Spectral and Persistence Properties of Cyclical Growth


Spectral and Persistence Properties of Cyclical Growth
Fabrice Collard CNRS{CEPREMAP First draft { February 1994 This draft { October 1997
Abstract

This article investigates the cyclical properties of an endogenous growth model. An explicit solution is derived which permits the full characterization of the cyclical as well as the long{run properties of the model. Using spectral, autocorrelation and impulse response functions, we show that introducing endogenous growth overcomes some of the main shortcomings of RBC models. In particular, the model can generate positive autocorrelation of output growth when endogenous growth dominates the dynamics. Moreover, the model exhibits the hump{shaped pattern of impulse response functions of the trend{reverting component of output.
Keywords:

Endogenous growth, Business Cycle, Persistence, Spectral Analysis C32, E32, O41

JEL Classi cation:

CEPREMAP, 142 rue du Chevaleret, 75013 Paris. E-Mail: fcollard@univ-paris1.fr. I am grateful to two anonymous referees for their comments. I have also bene ted from discussion with P. Feve, J.O. Hairault and J.F. Jacques. I am also indebted to P.Y. Henin and T. Kollintzas and especially H. Dellas for numerous remarks. Obviously, I remain responsible for all mistakes and insu ciencies. Part of this research was undertaken while I was at MAD{Universite de Paris I and at IRES{Universite catholique de Louvain.

1

1 Introduction
Since the seminal papers by Kydland and Prescott (1982) and Long and Plosser (1983), Real Business Cycle (RBC) models have been used extensively to study cyclical behavior. While these models have proved quite successful in explaining many features of the business cycle, they have been unable to account for some important features of output dynamics. In particular, they cannot reproduce the shape of the spectrum of output that is observed in the real world. More precisely, they cannot mimic low frequencies patterns or generate a peak at business cycle frequencies. Cogley and Nason (1995) note that the RBC models fail to reproduce two important stylized facts of output dynamics in the United States, namely that: (i) \GNP growth is positively autocorrelated over short horizons and has weak and possibly insigni cant negative autocorrelation over longer horizons", (ii) \GNP appears to have an important trend{reverting component that has a hump-shaped impulse response function." (Cogley and Nason (1995), p.492). Moreover, Cogley and Nason argue that RBC models have weak internal propagation mechanisms and as a result they cannot generate a satisfactory pattern of output dynamics unless they rely heavily on exogenous sources of dynamics. The purpose of this paper is to address these weaknesses of the RBC models. This is done within the context of a model that encompasses both cycles and endogenous growth. The source of growth is human capital accumulation which results from learning by doing on the job (the model is based on Stadler (1990)) and \labor quality" (through aggregate productivity). Bringing in human capital accumulation endows the standard RBC model with an additional propagation mechanism. Allowing for endogenous growth creates an interaction of macroeconomic uctuations with growth, making the dynamics of output richer. In order to provide greater insights we follow Hercowitz and Sampson (1991) in using log{linear speci cations for preferences and technology. This allows us to describe analytically the process of output growth. The paper then proceeds as follows. Section 2 describes the model. It focuses on the properties of human capital accumulation process, and aims at showing how our model 2

can account for both endogenous and exogenous growth. We then derive an analytical solution to the problem and show how endogenous growth is related to unit roots. We nally discuss the cointegration and common trend properties of the model and show how endogenous growth can a ect standard procedures to identify demand and supply shocks. Section 3 assesses the ability of the model to reproduce the business cycle features of output, with a focus on how endogenous growth may help solve the Cogley{Nason puzzle. Based on the spectral properties of output growth, we show that the endogenous growth hypothesis can improve the ability of the model to mimic low frequencies properties of output dynamics. We further nd that the model can generate a peak in the spectrum. We then study the autocorrelation function of output growth and impulse response functions for the trend{reverting component in output and show that the model is consistent with the Cogley{Nason (1995) observations. The last section o ers some conclusions.

2 The stochastic endogenous growth model
The model combines aspects of endogenous growth and Real Business Cycle (RBC) models. Unlike the standard RBC model, however, the production function depends on both physical and human capital. The production function exhibits constant returns to factors that can be accumulated. When the laws of motion of human and physical capital also exhibit constant returns to factors that can be accumulated, the model displays endogenous growth. Human capital is accumulated via learning by doing which is assumed to be a function of the level of economic activity as well as the level of labor productivity (labor quality). The business cycle has two e ects on human capital accumulation. First, a low level of economic activity implies a lower rate of learning which has a negative e ect on long term growth. At the same time, labor productivity is higher during bad aggregate times, which has a positive e ect on human capital accumulation and growth (the latter factor has been emphasized in the literature on the \virtue of bad times"). Following Hercowitz and Sampson (1991) we use a log{linear speci cation of the model in order to obtain a closed{form solution. 3

2.1 The model
The economy is populated by many, identical, in nitely-lived individuals and rms acting in a competitive environment. The representative agent maximizes expected utility:

U = E0
(0) = 0;

+1 X
t=0

t

u(Ct; Ht?1 (Nt); t) 0 < < 1
00(:) > 0

(1)

0 (:) > 0;

where is the discount factor. E0 is the mathematical conditional expectation operator1 . Ct denotes consumption good, Nt is work e ort and Ht?1 is the end-of-period human capital. As previously noted by Hercowitz and Sampson (1991), the convexity of function (:) alters the utility of leisure2. Further, human capital alters the decisions of the household concerning leisure: the higher the level of human capital, the less willing the household to supply labor. Besides its economic dimension, this hypothesis is necessary from a technical point of view. Since Ct grows and consumption and leisure enter the utility function in a non-separable fashion, leisure also has to grow to render utility compatible with the existence of a balanced growth path. As in Baxter and King (1991), we introduce a taste shock, t, as a means of incorporating demand shocks3 . t4 a ects the marginal rate of substitution between leisure and consumption, so that an individual supplies more hours after a positive taste shock. ut takes the following form: log( tCt ? Ht?1Nt1+ ) where > 0. This hypothesis ensures the stationarity of work e ort as long as the real wage rate and the human capital share the same stochastic trend, which is guaranteed in this model as this will become clear later. The peculiar form we adopt for preferences implies that the marginal rate of substitution between consumption and labor supply will not depend on consumption in the RBC version of this model. Hence, labor supply is entirely determined by the price
e ect, since it alters labor supply behavior. 4We will come back to the stochastic properties of t .
1Et (Xt+i ) = E (Xt+i jIt) where It = fKt?1; Ht?1; At ; tg as it will become clear later. 2Hercowitz and Sampson (1991) read this convexity as a fatigue e ect 3Nevertheless, even if this shock relies on a household's behavior it transits only through supply side

4

ratio and does not depend on income5. Output is produced by means of a Cobb-Douglas production function:

Yt = AY AtKt?1 (NtHt?1 )1?

0< <1

(2)

where Kt?1 denotes the end-of-period physical capital stock, AY is a scaling factor and At is a technological shock. The production function exhibits constant returns to the factors that can be accumulated: physical and human capital. This latter hypothesis is essential in order to generate endogenous growth6. The physical capital accumulates according to: ! It Kt = AK Kt?1 K t?1 0(:) > 0; 00(:) < 0 This law of motion is based on Lucas and Prescott (1971) and Hercowitz and Sampson (1991). It di ers from the traditional equation Kt = (1 ? )Kt?1 + It in that it exhibits decreasing returns to investment. This can be interpreted as re ecting the presence of adjustment costs in physical capital accumulation. The choice of this speci cation also allows for incomplete capital depreciation assumption7 and a closed{form solution at the same time. 1? t Further, when we set: KIt?1 = KItt?1 , where 0 < < 1 then:

Kt = AK Kt?1 It1?

(3)

As before, AK is a scale parameter. Physical capital is the geometric mean between old installed capital stock and new investment. Unlike Hercowitz and Sampson (1991) who obtained endogenous growth by relying on an externality a la Romer (1986): Ht = Kt where Kt is aggregate physical capital, we allow for a more general accumulation for human capital:
5Just note that U` (:) = (1+ )Ht?1 Nt = wt. Uc (:) 6This will be demonstrated in tsection 3. 7An alternative hypothesis, imposing complete depreciation, was adopted in earlier works, such as

Long and Plosser (1983) and McCallum (1989), who also got an analytical solution.

5

1 0 ! Yt (N H )(1? ? )A ? (4) Ht = AH Ht1?1 @ N t t?1 t where 0 1, 0 < < 1 and 0 (1 ? ). Ht is homogenous of degree one with respect to (Yt; Ht?1 ) when = 0. Then, since Yt displays constant returns to (Kt?1; Ht?1), the law of motion of human capital also exhibits constant returns to factors that are accumulated. Subsequently, the model can generate endogenous growth as long as 0 < 1. When > 0; growth is exogenous (See section 2.3). t Denoting gt = HH?1 , (4) can be written: t 0 Y 1 (1? ? ) t (NtHt?1) C gt = AH B Nt @ A H
t?1

Setting = 0, we obtain:

gt = AH

As soon as AH is set to the average growth rate of technology, the model reduces to a (non{standard) RBC model with exogenous growth. Then can be interpreted as the relative weight of endogenous growth in the growth process. As soon as > 0 (and = 0) the model displays endogenous growth, and the weight of human capital investment increases with . Contrary to Rebelo (1991) and Lucas (1988), human capital accumulation is not produced in a separate sector in the economy. It is simply the by{product of work activities. The term in NtHt?1 re ects the learning{by{doing mechanism. Its impact on knowledge accumulation is positive: The higher the level of employment, the higher the rate of knowledge increase. Unlike the standard approach (see Arrow (1962)), this e ect is not treated as an externality and as a result it directly a ects labor supply as the household realizes that o ering an additional unit of work will increase its future human capital. Another mechanism linking economic activity to human capital accumulation often appears in the literature. It relies on the so{called \opportunity cost" approach (See e.g. Saint-Paul (1992)). For instance, Dellas (1992), Gali and Hammour (1991), SaintPaul (1992) and Aghion and Saint-Paul (1991) assume that recessions are a good time for undertaking activities that enhance long term productivity (for instance, undertaking 6

Y schooling). In this model, this e ect is captured by the term ntt , which is a measure of aggregate productivity per worker. Note that this e ect is external to individual workers. During bad times, the reduction in employment allows the economy to maintain a high level of average productivity so that human capital keeps on growing. As Gali and Hammour (1991) note:

\Recessions have a \cleaning-up" e ect that causes less productive jobs to be closed down. This can happen either because those jobs become unpro table, or because recessions provide an excuse for rms to close them down in the context of formal or informal worker- rms arrangements. As a consequence, the average productivity of jobs will rise." (Gali and Hammour (1991), p. 15) As table (1) shows the net impact of each e ect on human capital accumulation relies on parameter . Table 1: Net impact of hours worked at the symmetric equilibrium =
1? < 1+
@Ht+1 @nt @Ht+1 @nt @Ht+1 @nt

>

1? 1+ 1? 1+

> 0 domination of learning by doing
=0 <0 neutrality of hours worked domination of rationalizing

We nally describe the properties of the exogenous stochastic disturbances. The vector of shocks, expressed in logarithm, t = (at; !t)0 is assumed to be a stationary VAR process:

D(L) t = "t

(5)

where E ("t) = 0 and E ("t"0t) = . D(L) is a polynomial matrix of degree p and dimension (2,2) so that D(0)=I2. This implies that the stochastic process for t is canonical.

2.2 The Solution to the model
The representative agent's problem is to maximize(1) subject to (3), (4), (2) and the goods market clearing condition. This is equivalent to solving the problem of a central planner, Y Y with the supplementary condition that Ntt = Ntt (See e.g. King, Plosser and Rebelo (1988) 7

p.327-335 for more). The problem can be stated recursively and the solution satis es the following Bellman equation: n o V (Kt?1; Ht?1; t) = maxt log( tCt ? Ht Nt1+ ) + Et V (Kt; Ht; t+1)] Ct ;N

8 > Kt = AK Kt?1 AY AtKt?1(NtHt?1)1? ? Ct 1? > < s:t: > > Ht = AH H 1? N (1? ? ) Yt : t t?1 Nt The optimality conditions associated with this problem are given by the rst order conditions, the envelope conditions, the laws of motion of both capital stocks, and the transversality conditions8. " #
@ut @Ct =
0 Et (1 ? )VK (t + 1) Kt I

@ut @Nt =
0 VK (t) = 0 VH (t) =

Et (1 ? )(1 ?
0 Et VK (t + 1)

"

t

"

0 )VK (t + 1) Kt Yt + (1 ?

It Nt

Et (1 ? )(1 ? @u ? @H t
t?1

"

Kt Kt Yt Kt?1 + (1 ? ) It Kt?1
0 )VK (t + 1) Kt HYt + (1 ? It t?1

#

?

0 )VH (t + 1) Ht

#

(6)

Nt

(7) (8)

?

H 0 )VH (t + 1) H t t?1

#

(9)

With the transversality conditions: lim E t!+1 0 lim E t!+1 0

h h

t 0 ( K

V t + 1)Kt = 0 V t + 1)Ht = 0
i

i

(10) (11)

t 0 ( H

Equation (6) together with (8) mean that each additional unit of consumption increases the household's current utility by uc(t) but lowers its future utility by Et((1+ rt+1)uc(t + 1)). These equations thus describe the consumption{savings decision.
8Hereafter we will note Vx (t + 1) = @ V (Kt ;Ht; t+1 ) , where x 2 fK; H g 0 @x

8

Equation (7) describes the labor supply decision. It states that an additional unit of work increases an individual's future wealth by the rst term in the right hand side and his human capital by the second term. But it also lowers his utility by @u(tt) . @N Equation (9) says that increasing e ort in human capital formation leads to higher future wealth through both the productive process and the human capital formation process. But this is achieved at the price of an instantaneous loss in the marginal value of human capital and utility. Finally, equations (10) and (11) provide terminal conditions to the two types of capital. Since aggregate productivity is introduced as an externality in the law of motion of human capital, the individual treats the process of productivity as exogenous9 . The log{ linear structure of the model allows us to guess the following form for the value function (See e.g. Hercowitz and Sampson (1991) and also the Appendix): ! X Yt + p?1 D? V (Kt?1; Ht?1; t) = D0 + D1 log Kt?1 + D2 log Ht?1 + D3 log N j t?j
t j =0

p denotes the number of lags in the process of shocks. 0 0 Computing VK (t) and VH (t), and plugging the result in equations (6)-(9) and combining (7) and (9) we get: It = 1 (1 ? ) Yt = siYt (12) ?

Ct = (1 ? si)Yt

(13)

The model implies a perfect instantaneous investment{output and consumption{output correlation. This can be attributed to the physical capital accumulation which, through the envelope condition (8), constraints investment to be proportional to output. Y Y In equilibrium we have Ntt = Ntt . Combining (6) and (7) we obtain the following decision rule for labor10: 1 1 nt = bn + + kt?1 ? + ht?1 + + at + + !t (14)

9On a technical point of view, individuals take the policy rule of aggregate productivity as given by a function (Kt?1; Ht?1; At; t). 10Lower cases denote logarithms

9

Hence, contrary to models a la Long and Plosser (1983) with complete depreciation, the labor process can deviate from its steady state value. This result is due to the particular speci cation of the utility function. Income e ects are low with this type of utility function and cannot counter the substitution e ects. Using nt in the production function and the laws of motion, we obtain the following solutions:
? + yt = by + (1+ ) kt?1 + (1+ ) ht?1 + 1 + at + 1 ? !t + + )(1 )(1 kt = bk + 1 ? (1 ? + ? ) kt?1 + (1 ? + ? ) ht?1 + (1 + )(1 ? ) at + )(1 + (1 ? + ? ) !t
(15)

(16)

? ht = bh + (1 ? + + ) kt?1 + 1 ? (1 ? ? ++ ) + ( + ) ht?1 + (1 ? ? + ) at + (1 ? ? ? ) !t + +

(17)

The model also allows the computation of the dynamics of the Solow residual:

rest = log AY + (1 ? )ht?1 + at
or, substituting for ht in terms of the solution above:
rest = br +
(1 ? )(1 ? ? + ) k + 1 ? (1 ? ? + ) + ( + ) res t?2 t?1 + + ) +at ? 1 ? (1 ? + + + ( + ) at?1 + (1 ? )(1 ? ? ? ) !t?1 (18) +

Unlike the technological shock, the taste shock takes one period to operate. Its direct 1? impact is positive as long as 1+ |i.e. as long as the learning{by{doing e ects are strong. This is so because a unit taste shock leads the household to increase its labor supply. Since the learning by doing mechanism dominates the dynamics of total factor productivity, more hours worked implies a higher level of total productivity. We now turn to the evaluation if the long{run e ect of such shocks.

10

Using equations (16) and (17), we have11:
1 ? (1 ? k )L ? kL ? hL 1 ? (1 ? h ? )L

2.3 Unit root and endogenous growth
kt ht
=

bk + bh

ka ha

k! h!

at !t

(19)

Or in vector form:

A(L)Xt = C + B (L)"t

h? with detA(L) = (1 ? 1L)(1 ? 2 L), where ?1 = (2? ? 2(1?k ) ? h? ++k + kh)) ?4 k . i k When = 0, the model trivially reduces to an exogenous growth model 12 as shown in section (2.1). In this section, we restrict ourselves to situations in which > 0. From the computation of i it appears that if = 0, the system contains a unit root since in this case 1 = 1 and 2 = (1 ? h ? k )?1. The model thus generates endogenous growth since the unit root property implies that transitory shocks a ect the level of the long{run balanced growth path. As is well known, sustained growth in this model is the result of postulating constant returns with regard to the factors that are accumulated13. We will focus on this case in what follows. The associated stationary system, in rst di erence form, is then given by:

p(

(20)
2

(1 ? (1 ? k ? h )L)4Xt = A?(L)(C + B (L)"t)

(21)

where A?(L) is such that A?(L)A(L) = detA(L). Let us now consider the system in centered variables and examine its dynamic properties. Let Zt = Xt ? X , where X denotes the average of the process. X is given
)(1 (1+ )(1? ) (1? )(1? ) = (1? + ? ) ka = k! = + + (1? ? + ) (1? ? + ) (1? ? ? ) = ha = h! = + + + n xt = xt?n and L denotes the lag operator such that L 12It must be emphasized that the dynamics of physical capital are a ected by demand shocks in the short{run, but not in the long{run. 13This can be understood considering a simpler version of the model. Let output be given by Yt = Xt Kt 0 , 0 < 0 < 1 and Xt = AK 1 , 0 < 1 < 1 and 0 + 1 1. In equilibrium the interest rate is given by rt = 0AKt 0 + 1 ?1 . As long as 0 + 1 < 1, the interest rate will react to changes in capital, and will adjust in order to render capital accumulation compatible with an exogenously given rate of growth. But as soon as 0 + 1 = 1, the interest rate is constant, so that it is the rate of growth that will adjust. Growth is then endogenous.
k h

11where:

11

by:

X=

It is worth noting that this shows that the deterministic part of the model generates a balanced growth path, since the deterministic growth rate is the same for the two variables that are responsible for growth. The system becomes: (1 ? (1 ? k ? h)L) Zt = A?(L)B (L)"t and has the following MA(1) representation:
? L Zt = (1 ? A (? )B (L) )L) "t = H (L)"t (1 k ? h

h bK + h+ h bK + h+

k bH k k bH k

!

(22)

(23)

According to the MA(1) representation, the process can be rewritten:

?H Zt = H (1)"t + H (L1)? L (1) "t = H (1)"t + H ?(L) "t

(24)

If there exists 2 R2 such that 0H (1) = 0, then 0Zt = 0H ?(L)"t, i.e. there exist linear combinations of the I(1) variables that are stationary. Indeed, rank(H (1)) = 1, so there exists one cointegrating relationship in the model. Furthermore, it is straightforward to show that KerH (1)0 , the cointegrating space, is spanned by (1; ?1) so that in the long{ run kt ? ht is stationary. We then get the following proposition.
Proposition 1 Both human and physical capital share the same common trend in the

long{run. We then have the following Stock and Watson (1988) common trend representation: ( Zt = t + Ct t = t?1 + H (1)"t

This property is common to all endogenous growth models. Since the model satis es the balanced growth property, each non{stationary variable is cointegrated with Ht . So Yt, Ct and It are all cointegrated, with the cointegration vector being (1; ?1). It is noteworthy that if human and physical capital are cointegrated, so are physical capital and the Solow residual, with a cointegrating space spanned by ( ? 1; 1) so that in the long{run rest ? (1 ? )kt is a stationary variable. 12

Now, de ne vt = kt ? ht (the cointegrating relationship) and consider the model in the following VECM form: ! ? k vt?1 + B"t (25) Zt = This formulation means that after a shock, the growth rate goes back to its steady state level according to a feedback relation that is due to cointegration. It then appears that the model generates adjustment dynamics following a shock, independent of the shock persistence. Moreover, the cointegrating relationship shows that these shocks not only a ect the short{run dynamics of aggregates, but also their long{run level. The model thus contains an integration of growth and macroeconomic cycles. Moreover, there are additional important implications for modelling. The innovation of the trend component of the Stock and Watson (1988) representation is given by H (1)"t, where H (1) is: 0 1 h ka + k ha h k! + k h! ( k + h )(1? a)(1? ! ) ( k + h )(1? a)(1? ! ) C B B C @ A h k! + k h! h ka + k ha This means that when > 0 and = 0, each shock a ects the trend component. This implies that either shock may a ect the long{run dynamics of the model, and in particular, that demand shocks may alter the long{run growth path of output. Thus we cannot identify in such a setting demand shocks as disturbances that have no e ect on the long{run dynamics of output as Blanchard and Quah (1989) did. For empirical evidence on this relation see Gali and Hammour (1991), Saint-Paul (1992) and Evans (1989) who nd evidence supporting the thesis that demand shocks have long{run e ects. In an output{unemployment V.A.R., Evans identi es output shocks, corresponding to demand shocks in a Blanchard and Quah framework, assuming that output growth is causally prior to unemployment. He nds that the long{run e ect of a one unit output shock on output is 0.338 (with a standard deviation of 0.168). Thus there appears to exist a small but signi cant positive long{run e ect of demand shocks. These results create doubts on the validity of using the Blanchard and Quah (1989) approach to identify supply and demand shocks14.
14The approach is still valid for identifying trend versus cyclical components.
h

( k + h )(1? a)(1? ! ) ( k + h )(1? a)(1? ! )

13

3 Can Endogenous Growth Explain Output Dynamics ?
This section is devoted to the exposition of the main dynamic properties of the model. We investigate whether introducing endogenous growth allows the model to resolve the Cogley and Nason (1995) puzzle.

3.1 Calibration
The model is calibrated on U.S. quarterly data. In order to keep RBC model as a benchmark, most of parameters are close to the ones used in the RBC literature. , the discount factor, is set at 0.988 according to the standard value used in the RBC literature. , the inverse of elasticity of substitution of the labor supply, is set at 3.65. This value is in accordance with estimates by Hercowitz (1986) and lies within the range of estimates by McCurdy (1981). The labor share of output, 1 ? , is xed at a value of 0.58, which corresponds to the average value of the wage output ratio in the post World War II period. The parameter is set at a value of 0.98, which is consistent with Eckstein, Foulides and Kollintzas' 1994] estimates. Since we do not have any information on the parameters of the human capital process, we estimate and . Using the law of motion of human capital and using the fact that rest = log AY + (1 ? )ht?1 + at, we obtain:

rest = b0 + (1 ? )rest?1 + (1 ? )pt?1 + (1 ? )(1 ? )nt?1 + at ? (1 ? )at?1
where b0 is a constant and pt is the logarithm of average productivity. Letting ut = at ? (1 ? )at?1, the previous equation is rewritten:

rest = b0 + (1 ? )rest?1 + (1 ? )pt?1 + (1 ? )(1 ? )nt?1 + ut ut then denotes the residual. It must be emphasized that this equation will generate stationary residuals since rest and rest?1 share the same stochastic trend and (1 ? )pt?1 ? rest?1 is a cointegrating relationship. We then construct the technological shock as: at = (1 ? )at?1 + ut
14

Then the autoregressive process for at can be estimated15. Using the rst order conditions, the taste shock can be identi ed as:

Ht?1 nt t = B1 Y
t

1+

where B1 is a constant that relies on the model parameters. Ht?1 is obtained using the equation de ning the Solow residual. Then the rst order autoregressive process for !t can be estimated. Table 2: Estimated Parameters 0.119565 0.449751 0.950234 0.010438 0.939130 0.063789
(0.063961) (0.295788) (0.002129) Standard errors in parenthesis (0.000705) (0.007768) (0.004039)
a a ! !

The results obtained are displayed16 in table (2).

3.2 The Spectrum of Output
It is widely accepted that one of the main shortcomings of the RBC models is their inability to reproduce the shape of the spectrum of output. In particular, they can neither mimic the low frequency pattern nor generate a peak in the spectrum at business cycle frequencies. This section aims to show that an endogenous growth model can overcome both problems. As we showed previously, one of the key parameters is . As ! 0, the returns to scale with regard to the factors that are accumulated tend towards unity and the model tends towards an endogenous growth model. As long as the endogenous growth property is not satis ed, the theoretical spectrum17 is null at frequency zero, which does not agree with the empirical evidence18. If = 0, that is as soon as the model exhibits endogenous
independent. 16The data are taken from the Citibase dataset. Hours worked are taken from Hansen's study. Capital data were kindly provided by Tryphon Kollintzas. 17Technical details on the computation of the theoretical spectrum are laid out in the appendix. 18The historical spectrum was estimated by smoothing the periodogramm using a Bartlett window and a bandwidth parameter equal to 60.
15We assume that both technological shocks and taste shocks follow an AR(1) process and are

15

growth, the spectrum at frequency zero shifts and mimics in quite a good way the actual world. Figure (1) illustrates the results for = 0:119565 and = 0:449751. The solid line shows the spectrum of the actual series and the marked lines show the spectrum generated by the model using di erent values for . One of the key results of the analysis is that f y (0) exhibits a discontinuity with respect to as tends towards 0. This illustrates that endogenous growth is a \razor's edge" property. But as frequency increases, the theoretical spectra deviate from the empirical one. So, even if the endogenous property helps to solve the zero frequency puzzle, problems remain at higher frequencies for a \small" contribution of endogenous growth. More precisely, it appears that the model{generated spectrum is much higher at all frequencies than the actual spectrum. This problem is partly related to the scaling of shocks. As soon as we reduce the volatility of both technological shocks and taste shocks, the model{generated spectrum shifts downwards19 and mimics quite well the actual data spectrum for = 0:116595 and = 0. But as tends to 1, the model{ generated spectrum remains higher than the sample spectrum, even for low volatility. This is due to the fact that the propagation mechanisms are magni ed when increases. Indeed, as tends to 1, the rate of growth reacts more to shocks, thus providing a new channel of propagation that reinforces the volatility generated by the model. Figure 1: Low frequencies implications of endogenous growth

19The corresponding graphs are not reproduced here but are available from the author upon request.

16

Another nding emerges from the analysis20: technological shocks always dominate demand shocks in the behavior of the spectrum. This result reveals that even if demand shocks intervene in the dynamics of output, output is mainly driven by technological conditions. But, it is noteworthy that if we focus only on demand shocks, the spectrum at frequency zero is not null. This means that demand factors exert a long{run e ect on output, implying that they are a component of the common trend of the model. What do we gain in using an endogenous growth model, since a simple RBC model with a random walk in the technology shock may have generated the same property ? The answer to this question is found in gure (2). As we previously noticed, the RBC model generates a at spectrum, in the sense that it cannot display a peak at business cycle frequencies (this is the case for = 0). Figure 2: E ects of the size of endogenous growth

As increases, the model is able to generate a peak in the spectrum. Nevertheless, it must be noted that this peak does not appear at pure business cycle frequencies, but rather at much higher periodicity (40{60 quarters). Furthermore, one must recall that high persistence in exogenous shocks is important to obtain this peak. Cogley and Nason indicate that adjustment costs, capital or labor hoarding, or time{to{build may not be necessary for RBC models to mimic the spectral properties of output. This objective can
20Figures are not reported here, but are available from the author upon request.

17

be accomplished through mechanisms such as human capital formation or inter{sectorial propagation. It appears that taking into account endogenous growth mechanisms helps to reproduce the spectral properties of output, and in particular, its low frequencies properties (the model is less successful in reproducing the correct shape of the spectrum at business cycle frequencies). It also helps reproduce the autocorrelation function of output growth, and more generally, the short{run persistence properties of output.

3.3 Short-Run Persistence Analysis
This section21 deals with the issues raised by Cogley and Nason (1993) and Cogley and Nason (1995). In the former, the authors evaluate the ability of RBC models to pass the Nelson and Plosser (1982)'s tests, i.e. to reproduce: The unit root property. The autocorrelation function of output growth. By assumption, this model passes the rst Nelson-Plosser test, since endogenous growth implies the unit root property as the previous section showed. However, Cogley and Nason (1993) argued that: \This unit root] test is not very demanding. Any model which has persistent impulse dynamics will pass. ...] ADF tests have low power against near unit root alternatives, so they also fail to reject the trend stationary speci cations." (Cogley and Nason (1993), p. 11) The second Nelson and Plosser (1982) proposition is that output growth is positively serially correlated at lags of one and two quarters while its autocorrelation function is mostly negative and insigni cant at higher lags. This fact is not well reproduced by standard RBC models. In their 1995 paper, Cogley and Nason showed that even after introducing gestation lags or time to build, the RBC model fails to mimic the autocorrelation function of output growth. In this section we show that taking into account
21We restrict our analysis to the case = 0

18

endogenous growth improves the ability of the model to mimic the short{run persistence of output growth. Figure (3) displays the autocorrelation function of historical output growth versus that generated by the model. Solid lines show historical autocorrelations, and marks show the theoretical autocorrelation function for di erent values for . A striking fact appears in the left panel. As long as the weight of endogenous growth is null, the model behaves like a standard R.B.C model. It cannot display enough persistence, and output growth appears to be white noise. But as soon as endogenous growth is introduced, the autocorrelation becomes slightly positive. One must reach the value = 0:5 to get an autocorrelation function consistent with empirical estimates. When = 1, that is, when human capital accumulation only relies on investment in human capital, the autocorrelation function is very well reproduced in the sense that each theoretical autocorrelation lies within the con dence interval. Indeed, as increases, the relative weight of investment in human capital increases. This leads to higher persistence because each shock has persistent e ects arising through the law of motion of human capital. Figure 3: Output Growth Autocorrelation Function

It then appears that a model integrating growth and cycles can account for the two Nelson and Plosser stylized facts. But, it must be emphasized that this still requires persistence in the exogenous shock. If we set 0 = ! = 0, then the autocorrelation at one and two lags remains negative, even for = 1. This means that even if the growth{ cycles integration is needed to mimic the autocorrelation function, high persistence in the exogenous shocks remains necessary. 19

The right panel shows the autocorrelation function for = 1 and di erent values of . If we look at the entire autocorrelation function, we cannot discriminate between the two human capital accumulation mechanisms (learning by doing and average quality of labor) on the basis of the ability of this function to mimic the data. But, if we focus on autocorrelation at lag 1, which is perhaps the hardest to mimic, it appears that a pure learning{by{doing mechanism cannot account for short{run persistence even if is set to 1. But as soon as we introduce the \quality" of labor e ect the model behaves in a much better way. This is consistent with the empirical studies that have claimed that reallocation e ects can explain the output dynamics (Gali and Hammour (1991), SaintPaul (1992)). Once more, this relies on the fact that technological shocks fully come into play twice through aggregate productivity.

3.4 Impulse Response and Long-Run Persistence Analysis
There exists an important stylized fact concerning output. GNP has an important trend{ reverting component that exhibits a hump{shaped impulse response function (IRF). Cogley and Nason (1995) investigated whether di erent RBC models are consistent with that feature. Their analysis shows that RBC models have weak propagation mechanisms and cannot reproduce this feature. They also show that incorporating labor adjustment costs, or an AR(2) process for government spending, helps to reproduce the hump{shaped pattern. This section illustrates that endogenous growth may be a good way to circumvent this shortcoming of RBC models. We estimate a VAR(4) process for output growth and hours. We then obtain a trend{cycle decomposition using a Blanchard and Quah decomposition22 and nally compute the IRF of output to a shock on the trend{reverting component23. We also compute the corresponding model{generated IRF. Figure (4) reports the results for di erent values for . We assume that > 0 since the trend{cycle decomposition is irrelevant in the exogenous growth case ( = 0) as long as shocks are transitory. Figure (4) shows that the model can mimic the IRF of output to a shock in the
a trend{cycle decomposition. 23We provide a con dence interval obtained using a Monte{Carlo procedure (5000 replications).
22This decomposition is no longer valid for identifying demand shocks but remains valid for performing

20

Table 3: The VAR Process

yt 0.274 0.133 -0.050 -0.007 0.005 -0.054 0.011 0.018 0.112
(0.132) (0.139) (0.138) (0.128) Standard deviation in parenthesis.

yt?1

yt?2

yt?3

yt?4 nt?1

nt?2

nt?3

nt?4

R2

nt

0.523 0.457 0.219 0.182 0.468 0.001 0.106 0.345 0.838
(0.088) (0.098) (0.099) (0.088)

(0.097)

(0.102)

(0.101)

(0.094)

(0.065)

(0.072)

(0.073) (0.065)

Figure 4: Trend{reverting component in output

21

trend{reverting component independent of the value of used. It is nevertheless worth noting that must be strictly greater than zero otherwise the model would not have any permanent component. It appears that for su ciently high values of | that is when investment in human capital dominates the law of motion of human capital | the model is able to generate the hump{shaped pattern of the trend reverting component of output. Indeed, this e ect relies on two key points. First, as the autocorrelation analysis showed, the higher , the more persistent output growth. Second, persistence in the exogenous shock is essential. If we set a and ! to low values (below 0.8) the hump{shaped pattern also disappears. Thus hump{shaped pattern and persistence are fundamentally linked. Finally, we look at the long{run e ect of shocks on the system. This measure relies on the Campbell and Mankiw (1987) A(1): 1 0 (1? )(1? + ) (1? a)( (1? )(1? )+ (1? + )) C A(1)ya B A A(1)y! = @ (1? )(1? )(1? )
(1? ! )( (1? )(1? )+ (1? + ))

Figure (5) shows the di erent A(1) obtained for di erent pairs ( ; ). Figure 5: Long{run e ects of shocks

It appears that the more dominant the learning by doing e ect, the higher the long{ run impact of technology and taste shocks. With internalized learning by doing e ects the individual takes into account the impact of his labor supply behavior on his future wealth. For instance, a positive taste shock induces the individual to supply more labor. This generates future productivity gains that will be re ected in future output dynamics because of constant returns to capital stocks. When the labor quality e ect dominates, the impact is smaller because the individual does not internalize the impact of his labor 22

supply on future productivity. It is zero when human capital accumulation depends only on labor quality.24.

4 Concluding remarks
This paper attempts to characterize the dynamics of an endogenous cyclical growth model. Unlike Arrow (1962), Romer (1986), Saint-Paul (1992) or Aghion and Saint-Paul (1991) we do not a priori impose the theoretical sign of persistence25. The particular speci cation used for utility and technology allows one to analytically compute the dynamics of the economy. The model also helps shed light on the mechanisms of persistence in endogenous growth models. An important implication of the model is that we cannot identify supply and demand shocks on the basis of Blanchard and Quah's decomposition because each shock may alter the long{run growth path of the economy. We adopted the Cogley and Nason (1993) and Cogley and Nason (1995) methodology in order to assess for the ability of the model to mimic the business cycle. We found that the model is able to mimic the pattern of output at frequency 0. While the model is also able to generate a peak in the spectrum, such a peak does not occur at actual business cycle frequencies. Nevertheless, endogenous growth provides a new channel of propagation that can improve the ability of the model to reproduce the autocorrelation function of output growth. Indeed, the model is able to exhibit positive serial correlation in output growth at lag 1 and 2 as long as the size of endogenous growth relative to the exogenous growth component is high enough. We also partially solve the Cogley and Nason impulse response puzzle. The model is able to generate a hump{shaped IRF in the trend{reverting component in output if endogenous growth is important. Moreover, the IRF is statistically similar to the one obtained from the sample VAR. It thus appears that integrating cycles and growth leads to a substantial improvement of the RBC models' ability to account for the autocorrelation of output growth.
in the case of = 1. (See e.g. Collard (1995)). 25Even if the particular form we retain constrains the long{run e ect of shocks to be positive
24This property does not hold anymore in the case of traditional accumulation: Ay! (1) is then negative

23

A The value function
The program the individual has to solve admits the following Bellman equation: n o V (Kt?1; Ht?1; t) = maxt log( tCt ? Ht Nt1+ ) + Et V (Kt; Ht; t+1)] Ct ;N 8 > Kt = AK Kt?1 AY AtKt?1(NtHt?1)1? ? Ct 1? > < s:t: > > : H = A H 1? ( + )N (1? ? ) Yt
t H t?1 t Nt

Individuals treat productivity process as exogenous. Let us guess the form of the value function: ! X Yt + p?1 D? V (Kt?1; Ht?1; t) = D0 + D1 log Kt?1 + D2 log Ht?1 + D3 log N j t?j
t j =0

substituting this expression in the Bellman equation and di erentiating, we obtain the policy rules. Noting that Ct = scYt and tCt ? Ht?1Nt1+ = y Yt we get Ht?1Nt1+ = ny Yt. Then identifying the R.H.S to the L.H.S of Bellman's equation, we get:
D1 = D2 =
((1? )(1+ )+ (1+ ( + )) (1? )(1? ? )+(1? + ( + ))(( + )(1? )? (1? )(1+ )) (1? )(1? ) (1? )(1? ? )+(1? + ( + ))(( + )(1? )? (1? )(1+ ))

We proceed using the same method for the other parameters.

Using equations (19) and (15), we get:
1 ?( y L + ( h ? y )L2 ) 0 1 ? (1 ? h ? k )L

B Computation of the output spectrum
yt zt
=
ya + ( ha ? ya )L ka ? ha y! + ( h! ? y! )L k! ? h!

at !t

with y = (1+ ) ya = 1+ y! = 1? + + + Assuming that at and !t are AR(1) processes, and inverting the VAR and using the MA(1) representation of disturbances we get: X 0{ ? 1{L ? 2{L2 { yt = (1 ? 1L)(1 ? 2L)(1 ? {L) "t = Hy (L)"t {=fa;!g where 1 and 2 are the same as previously de ned, and ({ = fa; !g) and: 24
{

denotes the AR(1) parameter

8 < :

0{ = 1{ = 2{ =

y{ y{ ( 1 + 2 ) ? ( y k{ + (1 ? y ) h{ ) y ( k{ (1 ? h ? ) ? k h{ + (1 ? y )( h{ (1 ? k ) ? h k{) ? y{ 1 2

{ 2 fa; !g

This allows one to compute the spectrum of the rate of growth of output: 8 > 2(1 ? cos( )) 1 P{=fa;!g (a0{+a1{ cos( )+a2{ cos(2 )) "2{ >0 > 2 (b0{ +b1{ cos( )+b2{ cos(2 ))+b3{ cos(3 )) < f y( ) = > 1P a : 2 {=fa;!g (a(0c{0+{+1c{1cos( )+a2c{2{cos(2 ))))"2{ =0 { cos( )+ cos(2 where:

8a = 2+ 2+ 2 < 0{ 0{ 1{ 2{ a1{ = 2( 1{ 2{ ? 0{ 1{ ) { 2 fa; !g : a2{ = ?2 0{ 2{

and 8b > 0{ < b1{ > b2{ : b3{

= = = =

1 + ( 1 + 2 + { )2 + ( 1 2 + { ( 1 + 2 ))2 + ( { 1 2)2 ?2 f( 1 + 2 + { )(1 + 1 2 + { ( 1 + 2 )) + { 1 2 ( 1 2 + { ( 1 + 2 ))g 2 f 1 2 + { ( 1 + 2 ) ? { 1 2( 1 + 2 + { )g ?2 { 1 2
1 = 1, and factorizing by 2(1 ? cos(

{ 2 fa; !g

c|{ are obtained imposing

)).

25

References
Aghion, P. and G. Saint-Paul, 1991, On the Virtues of Bad Times, Working paper 578, C.E.P.R. Arrow, K.J., The Economic Implications of Learning by Doing, 1962, Review of Economic Studies, 80, 155{173. Baxter, M. and R.G. King, Productive Externalities and Business Cycles, 1991, Discussion paper 53, Institute for Empirical Macroeconomics, Federal Reserve Bank of Minneapolis. Blanchard, O.J. and D. Quah, The Dynamic E ects of Aggregate Demand and Supply Disturbances, 1989, American Economic Review, 79 (4), 655{673. Campbell, J. and N. Mankiw, Are Ouput Fluctuations Transitory?, 1987, Quarterly Journal of Economics, 102 (4), 857{880. Cogley, J. and T. Nason, Do Real Business Cycle Models Pass the Nelson-Plosser Test?, 1993, Manuscript, University of British Columbia. and , Output Dynamics in Real Business Cycles Models, 1995, American Economic Review, 85 (3), 492{511. Collard, F., 1995, Business Cycle and Endogenous Growth: Learning by Doing Versus Rationalizing, in P.Y. Henin ed., Advances in Business Cycle Research (SpringerVerlag). Dellas, H., 1992, Schooling and the Business Cycle, Manuscript, Catholic University Leuven. Eckstein, Z., F. Foulides, and T. Kollintzas, 1994, Can Exact Log{Linear Growth Models Account for U.S. Growth ?, Manuscript. Evans, G.W., Output and Unemployment Dynamics in the United States: 1950{1985, 1989, Journal of Applied Econometrics, 4, 213{237. 26

Gali, J. and M. Hammour, 1991, Long Run E ects of Business Cycles, Working paper, Columbia University. Hercowitz, Z., 1986, The Real Interest Rate and Aggregate Supply, Journal of Monetary Economics, 18, 121{145. and M. Sampson, 1991, Output Growth, the Real Wage, and Employment Fluctuations., American Economic Review, 81 (5), 1215{1237. King, R., C. Plosser, and S. Rebelo, 1988, Production, Growth and Business Cycles II: New Directions, Journal of Monetary Economics, 21 (2/3), 309{342. Kydland, F. and E. Prescott, 1982, Time to Build and Aggregate Fluctuations, Econometrica, 1982, 50, 1345{1370. Long, J. and C. Plosser, 1983, Real Business Cycle, Journal of Political Economy, 91 (1), 39{69. Lucas, R., On the Mechanisms of Economic Development, 1988, Journal of Monetary Economics, 22, 3{42. and E.C. Prescott, Investment under Uncertainty, 1971, Econometrica, 39, 659{681. McCallum, B., 1989, Real Business Cycles Models, in R. Barro ed., Modern Business Cycle Theories (Harvard University Press, Boston), pp. 16{50. McCurdy, T., 1981, An Empirical Model of Labour Supply in a Life Cycle Setting, Journal of Political Economy, 89 (61), 1059{1085. Nelson, C. and C. Plosser, 1982, Trend and Random Walks in Macroeconomic Time Series, Journal of Monetary Economics, 10 (1), 139{162. Rebelo, S., 1991, Long-run policy analysis and long-run growth, Journal of Political Economy, 99, 500{521. Romer, P., 1986, Increasing Returns and Long Run Growth, Journal of Political Economy, 94, 1002{1037. 27

Saint-Paul, G., 1992, Productivity Growth and the Structure of the Business Cycle, European Economic Review, 37. Stadler, G.W., 1990, Business Cycle Models with Endogenous Growth, American Economic Review, 80 (4), 763{778. Stock, J.H. and M.W. Watson, 1988, Testing for common trends, Journal of the American Statistical Association, 83, 1097{1107.

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