When Can Trend-Cycle Decompositions Be Trusted?

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. When Can Trend-Cycle Decompositions Be Trusted? Manuel Gonzalez-Astudillo and John M. Roberts 2016-099 Please cite this paper as: Gonzalez-Astudillo, Manuel, and John M. Roberts (2016). “When Can Trend- Cycle Decompositions Be Trusted?,” Finance and Economics Discussion Series 2016-099. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2016.099. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

When Can Trend-Cycle Decompositions Be Trusted? Manuel Gonzalez-Astudillo† John M. Roberts manuel.p.gonzalez-astudillo@frb.gov john.m.roberts@frb.gov Federal Reserve Board Federal Reserve Board Washington, D.C., USA Washington, D.C., USA December 19, 2016 Abstract In this paper, we examine the results of GDP trend-cycle decompositions from the estimation of bivariate unobserved components models that allow for correlated trend and cycle innovations. Three competing variables are considered in the bivariate setup alongwithGDP:theunemploymentrate,theinflationrate,andgrossdomesticincome. We find that the unemployment rate is the best variable to accompany GDP in the bivariate setup to obtain accurate estimates of its trend-cycle correlation coefficient and the cycle. We show that the key feature of unemployment that allows for precise estimates of the cycle of GDP is that its nonstationary component is “small” relative to its cyclical component. Using quarterly GDP and unemployment rate data from 1948:Q1to2015:Q4, weobtainthetrend-cycledecompositionofGDPandfindevidence ofcorrelatedtrendandcyclecomponentsandanestimatedcyclethatisabout2percent below its trend at the end of the sample. Keywords: Unobserved components model, trend-cycle decomposition, trend-cycle correlation JEL Classification Numbers: C13, C32, C52 ∗Theviewsexpressedinthispaperaresolelytheresponsibilityoftheauthorsandshouldnotbeinterpreted as reflecting the views of the Board of Governors of the Federal Reserve System. †Corresponding author. We thank Timothy Hills for speedy and accurate research assistance.

1 Introduction When can we trust trend-cycle output decompositions? Univariate studies, such as Morley, Nelson and Zivot (2003) (MNZ hereafter), find a large negative correlation between the innovations of output’s trend and cycle as well as small and economically unimportant business cycles. By contrast, studies that include additional variables such as inflation or the unemployment rate typically find estimates of the cyclical component of output that are to a large extent conventional, closely resembling, for example, estimates published by the Congressional Budget Office (CBO). Early examples include Clark (1989), who added the unemployment rate, and Roberts (2001), who incorporated inflation and hours in the analysis. Moreover, both these studies found the output trend-cycle correlation to be small and statistically insignificant. Other bivariate studies, such as Basistha and Nelson (2007) and Basistha (2007), using inflation as an additional variable, find output trend-cycle correlations that are negative and statistically significant, but smaller than in MNZ; they find conventional business cycles estimates. Basistha (2007) sheds some light on the source of these disparate results using a Monte Carlo study. Basistha shows that a strong estimated trend-cycle correlation can be spurious in a univariate setup. In particular, the correlation can be estimated to be large even when it is zero in the data generating process. By contrast, in a bivariate setup with a variable that resembles inflation, the estimated trend-cycle correlation coefficient is close, on average, to the true correlation of zero. In this paper, we use Monte Carlo experiments to explore under what conditions an auxiliary variable in a bivariate unobserved components (UC) model of trend-cycle decompositions of output will be helpful for correctly estimating the trend-cycle correlation of output and for identifying more precisely its cyclical component. We start by reviewing the univariate estimation proposed by MNZ, which allows for correlated trend-cycle output innovations. We then assess the conditions under which an auxiliary variable in the UC model will be helpful for estimating accurately the output trend-cycle correlation, for identifying the business cycle, and for testing hypotheses with respect to the trend-cycle correlation coefficient. We examine three specific set-ups, one using an auxiliary variable designed to resemble the unemployment rate, another with a variable resembling inflation, and a third that amounts to using two readings on output, meant to resemble the use of gross domestic income (GDI) as an auxiliary variable. We find that the univariate model’s estimation delivers an output trend-cycle correlation sampling distribution that piles up at -1 and +1 and that the properties of the cycle are not accurately obtained. We also find that there is considerable variation in the ability of the auxiliaryvariablestodistinguishthetrend-cyclecorrelationcoefficientandthebusinesscycle. In particular, for some auxiliary variables, the econometrician can obtain spurious estimates of the correlation between trend and cycle, similar but not as bad as those obtained with the univariate setup. For example, if both variables are nonstationary and resemble GDP—as when the bivariate model included both GDP and GDI—spurious correlation results are somewhat likely. We also consider a variable that resembles inflation. As with GDI, we find that spurious correlations can obtain for parametrizations of the model that accord with empirical results in the literature. Thus, simply adding an auxiliary variable does not 1

appear sufficient to allow the proper identification of trend and cycle. We find, however, that if the auxiliary variable in the bivariate analysis resembles the unemployment rate, the estimation results can be trusted. Based on our experiments, it appears that the key reason the unemployment rate is well-suited to help distinguish the trendandthecycleisthatthevarianceofitsunitrootcomponentisrelativelysmallcompared to the variance of its cyclical component. Motivated by our Monte Carlo results, we use GDP and unemployment rate data to estimate a bivariate UC model. As in other studies using the unemployment rate, we find a conventional cyclical component of GDP, similar to that published by the CBO. The estimated cycle has a pronounced hump-shaped pattern and complex roots, with a period of 11.1 years. Our empirical results suggest that there is a statistically significant correlation between the output trend and cycle. However, unlike MNZ, we find that the correlation is positive, not negative. That result is suggested by standard statistical tests, and we find in our Monte Carlo work that the size of these tests is approximately correct and that we have sufficient power to reject incorrect parameter values. The resulting business cycle estimates are conventional and similar to those of CBO. The paper is structured as follows: Section 2 presents a review of the literature on trend-cycle decompositions with UC models. In Section 3, we present the characteristics of the bivariate UC models we will examine. Section 4 presents the results of our Monte Carlo experiments with respect to the estimators of the output trend-cycle correlation and decomposition. The size and power of the LR test of hypothesis on the output trend-cycle correlation coefficient appear in Section 5. Section 6 summarizes the results from the Monte Carlo simulations. In Section 7, we estimate a bivariate UC model including GDP and unemployment data for the U.S. and test for significance of the correlation between trend and cycle components. Section 8 concludes. 2 Contacts with the Literature Watson (1986) and Clark (1987) were among the first to use a UC model to decompose GDP into independent nonstationary trend and stationary cycle components. The estimates implied that much of the quarterly variability in U.S. economic activity can be attributed to a stationary cyclical component. By contrast, Nelson and Plosser (1982), found that most of the variation in U.S. economic activity can be attributed to a nonstationary trend component. A central assumption of Clark’s estimation was the orthogonality between trend and cycle components; the method of Nelson and Plosser (1982) places no restrictions on the correlation between trend and cycle. In a subsequent paper, Clark (1989) proposed considering GDP and the unemployment rate in a bivariate UC model to decompose GDP into trend and cycle components, allowing a nonzero correlation between trend and cycle innovations. In this case, the trend and cycle disturbances are disentangled by assuming that the cyclical component of output also affects the unemployment rate through an Okun’s law relationship. Clark’s results provide evidence consistent with the hypothesis that innovations in the trend and cyclical components are independent: the 90 percent confidence interval for the correlation is [-0.4,0.3]. Kuttner (1994) pursued an alternative bivariate approach, adding inflation as an ob- 2

servable and linking inflation and the cycle through a Phillips curve relationship. Kuttner found a business cycle that was similar to Clark (1989). However, Kuttner did not allow for correlation between trend and cycle. Following Kuttner, Roberts (2001) also included a Phillips-curve relationship and further decomposed output into hours and productivity components. Hours and output per hour are each divided into trend and gap components, and the gap affects inflation through a Phillips curve relationship. Both Roberts and Kuttner found that estimates of the trend-cycle decomposition were not much affected by the addition of inflation. In addition, Roberts found that the correlations between trend shocks and the cycle were not statistically significant at conventional levels. Morley, Nelson and Zivot (2003) carefully explored identification in the univariate UC model. They showed that an unrestricted ARIMA(2,1,2) model implies second moments that can be matched uniquely to the second moments of the UC model. The estimation of the cycle through both the Beveridge-Nelson decomposition of the ARIMA(2,1,2) model and a univariate UC model allowing for correlation between trend and cycle yield estimates in which the cycle is mostly noise and most of the variability in GDP occurs through its trend component, similar to Nelson and Plosser (1982). Basistha and Nelson (2007) estimate a bivariate UC model with inflation and GDP as observable variables. They introduce the spread between inflation and a survey measure of expectations, motivated by the New Keynesian Phillips curve. They allow for a dense variance-covariance matrix of the shocks and find that the GDP trend and cycle innovations are negatively correlated, as obtained by MNZ, albeit with a smaller absolute magnitude. Their estimated cycle is nonetheless conventional. The authors extend the model to include the unemployment rate as an additional observable via an Okun’s law relationship. Results are similar to those when only inflation is included as an additional observable. As noted in the introduction, Basistha (2007) performed a set of Monte Carlo simulations that showed that while a univariate UC specification is not able to identify the correlation coefficient between trend and cycle innovations, a bivariate setup similar to Basistha and Nelson (2007) yields an estimated correlation coefficient that is close on average to the true correlation. The work of Perron and Wada (2009) is also related to our paper. As in our work, Perron and Wada (2009) consider the estimation of the correlation between trend and cycle when no correlation exists. They, however, look at a model with a deterministic trend whereas our focus is on stochastic trends. The find that when the trend component is deterministic, the estimated correlation between trend and cycle shocks is either -1 or +1, depending on parameter configurations, because the correlation coefficient is not identified when the trend is deterministic. Wada (2012) explores this issue further to offer a more detailed explanation of the lack of identification. The literature cited above emphasizes a common cycle linking GDP and other variables. Sinclair (2009) pursues an alternative approach in which each variable is allowed to have its own trend and cycle, specifically estimating the trends and cycles for GDP and the unemployment rate in a bivariate UC model. Sinclair finds a statistically significant negative correlation between the trend and cycle components of GDP, as well as in the corresponding innovations of the unemployment rate. The resulting business cycle resembles that of MNZ, with high volatility and relatively small amplitude. In our work, we follow the main body of the literature, which emphasizes a common cycle linking GDP and other variables. 3

3 UC Models for Trend-cycle Decompositions We first present the basic structure of the unobserved components model for trend-cycle decomposition. We then examine three specific bivariate extensions and posit a common, nesting framework. 3.1 The basic UC model y = τ +c (1) t yt t τ = µ +τ +η (2) yt y y,t−1 yt c = φ c +φ c +ε , (3) t 1 t−1 2 t−2 t In our analysis, {y } is the log of GDP, {τ } is its unobserved trend, assumed to be a random t yt walk with mean growth rate µ , and {c } is the unobserved stationary cycle, assumed to y t follow an AR(2) process. The roots of 1 − φ z − φ z2 = 0 are outside the unit circle, and 1 2 {η } and {ε } are potentially correlated disturbances with variance-covariance matrix given yt t by: ε σ2 ρ σ σ t ∼ iid N 0 , ε ηyε ηy ε . (cid:20)η yt (cid:21) (cid:18) 2×1 (cid:20)ρ ηyε σ ηy σ ε σ η 2 y (cid:21)(cid:19) As noted in the introduction, our main focus is the value of the correlation coefficient ρ —that is, the correlation between the trend and cycle for output. When it is imposed ηyε to be zero—as in Clark (1987) and Watson (1986)—or estimated to be near zero—as in Clark (1989)—the resulting estimates of the cycle are conventional. By contrast, in many univariate studies, such as Nelson and Plosser (1982) and MNZ, it is estimated to be close to -1 and the resulting estimates of the cycle are unconventional, relative, for example, to the output gap estimates of CBO. While the simulation results of Basistha (2007) have shown that these univariate results can be spurious, a key question is when we can trust bivariate results. We therefore consider several bivariate models for trend-cycle decomposition in the presence of nonzero correlation between trend and cycle disturbances and estimation of the correlation coefficient. 3.2 Bivariate UC Model: GDP and the unemployment rate Clark (1989) proposed extending the univariate UC model for GDP in equations (1)-(3) to include the unemployment rate in the following fashion: u = τ +θ c +θ c (4) t ut 1 t 2 t−1 τ = τ +η , (5) ut u,t−1 ut 4

and variance-covariance matrix: ε σ2 ρ σ σ ρ σ σ t ε ηyε ηy ε ηuε ηu ε η yt  ∼ iid N 0 3×1 ,ρ ηyε σ ηy σ ε σ η 2 y ρ ηyηu σ ηy σ ηu . (6) η ρ σ σ ρ σ σ σ2  ut    ηuε ηu ε ηyηu ηy ηu ηu  In equation (4), {u } is the unemployment rate, which is decomposed into a trend, {τ }, t ut assumed to be a random walk with zero drift, and a cyclical component. The cycle of output is allowed to affect the unemployment rate both contemporaneously and with a lag, reflecting the well-known characterization of the unemployment rate as a lagging indicator of the business cycle (see Stock and Watson, 1998). The model allows the correlations between the cycle and trend innovations of GDP, ρ , and the unemployment rate, ρ , to ηyε ηuε be nonzero, as well as the correlation between the two trend shocks, ρ .1 ηyηu There are a number of ways to interpret this model. One is that the system of equations (1)-(3) and (4)-(6) forms a factor model, with {c } the common factor, normalized t so that its effect on {y } is contemporaneous with a coefficient of one. Another interpretat tion of equation (4) is Okun’s Law, with the unemployment gap related to the output gap contemporaneously and with a lag. 3.3 Bivariate UC Model: GDP and Gross Domestic Income Gross Domestic Income (GDI) is another plausible cyclical indicator. GDI is an alternative measure of exactly the same concept as GDP, but based on (largely) independent sources of data. Fixler and Nalewaik (2007) and Nalewaik (2010) have shown that GDI is at least as good a measure of aggregate economic activity as GDP; Fleischman and Roberts (2011) find similar results in their multivariate trend-cycle model. We introduce GDI into the UC model in equations (1)-(3) as follows: z = τ +c (7) t zt t τ = µ +τ +η , (8) zt z z,t−1 zt and ε σ2 ρ σ σ ρ σ σ t ε ηyε ηy ε ηzε ηz ε η yt  ∼ iid N 0 3×1 ,ρ ηyε σ ηy σ ε σ η 2 y ρ ηyηz σ ηy σ ηz . (9) η ρ σ σ ρ σ σ σ2  zt    ηzε ηz ε ηyηz ηy ηz ηz  In this specification, {z } is (the log of) GDI, {τ } is its unobserved trend, assumed t zt to be a random walk with mean growth rate µ , which is the same for GDP, and {c } z t is the common unobserved stationary cycle. The correlation coefficient ρ captures the ηyηz co-movements between the trends, and the coefficients ρ and ρ allow for separate trend- ηyε ηzε cycle correlations for GDP and GDI. 1When testing for statistical significance of the trend-cycle correlation coefficient of GDP, ρ , Clark ηyε (1989) assumed that the other two correlation coefficients where zero, implying a less general framework than the one presented here. 5

3.4 Bivariate UC Model: GDP and the inflation rate Another alternative introduces inflation in the UC model in equations (1)-(3): π = τ +θc (10) t πt t τ = µ (1−α)+απ +η , (11) πt π t−1 πt with α ∈ (0,1], and ε σ2 ρ σ σ ρ σ σ t ε ηyε ηy ε ηπε ηπ ε η yt  ∼ iid N 0 3×1 ,ρ ηyε σ ηy σ ε σ η 2 y ρ ηyηπ σ ηy σ ηπ . (12) η ρ σ σ ρ σ σ σ2  πt    ηπε ηπ ε ηyηπ ηy ηπ ηπ  This model incorporates a Phillips curve relationship (equation (10)), where the cyclical component of output helps predict the deviation of inflation, {π }, from trend inflation, τ . t πt The specification for trend inflation nests several alternatives. Kuttner (1994) assumed that inflation followed a unit root process, and hence that α = 1. Kuttner also assumed that the correlations between innovations were zero. Roberts (2001) also assumed that α = 1 but allowed for correlations between innovations. Basistha (2007, 2009), on the other hand, allowed α < 1; he also allowed for correlated trend-cycle innovations.2 Stella and Stock (2016) take a different approach and specify the inflation trend as a unit root process, similar to the unemployment rate trend in the GDP-unemployment rate bivariate UC model of Section 3.2, as follows:3 τ = τ +η ,, (13) πt π,t−1 πt with the assumption of orthogonal disturbances. They also introduce measurement errors in the observation equations and stochastic volatilities in all the error terms. The addition of measurement errors makes the model of Stella and Stock (2016), strictly speaking, no longer nested in the class of models we have been considering so far. We nonetheless consider a variant of this specification as one of the alternative models for the sake of completeness; we find that augmenting the model with measurement error does not have an important impact on the results. Similarly, we do not formally consider stochastic volatility but rather examine the implications of the model for the ability to discriminate trend and cycle under different (constant) assumptions about the volatility of trend inflation. 3.5 Nesting the bivariate models Here, we present a single bivariate UC model specification that encompasses the three models laid out above. In our general specification, x is the accompanying variable (the t 2Another paper that incorporates a Phillips curve relationship in the estimation of the output gap is Basistha and Nelson (2007). These authors included a survey measure of inflation expectations in equation (11) along with lagged inflation. 3In fact, the model that Stella and Stock (2016) propose has the unemployment and the inflation rates as observables, as opposed to GDP and the inflation rate. The aim, however, is the same as in the models with GDP as observable: to disentangle the cyclical component of GDP. 6

unemployment rate, GDI, or the inflation rate) whose trend can be modeled under two alternatives, as shown below: y = τ +c , (14) t yt t x = τ +θc , (15) t xt t τ = µ +τ +η , (16) yt y y,t−1 yt c = φ c +φ c +ε , (17) t 1 t−1 2 t−2 t • Exogenous Auxiliary Trend: τ = µ +τ +η , (18) xt x x,t−1 xt • Endogenous Auxiliary Trend: τ = µ (1−α)+αx +η , (19) xt x t−1 xt where α ∈ (0,1], and ε σ2 ρ σ σ ρ σ σ t ε ηyε ηy ε ηxε ηx ε varη yt  = ρ ηyε σ ηy σ ε σ η 2 y ρ ηyηx σ ηy σ ηx . η ρ σ σ ρ σ σ σ2  xt   ηxε ηx ε ηyηx ηy ηx ηx  The Exogenous Auxiliary Trend alternative encompasses the bivariate specifications of Sections 3.2 and 3.3, where GDP is accompanied by the unemployment rate and GDI, respectively. It also encompasses the specification of the inflation trend in Stella and Stock (2016). Notethat this specification includes only onelag of thecycle entering x , whereas the t GDP-unemployment rate case outlined inSection 3.2 includes two. To facilitate comparisons across the bivariate models proposed in the literature, we keep one lag only. (In work not presented, we explored using two lags instead of one; the results were very similar.) When the inflation trend is specified as in equation (11), the Phillips curve cannot be nested with the other models (because the trend is affected by the observation variable). We therefore also consider the Endogenous Auxiliary Trend alternative, which is chosen to encompass the bivariate specification of Section 3.4 where GDP is accompanied by the inflation rate. Depending on the value of the parameter α, inflation can be a stationary process or a process with a unit root. 4 Monte Carlo Exercises In this section, we assess the properties of the different specifications described above using Monte Carlo exercises. In our Monte Carlo experiments, we repeatedly simulate the various UC models of Section 3 and estimate them by maximum likelihood using the Kalman filter. We will evaluate the specifications along two main dimensions. First, we examine the properties of the estimated trend-cycle correlation for output, ρˆ , in particular its sam- ηyε pling distribution. We look at the sampling distribution because, as emphasized by Basistha 7

(2007), standard univariate techniques can find large estimates of the correlation between trend and cycle innovations, even when the true value is zero. The second dimension along which we evaluate the specifications is the properties of the estimated cycle, in particular, its period, which is computed from the estimates of φ and φ , and the estimated proportion 1 2 of the variance of output that is accounted for by the variance of the cycle. We look at the properties of the cycle because these properties can differ depending on the correlation between trend and cycle. For example, in MNZ’s baseline estimation, which allows correlation between trend and cycle, the period of the cycle for the U.S. economy is about 10 quarters. In contrast, the estimated periodicity of the cycle is considerably longer in bivariate specifications that allow for correlation between trend and cycle. In Clark (1989), the period of the cycle is estimated to be about 28 quarters; in Basistha and Nelson (2007), the period of the cycle is infinite in their bivariate setup that includes GDP and inflation only, and about 20 quarters in their trivariate setup that also includes the unemployment rate. Wealsolookathoweachspecificationcharacterizesthevariancedecompositionofoutput. It is useful to consider the variance decomposition in this framework because it encompasses several dimensions of the estimation that can be affected by the choice of one specification or another, namely the proportion of the variance of output growth attributed to variations in the trend or to variations in the cycle, as well as the impact of the correlation between disturbances. We obtain the variance decomposition of output in the presence of potentially correlated disturbances as follows: var( E (∆y |∆c )) t t % of var(∆y ) explained by var(∆c ) = 100× t t var(∆y ) t var(∆c ) 1+ ρηyεσηy σε = 100× t (cid:16) var(∆ct) (cid:17) , (20) σ2 +var(∆c )+2ρ σ σ ηy t ηyε ηy ε where the estimated parameters replace their theoretical counterparts to compute the estimated variance decomposition. As can be seen, the output trend-cycle correlation influences the variance decomposition. The central question we aim to answer is whether standard maximum-likelihood techniques canrecover thecorrect valueofthecorrelationbetween thetrendandcycle foroutput, ρ , as well as an accurate decomposition of output into trend and cycle. In particular, we ηyε are interested in the features of the accompanying variable, x , that make the estimation t of the correlation coefficient and the properties of the cycle precise. In discriminating these features, it is useful to look at the variance decomposition of x , that is, the percent of its t variance that is due to the variance of cycle. The variance decomposition for each of the specifications can be written as follows: • Exogenous Trend: var(E(∆x |∆c )) % of var(∆x ) explained by var(∆c ) =100× t t t t var(∆x ) t var(∆c ) θ+ ρηxεσηy σε 2 =100× t (cid:16) var(∆ct) (cid:17) (21) σ2 +θ2var(∆c )+2θρ σ σ ηx t ηxε ηx ε 8

• Endogenous Trend: – Nonstationarity [α = 1]: var(E(∆x |c )) % of var(∆x ) explained by var(c ) =100× t t t t var(∆x ) t var(c ) θ+ ρηxεσηy σε 2 =100× t (cid:16) var(ct) (cid:17) , (22) σ2 +θ2var(c )+2θρ σ σ ηx t ηxε ηx ε – Stationarity [α ∈ [0,1)]: var(E(∆x |c )) % of var(∆x ) explained by var(c ) =100× t t t t var(∆x ) t 2 =100× var(c t ) (cid:16) θ+ ρη v x a ε r σ ( η ct x ) σε +(α−1)cov v ( a x r t ( − c 1 t) ,ct) (cid:17) , var(∆x ) t (23) where 2 var(∆x ) = σ2 +θ2var(c )+2θρ σ σ +θ(α−1)cov(x ,c ) , t 1+α ηx t ηxε ηx ε t−1 t (cid:0) (cid:1) ∞ ∞ ∞ cov(x ,c ) = ρ σ σ αiψ +θσ2 αi ψ ψ , t−1 t ηxε ηx ε i+1 ε j j−i−1 Xi=0 Xi=0 jX=i+1 and ψ , for j = 0,1,2,..., are the coefficients of the Wold representation of the cycle.4 Notice j that as α → 1, the variance decomposition in Equation (23) converges to the variance decomposition in Equation (22). Toexplorethefeaturesofx thatmakesitagoodcandidatetofindanaccuratetrend-cycle t decomposition of output, we vary two parameters of the processes under the exogenous and endogenous trend alternatives. First, when x is nonstationary, we consider several values of t the variance ofthe innovation to thetrend, σ2 , which will affect the variancedecompositions ηx inequations(21)and(22). Second, whenx isstationary, andapersistent trendisabsent, we t vary the persistence coefficient, α, which will change the variance decomposition in equation (23). Given that our benchmark parametrizations assume that ρ = 0, we can achieve any ηxε variance decomposition of x by varying σ2 and keeping all the other coefficients fixed. t ηx ThemodelswereestimatedusingthefminconfunctioninMatlab2013a. Akeyconstraint on the correlation coefficients to guarantee positive definiteness of the variance-covariance matrix of the innovations ε ,η , and η is given by ρ2 +ρ2 +ρ2 +2ρ ρ ρ ≤ 1. t yt xt ηyε ηxε ηyηx ηyε ηxε ηyηx We used the interior-point algorithm. 4.1 Monte Carlo Exercises: x nonstationary t The parametrizations of the nonstationary models we consider appear in Table 1. Numbers in curly brackets indicate the possible values that the parameters can take over the 4ψ =1,ψ =φ ,ψ =φ ψ +φ ,ψ =φ ψ +φ ψ for j ≥3. 0 1 1 2 1 1 2 j 1 j−1 2 j−2 9

Table 1: Parameter Values - x nonstationary t Parameter Value µ 0.8 y φ 1.5 1 φ −0.6 2 σ 0.6 ε σ 0.7 ηy ρ {−0.9,0,0.9} ηyε θ 0.5 µ 0 x σ {0.01,0.1,0.2,0.35,0.5,0.7,1.2,1.7,2.5,3.0} ηx ρ 0 ηxε ρ 0 ηyηx T = {50,100,200,500,1,000} simulations; bold numbers indicate our benchmark parametrization. The combinations of these values will define the different cases to be explored in the Monte Carlo experiments. This calibration for y is similar to the results obtained by MNZ when they assumed no t correlation between trend and cycle (labeled UC-0 in MNZ). In particular, the values of φ 1 and φ in the cycle process, c , imply that it will display a hump-shaped pattern; in this 2 t case, there are complex roots resulting in a duration of the cycle of about 25 periods. Under the baseline variance decomposition, the variance of the cycle explains about 54 percent of the variations of ∆y . The mean growth rate, µ , is also chosen to match the MNZ results. t y Notice that the coefficients of the process for y , which is the process that represents (the log t of) GDP, are the same across experiments, keeping the properties of the cycle fixed across simulations. For convenience, we assume that the cyclical component of y enters x with t t a loading coefficient θ = 0.5.5 We also assume that the drift component of the auxiliary variable is zero—that is, that µ = 0.6 x As Basistha (2007) has emphasized, standard univariate techniques can find large estimates of the correlation between trend and cycle innovations even when the true value is zero. We therefore begin by assuming that in the data generating process, ρ is zero and ηyε then change it to values in its boundary. In addition, because our emphasis is on the correlation between the trend and cycle for output, we will assume throughout the experiments that ρ and ρ are zero. ηyηx ηxε 5Atfirstglance,itmay appearthatthis loadingfactoris onlyappropriatewhen x is the unemployment t rate (up to a sign change). However,an examinationofequations (21) and(22) indicates that, for ρ =0, ηxε we can replicate any variance decomposition of x through the appropriate choice of σ . We are therefore t ηx able to encompass the parametrizations of inflation or GDI. That is what we do in the simulations, as can be seen from Table 1 where σ takes on a large set of parameter values and θ is fixed. ηx 6Whenx correspondstounemploymentorinflation,thisassumptionisappropriate. InthecaseofGDI, t this coefficient should be equal to µ , but the results are not affected by the choice of µ . y x 10

Figure 1: Frequency Distribution of ρˆ under ρ = 0 ηyε ηyε (a) Bivariate model (b) Univariate model 0.35 0.7 T = 50 T = 50 T = 100 T = 100 0.3 T = 200 0.6 T = 200 T = 500 T = 500 T = 1000 T = 1000 0.25 0.5 0.2 0.4 0.15 0.3 0.1 0.2 0.05 0.1 0 0 -1-.9-.8-.7-.6-.5-.4-.3-.2-.10 .1.2.3.4.5.6.7.8.9 1 -1-.9-.8-.7-.6-.5-.4-.3-.2-.10 .1.2.3.4.5.6.7.8.9 1 ρˆηyǫ ρˆηyǫ 4.1.1 Base case: Exogenous auxiliary trend The first set of Monte Carlo experiments is designed to mimic the GDP-unemployment rate setup of Section 3.2 and the GDP-GDI setup of Section 3.3; it can also be used to interpret the Stella-Stock model of trend inflation discussed in Section 3.4. As stated before, we assume that the trend component of the accompanying variable follows a random walk process. Figure1ashowsthedistributionoftheestimatedcorrelationcoefficientρˆ obtainedfrom ηyε thesimulationswithbenchmarkparametervaluesforsamplesizesT = 50,100,200,500,1,000. Under these assumptions, the trend and cycle for output are orthogonal (ρ = 0) and the ηyε auxiliary variable has trend variability similar to that found for the unemployment rate (σ = 0.1) (see Clark, 1989; Fleischman and Roberts, 2011, for example). ηx The distribution of the maximum likelihood estimator of the correlation coefficient between the trend and cycle innovations of y has a conventional shape, with the mass of the t distribution concentrated around the true coefficient value as the sample size increases. For typical U.S. quarterly sample sizes of around 200, results are reasonably precise. The shape of the distribution of the estimator of the trend-cycle correlation coefficient under this bivariate setup contrasts sharply with the distribution of the same correlation coefficient when a univariate estimation is used—as in MNZ. In Figure 1b, results are reported based on data generated with the same benchmark parameter values as in Table 1, but omitting x as an observable in the estimation of the UC model. As can be seen, the t maximum likelihood estimation of the univariate model implies an estimated trend-cycle correlation coefficient that has a distribution with masses close to -1 and +1. Even with very large sample sizes, the fat tails of the distribution are evident. The results in Figures 1a and 1b suggest that estimation of the bivariate model that includes an x process with the t baseline features substantially improves the small sample properties of the estimator of the trend-cycle correlation coefficient. Table 2 reports some key statistics about the estimated cyclical properties—in particular, the median estimated period of the cycle, the percent of simulations that the estimation 11

Table 2: Features of theEstimated Period and Variance Decomposition under ρ = 0 ηyε Bivariate model Univariate model simulated var(E(∆yt|∆ct)) simulated var(E(∆yt|∆ct)) Sample Size Period % finite var(∆yt) Period % finite var(∆yt) true var(E(∆yt|∆ct)) truevar(E(∆yt|∆ct)) var(∆yt) var(∆yt) T =50 17 91 1.10 17 79 1.74 T =100 21 94 1.07 19 67 1.69 T =200 23 92 1.07 21 67 1.60 T =500 25 96 1.05 23 81 1.28 T =1,000 25 98 1.02 24 91 1.12 Note: Thetrueimpliedperiodis24.8quarters. Thetrueimpliedvariancedecomposition var(E(∆yt|∆ct)) is54.2%. var(∆yt) % finite denotes the percentage of times that a finite period was obtained. Results based on 1,000 simulations. Medianstatisticsarereported,exceptfor%finite. delivers correctly a finite period, and the ratio between the median of the estimated variance decomposition var(E(∆yt|∆ct)) and its true counterpart for different sample sizes. The results var(∆yt) in the second, third and fourth columns show that, in our baseline bivariate setup, the estimated duration of the cycle is close to the implied true duration of about 25 periods with a sample of 200 periods. The percentage of times that the estimation correctly obtains a finite period is above 90 percent. The variance decomposition is reasonably similar to that implied by the true parameters. As the sample size increases, both the period and the variancedecompositionapproachmonotonicallytheirtheoreticalvalues, whilethepercentage of times that a finite period is obtained approaches 100 percent. The fifth, sixth and seventh columns of Table 2 report the results from the estimation of the univariate setup that includes data on y only. The univariate estimation tends t to deliver a shorter cyclical period, and the percentage of times that it correctly obtains a finite period is substantially reduced. It also tends to overestimate the fraction of the variation of ∆y that is explained by the cycle. In particular, with a sample size of 200 t observations, the estimated period is 16 percent lower than its theoretical value, compared with 8 percent in the bivariate case. Also, while the median variance decomposition of the bivariate specification is 7 percent above its theoretical counterpart, the univariate model delivers a cyclical contribution that isabout 60percent higher. While theperiodrises toward its theoretical counterpart as the sample size increases, it does it a slower rate than in the bivariate model, and the estimated variance decomposition still overshoots its theoretical value at the largest sample size considered. Thus, the addition of a nonstationary variable, x , with features implied by the benchmark parametrization in Table 1 not only helps reduce t the bias in the estimation of the trend-cycle correlation coefficient but also helps reduce the overall bias in the estimation of the period and the amplitude of the cycle. We also simulate the bivariate and univariate models of this section assuming that there is important correlation between trend and cycle—that is, that the results of MNZ hold. In particular, we simulate the model with the benchmark parametrization of Table 1, except thatwealsoassumethatρ = −0.9(MNZ’scase)andthatρ = 0.9. Figure2reportsthe ηyε ηyε results of the bivariate and univariate models estimations. The plots show that the bivariate specification also does a good job at estimating the trend-cycle correlation coefficient of y t when the trend and cycle are either negatively or positively correlated, as the peak of the 12

Figure 2: Frequency Distribution of ρˆ under ρ ∈ {−0.9,0,0.9} ηyε ηyε (a) Bivariate model (b) Univariate model 0.9 0.7 Bivariate,ρηy,ǫ=0 Univariate,ρηy,ǫ=0 0.8 B B i i v v a a r r i i a a t t e e , , ρ ρ η η y y , , ǫ ǫ = = - 0 0 .9 .9 0.6 U U n n i i v v a a r r i i a a t t e e , , ρ ρ η η y y , , ǫ ǫ = = - 0 0 .9 .9 0.7 0.5 0.6 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -1-.9-.8-.7-.6-.5-.4-.3-.2-.10 .1.2.3.4.5.6.7.8.9 1 -1-.9-.8-.7-.6-.5-.4-.3-.2-.10 .1.2.3.4.5.6.7.8.9 1 ρˆηyǫ ρˆηyǫ distribution occurs close to the true values of -0.9 and 0.9, respectively. The univariate specification, on the other hand, continues to give probability masses that are excessively weighted in the tails. Thus, if the trend and cycle were correlated, the bivariate specification would very likely detect it. To assess how the estimates are affected by the relative contribution of trend and cycle to the variability of the accompanying variable, we conduct a set of Monte Carlo exercises with the benchmark parametrization of Table 1, except that σ varies in the set ηx {0.01,0.1,0.2,0.35,0.5,0.7,1.2}. Figure 3 shows the results.7 As can be seen, the simulation results imply that increasing the standard deviation of the trend innovation of x makes the estimation of ρ less accurate. In particular, the t ηyε distribution of the correlation estimates tends to pile up towards +1. The cyclical properties of the model are also distorted: The cycle period tends to be underestimated; the percentage of times that the estimation correctly delivers a finite period decreases significantly; and the estimated fraction of the variation of ∆y that is explained by variations in the cycle is t further above their theoretical counterparts. A key implication of these results is that adding a variable with large trend variability would not help recover accurately the correlation coefficient or to give precise estimates of the cycle. With these results, we can also explain why the correlation coefficient and the trend-cycle decomposition would not be accurately estimated using the GDP-GDI bivariate model of Section 3.3. In that setup, θ = 1 and µ = µ = µ . However, we know that, since GDI is x y z intended to measure the same concept of overall economic activity as GDP, it should have a variancedecompositionverysimilar tothatofGDP.Hence, wecankeepthecoefficients θ and µ at their benchmark values and modify σ to achieve the same variance decomposition x ηx of ∆y . That parametrization corresponds to the case where σ = 0.35 in the table of t ηx Figure 3. The results in this case are inferior to those with σ = 0.1, our benchmark case. ηx Inparticular, thesimulated distributionofρˆ hasnegative skewness andstartsto gainmass ηyε 7In the benchmark parametrization, with σ = 0.1 and ρ = 0, the variance decomposition implies ηx ηxε that the variance of θ∆c explains about 94 percent of the variance of ∆x . t t 13

Figure 3: Frequency Distribution of ρˆ under σ ∈ {0.01,0.1,0.35,0.7,1.2} ηyε ηx (Exogenous Trend) 0.25 σηx =0.1 σηx =0.01 0.2 σ σ η η x x = = 0 0 . . 3 7 5 σηx =1.2 0.15 0.1 0.05 0 -1-.9-.8-.7-.6-.5-.4-.3-.2-.10 .1.2.3.4.5.6.7.8.9 1 ρˆηyǫ Standard Deviation true 100× var(E(∆xt|∆ct)) Period % finite simulated var( v E a ( r ∆ (∆ yt y | t ∆ ) ct)) var(∆xt) true var( v E a ( r ∆ (∆ yt y | t ∆ ) ct)) σ = 0.01 99.9 22 96 1.02 ηx σ = 0.1 94.0 23 92 1.07 ηx σ = 0.2 78.4 23 84 1.16 ηx σ = 0.35 54.2 22 71 1.26 ηx σ = 0.5 36.7 21 69 1.37 ηx σ = 0.7 22.9 21 64 1.39 ηx σ = 1.2 9.2 20 67 1.45 ηx Note: Thetrueimpliedperiodis24.8quarters. Thetrueimpliedvariancedecomposition var(E(∆yt|∆ct)) is54.2%. var(∆yt) %finite denotesthepercentageoftimesthatafiniteperiodwasobtained. Medianstatisticsarereported,except for%finite. Resultsbasedon1,000simulationswithsamplesizeT =200. 14

atpositivevaluesoftheestimatedcorrelationcoefficient. Also, thefourthcolumnofthetable shows that the percentage of times that the estimation delivers a finite cycle is 71 percent, compared with 92 percent in the benchmark case. Additionally, the variance decomposition attributes 26 percent more of the variations of ∆y to the cycle than it should, as can be t seen in the last column, compared with only 7 percent in the benchmark parametrization. These results suggest that using the unemployment rate as an auxiliary variable should lead to superior results relative to using GDI. Nonetheless, it is worth noting that the bivariate results with GDI would be more accurate than the univariate results. In particular, the mass of extreme results is significantly smaller than in the univariate case, and the overall performance of the estimates of the periodicity and variance decomposition is better. Wenext turnto calibrationsinwhich x hasproperties similar to inflationasinStella and t Stock (2016). As discussed above, Stella and Stock (2016) assume an exogenous inflation trend, so that with appropriate unit transformations, we can use Figure 3 to assess the usefulness of inflation as an auxiliary variable. In particular, their estimated Phillips curve slope coefficient is about −0.37, using the unemployment rate as their cyclical variable. Adjusting for an Okun’s law coefficient of −0.5 would imply θ = 0.19 in our setup. In terms of the volatility of the trend component of the auxiliary variable, Stella and Stock (2016) allow the variance of inflation to vary over time. They find that, for the period since 2010— near the end of their sample—the variance of the permanent component of inflation has been about 0.15, implying a standard deviation of a bit less than 0.4 percentage points. By contrast, inthe1970s, thevarianceofthepermanent component ofinflationwasconsiderably larger, reaching as high as 13/4 percent in the middle of the decade, implying a standard deviation of the permanent component as high as 1.3 percentage points. Accordingtoequation(21), thecyclewouldaccountforaround12percent ofthevariation in inflation based on Stella and Stock (2016)’s recent estimate, and 11/4 percent when the inflation trend was more volatile. Those values would put us at the bottom rows (and beyond) of the table of Figure 3, where the variance decomposition of x is between 23 and t 9 percent. As can be seen, with these values of the standard deviation of the trend of x , the t correlation between trend and cycle would be poorly estimated, as the sampling distribution shows, and the features of the cycle would imply a lower estimated period and a much higher contribution of the cycle to the variations of ∆y , compared to their theoretical counterparts. t Additionally, the percentage of times that the estimation delivers a finite period is relatively low.8 4.1.2 Endogenous auxiliary trend Our second assumption about the trend in the auxiliary variable accommodates another common model of the trend component of inflation, in which the cyclical component affects the evolution of the trend (see Section 3.4). In this section, we consider the nonstationary variant of the trend component of x , assuming that α = 1; this specification of the inflation t trend has been used by Kuttner (1994) and Roberts (2001). As in Figure 3, we examine the consequences of this particular UC model setup on the estimators of the trend-cycle 8As discussed in Section 3.4, Stella and Stock’s baseline specification included measurement error in inflation. We exploredadding measurementerrorto the auxiliaryvariable andfound that, qualitatively,the results still hold. 15

Figure 4: FrequencyDistributionofρˆ underσ ∈ {0.2,0.7,1.7,2.5,3} (En- ηyε ηu dogenous Trend) 0.25 σηx =0.2 σηx =0.7 0.2 σ σ η η x x = = 1 2 . . 7 5 σηx =3 0.15 0.1 0.05 0 -1-.9-.8-.7-.6-.5-.4-.3-.2-.10 .1.2.3.4.5.6.7.8.9 1 ρˆηyǫ Standard Deviation true 100× var(E(∆xt|ct)) Period % finite simulated var( v E a ( r ∆ (∆ yt y | t ∆ ) ct)) var(∆xt) true var( v E a ( r ∆ (∆ yt y | t ∆ ) ct)) σ = 0.1 99.1 22 94 1.02 ηx σ = 0.2 96.7 23 93 1.02 ηx σ = 0.5 82.3 23 91 1.07 ηx σ = 0.7 70.3 23 90 1.08 ηx σ = 1.2 44.6 23 85 1.13 ηx σ = 1.7 28.7 22 83 1.12 ηx σ = 2.5 15.7 22 78 1.23 ηx σ = 3.0 11.4 22 76 1.25 ηx Note: Thetrueimpliedperiodis24.8quarters. Thetrueimpliedvariancedecomposition var(E(∆yt|∆ct)) is54.2%. var(∆yt) %finite denotesthepercentageoftimesthatafiniteperiodwasobtained. Medianstatisticsarereported,except for%finite. Resultsbasedon1,000simulationswithsamplesizeT =200. correlation and the cycle by varying σ in the set {0.1,0.2,0.5,0.7,1.2,1.7,2.5,3.0} and ηx leaving the other coefficients at their benchmark values. Figure 4 shows the results. The parametrization of σ in Figure 4 includes higher values than in Figure 3 to capture ηx the small contribution of the cycle to the variability of inflation that is found in the results of the existing literature. For example, rough calculations of the variance decomposition based on the results of Kuttner (1994) and Roberts (2001) imply numbers lower than 15 percent, which correspond to the results reported in the last rows of the table in Figure 4. For those values of σ , the sampling distribution of ρˆ is relatively flat and tail events gain ηx ηyε in importance, as can be seen in the figure. The fourth column of the table shows that the percentage of times that the estimation delivers a finite period, between 76 and 78 percent, is higher than in the univariate case, but much lower than in our benchmark bivariate setup. The last column shows that the cycle contribution to the variability of y is overestimated in t the order of around 23 to 25 percent, again better than the univariate case but worse than with a variable that resembles the unemployment rate. 16

Table 3: Parameter Values - x stationary t Parameter Value µ 0.8 y φ 1.5 1 φ −0.6 2 σ 0.6 ε σ 0.7 ηy ρ 0 ηyε θ 0.5 µ 0 x α {0.1,0.3,0.5,0.7,0.9,0.95,0.99} σ {1,3,7} ηx ρ 0 ηxε ρ 0 ηyηx T = 200 4.2 Monte Carlo Exercises: x stationary t In this case, we assume that the inflation trend is given by τ = µ (1 − α) + αx + xt x t−1 η , as in the Endogenous Trend alternative, except that here, we allow α ∈ (0,1). We vary xt the persistence coefficient α inside the unit circle and examine the properties of the trendcycle correlation and cycle estimates. We also consider three values for the variability of the auxiliary variable, σ . The parametrizations used appear in Table 3. Setting σ = 3 makes ηx ηx the results as α → 1 compatible with the results under the specification of the Endogenous Trend in Section 4.1.2. As discussed in that section, our reading of the literature suggested a relatively high value of σ . To be consistent with the Monte Carlo exercise in Basistha ηx (2007), we also consider a value of σ = 1. Basistha (2007) assumed α = 0.5, σ = 1 and ηx ηx that the influence of the cycle on the auxiliary variable, θ, is 0.4. In Figure 5, σ = 1. With this setting, the results largely confirm the findings in the ηx simulation exercises of Basistha (2007): The distribution of ρˆ is well-behaved over a wide ηyε range of values for α. As can be seen in Figures 6 and 7, however, as σ increases and a ηx lower proportion of the variability of the auxiliary variable—inflation—is attributed to the cycle, the distribution becomes less well behaved.9 The figures indicate that the distribution of the estimated trend-cycle correlation coefficient tends to pile up at +1. Three additional features of the estimation can be distinguished as σ increases. First, the estimated period ηx of the cycle tends to decline. Second, the percentage of correctly estimated finite periods also tends to decline. This percentage worsens as the persistence of the auxiliary variable 9Notice that the second column of the table in Figure 6 shows that, as the persistence coefficient approachesone,thecontributionofthevarianceofthecycletothevarianceofx tendstothevaluewhenα=1, t which can be seen in the second column of the last row of the table of Figure 4. In fact, the contribution of the variance of the cycle to the variance of x is 11.35% when α=0.999. t 17

Figure 5: Frequency Distribution of ρˆ - x stationary, σ = 1 ηyε t ηx 0.14 α=0.1 α=0.5 0.12 α=0.7 α=0.95 0.1 0.08 0.06 0.04 0.02 0 -1-.9-.8-.7-.6-.5-.4-.3-.2-.10 .1.2.3.4.5.6.7.8.9 1 ρˆηyǫ Persistence true 100× var(E(∆xt|ct)) Period % finite simulated var( v E a ( r ∆ (∆ yt y |∆ t) ct)) var(∆xt) true var( v E a ( r ∆ (∆ yt y |∆ t) ct)) α = 0.1 0.4 22 88 1.11 α = 0.3 1.0 23 87 1.14 α = 0.5 2.9 22 87 1.12 α = 0.7 9.2 22 89 1.13 α = 0.9 30.3 23 83 1.12 α = 0.95 40.6 23 81 1.08 α = 0.99 50.9 23 75 1.07 Note: Thetrueimpliedperiodis24.8quarters. Thetrueimpliedvariancedecomposition var(E(∆yt|∆ct)) is54.2%. var(∆yt) %finite denotesthepercentageoftimesthatafiniteperiodwasobtained. Medianstatisticsarereported,except for%finite. Resultsbasedon1,000simulationswithsamplesizeT =200. 18

Figure 6: Frequency Distribution of ρˆ - x stationary, σ = 3 ηyε t ηx 0.08 α=0.1 α=0.5 0.07 α=0.95 α=0.99 0.06 0.05 0.04 0.03 0.02 0.01 0 -1-.9-.8-.7-.6-.5-.4-.3-.2-.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 ρˆηyǫ Persistence true 100× var(E(∆xt|ct)) Period % finite simulated var( v E a ( r ∆ (∆ yt y |∆ t) ct)) var(∆xt) true var( v E a ( r ∆ (∆ yt y |∆ t) ct)) α = 0.1 0.04 22 77 1.20 α = 0.3 0.1 21 76 1.25 α = 0.5 0.4 21 77 1.24 α = 0.7 1.4 22 77 1.22 α = 0.9 5.6 22 72 1.21 α = 0.95 8.1 22 71 1.21 α = 0.99 10.7 22 68 1.21 Note: Thetrueimpliedperiodis24.8quarters. Thetrueimpliedvariancedecomposition var(E(∆yt|∆ct)) is54.2%. var(∆yt) %finite denotesthepercentageoftimesthatafiniteperiodwasobtained. Medianstatisticsarereported,except for%finite. Resultsbasedon1,000simulationswithsamplesizeT =200. 19

Figure 7: Frequency Distribution of ρˆ - x stationary, σ = 7 ηyε t ηx 0.15 α=0.1 α=0.5 α=0.7 α=0.95 0.1 0.05 0 -1 -.9-.8-.7-.6-.5-.4-.3-.2-.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 ρˆηyǫ Persistence true 100× var(E(∆xt|ct)) Period % finite simulated var( v E a ( r ∆ (∆ yt y |∆ t) ct)) var(∆xt) true var( v E a ( r ∆ (∆ yt y |∆ t) ct)) α = 0.1 0.01 20 70 1.38 α = 0.3 0.02 21 68 1.39 α = 0.5 0.07 20 70 1.37 α = 0.7 0.3 20 68 1.43 α = 0.9 1.1 21 66 1.41 α = 0.95 1.6 21 64 1.45 α = 0.99 2.2 20 63 1.42 Note: Thetrueimpliedperiodis24.8quarters. Thetrueimpliedvariancedecomposition var(E(∆yt|∆ct)) is54.2%. var(∆yt) %finite denotesthepercentageoftimesthatafiniteperiodwasobtained. Medianstatisticsarereported,except for%finite. Resultsbasedon1,000simulationswithsamplesizeT =200. 20

increases for any value of σ . Finally, the proportion of the variance of output attributed ηx to the cycle tends to be more overestimated. WebelievethattherelevantdistributionistheoneshowninFigure7. Whiletheempirical estimates in Basistha (2007) of α = 0.75 and σ = 1.35 are not far from the values assumed ηx in his Monte Carlo exercise, crucially, he finds an estimate of θ = 0.11. This small value for the influence of the cycle on the auxiliary variable, inflation in this case, dramatically reduces its value as a cyclical indicator relative to his Monte Carlo exercise; the impact of the parameter θ on the variance decomposition can be seen in Equation (23).10 In particular, Basistha’s empirical evidence suggests that the contribution of the cycle to the variance of inflation is on the order of 1 ⁄ percent.11 Our simulations would be able to reproduce 4 that order of the variance decomposition for α = 0.7—close to the persistence obtained by Basistha—with a standard deviation of the auxiliary variable’s trend, σ = 7. In that case, ηx we would obtain results similar to those in the α = 0.7 row of the table in Figure 7. Those results indicate that the distribution of ρ would be seriously distorted, the period would ηyε be underestimated, only 68 percent of the times a finite period would be obtained, and the cycle explains 43 percent more of the variations in GDP than it should. It is an open question whether inflation should be modelled as a stationary process, especially over long samples. In Basistha (2007)’s empirical application for the Canadian economy, the results indicate a persistence coefficient of 0.75 with a conventional confidence interval that does not include unity. However, Basistha requires three mean breaks in inflation to obtain that result. Other authors have favored a unit-root specification for inflation. For example, Basistha and Nelson (2007) find that the autoregressive coefficient on U.S. inflation is about 0.88 and the 95% confidence interval includes unity. Along the same lines, Stock and Watson (2007) find that U.S. inflation probably has a unit root.12 Hence, at least for the case of the United States, inflation is probably nonstationary, in which case the results imply that the trend-cycle correlation and the cycle using inflation as the auxiliary variable would not be accurately estimated. 5 Size and Power of the Likelihood Ratio Test of Hypotheses about ρ ηyε In this section, we investigate the performance of the Likelihood Ratio (LR) test under nonstationarity of the accompanying variable and the three following assumptions: • Univariate model and benchmark parametrization. • Bivariate model under Exogenous Auxiliary Trend and benchmark parametrization. 10Basistha’s estimation also finds a statistically significant MA(1) term, which, along with the other estimated parameters, implies a lower contribution of the variance of the cycle to the variance of inflation compared with the case in which there is not such a term. 11ThevariancedecompositionoftheauxiliaryvariableunderthebenchmarkparametrizationinBasistha’s Monte Carlo exercise is 5.8%. 12Otherpaperssimplyassumeaunitrootininflation,suchasKuttner(1994),Roberts(2001)orBasistha and Startz (2008) 21

Under this parametrization, the auxiliary variable has properties similar to the unemployment rate. • Bivariate model under Endogenous Auxiliary Trend and high trend shock variance of the accompanying variable. Under this parametrization, the accompanying variable resembles inflation. 5.1 Size of the LR Test of Hypotheses about ρ η ε y We first compute the frequency with which the LR test incorrectly rejects a true hypothesized value—that is, the size of the test. We simulate the bivariate model 1,000 times according to the specification in Table 1, except that we set ρ = ρ0 for ρ0 ∈ {−0.95, ηyε ηyε ηyε −0.9,−0.8,...,−0.1,0,0.1,...,0.8,0.9,0.95}. We consider the null hypotheses in the univariate and bivariate estimations as H : ρ = ρ0 . 0 ηyε ηyε Figure 8 plots the size of the likelihood ratio test for the univariate and bivariate models using a 5 percent significance level. Recall that, ideally, the test should lead to a horizontal line at a size of 0.05—that is, for all values of ρ , the hypothesis would be rejected 5 percent ηyε of the time. As can be seen, none of these models meets that ideal, although of the three, the BivariateExogenousTrendalternativecomesclosest. Inparticular, theunivariateestimation delivers very low size for hypothesized values of the correlation coefficient above -0.3, such that the test will not reject the hypothesized true value often enough. For correlations below -0.5, the size is too large, meaning that the test of the null hypothesis will be rejected too often. Based on their simulations, MNZ argue that the size of the LR test of the hypothesis H : ρ = 0 is approximately correct. This interpretation is broadly consistent with our 0 ηyε findings because we find that at H : ρ = 0, the null hypothesis will not be rejected often 0 ηyε enough, making MNZ’s rejection of the hypothesis all the more convincing. In the case of the Bivariate Endogenous Trend model, the size is uniformly too large, meaning that the LR test under this alternative would reject the null hypothesis too often, in particular the hypothesis H : ρ = 0. While the size of the test under the Bivariate 0 ηyε Exogenous Trend alternative is too large for values of the correlation coefficient lower than -0.7 and too small for values larger than 0.3, it performs almost uniformly better than the other two cases contemplated. 5.2 Power of the LR Test of Hypotheses about ρ η ε y We now consider the ability of the LR test to reject various false hypothesized values for ρ —that is, the power of the test. Three exercises are performed in which we simulate the ηyε bivariate model 1,000 times. First, we simulate the model according to the benchmark parameter specification in Table 1, that is, assuming that ρ = 0, and set the null hypotheses ηyε in the univariate and bivariate estimations as H : ρ = ρ0 , where ρ0 ∈ {−0.95,−0.9, 0 ηyε ηyε ηyε −0.8,−0.7,...,−0.1,0.1,...,0.7,0.8,0.9,0.95}. In the second exercise, we set ρ = −0.9 ηyε to simulate the model and test the null hypotheses H : ρ = ρ0 , where ρ0 ∈ {−0.95, 0 ηyε ηyε ηyε −0.85,−0.8,−0.7,...,−0.1,0,0.1,...,0.7,0.8,0.9,0.95}. Finally, we simulate the model setting ρ = 0.9 and test the null hypotheses H : ρ = ρ0 , where ρ0 ∈ {−0.95,−0.90, ηyε 0 ηyε ηyε ηyε −0.8,−0.7,...,−0.1,0,0.1,...,0.7,0.8,0.85,0.95} 22

Figure 8: Size of the Likelihood Ratio Test 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -.95 -.8-.7-.6-.5-.4-.3-.2-.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .95 ρ0 η,ǫ y eziS Univariate Bivariate Exog. Trend Bivariate Endog. Trend Note: Thesizecorrespondstoasignificancelevelα=0.05. Figure 9 reports the fraction of times the LR test rejects the false hypothesized value, for the univariate and bivariate models, in the three exercises performed. When the true value of ρ is zero, Figure 9b shows that the test based on univariate estimation has virtually ηyε no power to reject hypothesized values of ρ greater than -0.5. The power increases as the ηyε value of the hypothesized correlation coefficient approaches the left tail but still reaches only 37 percent for the null hypothesis H : ρ = −0.95. Hence, the test based on the univariate 0 ηyε estimation has very low power to reject the false null hypothesis of negative correlation between trend and cycle. Better performance is obtained under a Bivariate Endogenous Trend. The maximum power of the LR test is about 70 percent, and it is reached when ρ0 = −0.95. Better still is the Bivariate Exogenous Trend alternative, which has power ηyε increasing to 100 percent as the hypothesized value moves away from the true correlation coefficient of zero. Figure9aconsiders thecase when thetruevalueof ρ is-0.9. Inthiscase, the univariate ηyε estimation and Bivariate Endogenous Trend alternatives lack the ability to reject almost any false hypothesized value, whereas the Bivariate Exogenous Trend alternative rapidly approaches a rejection probability of one as the hypothesized values move above the true value of the correlation. In particular, the Bivariate Exogenous Trend model would reject with probability one a hypothesized value of zero for the correlation between trend and cycle innovations if the true correlation were -0.9, compared with a power of around 10 percent in the univariate estimation and 55 percent in the Bivariate Endogenous Trend assumption. A similar performance of the LR test is obtained when the true value of ρ is 0.9, as can ηyε bee seen in Figure 9c. Once again, the Bivariate Exogenous Trend alternative yields the highest power of the test almost uniformly, and the worst performance is obtained from the univariate specification. 23

Figure 9: Power of the Likelihood Ratio Test (a) ρ =−0.9 ηyε 1 0.8 0.6 0.4 0.2 0 -.95 -.8-.7-.6-.5-.4-.3-.2-.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .95 ρ0 η,ǫ y rewoP Univariate Bivariate Exog. Trend Bivariate Endog. Trend (b) ρ =0 ηyε 1 0.8 0.6 0.4 0.2 0 -.95 -.8-.7-.6-.5-.4-.3-.2-.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .95 ρ0 η,ǫ y rewoP Univariate Bivariate Exog. Trend Bivariate Endog. Trend (c) ρ =0.9 ηyε 1 0.8 0.6 0.4 0.2 0 -.95 -.8-.7-.6-.5-.4-.3-.2-.1 0 .1 .2 .3 .4 .5 .6 .7 .8 .95 ρ0 η ,ǫ y rewoP Univariate Bivariate Exog. Trend Bivariate Endog. Trend Note: Thepowercorrespondstoasignificancelevelα=0.05. 24

6 Summary of Monte Carlo Findings Before proceeding to the empirical application to obtain the trend-cycle decomposition of GDP for the U.S., we summarize the findings of the Monte Carlo exercises: • Adding a second variable improves the econometrician’s ability to distinguish the trend and cycle components of output correctly. • For example, adding GDI—a second measure of the same concept as GDP—reduces the distortion in the sampling distribution of the correlation between trend and cycle. Nonetheless, in the case of GDI, the distortion in the estimated period remains substantial. • Performance improves as the variability of the auxiliary variable’s trend falls. • In particular, when the variability of the trend is about the same as that typically found for the unemployment rate, the distortion in the sampling distribution of the correlation between trend and cycle for GDP is practically absent. • When a stationary accompanying variable is considered with a parametrization that leads to properties similar to inflation—the leading candidate considered in the literature (see Basistha, 2007; Basistha and Nelson, 2007)—the sampling distribution of the correlation coefficient is distorted and the features of the estimated cycle are somewhat far from its theoretical counterpart. • When we consider treatments that correspond to nonstationary inflation, whether as part of an “accelerationist Phillips curve” as in Kuttner (1994) or Roberts (2001), or as anexogenoustrend, asinStellaandStock(2016),wefindthatinflationperformspoorly as an auxiliary variable. At best—when the variability of trend inflation is relatively low, asStella andStock(2016)suggest isthecase inrelatively recent periods—inflation performs better than GDI. In other settings, such as when the variability of trend inflation is greater, its performance as an auxiliary variable deteriorates. • The model with the best performance of the LR test of hypothesis on the trend-cycle correlation coefficient in terms of size is the bivariate specification under an exogenous trend and the benchmark parametrization—the specification that resembles the unemployment rate. It is followed by the (realistically calibrated) bivariate specification under an endogenous trend and the univariate model, in that order. The same ranking is obtained when considering the power of the LR test. • On balance, we find that the best of the three available auxiliary variables we consider is the unemployment rate, with its relatively low trend volatility being the key to its success. 25

Table 4: Log-likelihood and BIC for Different Restrictions Restriction Log-likelihood BIC 1 None -326.52 714.66 2 ρ = 0 -329.13 714.28 ηyε 3 ρ = 0 -326.89 709.80 ηuε 4 ρ = 0 -328.22 712.46 ηyηu 5 ρ = ρ = 0 -329.56 709.54 ηuε ηyε 6 ρ = ρ = 0 -330.64 711.70 ηyε ηyηu 7 ρ = ρ = 0 -328.37 707.16 ηuε ηyηu 8 ρ = ρ = ρ = 0 -330.71 706.24 ηuε ηyε ηyηu Note: BICisBayesianInformationCriterion 7 Estimation Results of the GDP-Unemployment Bivariate Model Based on the Monte Carlo exercises, we conclude that, among the three considered, the nonstationary variable that would deliver the most reliable trend-cycle decomposition of GDP is the unemployment rate. Consistent with this finding, we therefore adopt the model ofSection3.2thatincludestheunemployment rate, which wereproducehereforconvenience. y = τ +c t yt t τ = µ +τ +η yt y y,t−1 yt c = φ c +φ c +ε , t 1 t−1 2 t−2 t u = τ +θ c +θ c , t ut 1 t 2 t−1 τ = τ +η , ut u,t−1 ut ε σ2 ρ σ σ ρ σ σ t ε ηyε ηy ε ηuε ηu ε η yt  ∼ iid N 0 3×1 ,ρ ηyε σ ηy σ ε σ η 2 y ρ ηyηu σ ηy σ ηu . η ρ σ σ ρ σ σ σ2  ut    ηuε ηu ε ηyηu ηy ηu ηu  We use quarterly GDP and unemployment rate data from the St. Louis FRED database as observable variables for the period 1948:1-2015:4 to estimate the unrestricted unobserved components (UC-UR) model above to extract the trend and the cycle of GDP. Inoursimulationwork, weassumedthatthetwocorrelationsinvolvingtheunemploymentratetrend, ρ and ρ , were equal to zero. In Table 4, we assess whether this hypothesis is ηuε ηyηu correct. By comparing lines 1 and 7, we see that the evidence suggests this hypothesis is not rejected: Twice the difference in the log likelihood is 3.7, a difference that is not significant at the 5 percent level for two degrees of freedom, indicating that we are safe in assuming these correlations are zero. A comparison of lines 7 and 8 provides a test of the restriction ρ = 0, which is the ηyε hypothesis that is explored in detail in the main text. This restriction is strongly rejected. Based on our Monte Carlo analysis, we can be confident that this test is valid. We will 26

Table 5: Bivariate UC Model Estimates ρ = ρ = 0 ρ = ρ = ρ = 0 ηuε ηyηu ηyε ηuε ηyηu Estimate Standard Error Z-statistic Estimate Standard Error Z-statistic µ 0.80 0.04 21.13 µ 0.79 0.04 20.96 y y φ 1.62 0.06 27.61 φ 1.56 0.06 26.56 1 1 φ -0.67 0.06 -11.73 φ -0.61 0.06 -10.50 2 2 σ 0.41 0.07 5.76 σ 0.58 0.05 12.42 ε ε σ 0.60 0.04 14.20 σ 0.60 0.04 16.65 ηy ηy ρ 0.48 0.18 2.64 ρ 0 - - ηyε ηyε θ -0.51 0.10 -5.29 θ -0.36 0.04 -8.59 1 1 θ -0.17 0.04 -3.85 θ -0.18 0.03 -5.63 2 2 σ 0.16 0.02 9.98 σ 0.15 0.02 9.93 ηu ηu ρ 0 - - ρ 0 - - ηuε ηuε ρ 0 - - ρ 0 - - ηyηu ηyηu LogL = -328.37,BIC = 707.16 LogL = -330.71,BIC = 706.24 thus focus on the version of the model that does not impose the ρ = 0 restriction. We ηyε will nonetheless also consider the version of the model with the restriction imposed, in part because it is an important benchmark in the literature. In addition, as noted in the final column, the version of the model with the restriction imposed is preferred according to the Bayesian Information Criterion (BIC). (Although the difference relative to line 7 is “not worth more than a bare mention” (see Kass and Raftery, 1995).) The estimates of the two models appear in Table 5. In the results to the left, the correlation between the output trend and cycle is estimated to be positive, about 0.5. Based on our Monte Carlo results, we are confident that this result is robust. It stands in contrast to the univariate results of MNZ, who found a strongly negative correlation between the output trend and cycle. This result is also different from that of Clark (1989), who found a correlation that was negative but not statistically different from zero at conventional significance levels. The positive correlation between trend and cycle innovations of GDP means that when the UC model sees a surprising acceleration (deceleration) in GDP, it revises both the trend and the cycle upward (downward). Thus, a positive (negative) perturbation to the trend is likely associated with a positive (negative) perturbation to the cycle. Our model is, of course, purely statistical and there are a number of theoretical interpretations of the positive correlation between trend and cycle. One possibility is hysteresis: Long and pronounced periods of economic slack may affect the trend—for example, as the skills of unemployed workers atrophy (Haltmaier, 2012; Ball, 2014; Reifschneider, Wascher and Wilcox, 2015, see).13 13The finding of a positivecorrelationbetweentrendandcycle appears to be importantly affected by the experience after the Great Recession. In fact, if the estimation were to be conducted with data ending in 2009:4 (or earlier), little statistical evidence would have been found to reject the null hypothesis ρ =0. ηyε 27

Figure 10: Smoothed GDP Cycle (a) ρηuε=ρηyηu =0 8 8 6 6 4 4 2 2 0 0 -2 -2 -4 -4 -6 95% confidence interval -6 Smoothed cycle -8 CBO-implied cycle -8 1950 1960 1970 1980 1990 2000 2010 dnert morf noitaived % (b) ρηyε=ρηuε=ρηyηu =0 8 8 6 6 4 4 2 2 0 0 -2 -2 -4 -4 -6 95% confidence interval -6 Smoothed cycle -8 CBO-implied cycle -8 1950 1960 1970 1980 1990 2000 2010 dnert morf noitaived % With respect to the other model parameters, the autoregressive coefficients of the cycle imply a strong hump-shaped pattern of the responses to a business-cycle shock, similar to the results of Fleischman and Roberts (2011) and of MNZ in the estimation of the constrained univariate unobserved components model, the case they label UC-0. The estimated parameters imply complex roots, with a period of 11.1 years. The variance decomposition indicates that about 65 percent of the variation in GDP growth is explained by the cycle. The estimates of the coefficients that relate the cycle to the unemployment rate, θ and 1 θ , suggest a conventional Okun’s law relationship, with the unemployment rate reacting 2 to cyclical shocks with a lag relative to GDP, and a total effect after two quarters of -0.68, somewhat above in absolute terms from conventional estimates of -0.5 (see Abel, Bernanke and Croushore, 2013). Figure 10a plots the smoothed cycle obtained from the estimation of the model along with the 95 percent confidence intervals. For comparison, we also plot the CBO GDP gap. The CBO and our estimate of the cycle behave similarly, although the cycle obtained from our model shows somewhat less variability than CBO’s. At the end of the sample (2015:Q4), our bivariate model predicts that the output gap is still around negative 2 percent, in line with the CBO estimate. The results in the right-hand part of Table 5 consider the model with no correlations among the trends and cycle innovations. The parameter point estimates are broadly similar to those under the more-restrictive model. The point estimates of φ and φ indicate that 1 2 the period of the cycle would be 13.5 years. The variance decomposition would attribute about 60 percent of the variations in GDPgrowth to variations in the cycle, and the long-run Okun’s law coefficient would be about -0.54. The estimate of the cycle in the absence of trend and cycle correlations appears in Figure 10b. This estimate of the cycle is broadly similar to the estimate when the correlation is freely estimated; their unconditional correlation coefficient is 0.92. there are, however, two key differences. First, the variability of cycle is greater in this version of the model. Second, starting in the mid-1990s, the estimate of the cycle is shifted upwards. We view the smaller variability of the cycle in the first case as a natural consequence of the positive correlation 28

Figure 11: Observed real GDP and Unemployment Rate and Corresponding Trends (a) ρηuε=ρηyηu =0 10 12 Log of GDP (left axis) 11 Log of GDP trend (left axis) 9.5 10 9 9 8 7 8.5 6 5 8 4 3 7.5 2 11995500 11996600 11997700 11998800 11999900 22000000 22001100 % (b) ρηyε=ρηuε=ρηyηu =0 10 12 Log of GDP (left axis) 11 Log of GDP trend (left axis) 9.5 10 9 9 8 7 8.5 6 5 8 4 Unemployment rate (right axis) Unemployment rate trend (right axis) 3 7.5 2 11995500 11996600 11997700 11998800 11999900 22000000 22001100 % Unemployment rate (right axis) Unemployment rate trend (right axis) between trend and cycle, as a shock that pushes up the cyclical component of output will also tend to raise the trend, softening the cyclical increase. We view the recent higher level of the trend (lower cyclical component) in the model with the unconstrained trend-cycle correlation as being the consequence of the longer recoveries that have been the norm since the 1980s: A long recovery will be accompanied by upward revisions to trend output. Thus, by 2007, the level of trend GDP was about 2 percent higher in the model with correlated trend and cycle. It is interesting to note that in the succeeding recession, this difference in trends was almost entirely eliminated, and in both models, output was about 6 percent below trend at the trough. The gap re-emerged over the next several years, however, and at the end of the sample, the estimated cycle is around zero in the model with no correlation, compared with the 2 percent shortfall in the model with a correlated trend and cycle. Figure 11 shows the observed (log of) GDP and unemployment rate series along with their estimated trends for the two specifications whose results appear in in Table 5. Both modelspredict relatively smallincreases inthetrendcomponent ofGDPinthelatteryearsof the sample, as is apparent from the dashed blue lines in Figures 11a and 11b. The estimated trendof themodel with(out) correlated outputinnovations rises onlyat anannualized rateof 11/4 (1) percent on average from 2010 to 2015, compared with 21/4 (23/4) percent on average from 2007 to 2009 and 23/4 (21/2) percent per year in the four years before that. Figure 11 also shows the unemployment rate and its trend. The broad movements in the trend unemployment rate are similar across the two specifications (with and without trendcycle correlation in ouptut): The unemployment rate trend moves up fairly steadily from the beginning of the sample to the mid-1970s, reaching as high as 7 percent in the early 1980s. After a temporary decline in the mid-1980s and a subsequent increase, the trend moves down over the rest of the 1980s and through the 1990s, reaching as low as 51/2 percent in the model with correlation and 6 percent in the model without correlation. From the mid-1990s to 2007, the unemployment trend stays around 51/2 percent in the model with correlation and 61/2 percent percent in the model without correlation. The trend unemployment rate spikes in both models during the financial crisis but then moves down thereafter. As would 29

be expected given Okun’s law, at the end of 2015, the model with correlation estimates a trend unemployment rate of around 4 percent and thus an unemployment gap, whereas the model without correlations obtains a trend unemployment rate of around 5 percent. 8 Conclusions In this paper, we investigated the performance of different bivariate unobserved components models in estimating the trend and cycle of GDP. We found that the best variable to accompany GDP in the bivariate specification is the unemployment rate, which is superior in performance to two alternatives, namely inflation and gross domestic income. Our results suggest that the main reason the unemployment rate is especially helpful is that its unit root component (trend) has a relatively small variance relative to its cyclical component. We estimated the cycle using GDP and unemployment rate data for the U.S. and found that there is evidence of positively correlated trend and cycle innovations and that the cycle has a conventional shape, with a period of about 11 years. Overall, our Monte Carlo experiments suggest that the results of our statistical tests could be trusted when the unemployment rate is included in the model. 30

References Abel, AndrewB., BenBernanke, andDeanCroushore. (2013)Macroeconomics (8th Edition): Prentice Hall. Ball, Laurence. (2014) “Long-term damage from the Great Recession in OECD countries.” European Journal of Economics and Economic Policies: Intervention, 11, 149–160. Basistha, Arabinda. (2007) “Trend-Cycle Correlation, Drift Break and the Estimation of Trend and Cycle in Canadian GDP (Corrlation tendance-cycle, discontinuit, et estimation de la tendance et du cycle dans le PIB canadien).” The Canadian Journal of Economics / Revue canadienne d’Economique, 40, pp. 584–606. Basistha, Arabinda. (2009) “Hours per capita and productivity: evidence from correlated unobserved components models.” Journal of Applied Econometrics, 24, 187–206. Basistha, Arabinda and Charles R. Nelson. (2007) “New measures of the output gap based on the forward-looking new Keynesian Phillips curve.” Journal of Monetary Economics, Elsevier, 54, 498–511. Basistha, Arabinda and Richard Startz. (2008) “Measuring the NAIRU with Reduced Uncertainty: A Multiple-Indicator Common-Cycle Approach.” The Review of Economics and Statistics, 90, 805–811. Clark, Peter K. (1987)“The Cyclical Component of U. S. Economic Activity.” The Quarterly Journal of Economics, 102, pp. 797–814. Clark, Peter K. (1989) “Trend reversion in real output and unemployment.” Journal of Econometrics, 40, 15 – 32. Fixler, Dennis J. and Jeremy J. Nalewaik. (2007) “News, noise, and estimates of the “true” unobserved state of the economy.” Finance and Economics Discussion Series 2007-34, Board of Governors of the Federal Reserve System (U.S.). Fleischman, Charles and John M. Roberts. (2011) “From many series, one cycle: improved estimates of the business cycle from a multivariate unobserved components model.” FinanceandEconomics DiscussionSeries2011-46,BoardofGovernorsoftheFederalReserve System (U.S.). Haltmaier, Jane. (2012) “Do recessions affect potential output?” International Finance Discussion Papers 1066, Board of Governors of the Federal Reserve System (U.S.). Kass, Robert E and Adrian E Raftery. (1995) “Bayes factors.” Journal of the American Statistical Association, 90, 773–795. Kuttner, Kenneth N. (1994) “Estimating Potential Output as a Latent Variable.” Journal of Business & Economic Statistics, 12, pp. 361–368. 31

Morley, James C, Charles R Nelson, and Eric Zivot. (2003) “Why are the Beveridge-Nelson and unobserved-components decompositions of GDP so different?” Review of Economics and Statistics, 85, 235–243. Nalewaik, Jeremy J. (2010) “The Income- and Expenditure-Side Estimates of U.S. Output Growth.” Brookings Papers on Economic Activity, 41, 71–127. Nelson, CharlesRandCharlesRPlosser. (1982)“Trendsandrandomwalksinmacroeconmic timeseries: someevidence andimplications.” Journal of monetary economics,10,139–162. Perron, Pierre and Tatsuma Wada. (2009) “Let’s take a break: Trends and cycles in US real GDP.” Journal of Monetary Economics, 56, 749–765. Reifschneider, Dave, William Wascher, and David Wilcox. (2015) “Aggregate Supply in the United States: Recent Developments and Implications for the Conduct of Monetary Policy.” IMF Economic Review, 63, 71–109. Roberts, John M. (2001) “Estimates of the Productivity Trend Using Time-Varying Parameter Techniques.” The B.E. Journal of Macroeconomics, 1, 1–32. Sinclair, Tara M. (2009) “The Relationships between Permanent and Transitory Movements in U.S. Output and the Unemployment Rate.” Journal of Money, Credit and Banking, 41, 529–542. Stella, Andrea and James H. Stock. (2016) “A state-dependent model for inflation forecasting.” in Siem Jan Koopman and Neil Shephard eds. Unobserved Components and Time Series Econometrics: Oxford University Press, Chap. 3, 14–29. Stock, James H. and Mark Watson. (2007) “Why Has U.S. Inflation Become Harder to Forecast?” Journal of Money, Credit and Banking, 39, 3–33. Stock, James H. and Mark W. Watson. (1998) “Business Cycle Fluctuations in U.S. Macroeconomic Time Series.” NBER Working Papers 6528, National Bureau of Economic Research, Inc. US. Bureau of Economic Analysis. (2016) “Real Gross Domestic Product [GDPMC1].” retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/ series/GDPMC1. US. Bureau of Labor Statistics. (2016) “Civilian Unemployment Rate [UNRATE].” retrieved fromFRED,FederalReserveBankofSt.Louis; https://fred.stlouisfed.org/series/ UNRATE. Wada, Tatsuma. (2012) “On the correlations of trendcycle errors.” Economics Letters, 116, 396 – 400. Watson, Mark W. (1986) “Univariate detrending methods with stochastic trends.” Journal of Monetary Economics, 18, 49 – 75. 32