Fractional trends and cycles in macroeconomic time series
FFractional trends and cycles in macroeconomic timeseries
Tobias Hartl ∗ , Rolf Tschernig , and Enzo Weber University of Regensburg, 93053 Regensburg, Germany Institute for Employment Research (IAB), 90478 Nuremberg, GermanyMay 2020
Abstract.
We develop a generalization of correlated trend-cycle decompositions thatavoids prior assumptions about the long-run dynamic characteristics by modelling thepermanent component as a fractionally integrated process and incorporating a fractionallag operator into the autoregressive polynomial of the cyclical component. The modelallows for an endogenous estimation of the integration order jointly with the other modelparameters and, therefore, no prior specification tests with respect to persistence are re-quired. We relate the model to the Beveridge-Nelson decomposition and derive a modifiedKalman filter estimator for the fractional components. Identification, consistency, andasymptotic normality of the maximum likelihood estimator are shown. For US macroeco-nomic data we demonstrate that, unlike I (1) correlated unobserved components models,the new model estimates a smooth trend together with a cycle hitting all NBER reces-sions. While I (1) unobserved components models yield an upward-biased signal-to-noiseratio whenever the integration order of the data-generating mechanism is greater than one,the fractionally integrated model attributes less variation to the long-run shocks due tothe fractional trend specification and a higher variation to the cycle shocks due to thefractional lag operator, leading to more persistent cycles and smooth trend estimates thatreflect macroeconomic common sense. Keywords. unobserved components, fractional lag operator, long memory, trend-cycledecomposition, Kalman filter.
JEL-Classification.
C22, C51, E32 ∗ Corresponding author. E-Mail: [email protected] authors thank James Morley, the participants of the econometric seminar in Nuremberg, the depart-ment seminar at the Christian Albrechts University Kiel, the DAGStat conference 2019 in Munich, and theworkshop on high-dimensional time series in economics and finance 2019 in Vienna. Support through theprojects TS283/1-1 and WE4847/4-1 financed by the German Research Foundation (DFG) is gratefullyacknowledged. a r X i v : . [ ec on . E M ] M a y Introduction
Unobserved components (UC) models are widely applied in macroeconomic research, e.g.to decompose GDP and industrial production into trend and cycle (Harvey; 1985; Morleyet al.; 2003; Weber; 2011; Morley and Piger; 2012), to study cyclical consumption (Morley;2007), and to measure long-run investment (Harvey and Trimbur; 2003). While empiricalevidence supports a strong negative correlation of long- and short-run shocks for the afore-mentioned macroeconomic aggregates, the correlated UC model as proposed by Balke andWohar (2002) and Morley et al. (2003) frequently produces a volatile long-run componentestimate together with a noisy cycle (Weber; 2011; Kamber et al.; 2018), thereby contra-dicting macroeconomic common sense. In addition, the integration order of the long-runcomponent is subject to debate. While all aforementioned papers model the long-run com-ponent as an I (1) process, Clark (1987) and Oh and Zivot (2006) suggest specificationswith I (2) trends that nest the HP filter (cf. G´omez; 1999, 2001), and I (0) specificationswith structural breaks are considered in Perron and Wada (2009) and Wada (2012).In this paper, we argue that economically implausible cycle estimates are likely to resultfrom a too restrictive specification of the integration order: If a process is in fact integratedof order greater than one, then misspecifying the long-run component to be I (1) upward-biases the variance estimate of the long-run shocks, yielding a high signal-to-noise ratio.This results in a volatile trend estimate together with a noisy cycle. As will be shown,generalizing the persistence properties from integer integration orders to the fractionaldomain and estimating the integration order jointly with the other parameters of themodel can solve the problem.Focusing on key macroeconomic indicators, the assumption of integer integration orders (0,1, or 2) has been contested for several variables. For real GDP, M¨uller and Watson (2017)find that the likelihood is flat around d = 1, yielding a 90% confidence interval for d thatis given by [0 . , . d > d > I ( d )), d ∈ R + , whereas the fractional lag operator L d = 1 − ∆ d + of Johansen (2008), that is de-fined in (3), enters the lag polynomial of the cyclical component. In UC models fractionallyintegrated processes allow for richer dynamics of both, trend and cycle components, andnest a broader class of data-generating mechanisms. Contrary to I (1) UC models, theyallow for mean-reversion of the long-run component for d <
1, while d > I (1) UC models is upward-biased, yielding a high signal-to-noise ra-tio that causes volatile trend estimates together with noisy, implausible cycles. Conversely,fractionally integrated UC models are likely to adequately capture the true signal-to-noiseratio and produce reliable trend and cycle estimates, as they nest the data-generatingmechanism.In addition, the fractional lag operator L d yields a weighted sum of past realizations whenmultiplied to a contemporaneous random variable, and thus it qualifies as a lag operator(Johansen; 2008; Tschernig et al.; 2013). Since it preserves the integration order of aprocess for non-negative d , the fractional lag operator adds flexibility to the short-runproperties of a process rather than influencing the integration order. While the standardlag operator L = 1 − (1 − L ) subtracts an I ( −
1) process from a contemporaneous variable,the fractional lag operator L d = 1 − (1 − L ) d + subtracts an I ( − d ) process. Thus, forequality of the variances of the two generic processes Lz ,t = z ,t − , L d z ,t = (1 − ∆ d + ) z ,t ,i.e. Var( z ,t − ) = Var((1 − ∆ d + ) z ,t ), with z ,t , z ,t white noise, it must hold that Var( z ,t ) < Var( z ,t ) for d > d <
1. Consequently for d >
1, which will be therelevant case in our applications, the variance of the short-run shocks in the fractionallyintegrated UC model must be greater than in the I (1) case to arrive at the same varianceof the cyclical component, yielding a smaller signal-to-noise ratio and, therefore, a morepersistent cycle.Since d is defined on a continuous support and enters the likelihood function as an unknownparameter, our model allows for an endogenous estimation of the integration order jointlywith the other model parameters, avoids prior unit root testing and takes into accountmodel selection uncertainty with respect to d .The canonical (reduced) form of the fractional trend-cycle model exhibits a fractionalARIMA representation in the fractional lag operator L d , and directly relates to a gen-eralization of the Beveridge-Nelson decomposition to the fractional domain. We discussidentification and show that consistency and asymptotic normality carries over from theARFIMA estimator of Hualde and Robinson (2011) and Nielsen (2015). Contrary to thecorrelated UC model, that requires an autoregressive cycle of order p ≥ p ≥ d (cid:54) = 1. Finally,we assess the state space representation of the fractional trend-cycle model and propose acomputationally efficient modification of the Kalman filter for the estimation of the latentlong- and short-run components.When taking the new fractional UC framework to the data, we demonstrate that it makesan important difference for key questions in empirical macroeconomics. We contribute tothe empirical literature by applying our fractional trend-cycle decomposition to US GDP,2ndustrial production, private investment, and personal consumption. Our model nests awide class of UC models and allows to draw inference on the proper specification of the long-run component for the four series under study. We contrast the trend and cycle estimatesfrom integer-integrated UC models with the results from our fractional decomposition andstate differences regarding shape, smoothness, variance and importance of the differentcomponents. Especially for industrial production we obtain a decomposition that is inline with economic theory, as the cyclical component captures all NBER recession periods,whereas the correlated UC model clearly fails to produce a plausible cycle. For all timeseries under study, we estimate a continuous increase of the cyclical components in periodsof economic upswing, where the correlated UC model produces a noisy cycle that sharplyincreases before the economy is hit by a recession.The structure of the paper is as follows. Section 2 details the fractional UC model, relatesit to well-established trend-cycle decompositions, derives the reduced form, and discussesidentification. Section 3 derives the state space representation, proposes a computation-ally efficient modified Kalman filter for the estimation of the latent fractional trend andcycle, and discusses parameter estimation. In section 4 the model is applied to decomposemacroeconomic aggregates. Section 5 concludes. We define a fractional trend-cycle decomposition of a scalar time series { y t } nt =1 as the sumof a long-run component τ t and a cycle component c t y t = τ t + c t , t = 1 , ..., n. (1)The long-run component τ t is characterized by an autocovariance function that decaysmore slowly than with an exponential rate and, therefore, captures the long-run dynamicsof a time series, whereas the cycle component c t is I (0) and accounts for transitory fluctu-ations of a series around its trend. In contrast to the bulk of the literature on unobservedcomponents models that specifies the stochastic trend component typically as a nonsta-tionary process integrated of order 1 or 2 – most often as a random walk – we suggest amore general formulation. τ t is specified as a combination of a linear deterministic processand a fractionally integrated series τ t = µ + µ t + x t , ∆ d + x t = η t , (2)3here µ and µ are constants, η t ∼ i . i . d . N(0 , σ η ), and d ∈ R + . The fractional differenceoperator ∆ d is defined as∆ d = (1 − L ) d = ∞ (cid:88) j =0 π j ( d ) L j , π j ( d ) = j − d − j π j − ( d ) j = 1 , , ..., j = 0 , (3)and a +-subscript denotes a truncation of an operator at t ≤
0, e.g. for an arbitrary process z t , ∆ d + z t = (cid:80) t − j =0 π j ( d ) z t − j (Johansen; 2008, def. 1). The fractional long-run component x t adds flexibility to the weighting of past shocks for d ∈ R + and nests the classic integerintegrated specifications for d ∈ N . The memory parameter d determines the rate at whichthe autocovariance function of x t decays, and a higher d implies a slower decay. For d < x t is mean-reverting, while d ∈ [1 ,
2) yields the aggregate of a mean-reverting process.Throughout the paper, we adopt the type II definition of fractional integration (Marinucciand Robinson; 1999) that assumes zero starting values for all fractional processes, and, asa consequence, allows for a smooth treatment of the asymptotically stationary ( d < . d ≥ .
5) case. Due to the type II definition of fractional integrationthe inverse of the fractional difference operator ∆ − d + = (1 − L ) − d + exists for all d , such thatwe can write x t = ∆ − d + η t = t − (cid:88) j =0 ϕ j ( d ) η t − j ϕ j ( d ) = j + d − j ϕ j − ( d ) j = 1 , , ..., j = 0 . Turning to the transitory component, we allow for an AR( p ) process in the fractional lagoperator φ ( L d ) c t = ε t , (4)where φ ( L d ) = 1 − φ L d − ... − φ p L pd , L d = 1 − ∆ d + is the fractional lag operator (Johansen;2008, eq. 2), and ε t ∼ i . i . d . N(0 , σ ε ). For stability of the fractional lag polynomial φ ( L d )the condition of Johansen (2008, cor. 6) is required to hold. It implies that the roots of | φ ( z ) | = 0 lie outside the image C d of the unit disk under the mapping z (cid:55)→ − (1 − z ) d . Infractional models L d plays the role of the standard lag operator L = L , since (1 − L d ) x t =∆ d + x t ∼ I (0). For an arbitrary process z t , L d z t = − (cid:80) t − j =1 π j ( d ) z t − j is a weighted sumof past z t , and hence L d qualifies as a lag operator. Furthermore, by definition the filter φ ( L d ) preserves the integration order of a series since d > ρ = Corr( η t , ε t ) (cid:54) = 0, which directly implies E( η t ε t ) = σ ηε (cid:54) = 0. Fordifferent time indexes we restrict the cross-correlation to be zero, E( η t ε s ) = 0 ∀ t (cid:54) = s .4hus, the long- and short-run shocks are i.i.d. N distributed with non-diagonal variance Q (cid:32) η t ε t (cid:33) ∼ i . i . d . N(0 , Q ) , Q = (cid:34) σ η σ ηε σ ηε σ ε (cid:35) . Our model is very general in terms of its long-run dynamic characteristics, as it neststhe well-known framework of Harvey (1985) for d = 1, where the long-run component isa random walk with drift, and c t is an autoregressive process of finite order. Correlatedshocks as in Balke and Wohar (2002), Morley et al. (2003), and Weber (2011) are explicitlyallowed. For d = 2, one obtains the double-drift unobserved components model of Clark(1987), and a fractional plus noise decomposition as proposed in Harvey (2002) is obtainedby setting d ∈ R + , p = 0.As shown in (6) below, similar to the classic UC-ARMA model that exhibits an ARIMArepresentation (Morley et al.; 2003, eq. 2b) the canonical form of our fractional trend-cyclemodel is an ARIMA( p, d, n −
1) model in the fractional lag operator. To see this, notethat (1 − L d )[ φ ( L d ) − ] + ε t = θ ε + ( L d ) ε t is a stable moving average process in the fractionallag operator (cf. Johansen; 2008), such that we can write∆ d + ( y t − µ − µ t ) = η t + (1 − L d )[ φ ( L d ) − ] + ε t = θ u + ( L d ) u t , (5) φ ( L d )∆ d + ( y t − µ − µ t ) = φ ( L d ) η t + (1 − L d ) ε t = ψ + ( L d ) u t , (6)where ψ + ( L d ) = φ ( L d ) θ u + ( L d ) is a truncated moving average polynomial of infinite orderthat results from the aggregation of φ ( L d ) η t + (1 − L d ) ε t . Its existence together witha recursive formula for the coefficients ψ j is shown in appendix B. u t ∼ N(0 , σ u ) holdsthe disturbances and is Gaussian white noise with σ u = σ η + σ ε + 2 σ ηε , which followsfrom Granger and Morris (1976, p. 248f) for contemporaneously dependent ε t , η t , and θ u + ( L d ) = (cid:80) t − i =0 θ ui L id , θ = 1, θ ui = σ ε σ u θ εi for all i >
0. While aggregating MA processes inthe standard lag operator yields an MA process whose lag length equals the maximum lagorder of its aggregates, this does not hold in general for the aggregation of MA processesin the fractional lag operator L d , since L id u t , L jd u t are not independent for i, j > i (cid:54) = j .Only for p = 1 equation (6) becomes an ARIMA(1 , d,
1) in the fractional lag operator,since η t + ε t and L d ( φ η t + ε t ) are independent. For d ∈ N the model in (6) nests theinteger-integrated ARIMA models. Due to the inclusion of the fractional lag operator (6)differs from the standard ARFIMA model. Nonetheless, (6) exhibits an ARFIMA( n − d , n −
1) representation as φ ( L d ) can be written as an AR( n −
1) polynomial.Our fractional trend-cycle model can be seen as a generalization of the decomposition ofBeveridge and Nelson (1981) to the fractional domain. To see this, consider (5) from which5ne obtains directly(1 − L d )( y t − µ − µ t ) = θ u + ( L d ) u t = θ u + (1) u t − (1 − L d ) t − (cid:88) k =0 L kd u t t − (cid:88) j = k +1 θ uj , such that multiplication with ∆ − b + yields the long- and short-run components x BNt = ∆ − d + θ u + (1) u t = x t , c BNt = − t − (cid:88) k =0 L kd u t t − (cid:88) j = k +1 θ uj = c t . (7)Equality of the decomposition of Beveridge and Nelson (1981) and the UC model in (1), (2),and (4) was shown in Morley et al. (2003) for d = 1. Note that x BNt and c BNt are identicalto the unobserved components in (2) and (4) for any d , which follows immediately fromplugging L d = 1 in (5). Consequently, the fractional trend-cycle decomposition generalizesthe I (1) Beveridge-Nelson decomposition to the class of ARIMA models in the fractionallag operator.For d = 1 Morley et al. (2003) demonstrate that the integer-integrated UC model is notidentified for p = 1, d = 1, σ ηε (cid:54) = 0. In that case, imposing the restriction σ ηε = 0 yieldsa decomposition that is different to the one of Beveridge and Nelson (1981). The same isshown by Weber (2011) for the simultaneous unobserved components model identified byheteroscedasticity and by Trenkler and Weber (2016) for the multivariate UC model.In fact, d = 1 is the only case where the unobserved components model is not identified for p = 1. In any other case, where d ∈ R + , d (cid:54) = 1, we show in the following that the modelparameters φ ( L d ), d , σ η , σ ε , and σ ηε can be uniquely recovered from (6), which is sufficientfor identification. Since φ ( L d ) and d are obtained directly from the model in its canonicalform, we consider σ η , σ ε , and σ ηε on which identification of the unobserved componentsmodel crucially depends.For d (cid:54) = 1 the parameters σ η , σ ε , and σ ηε are obtained from the autocovariance functionof ψ ( L d ) u t for any p ≥
0, whereas p ≥ d = 1, as Morley et al. (2003)demonstrate. To see this, we consider p = 2, for which γ = Var [ ψ ( L d ) u t ] = σ η (cid:40) t − (cid:88) k =1 [( φ + 2 φ ) π k ( d ) − φ π k (2 d )] (cid:41) + σ ε t − (cid:88) k =0 π k ( d ) + 2 σ ηε (cid:40) t − (cid:88) k =1 [( φ + 2 φ ) π k ( d ) − φ π k (2 d )] π k ( d ) (cid:41) ,γ j = Cov[ ψ ( L d ) u t , ψ ( L d ) u t − j ] = σ η (cid:110) [( φ + 2 φ ) π j ( d ) − φ π j (2 d )]6 t − (cid:88) k = j +1 [( φ + 2 φ ) π k ( d ) − φ π k (2 d )] [( φ + 2 φ ) π k − j ( d ) − φ π k − j (2 d )] (cid:111) + σ ε t − (cid:88) k = j π k ( d ) π k − j ( d ) + σ ηε (cid:40) π j ( d ) + t − (cid:88) k = j [( φ + 2 φ ) π k ( d ) − φ π k (2 d )] π k − j ( d )+ t − (cid:88) k = j +1 [( φ + 2 φ ) π k − j ( d ) − φ π k − j (2 d )] π k ( d ) (cid:41) . Note that d = 1 implies π j ( d ) = 0 ∀ j > π j (2 d ) = 0 ∀ j >
2. If now φ = 0,then γ j = 0 ∀ j >
1, and hence the model is not identified, as also Morley et al. (2003)demonstrate. For d (cid:54) = 1 the model is identified for any p ≥
0, since γ (cid:54) = 0. Contrary tothe I (2)-model of Oh et al. (2008) with three shocks that requires p ≥
4, our model isidentified for any p ≥ d = 2, since γ , γ , γ are different from 0 which is sufficientfor the identification of σ η , σ ε , and σ ηε . In this section we derive a state space representation for the fractional trend-cycle decom-position together with a modified Kalman filter estimator for the unobserved components.Furthermore, we discuss the maximum likelihood estimator for the unknown model pa-rameters and show that consistency carries over from the ARFIMA estimator of Hualdeand Robinson (2011) and Nielsen (2015).There exists a finite-order state space representation of (2) and (4) for fixed sample size n ,since any type II fractionally integrated process exhibits an autoregressive representationof order n −
1. Thus, an exact state space representation of the fractionally integratedsystem yields a state vector of dimension k ≥ n for d / ∈ N . Estimation of τ t , c t via theKalman filter then involves the inversion of the k × k conditional state variance for each t = 1 , ..., n , which slows down the Kalman filter substantially for large n .Consequently, different approximations for fractionally integrated processes in state spaceform have been considered (cf. e.g. Chan and Palma; 1998; Palma; 2007, for truncated MAand AR approximations). Hartl and Weigand (2019) study ARMA( v , w ) approximationswith v, w ∈ { , , } for fractional processes. Hartl et al. (2020) then correct for theresulting approximation error of the ARMA approximations for fractionally integratedtrends. Their approach keeps the state dimension manageable and is feasible from acomputational perspective, as it only requires to calculate the inverse of Var[( y , ...., y n ) (cid:48) ]once. Furthermore, it yields the identical likelihood function as the exact state space modelbut is computationally superior. We generalize their method to the fractional trend-cycle7odel in the following, and thereby provide a computationally feasible exact Kalman filterestimator for τ t , c t .To begin with, collect all model parameters in θ = ( d, φ , ..., φ p , σ η , σ ηε , σ ε ) (cid:48) and define E θ ,Var θ , Cov θ as the moments given the parameter vector θ . Define F t as the σ -field generatedby y , ...., y t and let z t | s = E θ ( z t |F s ) for any random variable z t . Then the prediction errorof the Kalman filter for the exact state space model of (1), (2), and (4) is v t +1 = y t +1 − E θ ( y t +1 |F t ) = y t +1 − µ − µ ( t + 1) − x t +1 | t − c t +1 | t . (8)Let ˜ x t and ˜ c t denote the approximate long-run and cyclical components defined in detailin (12) and (13) below. We will show that the following relationships for the conditionalexpectations hold ˜ x t +1 | t = x t +1 | t − (cid:15) xt , (9)˜ c t +1 | t = c t +1 | t − (cid:15) ct , (10)where (cid:15) xt , (cid:15) ct denote the approximation errors of the Kalman filter estimates and are F t -measurable. Then (8) can be rewritten as v t +1 = y t +1 − µ − µ ( t + 1) − (cid:15) xt − (cid:15) ct − ˜ x t +1 | t − ˜ c t +1 | t = ¨ y t +1 − E θ (¨ y t +1 |F t ) , (11)with ¨ y t +1 = y t +1 − (cid:15) xt − (cid:15) ct as the approximation-corrected y t . Therefore, the predictionerrors of the approximation-corrected model and the exact state space model are the same.Next we derive (9) and (10).The long-run component x t in (2) is approximated by an ARMA( v, w ) process ˜ x t =[ a ( L, d ) − m ( L, d )] + η t = (cid:80) t − j =0 b j ( d ) η t − j where a = m = 1 and, therefore, b = 1. a ( L, d )is an AR polynomial of order v , whereas m ( L, d ) is a MA polynomial of order w . Thisyields an approximation error x t − ˜ x t = t − (cid:88) j =1 ( ϕ j ( d ) − b j ( d )) η t − j . (12)The ARMA coefficients are obtained beforehand by minimizing the mean squared errorbetween the Wold representations x t = (cid:80) t − j =0 ϕ j ( d ) η t − j and ˜ x t = (cid:80) t − j =0 b j ( d ) η t − j for a fixed d and sample size n . A continuous function that maps from the integration order d toits ARMA coefficients is then obtained by optimizing over a grid of d and smoothing theoutcomes using splines. Hence, optimization of the likelihood for the fractional trend-cycledecomposition is conducted over the scalar fractional integration order d , and does not8nvolve the estimation of any parameters in a ( L, d ), m ( L, d ), such that the dimension ofthe parameter vector θ is kept small during the optimization. Details together with a largesimulation study are contained in Hartl and Weigand (2019).The cyclical component c t can be expressed as an AR( n −
1) process in the standard lagoperator φ ( L d ) c t = δ + ( L, d, φ ) c t that is initialized deterministically with c t = 0 ∀ t ≤ δ + ( L, d, φ ) results from φ ( L d ) c t = p (cid:88) j =0 φ j (cid:32) − ∞ (cid:88) k =1 π k ( d ) L k (cid:33) j + c t = t − (cid:88) j =0 δ j c t − j . An approximation for the fractional cyclical component is obtained by truncating δ ( L, d, φ )after lag l , ˜ δ ( L, d, φ ) = (cid:80) lj =0 δ j L j , ˜ δ ( L, d, φ )˜ c t = ε t . Note that δ ( L, d, φ ), ˜ δ ( L, d, φ ) solelydepend on d and φ , ..., φ p . Define ω + ( L, d, φ ) = [ δ ( L, d, φ )] − , ˜ ω + ( L, d, φ ) = [˜ δ ( L, d, φ )] − as moving average lag polynomials of c t , ˜ c t in the standard lag operator L . The approxi-mation error is then given by c t − ˜ c t = (cid:32) t − (cid:88) j =0 δ j L j (cid:33) − − (cid:32) l (cid:88) j =0 δ j L j (cid:33) − ε t = t − (cid:88) j =0 ( ω j − ˜ ω j ) ε t − j . (13)With these approximations and an expression for the resulting approximation error athand, we are ready to derive the impact of the two approximations on the Kalman filterestimates of τ t and c t . Note thatCov θ ( y t , η t − j ) = ϕ j ( d ) σ η + ω j σ ηε , (14)Cov θ ( y t , ε t − j ) = ϕ j ( d ) σ ηε + ω j σ ε , (15)Cov θ ( y t , y t − j ) = t − j − (cid:88) k =0 ϕ k ( d ) ϕ k + j ( d ) σ η + t − j − (cid:88) k =0 ( ω k ϕ k + j ( d ) + ϕ k ( d ) ω k + j ) σ ηε + t − j − (cid:88) k =0 ω k ω k + j σ ε , (16)and define y t = ( y , ..., y t ) (cid:48) − (E θ [ y ] , ..., E θ [ y t ]) (cid:48) , η t = ( η , ..., η t ) (cid:48) , and ε t = ( ε , ..., ε t ) (cid:48) .Then the joint distribution can be stated as η t ε t y t ∼ N , σ η I σ ηε I Σ η t y t σ ηε I σ ε I Σ ε t y t Σ (cid:48) η t y t Σ (cid:48) ε t y t Σ y t , (17)9here Σ η t y t = Cov θ ( η t , y t ), Σ ε t y t = Cov θ ( ε t , y t ), and Σ y t = Var θ ( y t ) withentries from equations (14), (15), and (16).Let e j be a t -dimensional unit vector with a one at column j and zeros elsewhere. Thencomputing the conditional expectation for (12) and (13) respectively delivers˜ x t +1 | t = x t +1 | t + E θ [˜ x t +1 − x t +1 |F t ] = x t +1 | t − t (cid:88) j =1 ( ϕ j ( d ) − b j ( d )) E θ [ η t +1 − j |F t ] == x t +1 | t − t (cid:88) j =1 ( ϕ j ( d ) − b j ( d )) e t +1 − j Σ η t y t Σ − y t y t = x t +1 | t − (cid:15) xt , ˜ c t +1 | t = c t +1 | t + E θ [˜ c t +1 − c t +1 |F t ] = c t +1 | t − t (cid:88) j =1 ( ω j − ˜ ω j ) E θ ( ε t +1 − j |F t ) == c t +1 | t − t (cid:88) j =1 ( ω j − ˜ ω j ) e t +1 − j Σ ε t y t Σ − y t y t = c t +1 | t − (cid:15) ct , where (cid:15) xt , (cid:15) ct are the approximation errors in (9), (10), and the last step follows fromlemma 1 in Durbin and Koopman (2012). It is easy to see that the approximation errors (cid:15) xt = (cid:80) tj =1 ( ϕ j ( d ) − b j ( d )) e t +1 − j Σ η t y t Σ − y t y t and (cid:15) ct = (cid:80) tj =1 ( ω j − ˜ ω j ) e t +1 − j Σ ε t y t Σ − y t y t solely depend on the parameters θ and y , ..., y t . Hence, they are F t -measurable and canbe computed precisely.Define the approximation-corrected ¨ y t +1 = y t +1 − (cid:15) xt − (cid:15) ct . Then the prediction error of theexact state space model and of the approximation-corrected, truncated state space modelare identical, which proves (11). Thus they have the same conditional log likelihood givena set of parameters θ . Consequently, maximization of the conditional log likelihood ofthe approximation-corrected truncated model solves the same optimization problem as forthe exact state space representation but reduces the dimension of the state vector. Thestate space representation of the approximation-corrected truncated model is derived inappendix C.The exact choice of v , w for the ARMA approximation of the fractional trend componentand l for the truncation of the fractional lag operator does not affect the equality in (11),since the approximation-correction yields the exact likelihood function that is identicalwith a non-truncated model but is computationally superior. Nonetheless, numerical op-timization takes longer when v , w , l are chosen too big, as the Kalman filter then has toinvert high-dimensional covariance matrices. As a rule-of-thumb, we suggest v = w = 4,which keeps the dimension of the state vector small and is found to resemble the dynamicsof fractionally integrated Gaussian noise ∆ d + η t well, as it yields a better fit than autoregres-sive and moving average processes of order 50 (Hartl and Weigand; 2019). For the cyclical10omponent we suggest l = 10. We use this specification in all empirical applications thatfollow.Finally we comment on the estimation of θ via maximum likelihood (ML). Under theprerequisites derived in Hualde and Robinson (2011, Assumptions A.1-A.3) the conditionalsum-of-squares estimator of (6), that is asymptotically equivalent to the ML estimator, isconsistent and asymptotically normally distributed. As they show, imposing stationarityand invertibility on θ u + ( L d ) together with u t being white noise is sufficient for consistencyand asymptotic normality. Similar results are obtained by Nielsen (2015). Since the MLestimator has the same limit distribution as the conditional sum-of-squares estimator, andsince our model in its reduced form satisfies the conditions of Hualde and Robinson (2011)and Nielsen (2015), their asymptotic results hold for the ML estimator of the reduced formin (6).Under identification, the asymptotic results carry over from the reduced form in (6) to thestructural form in (1), (2), and (4). As shown in section 2, the parameters of the structuralform θ can be uniquely recovered from (6) for any p if d (cid:54) = 1 (and for any p ≥ d = 1 asshown in Morley et al. (2003)). Therefore, the maximum likelihood estimator of θ basedon the probability density function of the prediction errors in (8) is consistent.Consequently, three different model formulations of the fractional trend-cycle decomposi-tion yield consistent estimates for θ in (1), (2), and (4) via maximum likelihood, namelythe reduced form model in (6), the exact state space representation based on y t and theapproximation-corrected truncated model based on ¨ y t . The latter model allows to estimate τ t and c t directly via the Kalman filter and is computationally superior to the exact statespace representation. Therefore, it forms the basis of our empirical analysis in the nextsection. We apply our fractional trend-cycle decomposition in (1), (2), and (4) to extract long-runand transitory components from real GDP, industrial production, gross private domesticinvestment, and personal consumption expenditures for the US. Trend-cycle decomposi-tions of real economic output are typically conducted to estimate the cyclical deviation ofoutput from its long-run growth path. Examples are Harvey and Trimbur (2003); Gar-ratt et al. (2006); Perron and Wada (2009) for log US real GDP and Clark (1987); Stockand Watson (1999); Weber (2011) for log US real industrial production. Morley (2007)estimates the long-run component of personal consumption, whereas Harvey and Trimbur(2003) also consider US investment. Hence, our results from the fractional trend-cyclemodel can easily be compared and checked against widely used alternatives.11n our application several advantages of the fractional trend-cycle decomposition becomeapparent. From a methodological perspective the endogenous treatment of the integrationorder neither requires assumptions about the persistence of a series nor prior unit roottesting or differencing. Furthermore, fractional trends offer additional flexibility in mod-elling the permanent component, which directly affects the estimation of the transitorycycle. From an empirical perspective, we contribute to the literature by providing newinsights on the persistence of long-run output, investment and consumption, when thetrend component is not restricted to be I (1). In addition, we study cyclical adjustmentsduring economic recessions and comment on the correlation structure between permanentand transitory shocks. Finally, we investigate how establishing the fractional lag operatoraffects the estimate of the cyclical component. Since d > L d ε t is a weighted sum of past ε t that is I (0). Consequently, the fractional lag operator allows for a more flexible way ofmodelling the short-run properties of a series while preserving the integration order.The data was downloaded from the Federal Reserve Bank of St. Louis (mnemonics: GDPC1,INDPRO, PCECC96, GPDIC1), is in quarterly frequency and spans from 1961:1 to 2018:4.All series are seasonally and inflation adjusted and enter the dataset in logs.To estimate the unknown parameters θ in (1), (2), and (4) we draw 100 combinationsof starting values from uniform distributions with appropriate support and maximize thelog likelihood of the fractional trend-cycle model via the Nelder-Mead algorithm up to acertain relative tolerance. We ignore the approximation-correction, that has a negligibleimpact on the performance of the ML estimator as shown in Hartl and Weigand (2019), forthe estimation of the starting values to speed up the computations. Next, the parameterscorresponding to the greatest log likelihood are set as starting values for a finer maximiza-tion via the exact approximation-corrected method discussed in section 2. p is chosen viathe Bayesian Information Criterion (BIC).To study the impact of fractional trends and cycles we introduce a benchmark model thatrestricts d = 1 in (2) and (4). Hence, we contrast the fractional trend-cycle model withthe I (1) correlated unobserved components model studied in Morley et al. (2003) andWeber (2011). The restricted model is given in equation (18) below and will be called T-Cspecification in the following. We will refer to the unrestricted model, that is given in (19),as FT-FC specification τ ( T ) t = µ ( T )0 + µ ( T )1 t + ∆ − η ( T ) t , φ ( C ) ( L ) c ( C ) t = ε ( C ) t , (18) τ ( F T ) t = µ ( F T )0 + µ ( F T )1 t + ∆ − d + η ( F T ) t , φ ( F C ) ( L d ) c ( F C ) t = ε ( F C ) t . (19)Both models allow for correlated permanent and transitory shocks, ρ = Corr( η t , ε t ) (cid:54) = 0.Estimation results together with the log likelihoods are reported in table 1.12 DP: gradual cyclical upswing
For log GDP, empirical evidence for the exact value of the persistence parameter d is mixed.Diebold and Rudebusch (1989) and Tschernig et al. (2013) estimate d to be slightly smallerthan one, whereas M¨uller and Watson (2017) find that the likelihood is flat around d = 1,such that a 90% confidence interval yields d ∈ [0 . , . d EW = 1 .
24 and ˆ d GP H = 1 .
24 with tuning parameter α = 0 .
65 as inShimotsu and Phillips (2005).For the fractional trend-cycle model the ML estimator yields ˆ d F T − F C = 1 .
32, implying thatlog US real GDP is a non-stable, nonstationary fractional process. As figure 5 shows, thelog likelihood is considerably flat around ˆ d F T − F C , which explains the different results forthe persistence parameter in the literature and confirms the findings in M¨uller and Watson(2017). Nonetheless, most of the probability mass clearly lies at d ≥
1. Contrary to thebenchmark, the FT-FC specification attributes more volatility to the transitory shocks,whereas σ η is estimated to be smaller than in the T-C specification. Trend
Cycle FT−FC − − Cycle T−C − − Figure 1: Trend-cycle decompositions for log US real GDP with correlated innovations.The left plot sketches the trend component estimate from the restricted model (18) (T-C,with d = 1) in black, dashed, together with the trend component from the unrestrictedmodel (19) (FT-FC, with d (cid:54) = 1 allowed) in gray, solid. The plots on the right-hand sideshow the cyclical components for the unrestricted and the restricted model. Shaded areascorrespond to NBER recession periods.Figure 1 plots the decompositions from the T-C and the FT-FC specification in (18) and(19). At first glance, it demonstrates that the T-C and the FT-FC decomposition for logUS real GDP yield rather similar results, which may be due to the flat likelihood of the13T-FC model around d = 1 .
32, and the results coincide with the literature (cf. e.g. Mor-ley et al.; 2003; Sinclair; 2009). As economic theory suggests, both cyclical componentsdecline during the NBER recession periods. The FT-FC specification suggests a gradualcyclical upswing in non-recession periods and therefore captures an important feature ofthe business cycle, contrary to the cyclical T-C component that exhibits a steep increaseright before a recession period. Similar cycle estimates as from the FT-FC specificationare obtained from the nonlinear regime-switching UC-FP-UR model of Morley and Piger(2012). Thereby, the parsimonious parametrization of the FT-FC model together with itsability to resemble nonlinear dynamics foster its generality. Furthermore, the fractionaltrend component is smoother than its I (1) counterpart, as σ η in table 1 shows. As Kam-ber et al. (2018) demonstrate, forcing the signal-to-noise ratio to be small can yield cycleestimates via correlated I (1) UC models that are in line with economic theory. But asan inspection of their trend estimate shows, this comes with the cost of producing frac-tionally integrated long-run shocks that violate the white noise assumption. In contrast,fractionally integrated UC models directly estimate a small signal-to-noise ratio withoutrestricting parameters to a certain interval and yield long-run shocks that are I (0). Thus,a high signal-to-noise ratio in I (1) UC models can indicate a violation of the I (1) as-sumption for the long-run component. As explained in Weber (2011), the strong negativecorrelation between η t and ε t is typically interpreted as causal impact from long-run shocksto the transitory component, where a positive trend shift yields a negative cyclical adjust-ment that vanishes over time due to the stationary nature of the transitory component.However, Weber (2011) also finds significant negative effects in the reverse direction. Industrial production: plausible cycles in recessions
For log US industrial production we find ˆ d EW = 1 .
18 and ˆ d GP H = 1 .
26 which indicates aviolation of the I(1) assumption of the unobserved components model. This is confirmedby the fractional trend-cycle model, for which the ML estimator yields ˆ d F T − F C = 1 .
66. Asfigure 5 shows, the likelihood is steep around ˆ d .Figure 2 plots the unobserved components estimates from the T-C and the FT-FC spec-ification for log US industrial production. Since ˆ d F T − F C = 1 .
66 is considerably large,whereas ˆ σ F T − F Cη = 0 .
14 is relatively small compared to the benchmark ˆ σ T − Cη = 8 .
09, thefractional trend-cycle decomposition yields a smooth trend that only slightly drops dur-ing economic recessions, whereas the I (1) counterpart is more erratic. The small ratioˆ σ F T − F Cη / ˆ σ F T − F Cε may serve as an explanation for the differences between ˆ d F T − F C and thenonparametric estimates ˆ d EW , ˆ d GP H , since a small signal-to-noise ratio can downward-biasthe latter estimators (cf. Sun and Phillips; 2004).14 rend
Cycle FT−FC − Cycle T−C − Figure 2: Trend-cycle decompositions for log US real industrial production with correlatedinnovations. The left plot sketches the trend component estimate from the restrictedmodel (18) (T-C, with d = 1) in black, dashed, together with the trend component fromthe unrestricted model (19) (FT-FC, with d (cid:54) = 1 allowed) in gray, solid. The plots onthe right-hand side show the cyclical components for the unrestricted and the restrictedmodel. Shaded areas correspond to NBER recession periods.The cyclical component from the FT-FC specification is in line with the one obtained forlog US real GDP, as it captures the dynamics from the business cycle well. It sharply dropsduring the NBER recession periods and recovers continuously in the aftermath, whereasthe T-C cycle tends to increase during economic recessions, thereby contradicting economictheory. The results of Weber (2011), who shows that a sufficiently long AR polynomial(in his case p = 10 for monthly industrial production) can produce a more plausible cyclein an I (1) correlated UC setup, are in line with the fractional cycle specification that canbe interpreted as an autoregressive process of order n − Investment: strong cyclical variation
Turning to log US real gross private domestic investment, the nonparametric estimatorsyield ˆ d EW = 1 .
12 and ˆ d GP H = 1 .
15, whereas the maximum likelihood estimator for thefractional trend-cycle decomposition returns a slightly larger ˆ d F T − F C = 1 .
28 as shown intable 1. The likelihood is relatively steep around ˆ d F T − F C = 1 .
28, as figure 5 indicates.Figure 3 shows that the FT-FC specification produces a smoother trend than the T-Cbenchmark and attributes a larger fraction of total variation to the cyclical component.More in line with economic theory, long-run investment from the FT-FC model is almostlinear during economic upswings, whereas the T-C estimate peaks directly before the15 rend
Cycle FT−FC − − Cycle T−C − − Figure 3: Trend-cycle decompositions for log US real gross private domestic investmentwith correlated innovations. The left plot sketches the trend component estimate fromthe restricted model (18) (T-C, with d = 1) in black, dashed, together with the trendcomponent from the unrestricted model (19) (FT-FC, with d (cid:54) = 1 allowed) in gray, solid.The plots on the right-hand side show the cyclical components for the unrestricted andthe restricted model. Shaded areas correspond to NBER recession periods.NBER recession periods. This is especially striking in the 2000s. There, the T-C modelascribes a permanent character to development before and in the great recession. TheFT-FC model instead finds a strong cyclical upswing before the great recession, followedby a pronounced slump of the cycle. Regarding the debate on the nature and effects ofthe recession, this leads to clearly different conclusions. Consumption: smooth trend
Finally, for log US real personal consumption we estimate ˆ d EW = 1 .
40 and ˆ d GP H = 1 .
37 viathe nonparametric estimators. Similarly, the fractional trend-cycle model yields ˆ d F T − F C =1 .
44, as table 1 shows.Contrary to the results obtained for investment, the fractional decomposition attributesless variation to the cyclical component than the T-C benchmark. Hence, transitory con-sumption is estimated to be less volatile over the business cycle in the FT-FC framework.We find this more to be in line with economic theory than the results obtained from theT-C model, which indicate excessive overconsumption directly before a recession period.16 rend
Cycle FT−FC − − Cycle T−C − − Figure 4: Trend-cycle decompositions for log US real personal consumption expenditureswith correlated innovations. The left plot sketches the trend component estimate fromthe restricted model (18) (T-C, with d = 1) in black, dashed, together with the trendcomponent from the unrestricted model (19) (FT-FC, with d (cid:54) = 1 allowed) in gray, solid.The plots on the right-hand side show the cyclical components for the unrestricted andthe restricted model. Shaded areas correspond to NBER recession periods. Structural breaks and longer cycles
Since Perron and Wada (2009) find that the stochastic long-run component of US GDP iswell described by an I(0) process when a trend break in 1973:1 is introduced, we check theimpact of the Perron and Wada (2009) break on our fractional trend-cycle decomposition.Diebold and Inoue (2001) argue that structural breaks and fractional trends can easilybe confused. Hence, the robustness check clarifies whether the better performance of thefractional trend-cycle decomposition results from an ignored trend break.Table 2 reports the parameter estimates when a trend break in 1973:1 is allowed. As itshows, neither the integration order estimates ˆ d , nor the autoregressive parameters andvariance parameters differ substantially. The correlation between long- and short-runshocks is estimated to be slightly weaker when a trend break is introduced. The likelihoodratio (LR) test suggests that introducing a structural break in 1973:1 does not significantlyimprove the goodness of fit for GDP (p-value: 0 . . . . p = 4. In a non-fractional setting this implies that the cycle component contains lagged17nformation from four quarters, which we consider as the maximum lag length of a cyclicalcomponent for quarterly data. By adding additional lags to the cyclical polynomial, weinvestigate if an increased flexibility of the cycle yields the same integration order estimates,or if the estimated fractional integration orders ˆ d are just an artifact from a too restrictiveparametrization of c t . Estimation results are given in table 3 in appendix A. For industrialproduction, investment, and consumption additional lags have no significant impact. ForGDP, slightly different autoregressive coefficients for the cycle are obtained, but they donot increase the overall fit of the model significantly, as a comparison of the likelihoodsshows. Furthermore the estimated integration order is quite similar. The trend-cycledecompositions are sketched in figure 7. Since differences between the decompositionspresented above and those contained in figure 7 are negligible, we conclude that our resultsare robust to additional lags of the cyclical lag polynomials. We generalized unobserved components models to the fractional domain by modelling thelong-run component as a fractionally integrated series together with a cyclical componentwhere the fractional lag operator enters the lag polynomial. We derived the reducedform representation, related the model to the decomposition of Beveridge and Nelson(1981), and showed that the model is uniquely identified independent of the lag lengthof the cyclical polynomial for d (cid:54) = 1. With the modified Kalman filter for the truncated,approximation-corrected state space representation of our fractional UC model we proposeda computationally feasible exact estimator for the latent components.In an application to various macroeconomic series for the US, estimates for the cyclicalcomponent from the fractional trend-cycle model were often found to better capture thebusiness cycle dynamics than those of a benchmark correlated unobserved componentsmodel with an I (1) trend. E.g. for industrial production, the fractional trend-cycle modelwas shown to produce a cycle that is in line with economic theory. Furthermore, thefractional UC models estimated a smoother trend. The reason for the better performanceof fractionally integrated UC models compared to I (1) UC models is the smaller signal-to-noise ratio, i.e. the ratio of long- and short-run shock variances. For d >
1, as in ourfour applications, a violation of the I (1) assumption in I (1) UC models causes an upward-biased estimate of the long-run shock variance, which results in a high signal-to-noiseratio and, therefore, in a volatile trend estimate together with a noisy cycle. In contrast,allowing for fractional trends adequately captures the long-run dynamics of the trend andyields a consistent estimate of the long-run shock variance. In addition, the fractionallag operator L d attributes a higher variance to the short-run shocks to arrive at the same18yclical variance as in the I (1) benchmark for d >
1, thereby lowering the signal-to-noiseratio in the fractionally integrated UC model. Thus, the relatively small signal-to-noiseratio in the fractional model produces smooth trend estimates together with persistentcycles that reflect macroeconomic common sense.The fractional trend-cycle model offers a variety of opportunities for future research. Themodel may be generalized to the multivariate case, where fractional trends of differentpersistence with correlated innovations are allowed. A multivariate fractional trend-cyclemodel would then allow to estimate common fractional trends of cointegrated variablesand test for polynomial cointegration. Furthermore, inferential methods that test for thenumber of common trends or the equality of integration orders could be established. Asshown in Diebold and Inoue (2001), fractionally integrated processes and structural breaksare related, since the former class of processes can produce level shifts and since structuralbreaks can be misinterpreted as I ( d ) processes. Hence, combining both concepts, e.g. in afractional UC model with regime switching, can be a fruitful challenge for future research.To applied researchers, the model offers a flexible data-driven method to treat permanentand transitory components in macroeconomic and financial applications. It provides asolution for many issues of model specification that caused uncertainty and debates aboutrealistic trend-cycle decompositions and estimation of recessions. Based on that, also theinteraction of trends and cycles can be analyzed.19 Graphs and tables
GDP ind. production investment consumptionT-C FT-FC T-C FT-FC T-C FT-FC T-C FT-FCd 1.32 1.66 1.28 1.44(0.12) (0.18) (0.08) (0.07) φ φ -0.58 0.05 -0.67 -0.39 0.13(0.18) (0.04) (0.15) (0.23) (0.05) φ σ η σ ηε -0.95 -0.60 -4.71 -0.45 2.57 -2.11 -1.41 -0.39 σ ε ρ -0.98 -0.97 -1 -0.92 0.85 -0.74 -0.98 -0.99log L -261.01 -260.53 -370.50 -370.03 -629.06 -629.10 -208.12 -199.31Table 1: Estimation results for the trend-cycle decomposition for log US real GDP, logUS real industrial production, log US real gross private domestic investment, and logUS real personal consumption expenditures. T-C distinguishes between an I(1) trendand an autoregressive cycle, and
FT-FC between a fractionally integrated trend and anautoregressive cycle with fractional lag operator. ρ denotes correlation between permanentand transitory shocks. log L is the log likelihood. Standard errors are in parentheses.20DP ind. production investment consumptionEst. Std.Err. Est. Std.Err. Est. Std.Err. Est. Std.Err.d 1.26 0.16 1.65 0.10 1.27 0.09 1.43 0.07 φ φ φ σ η σ ηε -0.71 -0.37 -1.56 -0.41 σ ε ρ -0.97 -0.85 -0.60 -0.99log L -258.55 -368.81 -628.72 -197.89Table 2: Robustness check: Estimation results for the trend-cycle decomposition for log USreal GDP, log US real industrial production, log US real gross private domestic investment,and log US real personal consumption expenditures with a trend break in 1973:1. ρ denotescorrelation between permanent and transitory shocks. log L is the log likelihood.GDP ind. production investment consumptionEst. Std.Err. Est. Std.Err. Est, Std.Err. Est. Std.Err.d 1.22 0.09 1.71 0.14 1.05 0.41 1.47 0.08 φ φ φ -0.32 0.15 0.03 0.01 -0.01 0.08 0.07 0.03 φ -0.05 0.13 0.00 0.01 -0.06 0.08 0.01 0.01 σ η σ ηε -0.22 -0.52 -6.01 -0.37 σ ε ρ -0.64 -0.99 -0.84 -1log L -260.32 -365.04 -628.55 -198.88Table 3: Robustness check: Estimation results for the trend-cycle decomposition for log USreal GDP, log US real industrial production, log US real gross private domestic investment,and log US real personal consumption expenditures with four autoregressive lags. ρ denotescorrelation between permanent and transitory shocks. log L is the log likelihood.21 .8 1.0 1.2 1.4 1.6 1.8 − − − GDP d Log L i k e li hood − − − − − industrial production d Log L i k e li hood − − − − − − gross private domestic investment d Log L i k e li hood − − − personal consumption expenditures d Log L i k e li hood Figure 5: Log Likelihood of the fractional trend-cycle decomposition of log US real GDP( d ∈ [0 . , . d ∈ [1 . , . d ∈ [0 . , . d ∈ [0 . , . θ are fixed and given in table 1.22 rend G D P C Cycle − − − I ND P R O − − G P D I C − − P C E CC − − Figure 6: Robustness check: Trend-cycle decompositions for log US real GDP (GDPC1),log US real industrial production (INDPRO), log US real gross private domestic invest-ment (GPDIC1), and log US real personal consumption expenditures (PCECC96) withcorrelated innovations. The left plots sketch the trend component estimates from the un-restricted models (19) (FT-FC) with a trend break in 1973:1. The plots on the right-handside show the cyclical components for the model with structural break. Shaded areascorrespond to NBER recession periods. 23 rend G D P C Cycle − . − . . . I ND P R O − G P D I C − − P C E CC − − Figure 7: Robustness check: Trend-cycle decompositions for log US real GDP (GDPC1),log US real industrial production (INDPRO), log US real gross private domestic invest-ment (GPDIC1), and log US real personal consumption expenditures (PCECC96) withcorrelated innovations. The left plots sketch the trend component estimates from theunrestricted models (19) (FT-FC). The plots on the right-hand side show the cyclicalcomponents with four autoregressive lags. Shaded areas correspond to NBER recessionperiods. 24
Univariate moving average representation of aggre-gated model
We consider the aggregation of two moving average (MA) processes in the lag operator L d with generic lag polynomials h ( L d ) and ˜ h ( L d ) of order q and ˜ q , respectively, z t = h ( L d ) η t + ˜ h ( L d ) ε t , (20)with the white noise processes (cid:32) η t ε t (cid:33) ∼ i . i . d . (0 , Q ) , Q = (cid:34) σ η σ ηε σ ηε σ ε (cid:35) . In what follows, set p = max( q, ˜ q ) and let h i = 0 for all i > q , ˜ h i = 0 for all i > ˜ q . We firstderive the MA representation in the standard lag operator L = L . Next we derive theMA representation in the fractional lag operator L d which is in general not of finite order.To rewrite (20) in the conventional lag operator L define L kd = (1 − ∆ d + ) k = (cid:32) ∞ (cid:88) i = k ς k,i ( d ) L i (cid:33) + , insert it into (20), and rearrange terms z t = η t + ε t + p (cid:88) k =1 (cid:32) h k t − (cid:88) i = k ς k,i ( d ) η t − i + ˜ h k t − (cid:88) i = k ς k,i ( d ) ε t − i (cid:33) = η t + ε t + p (cid:88) k =1 t − (cid:88) i = k ς k,i ( d ) (cid:16) h k η t − i + ˜ h k ε t − i (cid:17) . Redefining the sum indexes we obtain z t = η t + ε t + t − (cid:88) l =1 η t − l (cid:32) l (cid:88) k =1 ς k,l ( d ) h k (cid:33) + t − (cid:88) l =1 ε t − l (cid:32) l (cid:88) k =1 ς k,l ( d )˜ h k (cid:33) (21)= t − (cid:88) l =0 g l η t − l + t − (cid:88) l =0 ˜ g l ε t − l , (22)with g = ˜ g = 1 and g l = (cid:80) lk =1 ς k,l ( d ) h k and ˜ g l = (cid:80) lk =1 ς k,l ( d )˜ h k , l = 1 , , . . . , t −
1. Notethat both moving average processes are of order n − n . If (22)can be aggregated, there exists a univariate moving average process of order less or equal25o n − z t = c ( L ) u t , u t ∼ i.i.d. (0 , σ u ) . To compute the coefficients c i , note that Cov( z t , c l u t − l ) = Cov( z t , g l η t − l + ˜ g l ε t − l ), whichgives c l σ u = g l σ η + ˜ g l σ ε + 2 g l ˜ g l σ ηε , l = 0 , , . . . , t − . (23)To make the dependence of c l on the parameters of the fractional moving average polyno-mials explicit insert g l and ˜ g l into (23). This delivers for l ≥ c l σ u = (cid:32) l (cid:88) k =1 ς k,l ( d ) h k (cid:33) σ η + (cid:32) l (cid:88) k =1 ς k,l ( d )˜ h k (cid:33) σ ε + 2 (cid:32) l (cid:88) k =1 ς k,l ( d ) h k (cid:33) (cid:32) l (cid:88) k =1 ς k,l ( d )˜ h k (cid:33) σ ηε = l (cid:88) k =1 l (cid:88) i =1 ς k,l ( d ) ς i,l ( d ) (cid:16) h k h i σ η + ˜ h k ˜ h i σ ε + 2 σ ηε h k ˜ h i (cid:17) , (24)with c = 1, σ u = σ η + σ ε + 2 σ ηε . Solving for c l yields the MA coefficients for u t .Next we derive the univariate moving average representation in the fractional lag operatorwhich is typically of infinite order z t = ψ + ( L d ) u t . (25)If (25) exists, then it can be rewritten similarly to (21) in the standard lag operator as z t = u t + t − (cid:88) l =1 u t − l (cid:32) l (cid:88) k =1 ς k,l ( d ) ψ k (cid:33) . For such a representation to exist, there must exist parameters ψ i , i = 1 , . . . , q u such that c l = l (cid:88) k =1 ς k,l ( d ) ψ k , l = 1 , , . . . , t − , while (23) holds. Solving for ψ l delivers ψ l = c l − (cid:80) l − k =1 ς k,l ( d ) ψ k ς l,l ( d ) . (26)Obviously, the order of the moving average polynomial in the fractional lag operator wouldonly be of finite order q u if c l = l − (cid:88) k =1 ς k,l ( d ) ψ k , l > q u . (27)26n general this is not the case. In order to represent the ψ l , l = 1 , ..., q u , in terms of theparameters h j , j = 1 , ..., q , and ˜ h k , k = 1 , ..., ˜ q , of the moving average polynomials in L d ,one inserts (24) into (26) and obtains ψ l = (cid:114)(cid:80) lk =1 (cid:80) li =1 ς k,l ( d ) ς i,l ( d ) (cid:16) h k h i σ η + ˜ h k ˜ h i σ ε + 2 σ ηε h k ˜ h i (cid:17) /σ u − (cid:80) l − k =1 ς k,l ( d ) ψ k ς l,l ( d ) . (28)Since only ψ , ..., ψ l − enter (28), ψ l can be calculated recursively, where the first coefficientis ψ = σ − u (cid:113) h σ η + ˜ h σ ε + 2 h ˜ h σ ηε and σ u = (cid:112) σ η + σ ε + 2 σ ηε . C State space representation
In this section we derive a state space representation of the univariate fractional trend pluscycle model in (1). Since for fixed sample size n every fractionally integrated process oftype II exhibits a finite-order autoregressive representation of length n −
1, an exact statespace form of the system (1), (2), and (4) exists, but is computationally infeasible for large n , as discussed at the beginning of section 3. As a solution, we derive an approximateversion of the system (1), (2), and (4) and directly correct for the resulting approximationerror. Define ˜ y t = ˜ τ t + ˜ c t (29)˜ τ t = µ + µ t + ˜ x t , ˜ x t = [ a ( L, d ) − m ( L, d )] + η t = b + ( L, d ) η t , (30)˜ δ + ( L, d, φ )˜ c t = ε t , ˜ c t = [˜ δ ( L, d, φ ) − ] + ε t = ˜ ω + ( L, d, φ ) ε t , (31)where the approximation errors for (30) and (31) are given in (12) and (13), a ( L, d ) and m ( L, d ) are the ARMA(v, w) polynomials that approximate the fractional difference op-erator in (2) and ˜ δ ( L, d, φ ) truncates the fractional lag polynomial φ ( L d ) = (cid:80) pi =0 φ i L id = (cid:80) ∞ i =0 δ i L i in (4) after lag l . Note that from (9), (10) it follows that E θ ( x t +1 − ˜ x t +1 |F t ) = (cid:15) xt ,E θ ( c t +1 − ˜ c t +1 |F t ) = (cid:15) ct , and consequently E θ (˜ y t +1 |F t ) = E θ ( y t +1 |F t ) − (cid:15) τt − (cid:15) ct = E θ (¨ y t +1 |F t )as defined in section 3.The state equation for the stochastic long-run component ˜ x t is then given by α xt = ˜ x t ˜ x t − ...˜ x t − u +1 ˜ x t − u = a · · · a · · · a u − · · · a u · · · ˜ x t − ˜ x t − ...˜ x t − u ˜ x t − u − + m ... m u − m u − η t = T x α xt − + R x η t , u = max( v, w + 1).The state equation for the cycle follows immediately α ct = ˜ c t ˜ c t − ...˜ c t − l +1 = ˜ δ · · · ˜ δ l − ˜ δ l · · · · · · ˜ c t − ˜ c t − ...˜ c t − l + ε t = T c α ct − + R c ε t . Deterministic terms are incorporated as usual (see, e.g. Durbin and Koopman; 2012, ch.3.2.1) via α µt .Finally, the observations equation is given by˜ y t = (cid:16) Z µ Z x Z c (cid:17) α µt α xt α ct = Zα t , where Z µ = (cid:16) (cid:17) , Z τ = (cid:16) · · · (cid:17) , and Z c = (cid:16) · · · (cid:17) . Since the Kalmanfilter estimates E θ (˜ y t +1 |F t ) = E θ (¨ y t +1 |F t ), and since the resulting prediction error is iden-tical to the one of the exact representation as shown in (11), the maximum likelihoodestimator for the unknown parameters θ based on the approximation-corrected truncatedstate space model (29) - (31) is identical to the one based on the exact representation (1),(2), and (4). 28 eferences Balke, N. S. and Wohar, M. E. (2002). Low-frequency movements in stock prices: Astate-space decomposition,
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