eth-25558-01

Institute for Empirical Research in Economics University of Zurich

Working Paper Series

ISSN 1424-0459

Working Paper No. 54 (Revised version)

Economic Growth and Business Cycles:

A Critical Comment on Detrending Time Series

Klaus Reiner Schenk-Hopp

May 2001

Economic Growth and Business Cycles:

A Critical Comment on Detrending Time Series

Klaus Reiner SchenkHoppe

Institute for Empirical Research in EconomicsUniversity of Zurich, Switzerland

[email protected]

May 1, 2001

Abstract

In this paper we pursue an approach based on economic theory toillustrate possible shortcomings of widely-used detrending methods. Weanalyze a simple model of economic growth and business cycles in whichinvestment and technical progress are stochastic. The Hodrick-Prescottand the Baxter-King lter are shown to detect spurious business cycleswhich are not related to actual cycles in the model. Our results castdoubts on the validity of commonly-accepted stylized business cycle facts.We also discuss the relation of business-cycle dating based on indicatorsof economic activity, as e.g. applied by the NBER, and the detrendingresults.

1 Introduction

Business cycles can be dened as deviations of macroeconomic data from anunderlying trend which, however, is not observable in general. Since the decom-position of a time series into a trend and a remaining cyclic part is in principlearbitrary, any attempt to identify, or approximate, business cycles has to bebased on economic theory. This way one can break open the above-mentionedcircularity of specifying one unobservable variable with the other.

There is a large strand of literature, in particular due to research associatedto the real business cycle school (Stadler [22]), which assumes that the trend is

Institute for Empirical Research in Economics, University of Zurich, Blumlisalpstrasse10, CH-8006 Zurich, Switzerland. Phone +41 (0)1 6343714, Fax +41 (0)1 6344907, [email protected]

I beneted from comments of the participants of the 8th Annual Meeting of the Society forNonlinear Dynamics and Econometrics. I am grateful to Hermann Garbers, Mordecai Kurz,and Rafael Lalive DEpinay for helpful discussions. The referees comments have helped todistinctly improve the rst draft of this paper.

1

smooth and all uctuations are driven by small transient productivity shocks.Starting from this assumption, aggregate data such as output and employmentare detrended using the Hodrick-Prescott lter and, more recently, the band-pass lter by Baxter and King [3]. The well-known stylized facts of the thusspecied business cycles provide a benchmark for any business cycle model,see e.g. King and Rebelo [14], Danthine and Donaldson [9], and Stock andWatson [23].

Both the assumptions imposed and the detrending methodology appliedhave undergone thorough inspections and recently face severe criticisms on twogrounds.

On the one hand, detrending of time series with the Hodrick-Prescott lter isshown to produce business cycle dynamics even if non are present in the original(articial) time series, see e.g. Cogley and Nason [8], Harvey and Jaeger [12],and Jaeger [13], who carry out spectral analyses of structural time-series models.Using actual US macroeconomic data, Canova [6] and Gregory and Smith [11]nd that most stylized facts are sensitive to the particular lter applied. Seealso the seminal paper by Nelson and Kang [16].

On the other hand, recent studies provide evidence that the time series ofU.S. GDP is not dominated by a smooth trend. Nelson and Plosser [17] andMurray and Nelson [15] claim that permanent shocks dominate. Fatas [10]questions the hypothesis of a smooth trend on the grounds of the empirical factthat long-term growth rates and persistence of output uctuations have a strongpositive correlation.

Blanchard and Fischer [4, p. 6] remarked that Macroeconomists are, andshould be, schizophrenic about the use of time series methods. Recent discus-sions, as manifested e.g. in Burnsides [5] comment on Canova [6] and Canovas [7]reply, cast doubts on the validity of the claim that the economics profession isfully aware of the potential pitfalls in using detrending methods.

This paper pursues a new direction in the study of the shortcomings ofwidely-used detrending methods. While in the above-mentioned literature ei-ther empirical macroeconomic data or structural time-series models have beenemployed, we propose and pursue a theory-based approach. The advantage ofour approach is two-fold. First, the statistical properties of the time series gener-ated by the underlying model are completely known while there is disagreementamong econometricians about the actual statistical properties of macroeconomictime series. Moreover, the trend and the cyclic component of the time seriesare explicitly given by the model. Second, the structure of the decompositionof the time series into trend and cycle is not merely assumed (as in structuraltime-series models) but based on economic theory. Moreover, we can comparethe qualitative dynamical behavior of the cyclic part of original stochastic eco-nomic model with that of the detrended time series of the model and thus gobeyond a mere study of the statistical properties.

The model considered here is a neoclassical growth model with stochastictechnical progress and stochastic investment. It takes the form of a stochasticdierence equation. The rst process is the main source of long-run growthwhile the latter is the main source of the short-run uctuations. Any correla-

2

tion of both sources of uctuations can be allowed for. Using results due toSchenkHoppe and Schmalfuss [20], we can completely determine the dynami-cal behavior and the statistical properties of the time series of this model. Inparticular, we can characterize the dynamics of model, which is stochastic andnon-linear, without any approximations such as log-linearization. In particular,for each realization of the exogenous stochastic processes, the sample paths ofall initial capital intensities are identical in the long-run. This property ensurese.g. that the numerical simulation of the long-run dynamics of the model yieldsreliable results.

Assuming that technical progress is driven by a stationary process of innova-tions, it turns out that the trend of the stochastic capital intensity is a dierencestationary process. We study dierent scenarios with respect to the statisticalproperties of the two sources of randomness, and thus can precisely quantify theerratic results of both Hodrick-Prescott and Baxter-King lter. Some qualita-tive properties of the detrending methods are also analyzed and compared withthe business-cycle-dating methodology based on indicators of economic activity,as is applied by the National Bureau of Economic Research.

We nd that even if technical progress is smooth and investment shocks aresmall and independent of the trend, these two detrending methods generatespurious business cycles. The higher the persistence of shocks, due to smallerrates of depreciation, the more pronounced is the misspecication of the businesscycles.

The remainder of the paper is organized as follows. Section 2 presents thestochastic model of economic growth and business cycles and provides an anal-ysis of its dynamics; in particular existence and uniqueness of a globally stablerandom xed point of the capital intensity is proved. A numerical study of themodel is carried out in Section 3. There, the actual business cycles of the modeland the result of detrending methods are compared. Section 4 concludes.

2 A Stochastic Economy

The starting point of our study of detrending methods is a theoretical eco-nomic model of growth and business cycles with stochastic technical progressand stochastic uctuations of investment. The model is strongly inuenced bythe seminal work of Solow [21] and Swan [24]. The technology is described bya neoclassical production function, technical progress is labor-augmenting, andthe investment-consumption decision of households is not explicitly modelledbut assumed. We enrich this basic model by allowing for stochastic technicalprogress and a stochastic saving rate.

The analysis of the model applies random dynamical systems theory, cf.Arnold [1]. The main result on the long-run dynamics relies of previous workdue to SchenkHoppe and Schmalfuss [20], and SchenkHoppe [18]. Under theassumption that the stochastic uctuations are ergodic and that the productionfunction satises an Inada-type condition, we can show that all sample paths ofcapital intensities asymptotically follow the same trajectory. This result enables

3

us to derive the true decomposition of the time series of the model into a trendand a cyclic part as well as the statistical properties of the two components.

We consider an economy in which a single homogeneous good is produced atany period in time. The good can be either consumed or used as capital input.Two factors, capital and labor, are needed in the production process, describedby the linear homogeneous production function

Yt = F (Kt, at Lt)

where Kt 0 is the capital stock at the beginning of period t, at Lt 0 is theecient labor supply, i.e. at is a measure of technical progress at time t and Ltis aggregate labor supply. Technical progress is labor-augmenting. We assumethat (K,L) F (K, at L) is neoclassical, exhibits constant returns to scale,and satises the Inada conditions for each possible realization of the exogenousvariable at, cf. Barro and Sala-i-Martin [2, Sec. 1.2.1]. Households do not havedisutility from work and inelastically supply their total endowment of labor. Wefurther assume a closed economy, i.e. the endowment of capital at the beginningof period t+ 1 is equal to the resources not consumed in the preceding period.Thus the law of motion of the capital stock is given by,

Kt+1 = F (Kt, at Lt) + (1 t)Kt Ct (1)where Ct denotes aggregate consumption in period t and t is the rate of depre-ciation.

Analogously to the standard Solow-Swan model we assume that each house-hold consumes a fraction 1 st of the total output in every period in time, i.e.Ct = (1 st)F (Kt, at Lt).

We make the following specic assumption on the process of technical inno-vations.

Assumption 2.1 The evolution of the ecient labor supply, atLt, is given byat+1Lt+1 = (1 + nt) atLt; and the exogenous variable (nt, t, st) is an ergodicprocess.

Appropriate assumptions on the range of values for these processes are im-posed below. For the moment it is sucient to assume at Lt > 0 for all t.

Dene the capital per ecient unit of labor kt = Kt/(at Lt), henceforthcalled capital intensity. Under assumption 2.1, (1) yields the following stochasticlaw for the capital intensity,

kt+1 =Kt+1

at+1 Lt+1=

(1 t)Kt + st F (Kt, at Lt)(1 + nt) at Lt

=(1 t) kt + st f(kt)

1 + nt

where f(k) := F (k, 1) is the intensity form of F (also a neoclassical productionfunction).

We model the ergodic process (nt, t, st) by an ergodic dynamical system(,F ,P, ). That is, the probability space (,F ,P) is the sample path space,and is the shift map. In this notation, the stochastic law becomes,

kt+1 = h(t, k) :=(1 (t)) kt + s(t) f(kt)

1 + n(t)(2)

4

For a given initial state k0 of the capital intensity and any sample path ofthe exogenous stochastic process, the random dierence equation (2) denes asample path of the capital intensity. (2) is henceforth called the stochastic Solowmodel.

Equation (2) generates a random dynamical system on the state space R+in the following sense. Dene,

(t, , k) =

h(t1) . . . h()k for t 1k for t = 0h(t)1 . . . h(1)1k for t 1

(3)

where h() := h(, ) : R+ R+. (t, , k) is the state of the stochasticsystem (2) at time t which has been started at k0 = k under the perturbationdetermined by .

The family of maps (t, , k) is called a random dynamical system. Thatis, : Z R+ R+, (t, , k) (t, , k) is a measurable mapping suchthat (0, ) = idR+ and (s+ t, ) = (t,

s) (s, ) for all s, t Z and all . Note that these properties replace the ow property of a deterministicdynamical system which is generated by the iteration of a map. Obviously,(t, ) inherits the regularities (such as continuity or smoothness) of h for t 0and of h1 for t 0.

We dene the concept of a xed point which is central to our subsequentanalysis of the model.

Denition 2.1 A random xed point of the random dynamical system gen-erated by the stochastic Solow model is a random variable k : R+ suchthat almost surely

k() = (1, , k()) := h(, k()). (4)

We are now in a position to state the main auxiliary result of this section.A proof can be found in SchenkHoppe and Schmalfuss [20].

Theorem 2.1 Assume that () [min, max] [0, 1], n() [nmin, nmax] ] 1,[, and s() [smin, 1] ]0, 1]. Assume further that f is non-negative,increasing, strictly concave, and continuously dierentiable.

Suppose that

(i) max + nmax > 0;

(ii) 0 limk

f (k) 0, |(t, , k)k(t)| 0 as ta.s.

The result ensures that the long-run behavior of all sample paths is uniquelydetermined by the random xed point k. For each initial capital intensity, thesample path asymptotically moves jointly with t k(t). The dynamics isthus governed by the ergodic process k(t).

Recall that the evolution of the aggregate capital stock is described by the(non-stationary) function Kt = atLtkt =

t1u=0(1+n(

u)) a0L0 kt for each ini-tial value of the ecient labor supply a0L0. The rst part is dierence stationaryafter taking logarithms because log(at+1Lt+1) = log(1+n(t))+ log(atLt), bydenition, and n(t) is ergodic and thus stationary.

We therefore obtain the following result.

Corollary 2.1 Fix any initial state of ecient labor supply a0L0 > 0. Thenthe sample path of the capital stock Kt = atLtkt is governed asymptotically bythe sample path t atLtk(t) for each initial state K0 > 0 and for almostall .

The asymptotic motion of the logarithm of the capital stock can therefore bedecomposed into a dierence stationary part and an ergodic part with expectedvalue zero:

logKt = [log(atLt) + E log k] +[log k(t) E log k] (5)

The two bracketed terms in (5) are referred to as trend and cycle, respectively.

For each realization of the exogenous process, a variation of the initial stateof ecient labor supply a0L0 results in a parallel translation of the samplepath of the capital stock. We therefore can and do assume a0L0 = 1. Thedecomposition (5) can be written as

logKt =

[t1u=0

log(1 + n(u)) + E log k]+[log k(t) E log k]

We discuss the properties of the two stochastic processes dened in thedecomposition (5) trend and cycle in turn. The trend is a non-stationary yetdierence stationary process. Its systematic contribution to the growth of thecapital stock is given by E logKt = tE log(1 + n) + E log k. The uctuationsof the trend are completely attributable to stochastic variations of the ecientlabor supply, i.e. to log(atLt).

The cycle is an ergodic process with mean zero. No systematic tendencyof the growth of logKt is caused by this part of the decomposition. The uc-tuations of the cycle are stationary and stem from the variation of the capitalintensity k. The cyclic part therefore depends on the process describing the

6

stochastic investment as well as on the process of innovation. The latter causesan indirect dependence of the cycle on the state of technical progress.

Put dierently, the ergodic investment process causes uctuations only inthe cycle part whereas the ergodic innovation process triggers uctuations ofboth trend and cycle.

The decomposition of the capital stock process dened in Corollary 2.1 alsoyields a decomposition of the total output. We have

log Yt = [log(atLt) + E log f(k)] +[log f(k(t)) E log f(k)] (6)

It is left to the reader as an easy exercise to derive the decomposition of theinvestment, the interest and wage rate (which are both ergodic), and the capitaland labor share (which have similar decompositions as given above). We willnot need these processes in the further study.

In summary, we have set up and completely analyzed the model in thissection. The main result is the decomposition of the motion of the capital stockinto trend and cycle. Both components are derived by the principles of economictheory and have clear-cut economic interpretations. The statistical propertiesof both components have also been described in detail.

3 Numerical Analysis

In this section we apply the Hodrick-Prescott and Baxter-King lter to time se-ries generated by our stochastic economic model. These widely-used detrendingmethods yield a decomposition of the stochastic aggregates into a trend and acyclic part. We examine the relation between this decomposition and the trendand the cycle which have been dened in the previous section using economictheory. In other words, we study numerically whether the Hodrick-Prescott orthe Baxter-King lter detects the actual business cycles in the time series gen-erated by the model. The main emphasis in this study is on qualitative ratherthan statistical properties of the cycle.

The software used in the simulations is a collection of MatlabR scripts.It is available on the web, SchenkHoppe [19]. Using the software the readerreproduce and check our results as well as analyze other interesting cases whichlack of space does not permit to present here.

Before presenting our numerical study, some remarks on the dierent usageof the terms trend and cycle in econometrics and economic theory are inorder. In econometrics, these two notions refer to dierent spectral properties,i.e. they are distinguished with respect to frequencies. The cycle is associatedto frequencies between 4 and 32 quarters whereas the trend is related to thelower frequencies in a time series.

In economic theory the trend refers to that part of a time series which isrelated to technical progress whereas the cycle corresponds to the business cycle.The trend is commonly believed to increase steadily and to show only sluggishvariations over time whereas the business cycle is associated to economic policiesthat have a short- or medium-run eect on the economy. This point of view is

7

manifested in the fact that the two components are studied in dierent elds,growth theory and business cycle theory, respectively. Only recently strongerties between these two elds started to develop due to empirical and theoreticalprogress. However, the above-mentioned distinction, or decomposition, does notrule out any possible dependencies between both factors and their impact onthe growth of an economy. Even in our simple model the innovation processexhibits an eect on the short-run uctuations.

We need to make specic assumptions on the stochastic processes governingthe evolution of innovation and investment as well as on the fundamentals.

Assumption 3.1 (i) Labor supply is xed and normalized to one, i.e. Lt 1. The rate of depreciation is constant.

(ii) The process of innovation n() is dened as follows. n() = 0.0075+(),where is the ergodic process generated by t+1 = At + t with i.i.d.process t being uniformly distributed on [, ].

(iii) The process of investment s() is dened as follows. s() = 0.25 (1 +z()), where z is the ergodic process generated by zt+1 = B zt + t withi.i.d. process t being uniformly distributed on [, ].

(iv) The technology is described by the CobbDouglas production function

f(k) = k 0 < < 1 (7)

We further assume that t and t are independent.

The expected value of the saving rate is 0.25 and expected value of technicalprogress is about 3% per year, where we interpret each period in time as onequarter of the year in the simulation.

We study three cases. The parameter settings are chosen as follows. We xA = 0.95, B = 0.95, = 5 104, and = 5 103 throughout the analysis.The other parameters are set to,

Case 1: = 0.75, = 0.9.

Case 2: = 0.25, = 0.9.

Case 3: = 0.25, = 0.1.

We rst need to ensure that the random xed point theorem 2.1 applies forthe above parameter-settings.

First note that () [/(1 A),+/(1 A)] and z() [/(1 B),+/(1 B)] for all . Second, elementary calculations yield the steadystate of the associated deterministic model,

k(max, nmax, smin) = (smin/(max + nmax))1/(1)

8

Using this expression, the contraction condition (iii) of Theorem 2.1 is satisedif and only if,

E log(1 + s() + nmax

smin

)< E log(1 + n()) (8)

since is deterministic. Validity of (8) in the cases 1-3 can be checked numeri-cally, see SchenkHoppe [19].

3.1 Case 1

We start with a case in which production is relatively capital intensive, = 0.75,and the rate of depreciation is high, = 0.9. The setting of the productionparameter is roughly in line with empirical studies employing the deterministicSolow model.

The simulation of the stochastic economy described in Section 2 is carriedout as follows. First the initial states of all processes are set to their expectedvalue. Second the model is iterated 600 periods to ensure that the sample path ofthe capital intensity is close to the path of the random xed point t k(t).Numerical studies show that this is indeed the case for the parameter settingsconsidered here. Third the model is simulated for 200 periods, where a period isunderstood as representing a quarter of a year. Thus the data generated in thelast 200 periods represent the time series of output and capital of our model-economy over a time-horizon of 50 years. Fourth we calculate the actual trendand cycle for the time series of logarithms of aggregate output and capital stock,log(Yt) and log(Kt), according to the denition given in (5) and (6). Finally,we apply the Hodrick-Prescott lter HP(1600) with parameter w = 1600 andthe band-pass lter BK(6,32,12), K = 12, introduced by Baxter and King [3],to the time series log(Yt) and log(Kt).

Figures 1 and 2 depict the results for case 1. We rst note that the Hodrick-Prescott and the Baxter-King lter show a close correspondence. This can beobserved in all simulations and is in agreement with the ndings of Baxterand King [3]. We will therefore mention only the Hodrick-Prescott lter in thesubsequent discussions. The BK(6,32,12) lter produces the smoother line dueto the fact that it is a band-pass lter and also removes the higher frequencycomponents from a time series which is not true for the HP(1600) lter.

In gure 1 the actual cycle of the logarithm of the total output exhibitsvalues of roughly between 4%. It is straightforward to check that this impliesa maximal deviation of total output Yt from the true trend of about 9.5%. Oneobserves a clear pattern of recurrent periods in which the logarithm of the totaloutput is above resp. below its trend, i.e. the cycle is positive resp. negative.The length of these periods as well as the magnitude of the cycle vary over time.However, the longer a period the larger the deviation from the trend. It is alsonoteworthy that the actual cycle is quite smooth. In summary, we can clearlydistinguish the dierent features of the actual business cycle in total output inour stochastic model.

9

0 5 10 15 20 25 30 35 40 45 50

0.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

0.05

Figure 1: [Case 1] Business cycles of GDP (deviation of log(Yt) from dierenttrends): HP(1600) (magenta), BK(6,32,12) (green), actual cycle (blue).

0 5 10 15 20 25 30 35 40 45 500.06

0.04

0.02

0

0.02

0.04

0.06

Figure 2: [Case 1] Business cycles of Capital Stock (deviation of log(Kt) fromdierent trends): HP(1600) (magenta), BK(6,32,12) (green), actual cycle (blue).

The Hodrick-Prescott lter predicts a business cycle which exhibits about thesame number of booms and recessions as existent in the actual cycle. However,the average magnitude of the HP cycle is smaller than that of the actual cycle.The dating of the cycles due to the Hodrick-Prescott lter is mainly erratic.Between year 10 and 30 the HP cycle shows a similar behavior as the actualcycle but predates the booms and recessions by 2 to 3 years.

Figure 2 depicts the results for the logarithm of the capital stock. The fea-tures of the actual cycle and the HP cycle are very similar to those discussedabove. This is due to the fact that the rate of depreciation is high. The varia-tions caused by the innovation and investment process aect output and capitalalmost in the same magnitude.

3.2 Case 2

We next consider a case in which production is relatively intensive in humancapital, = 0.25. We keep the same rate of depreciation as in case 1. Thus

10

only the production parameter is changed compared to the previous case. Thesimulations have been carried out exactly as explained above.

Figures 3 and 4 depict the results for case 2.

0 5 10 15 20 25 30 35 40 45 50

0.025

0.02

0.015

0.01

0.005

0

0.005

0.01

0.015

0.02

0.025


0 5 10 15 20 25 30 35 40 45 50

0.025

0.02

0.015

0.01

0.005

0

0.005

0.01

0.015

0.02

0.025


Due to the fact that physical capital is used less intensive in the productionprocess than human capital, the logarithm of the total output of the stochasticeconomy exhibits smaller deviations from the trend than in case 1. Aggregateoutput deviates from the actual trend by less than 1%. There are no pronouncedbooms and recessions in the stochastic economy. The Hodrick-Prescott lter,however, predicts business cycle of comparatively large magnitude. There is norelation between the HP and the actual cycle. The spurious business cycles asdated by the HP lter last roughly about 4 years.

The situation is somewhat dierent in the time series of the capital stock.There actual and HP cycle are of the same magnitude. The HP cycle exhibits acorrelation with the actual cycle to a certain degree. The business-cycle datingis not completely erratic though the agreement between the HP and the actual

11

data is quite bad. Again we observe a predating of the cycle by 1 to 2 years.

3.3 Case 3

We now study our stochastic economy with a comparatively low rate of depreci-ation, = 0.1. All other parameters are set as in case 2, in particular productionis relatively intensive in human capital, = 0.25. Again the simulations followthe same procedure as discussed above.

Figures 5 and 6 depict the results for case 3.

0 5 10 15 20 25 30 35 40 45 500.025

0.02

0.015

0.01

0.005

0

0.005

0.01

0.015

0.02


0 5 10 15 20 25 30 35 40 45 500.06

0.04

0.02

0

0.02

0.04

0.06

0.08


Due to the assumption that capital depreciates slower than in the previouscases, the time series of the logarithm of the capital stock, gure 6, exhibits largedeviations from its trend and is relatively smooth. The same pattern can beobserved for the actual cycle of the aggregate output, gure 5. Since productionis relatively intensive in human capital, the deviations from the trend are smallerthan for the capital stock.

12

In gure 5 the Hodrick-Prescott lter predicts business cycle of approxi-mately the true magnitude for total output. The dating as well as the lengthof the HP business cycle is erratic. The spurious business cycles have a lengthof about 3 years. There is almost no relation between the HP and the actualcycle.

The result of the Hodrick-Prescott lter is even worse for the time series ofthe capital stock, see gure 6. The magnitude of the HP cycle is about eighttimes too small. The dating of the cycle is also unrelated to the actual behavior.

3.4 Summary of the numerical results

We have analyzed numerically three dierent cases of our articial economy.In neither case have the Hodrick-Prescott or Baxter-King lter tracked downthe actual cycle. In fact the approximation of the true cycle is very poor. Inall cases both lters produced spurious cycles of an average length of about 3to 5 years. In most cases the magnitude of the deviation of the time seriesfrom the true trend is either over- or underestimated. In case 2 (resp. 3) inwhich production is labor intensive the uctuation of total output (resp. capitalstock) is predicted to be 5 times larger (resp. smaller) than it actually is for high(resp. low) depreciation. Any business-cycle dating based on these lters leadsto incorrect statements.

Of course it is mandatory to realize the causes of the observed shortcom-ings when applying these two widely-used lters. The qualitative results insection 2 ensured that the time series under study are integrated of order one(i.e. dierence-stationary). Together with the econometric results on the appli-cation of the Hodrick-Prescott lter to integrated time series to which we havealready pointed the reader in the introduction we have a good understandingof those causes.

3.5 NBER business-cycle dating revisited

In this section we focus on those causes of the detrending-problems that can bedetected by making use of the additional knowledge we have about the inno-vation and investment processes in our model. It will be shown that there is astrong link between the predictions of the Hodrick-Prescott (and Baxter-King)lter and the changes in economic activity in our model. Since the NationalBureau of Economic Research (NBER) business-cycle dating is based on indi-cators of economic activity, this observation highlights a possible explanationwhy the HP and NBER cycles exhibit a very close correspondence in empiricalstudies.

Figure 7 depicts the time series of the innovation process. Comparison of thisgure with the time series of total output and capital stock in case 3, gures 5and 6, yields the following observation. The sample path of either time seriesis above (resp. below) the actual trend if the sample path of the innovationprocess is below (resp. above) its expected value. This behavior is due to thedenition of the trend in our model: the trend is that part of the time series

13

0 5 10 15 20 25 30 35 40 45 50

0.99

0.995

1

1.005

1.01

Figure 7: [Case 3] Time series of technical progress 1 + n(t).

which is due to technical progress. If technical progress is slowing down thenthe slope of the trend decreases. If investment does not fall accordingly, thecapital per ecient unit of labor increases faster than the trend. Therefore thecycle is positive during these periods and we thus observe a boom. Clearly thisdenition of trend and cycle is not related to economic activity in a strict sense.

Interpreting economic activity in our model as (major) changes in the invest-ment process we can state the following observation. The Hodrick-Prescott (andBaxter-King) lter gives a clear-cut prediction on the changes in investment, cf.gures 5 and 7. The HP cycle is positive throughout year 5 to 10. In this periodinnovation is slowing down whereas investment is above its expected value andexperiences a temporary high in year 8. The actual cycle is negative. Similarpatterns can be observed also in the period from year 28 to 35. Both processesare below their expected value throughout this period. In year 30 investmenthas a local minimum whereas innovation experiences a local maximum. The HPcycle is negative but has a local minimum at year 30. The actual cycle positivethroughout this period. Summarizing we may state that the HP cycle is closelyrelated to changes in the investment process.

4 Conclusions

This paper illustrates the dangers of detrending non-stationary macroeconomictime series by lters. It provides a critical assessment of a common practice inempirical research, where econometric methods are employed without a soundtheoretical foundation. To this end we presented a model of stochastic economicgrowth in which the actual business cycles are not detected by the most com-monly applied lters in real business cycle theory. Both the Hodrick-Prescottand the Baxter-King lter generate spurious business cycles when applied tothe data generated by the model.

Our reasoning is based on economic theory, dening the trend as that part ofa time series that is caused by technical progress. Our criticisms gives further

14

0 5 10 15 20 25 30 35 40 45 50

0.2493

0.2494

0.2495

0.2496

0.2497

0.2498

0.2499

0.25

0.2501

0.2502

0.2503

Figure 8: [Case 3] Time series of investment s(t).

support to the purely econometric approaches due to Canova [6, 7], Cogleyand Nason [8], Harvey and Jaeger [12], Jaeger [13], and Nelson and Kang [16],among others. We suggest that the ndings of this paper and related work areunderstood as a motivation to strive for a new methodology (or the resuscitationof classical approaches) for determining the trend in economic growth paths.

References

[1] L. Arnold. Random Dynamical Systems. Springer-Verlag, New York, 1998.

[2] R. J. Barro and X. Sala-i-Martin. Economic Growth. McGraw-Hill, NewYork, 1995.

[3] M. Baxter and R. G. King. Measuring business cycles: Approximate band-pass lters for economic time series. Review of Economics and Statistics,81:575593, 1999.

[4] O. J. Blanchard and S. Fischer. Lectures on Macroeconomics. The MITPress, Cambridge (Mass.), 1989.

[5] C. Burnside. Detrending and business cycle facts: A comment. Journal ofMonetary Economics, 41:513532, 1998.

[6] F. Canova. Detrending and business cycle facts. Journal of MonetaryEconomics, 41:475512, 1998.

[7] F. Canova. Detrending and business cycle facts: A users guide. Journalof Monetary Economics, 41:533540, 1998.

[8] T. Cogley and J. M. Nason. Eects of the Hodrick-Prescott lter on trendand dierence stationary time series: Implications for business cycle re-search. Journal of Economic Dynamics and Control, 19:253278, 1995.

15

[9] J. P. Danthine and J. B. Donaldson. Methodological and empirical issuesin real business cycles theory. European Economic Review, 37:135, 1993.

[10] A. Fatas. Endogenous growth and stochastic trends. Journal of MonetaryEconomics, 45:107128, 2000.

[11] A. W. Gregory and G. W. Smith. Measuring business cycles with business-cycle models. Journal of Economic Dynamics and Control, 20:10071025,1996.

[12] A. C. Harvey and A. Jaeger. Detrending, stylized facts and the businesscycle. Journal of Applied Econometrics, 8:231247, 1993.

[13] A. Jaeger. Mechanical detrending by Hodrick-Prescott ltering: A note.Empirical Economics, 19:493500, 1994.

[14] R. G. King and S. T. Rebelo. Resuscitating real business cycles. In J. B.Taylor and M. Woodford, editors, Handbook of Macroeconomics Vol. 1B,chapter 14. Elsevier, Amsterdam, 1999.

[15] C. J. Murray and C. R. Nelson. The uncertain trend in U.S. GDP. Journalof Monetary Economics, 46:7995, 2000.

[16] C. R. Nelson and H. Kang. Spurious periodicity in inappropriately de-trended time series. Econometrica, 49:741751, 1981.

[17] C. R. Nelson and C. Plosser. Trends and random walks in macroeconomictime series. Journal of Monetary Economics, 10:139162, 1982.

[18] K. R. Schenk-Hoppe. Random dynamical systems in economics. Stochasticsand Dynamics, 1:6383, 2001.

[19] K. R. Schenk-Hoppe. Matlab scripts for business cycle simulationin the stochastic Solow model. Software available at the web pagehttp://www.iew.unizh.ch/home/klaus/numerics/, Institute for EmpiricalResearch in Economics, University of Zurich, May 2001.

[20] K. R. Schenk-Hoppe and B. Schmalfuss. Random xed points in a stochas-tic Solow growth model. Journal of Mathematical Economics. Forthcoming.

[21] R. Solow. A contribution to the theory of economic growth. QuarterlyJournal of Economics, 70:6594, 1956.

[22] G. W. Stadler. Real business cycles. Journal of Economics Literature,XXXII:17501783, 1994.

[23] J. H. Stock and M. W. Watson. Business cycle uctuations in US macroe-conomic time series. In J. B. Taylor and M. Woodford, editors, Handbookof Macroeconomics Vol. 1A, chapter 1. Elsevier, Amsterdam, 1999.

[24] T. W. Swan. Economic growth and capital accumulation. Economic Record,32:334361, 1956.

16

The Working Papers of the Institute for Empirical Research in Economics can be downloaded in PDF-format fromhttp://www.unizh.ch/iew/wp/

Institute for Empirical Research in Economics, Blmlisalpstr. 10, 8006 Zurich, Switzerland

Phone: 0041 1 634 37 05 Fax: 0041 1 634 49 07 E-mail: [email protected]

Working Papers of the Institute for Empirical Research in Economics

No.1. Rudolf Winter-Ebmer and Josef Zweimller: Firm Size Wage Differentials in Switzerland: Evidence from Job

Changers, February 19992. Bruno S. Frey and Marcel Kucher: History as Reflected in Capital Markets: The Case of World War II, February

19993. Josef Falkinger, Ernst Fehr, Simon Gchter and Rudolf Winter-Ebmer: A Simple Mechanism for the Efficient

Provision of Public Goods Experimental Evidence, February 19994. Ernst Fehr and Klaus M. Schmidt: A Theory of Fairness, Competition and Cooperation, April 19995. Markus Knell: Social Comparisons, Inequality, and Growth, April 19996. Armin Falk and Urs Fischbacher: A Theory of Reciprocity, July 20007. Bruno S. Frey and Lorenz Goette: Does Pay Motivate Volunteers?, May 19998. Rudolf Winter-Ebmer and Josef Zweimller: Intra-firm Wage Dispersion and Firm Performance, May 19999. Josef Zweimller: Schumpeterian Entrepreneurs Meet Engels Law: The Impact of Inequality on Innovation-

Driven Growth, May 199910. Ernst Fehr and Simon Gchter: Cooperation and Punishment in Public Goods Experiments, June 199911. Rudolf Winter-Ebmer and Josef Zweimller: Do Immigrants Displace Young Native Workers: The Austrian

Experience, June 199912. Ernst Fehr and Jean-Robert Tyran: Does Money Illusion Matter?, June 199913. Stefan Felder and Reto Schleiniger: Environmental Tax Reform: Efficiency and Political Feasibility, July 199914. Bruno S. Frey: Art Fakes What Fakes?, An Economic View, July 199915. Bruno S. Frey and Alois Stutzer: Happiness, Economy and Institutions, July 199916. Urs Fischbacher, Simon Gchter and Ernst Fehr: Are People Conditionally Cooperative? Evidence from a Public

Goods Experiment, July 200017. Armin Falk, Ernst Fehr and Urs Fischbacher: On the Nature of Fair Behavior, August 199918. Vital Anderhub, Simon Gchter and Manfred Knigstein: Efficient Contracting and Fair Play in a Simple

Principal-Agent Experiment, September 200019. Simon Gchter and Armin Falk: Reputation or Reciprocity?, September 199920. Ernst Fehr and Klaus M. Schmidt: Fairness, Incentives, and Contractual Choices, September 199921. Urs Fischbacher: z-Tree - Experimenters Manual, September 199922. Bruno S. Frey and Alois Stutzer: Maximising Happiness?, October 199923. Alois Stutzer: Demokratieindizes fr die Kantone der Schweiz, October 199924. Bruno S. Frey: Was bewirkt die Volkswirtschaftslehre?, October 199925. Bruno S. Frey, Marcel Kucher and Alois Stutzer: Outcome, Process & Power in Direct Democracy, November

199926. Bruno S. Frey and Reto Jegen: Motivation Crowding Theory: A Survey of Empirical Evidence, November 199927. Margit Osterloh and Bruno S. Frey: Motivation, Knowledge Transfer, and Organizational Forms, November 199928. Bruno S. Frey and Marcel Kucher: Managerial Power and Compensation, December 199929. Reto Schleiniger: Ecological Tax Reform with Exemptions for the Export Sector in a two Sector two Factor Model,

December 199930. Jens-Ulrich Peter and Klaus Reiner Schenk-Hopp: Business Cycle Phenomena in Overlapping Generations

Economies with Stochastic Production, December 199931. Josef Zweimller: Inequality, Redistribution, and Economic Growth, January 200032. Marc Oliver Bettzge and Thorsten Hens: Financial Innovation, Communication and the Theory of the Firm,

January 200033. Klaus Reiner Schenk-Hopp: Is there a Golden Rule for the Stochastic Solow Growth Model? January 200034. Ernst Fehr and Simon Gchter: Do Incentive Contracts Crowd out Voluntary Cooperation? February 200035. Marc Oliver Bettzge and Thorsten Hens: An Evolutionary Approach to Financial Innovation, July 200036. Bruno S. Frey: Does Economics Have an Effect? Towards an Economics of Economics, February 200037. Josef Zweimller and Rudolf Winter-Ebmer: Firm-Specific Training: Consequences for Job-Mobility, March 2000


No.


Institute for Empirical Research in Economics, Blmlisalpstr. 10, 8006 Zrich, Switzerland


38. Martin Brown, Armin Falk and Ernst Fehr: Contract Inforcement and the Evolution of Longrun Relations, March2000

39. Thorsten Hens, Jrg Laitenberger and Andreas Lffler: On Uniqueness of Equilibria in the CAPM, July 200040. Ernst Fehr and Simon Gchter: Fairness and Retaliation: The Economics of Reciprocity, March 200041. Rafael Lalive, Jan C. van Ours and Josef Zweimller: The Impact of Active Labor Market Programs and Benefit

Entitlement Rules on the Duration of Unemployment, March 200042. Reto Schleiniger: Consumption Taxes and International Competitiveness in a Keynesian World, April 200043. Ernst Fehr and Peter K. Zych: Intertemporal Choice under Habit Formation, May 200044. Ernst Fehr and Lorenz Goette: Robustness and Real Consequences of Nominal Wage Rigidity, May 200045. Ernst Fehr and Jean-Robert Tyran: Does Money Illusion Matter? REVISED VERSION, May 200046. Klaus Reiner Schenk-Hopp: Sample-Path Stability of Non-Stationary Dynamic Economic Systems, Juni 200047. Bruno S. Frey: A Utopia? Government without Territorial Monopoly, June 200048. Bruno S. Frey: The Rise and Fall of Festivals, June 200049. Bruno S. Frey and Reto Jegen: Motivation Crowding Theory: A Survey of Empirical Evidence, REVISED

VERSION, June 200050. Albrecht Ritschl and Ulrich Woitek: Did Monetary Forces Cause the Great Depression? A Bayesian VAR Analysis

for the U.S. Economy, July 200051. Alois Stutzer and Rafael Lalive: The Role of Social Work Norms in Job Searching and Subjective Well-Being, July

200052. Iris Bohnet, Bruno S. Frey and Steffen Huck: More Order with Less Law: On Contract Enforcement, Trust, and

Crowding, July 200053. Armin Falk and Markus Knell: Choosing the Joneses: On the Endogeneity of Reference Groups, July 200054. Klaus Reiner Schenk-Hopp: Economic Growth and Business Cycles: A Critical Comment on Detrending Time

Series, May 2001 Revised Version55. Armin Falk, Ernst Fehr and Urs Fischbacher: Appropriating the Commons A Theoretical Explanation, September

200056. Bruno S. Frey and Reiner Eichenberger: A Proposal for a Flexible Europe, August 200057. Reiner Eichenberger and Bruno S. Frey: Europes Eminent Economists: A Quantitative Analysis, September 200058. Bruno S. Frey: Why Economists Disregard Economic Methodology, September 200059. Armin Falk, Ernst Fehr, Urs Fischbacher: Informal Sanctions, September 200060. Rafael Lalive: Did we Overestimate the Value of Health?, October 200061. Matthias Benz, Marcel Kucher and Alois Stutzer: Stock Options: the Managers Blessing. Institutional Restrictions

and Executive Compensation, October 200062. Simon Gchter and Armin Falk: Work motivation, institutions, and performance, October 200063. Armin Falk, Ernst Fehr and Urs Fischbacher: Testing Theories of Fairness Intentions Matter, September 200064. Ernst Fehr and Klaus Schmidt: Endogenous Incomplete Contracts, November 200065. Klaus Reiner Schenk-Hopp and Bjrn Schmalfuss: Random fixed points in a stochastic Solow growth model,

November 200066. Leonard J. Mirman and Klaus Reiner Schenk-Hopp: Financial Markets and Stochastic Growth, November 200067. Klaus Reiner Schenk-Hopp: Random Dynamical Systems in Economics, December 200068. Albrecht Ritschl: Deficit Spending in the Nazi Recovery, 1933-1938: A Critical Reassessment, December 200069. Bruno S. Frey and Stephan Meier: Political Economists are Neither Selfish nor Indoctrinated, December 200070. Thorsten Hens and Beat Pilgrim: The Transfer Paradox and Sunspot Equilibria, January 200171. Thorsten Hens: An Extension of Mantel (1976) to Incomplete Markets, January 200172. Ernst Fehr, Alexander Klein and Klaus M. Schmidt: Fairness, Incentives and Contractual Incompleteness,

February 200173. Reto Schleiniger: Energy Tax Reform with Excemptions for the Energy-Intensive Export Sector, February 2001


No.


Institute for Empirical Research in Economics, Blmlisalpstr. 10, 8006 Zrich, Switzerland


74. Sandra Gth, Thorsten Hens and Klaus Schenk-Hopp: Evolution of Trading Strategies in Incomplete Markets.February 2001

75. Ernst Fehr and Klaus Schmidt: Theories of Fairness and Reciprocity Evidence and Economic Applications,February 2001

76. Bruno S. Frey and Alois Stutzer: Beyond Bentham Measuring Procedural Utility, April 200177. Reto Schleiniger: Global CO2-Trade and Local Externalities, April 2001

eth-25558-01

Documents

thusspecied business

business cycle model

business cycle dynamics

relation of business

detrending of time series

stylized business cycle

actual cycles

originalarticial time