states computed from the linear part of the policy
function only
(˜ yt ⊗ ˜ yt) =(G ⊗ G)(˜ xt ⊗ ˜ xt), (19)
(˜ xt ⊗ ˜ xt) =(H ⊗ H)(˜ xt−1 ⊗ ˜ xt−1) + σ2(N ⊗ N)(t ⊗ t). (20)
Rewrite the system in state space form using the above equations

yt
˜ yt ⊗ ˜ yt

=M1

xt
˜ xt ⊗ ˜ xt

+ K1, (21)

xt+1
˜ xt+1 ⊗ ˜ xt+1

=M2

xt
˜ xt ⊗ ˜ xt

+ K2 + ut+1. (22)
where
M1 =

G 1
2G∗
0 G ⊗ G

, M2 =

H 1
2H∗
0 H ⊗ H

,
K1 =
1
2gσ2
0

, K2 =
 1
2hσ2
σ2(N ⊗ N)vec(I)

,
ut =

σNt
σ2 (N ⊗ N (t ⊗ t − vec(I)))

.
Expected discounted lifetime utility conditional on an initial state vector with mean
7µx and covariance matrix Σx can then be expressed as
U(µ,Σ) ≈
u(y)
1 − β
+ E
"
∞ X
t=0
βt

∇u(y)yt +
1
2
vec(∇2u(y))0 (˜ yt ⊗ ˜ yt)
#
=
u(y)
1 − β
+

∇u(y),
1
2
vec
 
∇2u(y)
0

×

M1 [I − βM2]
−1

µ
vec(Σ + µµ0)

+
β
1 − β
K2

+
β
1 − β
K1

(23)
The difference to the case considered by Kim et al. (2003) manifests itself in the
matrices M1 and K1.
5 Conclusion
This note has shown that the second-order approximation yields Euler equation
residuals for the simple neoclassical growth model that are of an order of magni-
tude smaller than the residuals from the linear approximation. The den Haan and
Marcet (1994) test conﬁrms the superior accuracy of this solution method. For the
purpose of second-order accurate welfare rankings of policy, a simple way to com-
pute unconditional expectation of period utility is suggested. Such a measure is
essential, whenever the welfare function cannot be re-written in terms of quadratic
terms only.
Acknowledgements
I would like to thank Michael Massmann for helpful comments and the Deutsche
Forschungsgemeinschaft for ﬁnancial support.
References
denHaan, W.andMarcet, A., 1994. Accuracyinsimulations. ReviewofEconomic
Studies 61, 3–17.
Kim, J., Kim, S., Schaumburg, E., and Sims, C. A., 2003. Calculating and using
second order accurate solutions of discrete time dynamic equilibrium models.
manuscript, August 3, 2003.
Kim, J. and Kim, S. H., 2003. Spurious welfare reversals in international business
cycle. Journal of International Economics 60, 471–500.
Santos, M., 2000. Accuracy of numerical solutions using the euler equation resid-
uals. Econometrica 68, 1377–1402.
Schmitt-Groh´ e, U. and Uribe, M., 2004. Solving dynamic general equilibrium
models using a second-order approximation to the policy function. Journal of
Economic Dynamics and Control 28, 755–775.
8

Accuracy of numerical solutions using the euler equation residuals.

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Gains from second-order approximations

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Gains from second-order approximations

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