What are the Differences in Trend Cycle Decompositions by Beveridge and Nelson and by Unobserved Component Models?

Transcription

1 What are the Differences in Trend Cycle Decompositions by Beveridge and Nelson and by Unobserved Component Models? April 30, 2012 Abstract When a certain procedure is applied to extract two component processes from a single observed process, it is necessary to impose a set of restrictions that defines two components. One popular restriction is the assumption that the shocks to the trend and cycle are orthogonal. Another is the assumption that the trend is a pure random walk process. The unobserved components (UC) model (Harvey, 1985) assumes both of the above, whereas the BN decomposition (Beveridge and Nelson, 1981) assumes only the latter. Quah (1992) investigates a broad class of decompositions by making the former assumption only. This paper develops a convenient general framework in which alternative trend-cycle decompositions are regarded as special cases, and examines alternative decomposition schemes from the perspective of the frequency domain. In particular, we try to answer two questions: (i) Is the standard uncorrelated UC model a misspecification for describing the postwar US GDP? (ii) Can the correlated UC model proposed by Morley et al. (2003) be a better alternative to the standard UC model? Our empirical investigation appears to suggest negative answers to both questions. JEL Classification: E44, F36, G15 Key Words: Beveridge-Nelson decomposition, Unobserved Component models, frequency domain approach.

2 1 Introduction It has been a common practice in empirical macroeconomic analysis to treat a time-series process such as an output level as the sum of its long-run trend and the short run fluctuation from the trend. These two components are sometimes called the trend and cycle or the permanent and transitory components. This type of decomposition is clearly motivated by the modern macroeconomic theory, which is accustomed to separate analytically the long-run equilibrium of an economy from the short-run adjustment of the economy to an occasional shock. In the case of output, the long-run equilibrium process is treated by the economic growth theory, while the theory of business cycles investigates the short-run fluctuations of output from the trend and the impacts of government policies on its short-run behavior. When a certain procedure is applied to extract two processes from a single observed process, however, it is necessary to impose a set of restrictions that defines two component processes. When there are overidentified restrictions, they can be tested, which might be interpreted as specification tests. One popular restriction is the assumption that the shocks to the trend and cycle are orthogonal. Another is the assumption that the trend is a random walk process. The unobserved components (UC) model (Harvey, 1985) assumes both of the above, whereas the BN decomposition (Beveridge and Nelson, 1981) assumes only the latter. Quah (1992) investigates a broad class of decompositions by making the former assumption only. The orthogonality assumption has several desirable features to model the output process. First, when two shocks are orthogonal, the total variability is the sum of variability of two shocks, and hence the relative importance of each shock is easily calculated. Second, the dynamic response of y t to each shock can be computed and interpreted nicely. If two shocks were correlated, distinguishing the impact of one shock from the other would be diffi cult. Hence, some economists think that the orthogonality assumption is indispensable for the decomposition to be a useful analytical tool. The random walk assumption for the trend component has also some attractive features. First, such a trend is well defined and always identifiable without any additional assumptions. Second, it can be interpreted as the long-run forecast of y t. A potential drawback for this assumption applied to U.S. output is that it makes the trend too volatile from the traditional notion of the trend in the business cycle studies since Burns and Mitchell (1946). This paper examines the relationship among the trend-cycle decompositions suggested by Beveridge and Nelson (1981) and associated with UC models. 1 To do so, we proceed by two steps. First, it develops a general framework in which alternative trend-cycle decompositions are regarded as special cases, each of which corresponds to a particular set of restrictions on the general model. Second, the paper uses this framework to investigate the differences in decompositions associated with various UC models by examining the US GDP process from 1 The literature related to our paper includes Basistha (2007), Lippi and Reichlin (1992, 1994), Morley at al. (2003), Nagakura and Zivot (2007), Nagakura (2008), Nelson (2008), Oh et al. (2008), Proietti (2006), and Sinclair (2009) among others. 1

3 the perspective of the frequency domain as well as the time domain. We find that Morley et al. s (2003) claim that the standard uncorrelated UC model is misspecified would be correct only when US GDP is actually generated from ARIMA(2,1,2) reduced form model. In reality the model uncertainty and the diffi culty of accurately estimating the low frequency features of US GDP leads to a situation in which the fitted processes of the standard uncorrelated UC model and a variety of ARIMA reduced form models are hard to distinguish from one another except in the region of very low frequencies. The correlated UC model proposed by Morley et al. (2003) on the other hand, is found to fall short of the mark as a better alternative. The fitted model tends to be so close to the reduced form itself that its structural interpretation becomes less convincing. The organization of this paper is as follows. In the next section, we develop the general form of the UC model in which the reduced form ARIMA model, the standard (uncorrelated) UC model and the correlated UC model of Morley et al. (2003) are derived as special cases. We show the regression of the cycle shock on the trend shock as a link to connect the above three models. In section 3, we turn our attention to the US GDP process. Through this empirical examination of the alternative decomposition schemes, we ask two important questions in section 4: (i) Is the UC model misspecified? and if so, (ii) Can the correlated UC model provide a remedy? In section 5, we provide a brief conclusion. 2 UC and ARIMA Models 2.1 General Framework The trend-cycle decomposition of y t in the standard form of the unobserved component (UC) model (Harvey 1985) 2 is given by y t = τ t + c t (1) τ t = µ + τ t 1 + η t (2) c t = ψ c (L)ε t (3) where τ t is the trend component, c t is the cycle component of y t, and ψ c (z) = 1 + ψ c,1 z + ψ c,2 z 2 +,which has all roots outside the unit circle. τ t and c t are unobservable. Assume [ ηt ε t ] N(0, Σ), (4) 2 Clark (1989) uses this framework to analyze US GDP. Watson (1986) considers the UC model as a detrending method. 2

4 where Σ is diagonal. In the following we relax this standard UC assumption, and let two shocks η t and ε t be possibly correlated. 3 We write [ ] σ 2 Σ = η σ ηε (5) σ ηε σ 2 ε and consider the regression of ε t on η t ε t = βη t + e t (6) where β = σ ηε /σ 2 η. We can now express the general form of the UC model as (1), (2), (3) with [ ] [ ] [ ] ηt 1 0 ηt = (4 ) β 1 where [ ηt e t ] ε t [ 0 N( 0 e t ] [ 1 0, σ 2 η 0 γ 2 ] ) (5 ) where γ 2 = σ 2 e/σ 2 η so that σ 2 ε = (β 2 + γ 2 )σ 2 η. With the above parameterization, (5) can be expressed as [ ] 1 β Σ = σ 2 η β β 2 + γ 2 (7) and ρ ηε = β/ β 2 + γ 2. An advantage of this parameterization over that in (5) is, unlike (5), non-negative definiteness (n.n.d.) of Σ is automatically guaranteed in (7) without any extra restriction. 4 The model (1), (2), (3), (4 ), (5 ) is a general UC model for which the uncorrelated UC model as well as the ARIMA model are each as a special case. If β is set equal to zero, we have the uncorrelated UC model of Harvey (1985). If γ 2 is set equal to zero, the model becomes the single source of error (SSOE) UC model (Anderson et. al., 2006), equivalent to the ARIMA model. To see the point above, it is useful to have a convenient form of the general model. We take the first difference of (1) and use (2) and (3) to obtain y t = µ + η t + (1 L)ψ c (L)ε t = µ + [1 + β(1 L)ψ c (L)]η t + (1 L)ψ c (L)e t = µ + ζ(l)η t + (1 L)ψ c (L)e t. (8) We note here that [1 + β(1 L)ψ c (L)] η t can be expressed as a general linear process ζ(l)η t, where ζ(z) = 1 + ζ 1 z + ζ 2 z 2 +. To see this, note that 1 + β(1 z)ψ c (z) = (1 + β) β[(1 ψ c,1 )z + (ψ c,1 ψ c,2 )z 2 + (ψ c,2 ψ c,3 )z 3 + ] 3 This is the model discussed by Morley et al. (2003). See also Oh et al. (2008) for a survey of the literature. 4 If parameterization (σ 2 η, σ 2 ε, σ ηε ) rather than ( σ 2 η, β, γ 2) is used, one needs to have a nonlinear restriction σ 2 ησ 2 ε σ ηε > 0. 3

5 The expression of η t and { } ζ j may be found by letting j=0 and η t = (1 + β)η t N(0, (1 + β) 2 σ 2 η) (9) β ζ j = ( 1 + β )(ψ c,j 1 ψ c,j ) (10) for ( j = 1, 2, 3...with ψ c,0 = 1. In other words, there is a one-to-one mapping between } ( {ψc,j ψ, β, {ζj } ) c,j+1 j=0 η) σ2 and, j=0 σ 2 η. Note that the Wold representation of y t is given by y t = µ + ψ(l)u t (11) where u t i.i.d. N(0, σ 2 u) and ψ(z) = 1 + ψ 1 z + ψ 2 z 2 +, which has all roots outside the unit circle. In the following sections we discuss the relationship between (8) and (11). 2.2 Standard UC model The standard uncorrelated UC model developed by Harvey and others consists of (1)-(5) with β = 0 (or σ ηε = 0). In this case, (8) reduces to y t = µ + η t + (1 L)ψ c (L)e t, (12) An important question is whether the general linear process (11) can always be written in the form of (12). A simple answer is no. It is well known that the spectral density of y t in (12) is given by f y (ω) = f τ (ω) + f c (ω) = σ2 η 2π + σ2 ε 2π (1 e iω )ψ c (e iω ) 2 (13) which has the minimum σ 2 η/2π at ω = 0. It is easy to see that the spectrum of y t in (11) does not necessarily have this property. For example, if y t is either AR(1) process with φ > 0 or MA(1) process with θ > 0, then its spectrum has the maximum at ω = 0 and declines monotonically as ω gets larger in the range of 0 ω π. To illustrate this point more clearly, consider a simplest case of (12) where ψ c (z) = 1 + θz. Then the UC representation (12) becomes and the general linear representation (11) becomes y t = µ + η t + ε t (1 θ)ε t 1 θε t 2 (14) y t = µ + u t + ψ 1 u t 1 + ψ 2 u t 2. (15) We know that the class of processes given by (14) does not cover the whole class of processes given by (15). To investigate this relation in more detail, note that the autocovariances of 4

6 (14) are γ 0 = σ 2 η + [1 + (1 θ) 2 + θ 2 ]σ 2 ε (16) γ 1 = (1 θ) 2 σ 2 ε (17) γ 2 = θσ 2 ε (18) while those of (15) are γ 0 = (1 + ψ ψ 2 2)σ 2 u (19) γ 1 = ψ 1 (1 + ψ 2 )σ 2 u (20) γ 2 = ψ 2 σ 2 u (21) We now examine the mapping of ( σ 2 η, σ 2 ε, θ ) into (σ 2 u, ψ 1, ψ 2 ). Note first that calculating γ 0 + 2(γ 1 + γ 2 ) for (16) (18) and (19) (21) and equating them produces the equality σ 2 η = (1 + ψ 1 + ψ 2 ) 2 σ 2 u, which is independent of the choice of θ. This is a standard result and fixes the value of σ 2 u given the values of σ 2 η as well as ψ 1 and ψ 2. Hence, it is only necessary to consider the mapping of (σ 2 ε, θ) into (ψ 1, ψ 2 ). The shaded area in Figure 1 indicates the set of MA parameters (ψ 1, ψ 2 ) for the ARIMA(0,1,2) model that can be expressed as the uncorrelated UC(0,1) model. 5 Since the parameter space of (ψ 1, ψ 2 ) for the unrestricted ARIMA model is the whole area inside the (inverted) triangle connecting the points (2,1), (-2,1) and (0,-1) in the figure, the uncorrelated UC model specification implicitly imposes quite strong restrictions on the parameter space. For example, ψ 1 cannot take any positive value, and ψ 1 + ψ 2 cannot be positive. 6 Lippi and Reichlin (1992) show that ψ(1) 1 is a necessary condition for (11) to have a form (12). Since the postwar US GDP has ψ (1) > 1, this appears to rule out the standard UC model for the US business cycle analysis. This apparently reasonable conclusion prompts researchers to search for alternative UC models. One approach suggested by some authors relaxes the uncorrelatedness assumption between the trend shock and the cycle shock. This is what we discuss in the next subsection. 2.3 UC models with correlated shocks When the trend and cycle shocks are correlated or β 0, we can always find the UC representation (8) for any process written as (11). To do so, we simply set γ 2 = 0, which implies e t = 0 with probability one. This in turn implies from (6) that ε t = βη t. Therefore, (8) reduces to y t = µ + ζ(l)η t (22) Comparing (11) and (22), the uniqueness of the Wold representation implies 5 The derivation is provided in the Appendix. 6 Since ψ(1) = 1+ψ 1 +ψ 2, it follows that ψ 1 +ψ 2 > 0 is equivalent to ψ(1) > 1. Therefore, the uncorrelated UC(0,1) model cannot express ARIMA(0,1,2) model with ψ(1) > 1. This is an example of what Lippi and Reichlin (1992) point out. 5

7 u t = η t = (1 + β)η t (23) ψ(z) = ζ(z) = β [1 + β(1 z)ψ c(z)] (24) and (23), in turn, implies Setting z = 1 in (24), we obtain which, assuming ψ(1) 0, leads to σ 2 u = (1 + β) 2 σ 2 η. ψ(1) = β (25) β = 1 ψ(1). (26) ψ(1) Given ψ(z) and β, solving (24) for ψ c (z) yields ψ c (z) = 1 + β β = [ ψ(z) 1 ] /(1 z) 1 + β ψ(z) ψ(1) [1 ψ(1)] (1 z) = ψ+ (z) 1 ψ(1) (27) where ψ(z) ψ(1) = (1 z)ψ + (z). We now can write the UC model (1) (3) only with ψ j s and u t s. Noting that η t = u t /(1 + β) = ψ(1)u t by (23) and (25), while ε t = βη t = βψ(1)u t = [1 ψ(1)]u t by (6) and (26), we find (2) and (3) to be expressed as where ψ + (z) = [ψ(z) ψ(1)] /(1 z). τ t = µ + τ t 1 + ψ(1)u t (28) c t = ψ + (L)u t (29) The UC model (1), (28) and (29) is known as "single source of error" (SSOE) UC model of Anderson, Low and Snyder (2006). It is also known that this UC model is an alternative way to express the Beveridge-Nelson decomposition. This can be seen by writing this model in the form of (8) as y t = µ + ψ(1)u t + [ψ(l) ψ(1)]u t. Recalling γ 2 = 0, the correlation between the trend and cycle shocks is found to be β ρ ηε = β 2 + γ = β { 1 if β > 0 2 β = 1 if β < 0 or ρ ηε = { 1 if 0 < ψ(1) < 1 1 if ψ(1) < 0 or ψ(1) > 1. 6

8 The above result is well known. One important implication is that the likelihood value of the correlated UC model given by (1) (5) is maximized at γ 2 = 0 or equivalently ρ ηε = 1. This is simply because the correlated UC model is a one-to-one transformation of the ARIMA model (11) at γ 2 = 0. Unfortunately, this solution is too trivial to attract many empirical macroeconomists for an obvious reason. The UC model is originally proposed by Harvey and others as a "structural" time series model. It is not structural in the usual sense of the word used in the econometrics profession. But attempting to describe a time series as being generated from two uncorrelated shocks, each of which respectively drives its permanent and transitory component, the model can provide a useful interpretation to help get a structural understanding. In this sense the UC model with perfectly correlated shocks would lose all its attractiveness since it is diffi cult to attach any "structural" interpretation. The correlated UC model developed by Morley et al. (2003) may be regarded as an attempt to get around such a diffi culty. The idea is the following. Suppose y t and c t are both in ARMA form so that ψ(z) and ψ c (z) are expressed as ψ(z) = θ(z)/φ(z) and ψ c (z) = θ c (z)/φ c (z), where θ(z) = 1 + θ 1 z + + θ q z q, φ(z) = 1 φ 1 z φ p z p, θ c (z) = 1 + θ c1 z + + θ c q z q, and φ c (z) = 1 φ c1 z φ c p z p. Then, in order for models (8) and (11) to represent the same y t process, we first need p = p, q = max(p, q +1), and φ j = φ j for j = 1, 2,, p. Since AR parameters φ j s are always identified, we ignore them for a while. Then we have two sets of parameters: ( σ 2 η, β, γ 2, θ c,1, θ c, q ) for the UC model (8) and (σ 2 u, θ 1,, θ q ) for the general linear model (11). The number of parameters is q + 3 for the former model, and is q + 1 for the latter. The discussion in the previous section shows that if the value of γ 2 is set equal to zero, (8) and (11) can be expressed as a one-to-one transformation of each other. Note that we have q = q + 1 here, while in the previous discussion, we assume q = q =. When we set q = q + 2, however, the numbers of parameters in models (8) and (11) become the same. In this case, Morley et al. (2003) show that we can find a correlated UC representation of the general ARIMA model (11) with γ 2 0. It is a set of parameter values of the UC model (8) with γ 2 0 that yields the autocovariances γ 0, γ q equal to those of the model (11). There are a couple of points in need of attention. First, the uncorrelated UC model with the ARMA(2,0) cycle has the reduced form ARIMA(2,1,2), and is a restricted version of Morley et al. s correlated UC model with ARMA(2,0) cycle, with the restriction β = 0 (or ρ ηε = 0). However, this correlated UC model does not coincide with the original ARIMA(2,1,2) when γ 2 = 0. This point will be discussed in more detail in subsections 3.3 and 4.2. Second, the correlation of the trend and cycle shocks has the same sign in Morley et al s correlated UC model as in the BN decomposition. This is simply because the correlation ρ ηε is equal to β/ β 2 + γ 2, whose sign is same as that of β regardless of the size of γ 2. 7 For US GDP, for example, the BN decomposition leads to a perfect negative correlation of the trend and cycle shocks. Therefore, the shock correlation is always negative in the correlated UC model. Any "structural" interpretation of this sign would be diffi cult to justify. Third, the method of moment estimates of the UC model obtained by equating its autocovariances with that of the ARIMA model are not necessarily the MLEs. 7 Nagakura and Zivot (2007) use Kolmogorov s Lemma to prove ψ(1) > 1 implies β < 0. 7

9 3 US GDP In this section we examine a variety of trend-cycle models when applied to the US GDP process. Our special focus is on comparison between the BN decomposition and the UC models. Data used in our analysis are quarterly data on US real output from 1947Q1 through 2009Q Estimating three UC models In the following, the standard UC model with uncorrelated trend-cycle shocks whose cycle component is characterized by ARMA(p,q) will be denoted by UUC(p,q). If the two shocks are totally or perfectly correlated so that there is only a single shock, we denote it TCUC(p,q). If the two shocks are correlated but only partially, or equivalently, the absolute value of the correlation of two shocks is between 0 and 1, we denote it PCUC(p,q). The analysis in the previous section implies that TCUC(p,q) is identical to ARIMA(p,1,q+1), while UUC and ARIMA do not have one-to-one mapping. For example, both UUC(2,0) and UUC(2,1) imply ARIMA(2,1,2). But not all ARIMA(2,1,2) models have the exact UUC representation, as we illustrated in the previous section in ARIMA(0,1,2) case. The first column of Table 1 presents the relation between UUC(p, q) and ARIMA(p,1,q) for the values of p and q that are relevant for studying US GDP. The second and third columns display the pairs of the numbers of the AR and MA orders, respectively, of the corresponding TCUC and PCUC models. As you see, for any ARIMA model, there is always a corresponding TCUC model. This is not the case for the PCUC model. In the TCUC model, the number of parameters, apart from the AR parameters, is q+2 (θ c s, σ 2 η and β), while there is one additional parameter γ 2 in the PCUC model, so we need q+3. Unlike the TCUC(p,q-1) model, for which there exists a oneto-one mapping between q-1 parameters of θ c s with β = [1 ψ(1)] /ψ(1) and q parameters of θ s, not all ARIMA(p,1,q) models can be expressed as the PCUC(p,q-2) with β, γ 2 and (q 2)θ c s. In Table 1, we see that PCUC(p, q) can be found corresponding to ARIMA(p,1,q) as long as q 2. But we cannot regard those PCUC s as unrestricted versions of UUC s in all cases. For example, not only UUC(2,1) but also UUC(2,0) imply ARIMA(2,1,2), so PCUC(2,0) can be introduced as an unrestricted version of UUC(2,0). But UUC(1,0) is not a restricted version of PCUC(1,0) since the former corresponds to ARIMA(1,1,1) while the latter corresponds to ARIMA(1,1,2). In fact, characterizing PCUC as an unrestricted version of UUC is possible only in two cases in Table 1, ARIMA(2,1,2) and ARIMA(3,1,3). Table 2 reports the estimated parameters of three models: TCUC, UUC, PCUC models. Panel(a) shows the estimates of the models corresponding to ARIMA(2,1,2), namely TCUC(2,1), UUC(2,0), UUC(2,1) and PCUC(2,0). The estimates of the models corresponding to ARIMA(3,1,3) are reported in Table 2(b), namely TCUC(3,2), UUC(3,1), UUC(3,2) and PCUC(3,1). The most striking result in Table 2(a), for ARIMA(2,1,2) as the reduced 8 Output data are from the DRIBASIC (formerly CITIBASE) database (data series gdpq). 8

10 form, is a stark difference in the size of the variances σ 2 η of the trend shock in TCUC(2,1) (the second column) on one hand and in UUC(2,0) and UUC(2,1) (the third and fourth columns) on the other. The former is almost four times as large as the latter twos. A similar result is found in the case of ARIMA(3,1,3) as the reduced form reported in Table 2(b). As you recall, TCUC model is equivalent to the BN decomposition, and UUC model is the standard UC model. What puzzles many researchers is why two models widely accepted as standard tools produce such a large disagreement in measurement (e.g., Morley et al. 2003). This contrast can be seen in an even more dramatic fashion in panels (a) and (b) of Figure 2. Figure 2(a) displays the spectral density functions of y t based on ARIMA(2,1,2) model and the UUC(2,0) model (with the shaded area showing the business cycle frequencies). 9 The spectrum of y t implied by the TCUC(2,1) model or the BN decomposition is the same as that of the ARIMA model. The two functions ARIMA(2,1,2) and UUC(2,0) look quite different except over a high frequency region. The spectrum of y t based on the ARIMA model starts with about at frequency zero, peaks at frequency 0.203π, then declines quickly, and overlaps with the spectrum implied by the UUC(2,0) model after frequency 0.5π. In contrast, the spectrum for the UUC(2,0) model starts with at frequency zero, rapidly increases until it peaks at frequency 0.090π and then gradually declines. Since f y (0) = σ 2 η/2π, the difference in the height of the two spectra at frequency zero reflects the difference of σ 2 η in the two models. The entire empirical distributions for the height of the two spectral densities at frequency zero are displayed in Figure 2(b). The horizontal axis measures the size of the long-run variance of y t divided by 2π, namely σ 2 η/2π. The modes of the empirical distributions are reached at and The estimated value is smaller than the lower 2.5% quantile (0.090, marked by the vertical bar in Figure 2(b)) of the empirical distribution of the trend shock variance based on the ARIMA(2,1,2) model. Therefore, if ARIMA(2,1,2) is the true model, the long-run variance is very unlikely to be equal to the value 0.058x2π under the UUC(2,0) model. A natural question is what makes these two estimates so different. A quick answer might be that the standard UC (or UUC) model is misspecified. As we saw in subsection 2.2, the spectrum of y t in the standard UC form is the sum of the spectrum of a white noise ( τ t ) and I(-1) process ( c t ). The former is a positive constant ( σ 2 η/2π ) and the latter is a hump-shaped curve starting from the origin at frequency zero. It implies that the ordinate of the spectrum for y t is always higher than that at frequency zero, which contradicts the shape of y t in the ARIMA representation of US GDP (seen in Figure 2). Lippi and Reichlin (1992) show that ψ(1) < 1 is a necessary condition for y t to have the standard UC representation with uncorrelated shocks. The estimate of ψ(1) for US GDP under ARIMA(2,1,2) is 1.28 with our data. The estimated parameters for the uncorrelated UC model can be regarded as the maximizer of the misspecified likelihood function. The trend shock variance ( ση) 2 has to be significantly underestimated in order to give a room to let the spectrum of c t have a positive ordinate. But then c t has to have a near unit root to compensate the underestimation at frequency zero. According to this view, as is seen in 9 This comparison of two spectra is first shown by Proietti (2006). 10 These figures are quite close but not necessarily equal to our estimated values (0.058 and 0.222). 9

11 Figure 2(a), distortion of the spectrum shape appears substantial. A pitfall of this view when applied to the actual GDP data is that once the ARIMA(2,1,2) model is taken as a true model, we immediately find a pretty accurate estimate of the spectrum point at zero frequency even when we actually have little information about such a low frequency feature of the data. In fact, this accuracy comes from a specific choice of the parametric model rather than from the data itself. While the spectrum shapes of the unit-root process and the near-unit-root process are unmistakably different from each other at frequency zero or in its neighborhood, it is at the same time extremely diffi cult to distinguish the two processes with the actual series observations of a finite length. This is a special case of what Faust(1999) discusses regarding the confidence intervals of the spectrum points in a more general setting. We discuss this issue again in light of US GDP data later in subsection The cycle components of US GDP implied by various models One way to evaluate the practical usefulness of a particular procedure (or model) of the trend-cycle decomposition is to look at the spectrum shape of the cycle component. The right hand column of Figure 3 displays the plots of the cycle component implied by the models popular among macroeconomists: (a) the BN decomposition, (b) the standard uncorrelated UC model (denoted by UUC(p, q) model), (c) the Perron-Wada (2009) Trend-break model, (d) the HP filter (Hodrick and Prescott 1999), and (e) the Bandpass filter (Christiano and Fitzgerald 2003). To be precise, (d) and (e) are not originally proposed as procedures for decomposition. The shaded areas indicate the NBER recessions. The left hand column shows the corresponding spectral density of the cycle component of each model. The shaded area of each figure in this column indicates the range of business cycle frequencies (defined by the period of 6 to 24 quarters). A striking finding here is that, as seen in the left column of Figure 3(a), the BN cycle almost totally fails to capture the business cycle frequencies of US output. The right column shows that the BN cycle exhibits too many small fluctuations and does not accord with the NBER chronology in any sense. This fact has been pointed out by many authors, which some macroeconomists argue, discredits the BN approach as a way to separate the cyclical component of US output from the trend in a conventional sense. The cycles in the UUC(2,0) model (Figure 3(b)) and the Trend-break model (Figure 3(c)) are similar except that the trend after 1972 is adjusted down in the latter model (as emphasized in Perron & Wada 2009). Also the cycles in the HP filter and the Bandpass filter are similar. 11 Not surprisingly the cycle process obtained by the Bandpass filter captures the business cycle frequencies very well. So is the HP filter. The cycles implied by UUC(2,0), Trend-break models and the HP filter look more in accordance with the NBER business 11 Note that the spectral densities of these cycles are equal to zero at frequency zero, implying that they are I(-1) processes. In other words, both filters work when the GDP process is not only I(1) but also I(2). 10

12 cycle chronology than other models Partially correlated UC model suggested by Morley et al. (2003) As discussed in section 3.1, standard uncorrelated UC models, UUC(2,0) and UUC(2,1) both imply ARIMA(2,1,2) model as the reduced form. Autocovariances of (1 φ 1 L φ 2 L 2 ) y t give three moment conditions leading to the equivalence of two models, which help exactly identify three parameters ( σ 2 η, θ c, γ 2) of the UUC(2,1) model. If θ c is set equal to zero, we have UUC(2,0) model, which is overidentified. In this case, we can identify one more parameter, ( for example β, which is set equal to zero for the UUC models. It estimates σ 2 η, β, γ 2) instead of ( σ 2 η, θ c, γ 2). This is what Morley et al. (2003) estimate with the US GDP data, namely PCUC(2,0) model. The estimation result reported in the 5th column of Table 2(a) is remarkable in several respects. The PCUC(2,0) model is exactly identified as is the UUC(2,1) model, the latter of which is reported in the 4th column. Each of these two models can be regarded as an unrestricted model relative to the UUC(2,0) model. The UUC(2,1) model relaxes restriction θ c = 0, while the PCUC(2,0) model relaxes restriction β = 0. The two consequences are starkly different. The estimate of θ c is insignificant and the UUC(2,1) model does not improve the likelihood at all. The size of the estimated γ 2 becomes half as large as that for UUC(2,0) model and loses its significance. In contrast, the PCUC(2,0) model raises the likelihood up to the level of the unrestricted ARIMA(2,1,2) model. But if you compare the parameter estimates of the PCUC(2,0) and TCUC(2,1) models row by row, you can find that the two sets of estimates are almost identical except the values of θ c, β,and γ 2. In particular, the size of the variance of the trend shock, σ 2 η, gets much larger than the one estimated for the UUC(2,0) model and now becomes exactly equal to that for the TCUC model. The estimated size of γ 2 is much smaller. The value of parameter β, on the other hand, is found very similar to that for the TCUC(2,1) model but slightly smaller, resulting in the correlation between the trend and cycle shocks. This is important because the two shocks are still not perfectly correlated and hence not reduced to a single shock. Another remarkable aspect of this estimation result is that the likelihood of the estimated PCUC(2,0) model reaches the same level as that of the TCUC(2,1) model. The value of θ c is not necessary to set equal to zero in order to identify other parameters. The PCUC(2,1) model with θ c = (the 6th column of Table 2(a)) reproduces the TCUC(2,1) model (the 2nd column) pretty closely, which is not a surprising result. In this case, the estimated γ 2 becomes so small that the correlation of the trend and cycle shocks becomes In fact, the likelihood surface appears almost completely flat with θ c being in the range of (0, 0.082). 12 This might imply that inclusion of the frequency range a little lower than what is implied by 6 years period would help make cycles more in accordance with what we regard conventionally as the business cycle. The UUC(2,0) model does not look so bad in this respect. 11

13 The above result is impressive enough to make one to believe that if we relax the apparently misspecified zero correlation restriction of the standard UC model, we would obtain the PCUC model with no distortion in the size of the long run variance and without loss in the likelihood. Although strongly correlated, the PCUC model has two shocks so that we can use it for a structural analysis of what we observe. However, this does not appear to be the case when the reduced form is chosen to be ARIMA(3,1,3), as is shown in Table 2(b). The likelihood of the PCUC(3,1) model (the 5th column) does not reach the level of the TCUC(3,2) model (the 2nd column) and the maximum likelihood estimate of the correlation of the two shocks turns out to be negative one. In other words, the estimated PCUC(3,1) model reduces to a single shock model. 4 Discussion 4.1 Is the standard uncorrelated UC model really misspecified? As we see in section 3.1 (Figure 2(a) and 2(b) in particular), the spectrum shape of y t appears to be severely distorted under the UUC model specification at and near the frequency zero. However, we know that the spectral density values at very low frequencies are most diffi cult to estimate. Our popular practice of parsimonious parameterization tends to give a false impression regarding the accuracy of estimated low frequency features of data. In this section, we further investigate this issue more closely. Figure 4 displays the spectra of y t associated with various UUC models, namely UUC(1,0), UUC(1,1), UUC(2,0), UUC(2,1), and UUC(3,1). Except for two models with smallest parameterization, the spectra of y t implied by the UUC models with (2,0), (2,1) and (3,1) all exhibit a similar shape. Therefore, a large discrepancy of the spectrum of the UUC model from that of the reduced form ARIMA model is not particular to the UUC(2,0) model. We now investigate whether or not this discrepancy is particular to a choice of an ARIMA model. Figure 5 displays the spectra of the UUC(2,0) model versus various ARIMA models that reasonably fit to the US GDP series. Panels (a), (b), and (c) show the spectrum of y t of, respectively, ARIMA(1,1,q), ARIMA(2,1,q), ARIMA(3,1,q) models for q = 0, 1, 2, The spectrum of UUC(2,0) model is common in all these panels and can easily be distinguished in each panel from the spectra of ARIMA models because the spectral ordinate of UUC(2,0) at zero frequency is much lower than those for any ARIMA models. What we find particularly interesting in Figure 5 is that the spectral shapes of y t fitted to the parameterization with relatively low dimension are amazingly in close agreement with each other, including ARIMA(1,1,q) for q = 0, 1, 2, 3 as well as ARIMA(2,1,q) for q = 0, 1. As we move to higher dimensional parameterizations, the shape of the spectrum starts to fluctuate, in 13 As we mention elsewhere, we fail to obtain the stable estimates for ARIMA(2,1,3) and ARIMA(3,1,2) models. 12

14 particular, in the cases of ARIMA(3,1,q) for q = 0, 1, 3. The ARIMA(2,1,2) model, however, is a best choice based on the AIC (Akaike Information Criteria) and is the center of focus in Morley et al. (2003). As we see in the previous section, among the ARIMA(p,1,q) models for p, q 3, ARIMA(2,1,2) and ARIMA(3,1,3) are the only reduced form models in which PCUC(p,q-2) can be regarded as an unrestricted model relative to UUC(p,q-2). Our empirical focus is also on these two models, but they are also the two models that produce the largest discrepancy from the UUC models. Figure 5 displays this contrast. In the range of business cycle frequencies (the shaded area in each panel of Figure 5), the spectrum of y t under the UUC(2,0) model overlaps quite well with all those under ARIMA(1,1,0), (1,1,1), (1,1,2), (1,1,3), (2,1,0), and (2,1,1). As pointed out above, it deviates most from those under ARIMA(2,1,2) and (3,1,3). Since the AIC is not the only criteria for the model choice and the ARIMA(1,1,0), (1,1,1) and (2,1,0) all are often popular choices of macroeconomists to describe US GDP, the above point is important. Actually ARIMA(1,1,0), ARIMA(1,1,1) and ARIMA(2,1,0) models are the best choices under the Schwartz Criteria or BIC, 14 and are clearly more popular choices among empirical macroeconomists than ARIMA(2,1,2) or ARIMA(3,1,3), the latter two of which might be regarded as overparameterized. As far as the business cycle frequencies are concerned, the spectrum of y t under UUC(2,0) does not look so poor to describe the spectrum of actual y t. In fact, it does a quite good job as long as we are ready to accept ARIMA(1,1,0), (2,1,0), (1,1,1) models as the reasonable models for US GDP. The above issue is closely related to a well-known diffi culty in distinguishing a "near unit root" from a "unit root," which generated a heated discussion in the past. Since our available data are always finite, it is ultimately impossible to tell whether the autoregressive root of the data is exactly one or extremely close to but smaller than one. They are near observationally equivalent. We can see this phenomenon is carried over into the decomposition problem also. As we saw previously, in the standard UC model, the uncorrelatedness assumption of the trend and cycle shocks leads to a much smaller long-run variance estimate than that in the ARIMA model. This is necessary to give a room for the cycle component to have a positive ordinate in the spectrum. But then the cycle process has to have a near unit root 15 to quickly recover the initial loss of height in the spectrum, which is exhibited clearly in Figure 5. Just as we can hardly distinguish a "near unit root" from a "unit root," the low frequency behavior of the uncorrelated UC model is hard to distinguish from that of the ARIMA model with time series observations of a limited size. Figure 6 illustrates the above situation in a more dramatic fashion. Panel (a) of Figure 6 displays the spectra of variety of ARIMA models fitted to the artificial data generated from the UUC(2,0) model with estimated parameter values. 16 It also displays the original 14 The top 5 choices of ARIMA(p,1,q) models under AIC and BIC for the US GDP with 1 p 3 and 0 q 3 in the descending order are the following. AIC: (2,1,2), (3,1,1), (3,1,0), (1,1,0), (2,1,0); BIC: (1,1,0), (2,1,0), (1,1,1), (3,1,2), (3,1,0). 15 The cycle component in the UUC(2,0) model is AR(2) and its estimated AR parameters imply 1 ˆφ 1 ˆφ 2 = = 0.06, near unit root (see Table 2(a)). 16 The method used for bootstrapping UUC and ARMA models is a revised procedure in RATS based on Stoffer and Wall (1991). We would like to thank Tom Doan for his coding help. 13

15 UUC(2,0) model for comparison. Here we know that the true data generating process (DGP) is UUC(2,0). However, there is little resemblance between the spectrum of the true UUC model on one hand and a group of spectra for all fitted ARIMA models on the other in the region of low frequencies. In particular, there is a large discrepancy in two sets of spectra at frequency zero. The spectral value of the true UUC model at frequency zero is about 0.058, while those of the fitted ARIMA models all lie between 0.27 and In fact, the spectra of fitted ARIMA models bear a surprisingly close resemblance to those ARIMA spectra in Figure 5 that are estimated from the actual GDP series. Panel (b) of Figure 6 is the result of an exercise in the opposite direction. Here the artificial data are generated from the ARIMA(2,1,2) model with parameters estimated using US GDP. Figure 6(b) displays the spectrum of the UUC(2,0) model fitted to those artificial data, along with the spectrum of the original ARIMA(2,1,2) model. Here, the fitted UUC spectrum does not look like the original ARIMA spectrum in the low frequency region. Again the fitted UUC spectrum more resembles those UUC spectra in Figure 4 estimated from actual US GDP, but a little less successfully compared to panel (a). The lessons we learn from the above exercise are two folds. First, we cannot rule out the possibility that actual US GDP is generated from the UUC(2,0) model even though the reduced form ARIMA models seem to contradict the UUC model in features regarding the low frequencies. Second, the shape of the estimated spectrum depends more on what kind of models fitted to the data than on what kind of models to generate the data. But this suggests that data generated from a particular model do not contain much information about low frequency features of the model. 4.2 How do we interpret the correlation of shocks? International comparison In section 3.3 we observed that when the reduced form model is ARIMA(3,1,3), the PCUC model reduces to the TCUC model. In fact, this degeneration of PCUC models into a perfect correlation looks more common in other GDP processes. Table 3 presents the PCUC model estimates for the GDP processes of the G7 countries 17 during the same period as US data we examined. Panel (a) reports the estimates of the PCUC(2,0) model, while panel (b) reports those of the PCUC(3,1) model. The reduced forms of those two PCUC models are ARIMA(2,1,2) and ARIMA(3,1,3) models, which are not necessarily best fits for all countries (see panel (c) of Table 3). So some caution needs to be exercised when interpreting the result. However, simple counting of the cases reveals that the estimated shock correlations in the PCUC models for all countries except the US in panel (a) and except Germany and Italy in panel (b) 18, are In light of the model fit reported in panel (c), it probably makes 17 GDP and GDP deflator data of the G7 countries are from the International Financial Statistics (IFS) database published by the IMF. For Canada, data is from 1957 Q1 to 2010 Q1; France, 1970 Q1 to 2010 Q1; Germany, 1960 Q1 to 2010 Q1; Italy, 1980 Q1 to 2010 Q1; Japan, 1979 Q1 to 1995 Q1; UK, 1957 Q1 to 2010 Q1. 18 GDP series of France and Italy do not agree with ARIMA(2,1,2) or ARIMA(3,1,3) very well. US and Japan GDP are characterized clearly by ARIMA(2,1,2) and Germany is by ARIMA(3,1,3). 14

16 more sense to exclude France, Germany and Italy from panel (a) and France, Italy, Japan and US from panel (b). Even in this case, three out of four cases in panel (a) and two out of three cases in panel (b) involve perfect negative correlations. Hence, degeneration of PCUC models into a single shock model appears more common than otherwise. The above observation that the PCUC model often reduces to a single shock model provides an important insight that tends to be forgotten. First, the TCUC(2,1) model as well as the TCUC(2,0) model correspond to ARIMA(2,1,2). Second, the PCUC(2,0) model is the only identifiable PCUC model that corresponds to ARIMA(2,1,2). Third, the PCUC(2,0) model can degenerate into the TCUC(2,0) model but not the TCUC(2,1) model. The analogous relations hold for TCUC(3,2), TCUC(3,1) and PCUC(3,1). Table 4 displays comparison of TCUC(2,1), TCUC(2,0) and PCUC(2,0) for the US and Canada in panel (a), and comparison of TCUC(3,2), TCUC(3,1) and PCUC(3,1) for Germany and UK in panel (b). The difference between US and Germany on the one hand and Canada and UK on the other is in that the PCUC models in the first group, unlike those in the second, do not degenerate into single shock models. A closer look reveals that the likelihood reached by the PCUC model is as high as that of the TCUC(2,1) model and higher than the TCUC(2,0) model in the US case, whereas the likelihood of the PCUC(2,0) model in Canada has only the same height as that of the TCUC(2,0) model and lower than the TCUC(2,1) model. An exactly analogous relation is found for the PCUC(3,1) model in Germany and UK. In all four countries (and other countries not reported here), dropping an MA parameter reduces the likelihood, or the likelihood of TCUC(2,0) is lower than that of TCUC(2,1) and the likelihood of TCUC(3,1) is lower than that of TCUC(3,2). The difference of the degenerate case of Canada and UK from the non-degenerate case of US and Germany depends on the question whether or not the loss of the likelihood by omitting one MA parameter is compensated by allowing for two correlated shocks instead of a single shock through the introduction of nonzero parameter γ 2 (or equivalently allowing ρ to deviate from -1.0). The correlated two shocks do not restore the loss of likelihood for the degenerate case but they do for the non-degenerate case. This last point, in turn, appears to depend on whether or not an omission of one MA parameter would significantly reduce the likelihood. The loss of the likelihood values are small in the US-Germany case relative to the Canada-UK case. Characterizing the correlation of shocks in the PCUC model in the fashion described above would provide us with a new insight into understanding the UC models more deeply. Many macroeconomists frequently find that the correlation of the trend-cycle shocks of GDP and other macroeconomic series of the US as well as other countries to be negative and close to negative one. This is perfectly reasonable if we understand the shock correlation of the PCUC model in the context of the tradeoff between a MA parameter and γ 2 as shown above. It is better to think of a large negative correlation as a statistical artifact than reflecting a certain structural meaning contained in economic data. If this is the case, it would be misleading to read something structural from the findings of a negative correlation and attempt to interpret the result accordingly. 15

17 5 Conclusion This paper first developed a general framework of the UC model in which (i) the UUC (standard uncorrelated UC) model, (ii) the TC(totally correlated)uc model (equivalently, general ARIMA model or SSOE model), and (iii) the PC(partially correlated)uc model (proposed by Morley et al. (2003)) are special cases obtained by imposing a set of restrictions on the general model. This framework helps understand the link between different models and avoid unnecessary confusion. Equipped with this framework, we examined what makes the trend-cycle decomposition so different for different models. We lastly asked two questions: (i) whether the UUC model is a misspecification when applied to US GDP data, and if so, (ii) whether the PCUC model can provide a cure for this deficiency. If there were no model uncertainty and we were sure that US GDP is generated from the ARIMA(2,1,2) model, then it would be perfectly correct to say that the standard uncorrelated UC model with an AR(2) cycle component is a misspecification as Morley et al. (2003) argue. It is because the implications regarding the low frequency features by the two models are so different. In reality, however, uncertainty in model specification is a rule rather than an exception when one tries to fit a reduced form ARIMA model to time series data. Since observations with limited length usually available for macroeconomic variables contain very little information about the low frequency features, model uncertainty makes it extremely diffi cult with available data to distinguish the two models even when they have vastly different implications of low frequency features. The correlated UC model proposed by Morley et al. (2003), on the other hand, has two major problems. First, it has a technical limitation in application. Second, even when it applies, it would not be much useful as a device to investigate dynamics of macroeconomic data. A large negative correlation so frequently found by researchers in correlated UC analysis is better to be regarded as a statistical artifact rather than reflecting any structural information content in data. 16

18 Appendix: Derivation of admissible (ψ 1, ψ 2 ) set in Figure 1 Equating, respectively, (16) with (19), (17) with (20), and (18) with (21), we find three autocovariance conditions. σ 2 η + [ 1 + (1 θ) 2 + θ 2] σ 2 ε = ( 1 + ψ ψ 2 2) σ 2 u (A1) (1 θ)σ 2 ε = ψ 1 (1 + ψ 2 )σ 2 u (A2) θσ 2 ε = ψ 2 σ 2 u (A3) As discussed in section 2.1, we obtain, from the above three equations, a familiar condition σ 2 η = (1 + ψ 1 + ψ 2 ) 2 σ 2 u (A4) To find the restrictions on (ψ 1, ψ 2 ) space implied by the uncorrelated UC model, we substitute (A4) into (A1) and rearrange the terms to obtain 2(ψ 1 + ψ 2 + ψ 1 ψ 2 )σ 2 u = [ 1 + (1 θ) 2 + θ 2] σ 2 ε This implies ψ 1 + ψ 2 + ψ 1 ψ 2 < 0 or equivalently (ψ 1 + 1)(ψ 2 + 1) < 1 (A5) Second, directly from (A2), we also need ψ 1 (ψ 2 + 1) < 0 (A6) Third, multiplying (A3) by 2 and adding the result to (A2) yields (ψ 1 + 2ψ 2 + ψ 1 ψ 2 ) σ 2 u = ( 1 + θ 2) σ 2 ε which implies ψ 1 + 2ψ 2 + ψ 1 ψ 2 < 0, or equivalently (ψ 1 + 2)(ψ 2 + 1) < 2 (A7) Now, it is easy to show that the invertibility condition of y t is given by 1 + ψ 1 + ψ 2 > 0 (A8) 1 ψ 1 + ψ 2 > 0 (A9) and ψ 2 < 1 (A10) We want to know what values of (ψ 1, ψ 2 ) satisfy all the conditions (A8) (A10), but do not satisfy at least one of the condition of (A5) (A7). Alternatively, which subset of the set (ψ 1, ψ 2 ) satisfying (A8) (A10) also satisfy all (A5) (A7)?. The triangle connecting the points (2,1), (-2,1), and (0,-1) in Figure 1 is the set satisfying (A8) (A10). Any (ψ 1, ψ 2 ) pair in this set corresponds to some ARIMA(0,1,2) process that is invertible. Now, (A8) and 17

19 (A9) together imply that ψ is always positive in this set, so (A6) implies ψ 1 < 0 (A11) But then, when ψ 2 is also negative, (A6) and (A7) both are automatically satisfied. When ψ 2 is positive, (A7) implies (ψ 1 + 1)(ψ 2 + 1) < 2 (ψ 2 + 1) = 1 ψ 2 < 1 which is just (A5). So, if (A7) is satisfied, (A5) is no longer binding. Therefore, the set of points in (ψ 1, ψ 2 ) space that satisfy all the conditions (A5) (A10) need to satisfy (A7) (A11). This is given by the shaded area in Figure 1. 18

20 References [1] Anderson, H. M., C. N. Low, and R. Snyder (2006) "Single Source of Error State Space Approach to the Beveridge Nelson Decomposition," Economic Letters 91(1), [2] Basistha, A. (2007) "Trend-Cycle Correlation, Drift Break and the Estimation of Trend and Cycle in Canadian GDP," Canadian Journal of Economics, 40(2), [3] Beveridge, S. and C. R. Nelson (1981) "A New Approach," Journal of Monetary Economics 7, [4] Burns, A. F., and W. C. Mitchell (1946) "Measuring Business Cycles," National Bureau of Economic Research, New York, NY. [5] Christiano, L. J. and T. J. Fitzgerald (2003) "The Band Pass Filter," International Economic Review 44(2), [6] Clark, P.K. (1987) "The Cyclical Component of US Economic Activity," Quarterly Journal of Economics 102, [7] Faust, J. (1999) "Conventional Confidence Intervals for Points on Spectrum Have Confidence Level Zero," Econometrica 67, [8] Harvey, A.C. (1985) "Trends and Cycles in Macroeconomic Time Series," Journal of Business and Economic Statistics 3, [9] Hodrick, R.J. and E.C. Prescott (1999) "Postwar U.S. Business Cycles: An Empirical Investigation," Journal of Money, Credit, and Banking 29, [10] Lippi, M. and L. Reichlin (1992) "On Persistence of Shocks to Economic Variables: A Common Misconception," Journal of Monetary Economics 29, [11] Lippi, M. and L. Reichlin (1994) "Diffusion of Technical Change and the Decomposition of Output into Trend and Cycle," Review of Economic Studies 61, [12] Morley, J. C., R. C. Nelson, and E. Zivot (2003) "Why are the Beveridge-Nelson and Unobserved-Components Decompositions of GDP So Different," The Review of Economics and Statistics 85(2), [13] Nagakura, D. and E. Zivot (2007) "Implications of Two Measures of Persistence for Correlation Between Permanent and Transitory Shocks in U.S. Real GDP," unpublished manuscripts. [14] Nagakura, D. (2008) "A Note on the Two Assumptions of Standard Unobserved Components Models," Economic Letters 100(1), [15] Nelson, R.C. (2008) "The Beveridge-Nelson Decomposition in Retrospect and Prospect," Journal of Econometrics 146,