Harding, D., & Pagan, A. (2005). a Suggested Framework for Classifying the Modes of Cycle Research....

JOURNAL OF APPLIED ECONOMETRICSJ. Appl. Econ. 20: 151159 (2005)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jae.838

A SUGGESTED FRAMEWORK FOR CLASSIFYING THEMODES OF CYCLE RESEARCH

DON HARDINGa AND ADRIAN PAGANb*a Department of Economics, The University of Melbourne, Australia

b Centre for Applied Macroeconomic Research (CAMA), Australian National University, Australia

SUMMARYThe paper argues that it is important to realize that the concept of a cycle has rarely been precisely articulatedin empirical work and that often researchers are using very different definitions of it. We propose a two-foldclassification based upon what series one is measuring a cycle in and how one would recognize a cycle insuch a series. The paper illustrates how one can then categorize existing research based upon how it answersthese questions. It also shows that the existence and properties of a cycle differ greatly depending uponwhich of the categories the researcher is using. Copyright 2005 John Wiley & Sons, Ltd.

1. INTRODUCTION

Conferences that we have recently attended have suggested to us that there is the potential for agreat deal of confusion today in cycle research. This is caused by many different interpretationsof what is meant by a cycle and how one goes about measuring it. In order to bring some orderinto this literature we feel that it is useful to provide some classification scheme which emphasizesthis diversity. In doing so it is hoped that the scheme may foster a better understanding of themany modes of cycle research. In what follows we will be concerned with cycles in economicactivity, but it is the case that the principles extend to cycles in any series. Nevertheless, it isuseful to be able to concentrate upon a specific example that is probably the main object of mostcycle research.

Our suggested framework involves distinguishing between the following issues:

I. What series is it that we are looking for cycles in?II. How do we recognize and measure (describe) a cycle in the series once the previous decision

has been made?

Section 2 of this paper discusses the answers to the first of these questions in more detail, whileSection 3 does the same for the second question. Section 4 then comments upon the implicationsof our classification scheme for cycle research.

Correspondence to: Adrian Pagan, Economics Program (RSSS), Australian National University, Canberra, ACT0200, Australia. E-mail: [email protected]/grant sponsor: ARC; Contract/grant number: DP0342949.

Copyright 2005 John Wiley & Sons, Ltd. Received 15 May 2003Revised 25 June 2004

152 D. HARDING AND A. PAGAN

2. THE UNDERLYING SERIES

For cycles in economic activity we need to first define that concept. Doing so will suggest anappropriate series to examine as a measure of economic activity. Burns and Mitchell (1946) andthe NBER are quite clear that their preferred definition involved gross domestic product (GDP) ifone wished to investigate cycles at quarterly or annual frequencies.1 For monthly cycles howeverthe lack of GDP data generally means that a variety of monthly series must be combined in someway. For the moment we leave this technical issue of how to combine together a variety of seriesin order to measure economic activity and simply assume that it is quarterly economic activity,as represented by quarterly GDP, that we wish to investigate.2

Now we have only broadly answered the question of what series we wish to examine for acycle. When one becomes more specific there are generally three responses in the literature.

A. The level of GDP (or a monotonic transformation such as the log). We will designate this asyt and assume that it is the log of GDP.

B. The level of GDP less a permanent component Pt. We will designate this by zt D yt Pt.C. The growth rate in GDP. This could be either a quarterly, yt, or annual, 4yt, growth rate.

The first quantity is chosen by the NBER and Burns and Mitchell (1946) and is sometimesreferred to as the classical or business cycle. To operationalize the second, a stand has to betaken on the form of the permanent component that is to be subtracted off. Many answers havebeen given to this, mostly under the guise of filtering operations, e.g. the HodrickPrescott filterremoves a permanent component from the series, leaving a transitory component zt. But manyother filters are in use, e.g. BeveridgeNelson (1981), band-pass filter (Baxter and King, 1999),etc. Unfortunately, these filters often have associated with them the confusing terminology thatthe permanent component is a trend and the zt is a cycle. This arises from the term stochastictrend that is often used to describe an I(1) series. To see why it is unhelpful in cycle researchto call the permanent component a trend, consider looking for a cycle in GDP if it followed apure random walk yt D yt1 C et. Then Pt D yt, there is no transitory component and, adoptingthe terminology which equates transitory components and cycles, there can be no cycle. There arecountries in the world like this, i.e. which have no serial correlation in the growth rate of GDP.Yet we would rarely see the conclusion that there are no cycles in the economic activity of suchcountries. Consequently, if one removes a permanent component from a series, then the cycle inthe series zt is best given a special name, and generally it is referred to as a growth cycle.

It is worth observing here that quantities such as zt are also often called output gaps. Thelogic of this is that the permanent component is taken to represent supply side influences and thetransitory components are demand side, so that an output gap attempts to separate out demandand supply in a simple fashion. There is no necessary connection with a cycle. There may be onein the output gap series but the latter itself is not a cycle.

Finally, the choice of activity variable in C means that we would be studying a cycle in growthrates as distinct from the growth cycle featured in B. However, this distinction is perhaps notas clear-cut as it might first seem. One could always choose Pt D yt1 or Pt D yt4 in B and,

1 See The NBERs Recession Dating Procedure at http://www.nber.org/cycles/recessions.html2 Of course we might be looking at cycles in a yearly measure of economic activity or even averages of activity over 10 or20 years. The latter measures would be the type of input needed to look at what were called Jugler and Krondratieff cycles.

Copyright 2005 John Wiley & Sons, Ltd. J. Appl. Econ. 20: 151159 (2005)

CLASSIFYING THE MODES OF CYCLE RESEARCH 153

with these choices of the permanent component, zt D yt and zt D 4yt would be the respectivetransitory components.3 Moreover, it is possible that zt might actually be proportional to yt if ytfollowed processes that are empirically plausible and if the permanent component is measured bythe BeveridgeNelson (1981) approach. To see this suppose that the DGP for yt is an AR (1) ingrowth rates, viz

yt D yt1 C t

where t i.i.d.N0, 2 and jj < 1. Then Morley (2002) showed that the exact BeveridgeNel-son permanent component PBNt is given by

PBNt D yt C

1 yt

and so the transitory component will be

zBNt D

1 yt

Hence the cycle in zBNt will be the same as in yt. Essentially the BN decomposition in thisinstance uses yt1 C 11 yt as the permanent component, rather than yt1. Of coursethe quantity yt1 C C1jD0jytj will also be a permanent component provided that the jare square summable, and this suggests that other ways of extracting the permanent componentmay actually have transitory components that are combinations of ytj. To get some feel for thispossibility we know from King and Rebelo (1993) that the zt which would arise from HP filteringyt would in fact have the form

zHPt DTjD1

jtytj

Because the HP filter is a two-sided one with time-varying weights, it is useful to get someappreciation for what the nature of the associated zt is like through the following experiment. First,generate 60 000 observations on a series for yt, where the latter follows a pure random walk ofthe form yt D yt1 C et. Second, construct zHPt by applying the HP filter with D 1600. Finally,regress this series upon yt,yt1, . . . , yt4. Inspection of the estimates of the parametersattached to the ytj suggests that a good approximation would be 0.47.9j. From this result itseems that the value of zHPt can be expressed as a combination of current and lagged values ofyt with slowly declining weights. It is useful to contrast this outcome from the HP filter with thefact that zBNt D 0 if yt is a pure random walk, and this outcome makes it clear that the two ways ofeliminating a permanent component will produce transitory components with very different serialcorrelation properties.

Because of the connection just described we will be able to shorten our discussion by treatingC as a special case of B.

3 We thank Jan Jacobs for this observation.



3. THREE WAYS OF DESCRIBING THE CYCLENow all of the above simply defines what we are looking for a cycle in. The series yt and zt are notcycles, even though sometimes people will refer to quantities such as zt in this way. Although insome contexts it may be harmless to use such loose language, in many others it becomes confusingto do.

There are three ways in the literature of describing what we mean by a cycle.

(i) Those looking for a periodic cycle focus upon statistical models of either yt or zt. Thesestatistical models generally involve cosine and sine waves in some fashion, although in thenonlinear literature one can have periodic limit cycles.

(ii) Blinder and Fischer (1981, p. 227) offered the definition of a cycle as serially correlateddeviations of output from trend.

(iii) A different approach, and probably the most widespread given our tendency to present graphsof series when discussing cycles, is to recognize a cycle from the turning points in the seriesunder investigation. These turning points are peaks and troughs and the periods between themare classified as expansions and contractions.

3.1. Periodic Cycles

An early example of (i) is the rocking horse cycle in Frisch (1933), who asked whether theseries under investigation had an AR(2) representation with complex roots. But there are otherrepresentations that are now widely used, which also have cosine and sine wave foundations, e.g.the local linear trend plus cycle model in Harvey and Jaeger (1993) that is embodied in the STAMPprogram. The rocking horse view is also maintained in programs such as TRAMO-SEATS (seeKaiser and Maravall, 2001) and in books such as Arnold (2002). Underlying (i) therefore is a two-step strategy for extracting cycle information. First, fit (say) the cosinesine model to whateverseries in AC is chosen for investigation. Second, ask the question whether there is evidence of aperiodic cycle in the fitted model, e.g. are the roots in the AR(2) complex? If accepted the periodof the cycle is recorded.

There can be problems with this strategy unless the representations are sufficiently general toallow for the possibility of no periodic cycles. Indeed, this is an issue with Harveys models whichmake zt D t C t, where t is i.i.d. N0, 2 and t is governed by the following periodic process:[

t t

]D

[cos sin

sin cos] [ t1 t1

]C[

t

t

]1

In (1), 0 1 to impose stationarity, is the frequency in radians and t and t are mutuallyindependent i.i.d. N0, 2 and i.i.d. N0, 2 random variables, respectively. Then t will be anARMA(2,1) process with the AR(2) polynomial being 1 2 cosL C 2 cos2 L2, and inspectionof this shows that its roots are always complex, i.e. a periodic cycle is imposed upon the series,so that the only role for the data is to determine its period.

3.2. Serially Correlated Deviation Cycles

Blanchard and Fischer (1989) operationalized (ii) with the following refinement: We then think oftrends as that part of output which is due to permanent shocks.. . . That part of output that comes



from transitory shocks can be thought of as the cycle. Putting these together we might concludethat a cycle exists if there is serial correlation in the zt, although they do not indicate how onewould map the characteristics of any such cycle into the degree of correlation.

We will not spend much time on (ii) as it is too incomplete to be a useful definition of a cycle.Its fundamental weakness shows up when one recognizes how dependent it is upon the definitionof the permanent component. As mentioned earlier, since there are many ways of constructinga permanent component, each one will produce a different degree of serial correlation in zt.The simplest illustration of this is when yt follows a pure random walk, whereupon zBNt D 0.However, using the data from the experiment on such a series described earlier, the first-orderserial correlation coefficient of zHPt would be around 0.8. Thus, with the BeveridgeNelsondecomposition zBNt D 0 there is no cycle using definition (ii). In contrast, if one chooses theHodrickPrescott filter, there will be serial correlation in zt, and, under (ii), there is now a cycle.Clearly it is unsatisfactory that the existence of cycles can be dependent on the way we measurea permanent component. Hence, because the definition is so vague, it is hard to see it as a seriouscandidate for cycle analysis.

3.3. Turning Point Cycles

Under (iii) it is necessary to give some rules for locating the turning points. Because a turningpoint fundamentally involves the location of local maxima and minima in the chosen series, andthat is known to involve a change in sign of the first derivative, it is natural that the analogue of thederivative in discrete time is taken to be the first difference of the series under examination. Thusthe rules will involve studying quantities like yt (when the series being investigated is yt) andzt (when the series being investigated is zt). For the discussion that follows we will act as if weare looking for cycles in yt, but the methods are the same for any definition of economic activity.

Now, when we focus attention on such changes, we may want to estimate the derivative using(say) an average of the yt over some window around the point in time that is a candidate forbeing a turning point. Indeed this is exactly what the NBER methods of dating turning pointseffectively involve, with a peak in yt at time t being detected by examining whether the followingsequence holds:

f2yt > 0,yt > 0,ytC1 < 0,2ytC2 < 0gand with a trough being similarly defined.4 But there would be other ways of doing this and,within the academic literature, we can find quite a few. Once the dating rules are applied, theiroutput is a series of observations on a binary indicator St that takes the value unity when the periodinvolves an expansion, i.e. t is between a trough and a peak, and zero when it is a contraction.The St are then the basis for recognizing a cycle and measuring its characteristics (some of these,e.g. amplitudes, will use both St and yt).

Because it is the nature of yt that tells us about the cycle in yt, we might proceed in a moreparametric way than that described above. Hence we might first fit a parametric statistical modelto yt and then use this model to perform turning point dating for us. That is what Hamiltons

4 We have taken some liberties here, The BryBoschan (1971) program that is used to date turning points in series isactually one based on monthly data inputs and works with the sgnytj, j D 1, . . . , 5. In Harding and Pagan (2002) thiswas translated into a quarterly variant, with five months becoming two quarters. It was shown that this rule reproducedNBER peaks and troughs quite closely, certainly closely enough for most research work.



(1989) Markov switching approach does. He first fits an MS model to yt. In its simplestincarnation this has the form yt D aC bt C et, where et is n.i.d.(0, ) and t is a binary Markovprocess of latent states. After the model is fitted, the turning points in yt are located using

t D 1[Prt D 1jFt > 0.5]

where 1[A] is the indicator function giving the value unity when the event A is true and zero whenit is false, and Ft D fytjgTjD0. Clearly the t is a binary random variable and is the equivalentof the St coming from the NBER dating rules. Again the cycle should be described and measuredby the behaviour of the binary random variable t.5

Since Prt D 1jFt > 0.5 is based upon the DGP of yt, Harding and Pagan (2003a) use alinear approximation to this probability to show that Hamiltons approach detects turning pointsat t using weighted averages of ytj, j D 1, . . . , T. This contrasts with NBER methods thatuse only local information, i.e. j D 1, 2. Cast into non-parametric terms one might think of theweighting function used by Hamilton as like using a Gaussian rather than a uniform kernel (as theNBER procedure assigns zero weight outside the two-period window). In Hamiltons procedure theweights are chosen according to the estimated model. The latter is both a strength and a weakness.It is a strength as the weights vary by data set, but it is also a weakness since one does not wantto give too much weight to growth rates far away from the point t. There are well-documentedcases where the MS model has had to be changed in order to do this.

Finally we might ask what the relation between (i) and (iii) is. It is worth noting that the periodiccycle and turning point cycle approaches could be identical but, mostly, they are very different.There is little doubt that there is a widespread misconception about the equivalence, as exemplifiedby the following quote from Arnold (2002, p. 4): This view of business cycles as disturbed sinewaves is also consistent with the NBER studies, which are concerned with characterizing thelengths, amplitudes and turning points of observed cycles. The fallacy in this reasoning can beseen most simply by looking at yt D costC ut, where ut is some shock added on to reflectthe fact that it is known that economic cycles will never be strictly cosine or sine waves. Thenwe have

yt D costCutNow, if we set ut D 0, we would find that the turning points in cost would be found bylooking at the sign changes in cost and that the duration of such a cycle would indeedbe measured by . However, the turning points in yt depend on the sign of yt, i.e. uponwhether costCut > 0. The two coincide only if ut D 0 and the closeness of the matchwill depend upon the magnitude of ut. Thus the period of the periodic cycle indicated by willnot agree with that indicated by turning points. Indeed it is possible that there is no periodic cycle,although there will be a turning point cyclethe simplest example being when yt is AR(1). ThusSargent (1979, p. 240) simulated data from an AR(1) process and observed that This illustrateshow stochastic difference equations can generate processes that look like they have businesscycles even if their spectra do not have peaks. . .. To emphasize that the correspondence betweenturning point and periodic cycle characteristics is not close, we generated data from an AR(2),

5 Often a mistake is made and the DGP of t rather than that of t is used to describe cycle characteristics. In particularthis happens in reported calculations of the durations of expansions and contractions, where the transition probabilitiesfor t rather than those of t are employed.



yt D 1.4yt1 0.57yt2 C et, where et was n.i.d.(0, 1). Then the duration of the periodic cycle is23 quarters while the duration of the turning point cycle (using NBER rules) was 11 quarters.

4. MULTIVARIATE ISSUESSometimes more than one series is used to define economic activity. This may be because GDP isnot available or because other measures may become available much more quickly. Two approacheshave then been used to handle such multivariate information. In the first the series are combinedtogether to produce a single measure of economic activity, whereupon the analysis proceeds asoutlined earlier. Mostly the diversity encountered here involves how one should combine themultiple series together to produce a single one. Sometimes simple weighted averages are used.In other instances principal components of the series are constructed. More recently it has beenpopular to regard economic activity as a latent factor and to then fit parametric models that wouldenable one to extract estimates of this factor, e.g. Forni et al. (2001).

An alternative method is to retain all the series but to define the cycle in terms of somemultivariate measures. Thus the analogue of the periodic cycle methodology would be to computethe eigenvalues of a VAR system and to see if there are complex-valued ones. The multivariateequivalent of the turning points methodology, when there is no reduction to a single measureof activity approach, is to aggregate the turning point information provided by all the individualcycles into a single set of turning points. This provides what the NBER refers to as the referencecycle. Harding and Pagan (2003b) discuss the methods that the NBER employ to perform suchan aggregation.

In all these cases one either ends up with a single series whose cycle is examined or with asingle cycle if a multivariate approach is adopted. Thus, while the techniques used to performthese tasks will vary, the classification given above still serves to demarcate the various views oncycles and does not need to be extended.

5. SOME IMPLICATIONS OF THE ABOVE CLASSIFICATIONThe fact that there are six possible combinations of the various choices means that, at any givenconference and in published volumes on the cycle, we will have many of them being representedsimultaneously. To illustrate this we note that A(iii) is what the NBER and the recently formedCEPR (London) European business cycle dating committee work with. In contrast, the COIN indexof the European cycle, also published by CEPR, embodies a B(iii) philosophy. Many statisticalmodels of the cycle using threshold autoregressions utilize C(iii), since the regime switchingvariable is the annual growth rate in the cycle variable. B(i) is represented in the work by Kaiserand Maravall (2001) and by Harvey and Jaeger (1993). A(i) is probably not favoured much anymore but it can be argued that this was Tinbergens (1956) view, as seen from his attempt tofind an AR(2) with complex roots in a variable representing the level of economic activity. Thusthere is a great variety of definitions present in current cycle research and this can become veryconfusing since the researchers generally claim to be describing and measuring the business cycle,a term that seems best applied to A(iii), as that was its historical connotation.

Many problems arise in being unclear as to what cycle is being studied. A particularly perniciousone is that many of the stylized facts about cycles that are cited come from a category combinationthat is not that maintained by the citer. Two common examples of this error follow. First, all those



Table I. Cycle characteristics, various definitions

US GDP 1947/12002/4, durations in quarters

Expansion Contraction

A(i) No periodic cycle from fitted AR(3)A(iii) 20 3B(i) 2 2B(iii) 5 5

using HodrickPrescott filtered data are working with B(i), but they often cite business cyclestatistics from the NBER website that are gathered under A(iii). Second, asymmetry in cyclephasesexpansions lasting three to four times longer than recessionsis largely an A(iii) cyclecharacteristic. Whilst it can be present in cycle statistics gathered under the B(iii) category, itis generally a rather trivial phenomenon, with growth cycle expansions rarely being more thana quarter longer than recessions. These features can all be seen in Table I, which gives thecharacteristics of the durations of the phases of cycles in US GDP under definitions A, B and(i), (iii) over the period 1947/12002/4. The HodrickPrescott filtered series was used to find thepermanent component under B.

It is also the case that the analysis provided above shows that the type of permanent/transitorydecomposition that is performed can have a big impact upon the serial correlation propertiesof the zt. Consequently, as the characteristics of the turning point cycle depend upon the DGPof ztsee Harding and Pagan (2002)it is clear that these will change with the method ofextracting a permanent component, something evident from Canova (1998). It should not beregarded as any surprise that this happens.

6. CONCLUDING COMMENTS

Of course it is unlikely that everyone will adopt a single combination of the categories, as eachhas its purpose. But we feel that it would help a great deal if authors were clear about whichcombination of categories they think they are using when analysing the cycle and to providesome arguments for why this particular combination best matches their objectives. It should gowithout saying that, even when one settles on a definition, there are still many unsolved issuesin cycle analysis. Knowing what one is explaining is important, but ultimately we will want tobuild statistical and economic models of the variables used in AC in order to be able to describecharacteristics of the cycle and to provide economic stories about the causes of cycles.

ACKNOWLEDGEMENTS

We would like to thank Jan Jacobs for his comments on an earlier version of this paper. Researchsupported by ARC Grant DP0342949.

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