This paper investigates the cyclical behaviour of quarterly U.K. investment using an unobserved
components framework formulated in continuous time. Comparisons are made between
the results from two models that differ in the specification of the cyclical component

Joanne S. McGarry (7195733)

English

Loughborough University Institutional Repository

   This item was submitted to Loughborough’s Institutional Repository by the author and is made available under the following Creative Commons Licence conditions.      For the full text of this licence, please go to: http://creativecommons.org/licenses/by-nc-nd/2.5/  Cycles in U.K. Investment ∗Joanne S. McGarry †February 2003AbstractThis paper investigates the cyclical behaviour of quarterly U.K. investment using an unob-served components framework formulated in continuous time. Comparisons are made betweenthe results from two models that differ in the specification of the cyclical component.JEL Codes: C22, C51.Keywords: Differential-difference equations, unobserved components, cyclical behaviour.∗I am grateful to Marcus Chambers, Eric Pentecost and Terry Mills for helpful comments. Any remaining errorsare my own.†Address for correspondence: Department of Economics, Loughborough University, Leicestershire, LE11 3TU.Tel: 01509 222719. Fax: 01509 223910. Email : J.S.McGarry@lboro.ac.uk.1 IntroductionUnobserved components models, also known as structural time series models, can be used todecompose a time series into its most salient features. Here a continuous time unobservedcomponents framework is employed to investigate the nature of cycles in U.K. investment.The series used in this application is the logarithm of quarterly U.K. Total Gross CapitalFormation covering the period 1955q1 to 2001q2. Two models are estimated and both includetrend, seasonal and cyclical components. The models differ only in the specification of thenon-seasonal cycle. In one model the cyclical component is written as a differential equation,in the form specified in Harvey and Stock (1993), henceforth the H-S component. In the secondmodel the cyclical component is formulated as a differential-difference equation in the formspecified by Chambers and McGarry (2002), henceforth the DDE component. Each representscyclical behaviour in a very different way and here the primary interest are the estimatedinvestment cycle durations. The trend and seasonal components are the same in both modelsand are written as differential equations.Henceforth, the continuous time unobserved components models that incorporate the H-Sand DDE cyclical components will be referred to as the H-S model and the DDE modelrespectively.2 The Unobserved Components ModelsThe continuous time investment process y(t) is written here as a function of the three com-ponents y(t) = µ(t) + γ(t) + φ(t) where µ(t) represents the trend, γ(t) the seasonal and φ(t)the cycle. The two models differ only in the specification of φ(t), and we concentrate on thiscomponent here. The H-S specification is given by the following system of differential equationsd φ(t)φ∗(t) = ln ρ λc−λc ln ρ φ(t)φ∗(t) dt+ κ(dt)κ∗(dt) (1)2for which κ(dt) and κ∗(dt) are mutually and serially uncorrelated random measures withcommon variance σ2κdt, for stationarity 0 < ρ < 1 and λc is the frequency at which the cycleoscillates. The unknown parameters are ρ, λc and σ2κ. The discrete time representation of theH-S component is φtφ∗t = ρ cos(λc) sin(λc)− sin(λc) cos(λc) φt−1φ∗t−1+ κtκ∗t (2)where [κt, κ∗t ]′ is a moving average disturbance vector. The moving average representationarises because investment is a flow process. The non-stochastic form of this system is φt =αρt cos(tλc) + βρt sin(tλc), where φ0 = α and φ∗0 = β, and it exhibits a cyclical pattern with aperiod 2piλc . System (2) simply introduces stochastic behaviour into this cyclical pattern. Hencethe H-S component produces a single cycle with a period 2piλc . In contrast, the DDE componentcan generate an infinite number of cycles, although only those with longer durations (such asbusiness cycles) are likely to be of interest to economists. The simple form of DDE consideredhere is given bydφ(t) = [a0φ(t) + a1φ(t− ν)]dt+ κ(dt) (3)where again κ(dt) is a random measure with variance σ2κdt. The unknown parameters area0, a1, ν and σ2κ. This DDE specification was originally derived by Kalecki (1935) as a modelof the macrodynamic theory of business cycles. Variations of the DDE specification have arisenmore recently in, for example, Ioannides and Taub (1992) on time-to-build investment modelsand Boucekkine et al (1997) on vintage capital growth models.The cycle durations from the DDE component are calculated from a function of the es-timated values for a0, a1 and the lag parameter ν. The reader is referred to Chambers andMcGarry (2002) for a detailed discussion of the process of determining the cycle durations. Inpractice, the lag parameter ν in (3), which Kalecki (1935) interprets as a time-to-build factoror gestation period, will be of unknown value. However, the estimation procedure developed3in Chambers and McGarry (2002) will estimate ν as well as the coefficient parameters a0 anda1.The seasonal component, common to both H-S and DDE models, takes the form γ(t) =γ1(t) + γ2(t) where both γ1(t) and γ2(t) are specificied in a similar way to the H-S cyclicalcomponent (1) but with ρ = 1 and in terms of the respective frequencies λ1 = pi2 and λ2 = pi.These are the seasonal frequencies for quarterly data and γ(t) contains seasonal unit roots atthese frequencies. The common variance of the random measures in γ1(t) will be denoted σ2ω1dtand that for γ2(t) as σ2ω2dt. The trend component in both models takes the form dµ(t) = η(dt)where η(dt) is a random measure with variance σ2ηdt. The analogous representation in discretetime is as a random walk µt = µt−1+ ηt. The unknown parameters in the seasonal componentare σ2ω1 and σ2ω2 and in the trend it is σ2η.McGarry (2000) and Chambers and McGarry (2002) consider a frequency domain maxi-mum likelihood estimator for the estimation of the parameters in DDEs. This form of estimatoris employed here to estimate all of the unknown parameters in both H-S and DDE models.The reader is referred to Chambers and McGarry (2002) for details about this procedure.3 ResultsFor the H-S model, the parameter vector is θ = {ρ, λc, σ2κ, σ2η, σ2ω1 , σ2ω2 , σ2 } and for the DDEmodel it is θ = {a0, a1, ν, σ2κ, σ2η, σ2ω1 , σ2ω2 , σ2 }. The estimates of the structural parameters ofboth models are given in the table, where t statistics are given in bold. In both models, onlythe parameters that determine the cycle durations are statistically significant. The varianceparameters, although insignificant, are estimated with similar values in both models. Ofprimary interest however are the cycle duration estimates. Using the H-S model, a single cycleof length 2piλˆc= 13.5 quarters (3.4 years) is detected. As stated earlier, the DDE component canidentify more than one cycle and in this application several low frequency cycles are estimated.The durations of these cycles are not estimated directly and hence do not appear in the table.4Table 1: Parameter EstimatesParameter H-S model DDE modelλc 0.464479 -5.649774ρ 0.999984 -1.430622a0 - -1.593572-3.603930a1 - -2.015947-3.985691ν - 19.96940235.304580σ2κ 0.000002 0.0000010.099673 0.086071σ2η 0.000416 0.0004530.613775 0.650771σ2ω1 0.000054 0.0000500.480161 0.441680σ2ω2 0.000066 0.0000650.465939 0.434446σ2 0 00 0They are computed from a complex function of the estimated values for a0, a1 and ν. Thelargest of these cycles has a duration of 41 quarters (10.25 years). This fulfils the Kaleckidefinition of a business cycle, which must have length greater than twice the lag value, ν. Thelag value in this case is highly significant and represents a period of 19.97 quarters (5 years).Other shorter cycles that are found using the DDE model have lengths 5.1 years and 3.4 years,the latter being the same cycle as that estimated from the H-S model. The H-S representationhas not detected the longer cycles (10 and 5 years) in this investment series. It is likely thatthe longer durations are more difficult to estimate because there are fewer complete cycles inany fixed sample of data.The results from the DDE model compare favourably with those found in Reiter andWoitek (1999), in which the cyclical behaviour of fixed investment (amongst other variables)5is analysed for 15 OECD countries using spectral techniques. For the U.K. they find that thespectral density is concentrated in a cycle range of 5-7 years, although it also has some massin the ranges 7-10 and 10-15 years. The range 3-5 years has little spectral mass, which is notconsistent with the single cycle estimated by the H-S model. Wen (1998) estimates a cycle of7 years in the U.S. aggregate fixed investment to output ratio. It is suggested that the longperiod of this cycle is due to the length of time taken in the production of fixed capital andthat this time-to-build factor generates persistent demand for investment goods. This reflectsour findings from the DDE cyclical component, where the longer cycle of just over 10 years isgenerated by the long time-to-build factor of 5 years (the lag parameter).In both models, the cycle is non-stationary. In the H-S case, this is shown by the fact thatthe damping parameter is not statistically different from unity. This implies the presence of aunit root and that the cycle is actually undamped. The DDE parameter combination is alsonon-stationary, which is inferred also from damping factors of 1.25 for the major 10-year cycleand 1.2 for the cycle of 3.4 years.4 ConclusionIn conclusion, the H-S model has identified one cycle of duration 3.4 years and the DDE modelhas identified three important long length cycles of duration 10.25, 5.1 and 3.4 years. The10-year cycle accords with evidence from previous studies and has been detected by the DDEspecification despite just four of these cycles having been completed during the timespan ofthe dataset.References[1] Boucekkine, R., Licandro, O. and Paul, C. (1997), Differential-difference equations in eco-nomics: On the numerical solution of vintage capital growth models, Journal of Economic6Dynamics and Control, 21, 347-362.[2] Chambers, M. J. and McGarry, J. S. (2002), Modeling cyclical behavior with differential-difference equations in an unobserved components framework, Econometric Theory 18,387-419.[3] Harvey, A. C. and Stock, J. H. (1993), Estimation, smoothing, interpolation and dis-tribution for structural time series models in continuous time, in Models, Methods andApplications of Econometrics: Essays in Honour of A. R. Bergstrom, edited by P. C. B.Phillips, Blackwell, pp. 55-70.[4] Ioannides, Y. M. and Taub, B. (1992), On dynamics with time-to-build investment tech-nology and non-time-separable leisure, Journal of Economics Dynamics and Control 16,225-241.[5] Kalecki, M. (1935), A macro-dynamic theory of business cycles, Econometrica 3, 327-344.[6] McGarry, J. S. (2000), Estimating lag parameters in differential-difference equations, Eco-nomics Research Paper, Loughborough University, 00/14 (under revision for Journal ofEconometrics).[7] Reiter, M. and Woitek, U. (1999), Are there classical business cycles? Economics WorkingPapers Ref 398, Department of Economics and Business, Universitat Pompeu Fabra.[8] Wen, Y. (1998), Investment Cycles, Journal of Economics Dynamics and Control, 22 (7),1139-1165.7

Cycles in UK investment

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Cycles in UK investment

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