This note explores the persistence properties of a class of models proposed by Jones, Manuelli and Siu (2000) where growth stems from purposeful human capital accumulation. In doing so, we adopt Cogley and Nason's (1995) definition of output persistence. The propagation mechanism exhibited by this class of models appears unable to solve the output persistence puzzle.endogenous growth

Fabien Tripier

Tristan-Pierre Maury

English

Research Papers in Economics

Output persistence in human capital−based growth models 
Tristan−Pierre Maury Fabien Tripier
MODEM, University of Paris X − Nanterre MODEM, University of Paris X − Nanterre
Abstract
This note explores the persistence properties of a class of models proposed by Jones,
Manuelli and Siu (2000) where growth stems from purposeful human capital accumulation.
In doing so, we adopt Cogley and Nason's (1995) definition of output persistence. The
propagation mechanism exhibited by this class of models appears unable to solve the output
persistence puzzle.
We thank Julien Matheron for valuable comments. Any remaining errors are our own.
Citation: Maury, Tristan−Pierre and Fabien Tripier, (2003) "Output persistence in human capital−based growth models."
Economics Bulletin, Vol. 5, No. 11 pp. 1−8
Submitted: November 6, 2002.  Accepted: May 22, 2003.
URL: http://www.economicsbulletin.com/2003/volume5/EB−02E30010A.pdf1 Introduction
In their famous contribution Cogley and Nason (1995) (CN hereafter) brought to
light the inability of standard Real Business Cycle (RBC) models to reproduce
some crucial dynamic properties of output. Using several econometric methods
CN have deﬁned a set of properties that characterize the persistence of output
in postwar US data. Confronted with this set of properties, RBC models were
shown to inherently lack propagation properties. These puzzling output dynamics
have fostered a growing literature devoted to augmenting standard RBC models
with powerful persistence mechanisms capable of overcoming CN’s critique. Among
others, Jones, Manuelli and Siu (2000) (JMS hereafter) have put forward human-
capital-based endogenous growth as a way of generating strong persistence of output.
The aim of this note is to clarify the output persistence properties associated with
this mechanism1 within the empirical setup of CN. In doing so, we emphasize the
importance of preserving the whole of CN’s empirical setup.
The gap between JMS’s results and CN’s analysis lies in two points. First, both
the theoretical and empirical analyses considered by JMS are conducted at annual
frequency, in contrast with the standard practice in the RBC literature -and in CN’s
analysis- of considering quarterly data. Second, JMS consider only the ﬁrst order
autocorrelation of output growth as their measure of persistence. Though crucial,
this statistic does not encompass the whole set of properties highlighted by CN.
Thus, to clarify the persistence properties of the human capital accumulation
channel, we ﬁrst propose to study JMS’s model both at annual and at quarterly
frequency. Second, following CN2, we study the whole autocorrelation function
(ACF hereafter) as well as the spectrum of output growth.
We obtain the following results. At annual and quarterly frequencies, the model
succeeds in generating the ﬁrst positive value of the ACF, but fails to reproduce the
consecutive negative values, and, consequently, is unable to account for the well-
known peak of the spectrum at the business cycle frequencies. These results are
shown to be robust to alternative values of the intertemporal elasticity of substi-
tution. Hence, despite their strong propagation mechanism, human-capital-based
growth models do not solve the output persistence puzzle as deﬁned by CN.
The remainder is as follows. The model is described in section 2. Results are
presented in section 3. Section 4 brieﬂy concludes.
1We do not discuss the other advantages of this mechanism. These, such as the ampliﬁcation
of the volatility of total hours worked, are thoroughly described by JMS.
2We exclude from our analysis their results drawn from structural vector autoregressions. In-
deed, since the model fails on the overall behavior of output growth, it would be useless to adress
the issue of the sources of ﬂuctuations.
12T h e M o d e l
JMS propose a class of endogenous growth models featuring investment in both
human and physical capital. The model considered in this paper is their benchmark.
The social planner’s program writes
max E0
(
∞ X
t=0
β
tu(ct,n t)
)
(1)
subject to
ct + xh,t + xk,t = yt (2)
kt+1 =( 1− δ)kt + xk,t (3)
ht+1 =( 1− δ)ht + xh,t (4)
yt = stAk
α
t (ntht)
1−α (5)
where ct is consumption, nt is labor supply, xk,t is investment in physical capital, xh,t
is investment in human capital, yt is output, kt is the stock of physical capital, ht
is the stock of human capital, and st is the technological shock. We impose an equal
depreciation rate δ for both human and physical capital stocks. This assumption
corresponds to the benchmark model studied by JMS. A is a scale parameter, α
is the elasticity of output with respect to physical capital, and β is the subjective
discount factor. The shock evolves according to
log(st+1)=ρlog(st)+εt+1, εt ∼ iid(0,σε), (6)
where ρ is the persistence coeﬃcient. Finally, the instantaneous utility function
u(ct,n t) is speciﬁed as follows
u(ct,n t)=



1
1 − σ
[ct (1 − nt)
ψ]1−σ with σ > 0a n dσ 6=1
log(ct)+ψlog(1 − nt)w i t h σ =1
(7)
Once the ﬁrst order conditions are derived, the model is loglinearized in the
neighborhood of its deterministic steady-state. To simulate the model, we consider
two alternative calibrations which diﬀe ra c c o r d i n gt ot h es a m p l i n gf r e q u e n c y( a n n u a l
or quarterly). The calibration constraints are taken from JMS, except for the steady
growth rate which is taken from our data sample (described below). We impose
g =1 .02101/T, β =0 .9501/T, δ =1− (1 − 0.075)
1/T, ρ =0 .9501/T, α =0 .360,
n =0 .170, and σ =1 ,w i t hT = 1 for the annual calibration and T =4f o rt h e
quarterly calibration. The implied values for ψ are 8.707 for T =1a n d9 .028 for
T =4 .
23 Results
Our data set covers the period 1965-2001 for the U.S. Economy. Following CN,
we estimate a bivariate VAR of output growth and the consumption-output ratio.
The VAR is estimated twice, depending on the selected sampling frequency. The
optimal lag is chosen by minimization of the AIC criterion and is equal to 1 for
T =1a n d4f o rT = 4. In both cases, the associated ACF and spectrum of output
growth are computed from the estimated VAR coeﬃcients. Finally, the variance
of the theoretical shock σ2
ε is calibrated so as to reproduce the variance of output
growth implied by the VAR.
We begin our results description with the annual periodicity (which is the one
considered by JMS). Figure 1 reports the theoretical and empirical ACFs of output
growth. To begin with, output growth is positively autocorrelated at ﬁrst order in
the data. This property is also present in the theoretical model, and constitutes
the deﬁnition of output persistence retained by JMS. Thus, the model succeeds in
reproducing the positive ﬁrst point of the ACF of output growth. However, when
we look at higher orders, the theoretical ACF remains positive while its empiri-
cal counterpart becomes negative. This result comes from the endogenous growth
mechanism which induces permanent eﬀects of transitory shocks. Following a posi-
tive productivity shock, output monotonically rises until reaching its new long-run
level. Hence, the deviation of output growth with respect to its average level always
remains positive after the shock, thus explaining the ACF pattern. On the contrary,
as documented notably by CN, a hump-shaped response of output would have gen-
erated an ACF similar to that found in the data3.W en o wp r o p o s et oi n v e s t i g a t e
the implication of the discrepancy on ACFs in the frequency domain.
Figure 2 reports the theoretical and empirical spectra of output growth. The
vertical lines insulate the business cycle frequency band (i.e. movements whose
period of reproduction lies between 2 and 8 years). The empirical spectrum exhibits
a peak near business cycle frequencies. This pattern has been put forward by CN
(with quarterly data) as an element of the deﬁnition of output persistence. This
pattern is completely absent in the model. Indeed, the theoretical spectrum is
concentrated at very low frequencies, and underestimates the volatility of output
growth within the business cycle frequency band.
JMS show that the degree of intertemporal substitution of consumption is a
major determinant of the second moment properties implied by their model. In par-
ticular, the parameter σ deeply inﬂuences the ﬁrst point of the output growth ACF.
Hence, as the authors, we perform a sensitivity analysis by taking two alternative
values for σ = {0.9,3.0}. For each calibration, the variance of the shock is adjusted
to reproduce output growth volatility. Figures 3 and 4 depict the associated ACFs
3Notice that this property does not depend on the presence of permanent eﬀects of shocks.
3and spectra of output growth. As JMS, we conclude that the higher σ,t h el o w e r
the ﬁrst points of the ACF of output growth. However, the global pattern remains
the same: the ACF is always strictly positive and, consequently, the spectrum does
not exhibit the required peak at business cycle frequencies.
Finally, we propose to redo the previous analysis at quarterly frequency. As one
might argue, following CN, it seems more convenient to take a quarterly data set to
study the question of output growth persistence. Our results for T = 4, are reported
in ﬁgures 5 and 6. The empirical pattern of output growth dynamics is consistent
with CN estimates. The ACF of output growth is strictly positive for the ﬁrst four
points and negative beyond. The spectrum of output growth exhibits the well-known
peak at business cycle frequencies. On the theoretical side, the predictions of the
quarterly version of the model are very close to those of the annual version. The
model is unable to reproduce the negative values of the ACF for high orders. Hence,
it delivers a counterfactual spectrum pattern where the main part of output growth
volatility is concentrated at very low frequencies.
4C o n c l u s i o n
JMS showed that human-capital-based endogenous growth models outperform stan-
dard RBC models on several dimensions, especially when it comes to their ability
to reproduce the autocorrelation of output growth. Our objective was to gauge the
propagation properties of their models, when considered from the viewpoint of CN’s
analysis.
We obtained disappointing conclusions: despite its strong propagation mecha-
nism, JMS’s model does not satisfactorily reproduce the spectrum and the ACF of
output growth. This fact should not hide that JMS’s benchmark mechanism con-
stitutes a strong improvement over standard exogenous growth models. However, it
might prove useful to devote further research into the study of alternative speciﬁ-
c a t i o n so fg r o w t hw i t h i nt h i sc l a s so fm o d e l s .A tt h i sp r o s p e c t ,t h ea i mo ft h i sn o t e
was to emphasize the importance of preserving the whole CN’s empirical setup.
References
Cogley, T., and J.M. Nason (1995) ”Output dynamics in Real-Business-Cycle mod-
els” American Economic Review 85, 492-511.
Jones, L.E., Manuelli, R.E., and H.E. Siu (2000) ”Growth and business cycles”
NBER Working Paper 7633.
41 2 3 4 5 6
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Figure 1: ACF of annual output growth in the model (line with circle) and in the data
(line with star).
0 0.7854 3.1416
0
0.2
0.4
0.6
0.8
1
x 10
-3
Figure 2: Spectrum of annual output growth in the model (solid line) and in the data
(dotted line).
51 2 3 4 5 6
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Figure 3: ACF of annual output growth in the model with σ =0 .9 (line with circle),
with σ =1(line with triangle), with σ =3(line with square) and in the data (solid line).
0 0.7854 3.1416
0
0.2
0.4
0.6
0.8
1
x 10
-3
Figure 4: Spectrum of annual output growth in the model with σ =0 .9 (dotted line),
with σ =1(dashed line), with σ =3(dash-dotted line) and in the data (solid line).
61 2 3 4 5 6 7 8 9 10 11
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Figure 5: AFC of quarterly output growth in the model (line with circle) and in the data
(line with star).
0 0.1963 0.7854 3.1416
0
0.5
1
1.5
x 10
-4
Figure 6: Spectrum of quarterly output growth in the model (solid line) and in the data
(dotted line).
7

Growth and business cycles”

Output dynamics in Real-Business-Cycle models”

Output persistence in human capital-based growth models

http://www.accessecon.com/pubs/EB/2003/Volume5/EB-02E30010A.pdf

Output persistence in human capital-based growth models

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