GDP Trend-cycle Decompositions Using State-level Data

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. GDP Trend-cycle Decompositions Using State-level Data Manuel Gonzalez-Astudillo 2017-051 Please cite this paper as: Gonzalez-Astudillo, Manuel (2017). “GDP Trend-cycle Decompositions Using State-level Data,”FinanceandEconomicsDiscussionSeries2017-051. Washington: BoardofGovernors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.051. 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.

GDP Trend-cycle Decompositions Using State-level Data Manuel Gonzalez-Astudillo† manuel.p.gonzalez-astudillo@frb.gov Federal Reserve Board Washington, D.C., USA April 28, 2017 Abstract This paper develops a method for decomposing GDP into trend and cycle exploiting the cross-sectional variation of state-level real GDP and unemployment rate data. The model assumes that there are common output and unemployment rate trend and cycle components, and that each state’s output and unemployment rate are subject to idiosyncratic trend and cycle perturbations. The model is estimated with Bayesian methods using quarterly data from 2005:Q1 to 2016:Q1 for the 50 states and the District of Columbia. Results show that the U.S. output gap reached about -8% during the Great Recession and is about 0.6% in 2016:Q1. Keywords: Unobserved components model, trend-cycle decomposition, state-level GDP data JEL Classification Numbers: C13, C32, C52 ∗Theviewsexpressedinthispaperaresolelytheresponsibilityoftheauthorandshouldnotbeinterpreted as reflecting the views of the Board of Governors of the Federal Reserve System. †I thank Paul Yoo for speedy and accurate researchassistance.

1 Introduction Theoutputgap—thedifferencebetweenactualandpotentialortrendoutput—isacentral measure in policy making. Both monetary and fiscal policy look at the output gap to gauge inflationary pressures or the degree of labor market slack. Because the output gap is a nonobservable variable, several statistical and econometric techniques have been proposed toestimatethecyclical positionoftheeconomy. TheCongressional Budget Office(CBO),for example, offers a measure of potential output that helps forecast future federal revenue and spending, and from which the output gap can be obtained (see Congressional Budget Office, 2014, for a recent report). Central banks also construct estimates of the output gap that inform their monetary policy decisions (see Federal Reserve Bank of Philadelphia, 2017). Figure 1 plots the CBO-implied and Board of Governors (BOG) of the Federal Reserve System output gaps for the available periods.1 Figure 1: CBO-implied and BOG Output Gaps 8 6 4 2 0 -2 -4 -6 -8 1950 1960 1970 1980 1990 2000 2010 laitnetop/dnert morf noitaived % 8 6 CBO-implied Output Gap 4 2 0 -2 -4 -6 -8 1950 1960 1970 1980 1990 2000 2010 laitnetop/dnert morf noitaived % BOG Output Gap Among the statistical techniques used to discriminate the permanent component from the transitory or cyclical component is the Beveridge and Nelson (1981) decomposition of integrated time series. Other techniques involve filters of different natures to separate the cycle from the trend or long-run component, like the Hodrick and Prescott (1997) filter, or variations of a band-pass filter, as in Baxter and King (1999) and Christiano and Fitzgerald (2003), or the Kalman filter to estimate models with unobserved components (UC), as in Clark (1987). Another approach suggested by Stock and Watson (1989) uses multiple macroeconomic time series to extract a common cyclical component in the spirit of dynamic factor models. Even though there are existing methods to estimate the cyclical position of the economy or of a group of economies exploiting the variability of multiple variables, multiple cross- 1The BOGoutput gaparereal-time estimates andprojectionsofthe output gapusedby the staffof the Boardof Governorsof the Federal Reserve System in constructing its forecast. These estimates are released with a lag of 5 years and published by the Federal Reserve Bank of Philadelphia. Hence, the information ends in the fourth quarter of 2011, as of the date of publication of the current draft. 1

sectional units, or both, none of these methods has explored the estimation of the output gap—defined as the difference between actual and potential output—using multiple crosssectional units such as geographical regions or industries. In this paper, I propose a UC model that combines information about output and the unemployment rate from many cross-sectional units—the U.S. states—to estimate the aggregate GDP cycle and its trend. The proposed model exploits the panel structure of the state-level data and adds richness to the conventional UC approach in a way that resembles the dynamic factor models but that allows me to measure the output gap as the difference between actual and potential GDP. The setup assumes that (the log of) each state’s real GDP is the sum of state-specific trendandcyclecomponents. Eachofthestate-specifictrends, inturn, isalinearcombination of a trend common across states and an idiosyncratic trend. Likewise, each of the state’s cycles is a linear combination of a cycle common to all of the states and an idiosyncratic component. Lastly, an Okun’s law at the state level relates the cycle of GDP to the cycle of the unemployment rate. Hence, information from multiple cross sections and variables can be used to estimate the aggregate output cycle. The model is similar to the one suggested by Stock and Watson (2016) for estimating trend inflation, in which different inflation categories are used as the source of cross-sectional variation. The difference is that the model in this paper incorporates a bivariate analysis within the cross sections, including another dimension of variation that adds information to the estimation. Additionally, in the model of this paper, the cyclical components are assumed to have autocorrelated dynamics, while autocorrelation is absent from the setup of Stock and Watson. The aim of the paper is similar in spirit to that of Owyang, Rapach and Wall (2009), who extract business cycle factors using information on income and payroll employment at the state level. In the present paper, however, I use real GDP and unemployment rate data, so it is possible to interpret the cyclical component of GDP as a measure of the output gap. I estimate the model with quarterly information for the 50 states and the District of Columbia for the period from 2005:Q1 to 2016:Q1. The results are obviously influenced and characterized by the features of the Great Recession. In particular, the estimated cycle is very persistent, with a hump-shaped response of the cycle to an innovation in itself. The estimated smoothedcycle is-7.9%atthetroughoftherecession inthefourthquarter of 2009, slightly lower than the CBO-implied cycle, which reached about -7.3% two quarters before. The model’s estimates imply that the trend of GDP declined about 1.9% between the end of 2008 and the beginning of 2009, whereas the CBO estimate lowers its slope much more slowly. At the end of the sample, the estimated output gap is about 0.6% and trend output would be growing at an average annual pace of 3⁄ 4 percent, compared to 31⁄ 4 percent before the recession. Theproposedmodelalsoallowsonetoexaminethecyclicalfeaturesofthestateeconomies during the recession and at the end of the sample. For example, the states that were hit the hardest by the recession were those in the rust belt, as well as Arizona, California, Nevada, and Florida. At the end of the sample, states like West Virginia and Wyoming were experiencing negative output gaps, probably driven by low commodity prices. I find that adding cross-sectional information reduces the uncertainty around the estimator of the cycle compared with a model that uses aggregate data only. I also find that the model with many cross sections outperforms two competing models in predicting the growth 2

rate of real GDP. The first is a simple AR(1) process of real GDP growth. The second is a (bivariate) UC model that uses aggregate information on real GDP and the unemployment rate. The reduced uncertainty and the better forecasting ability shed light on the advantage of using cross-sectional information to obtain estimates of the cycle with a UC model. The paper is structured as follows: Section 2 reviews the existing literature. In Section 3, Iset upthe UC model with many cross sections. Section 4 presents the estimation technique. Section 5 describes the data at the state level and shows the features of the Great Recession. In Section 6, I present and discuss the results of the estimation. Section 7 evaluates the performance of the model. Section 8 concludes. 2 Contacts with the Literature This work is related to the literature on UC models of trend-cycle decompositions. It also relates to the literature on common components estimation with dynamic factor models and, indirectly, to the literature on extracting common stochastic trends with panel data. Introducing multidimensional variability into the UC model’s setup can be done in three ways. One is to incorporate many variables belonging to one unit of interest, where these variables provide information toextract common trends and/orcycles. Another possibility is to incorporate many units of interest for which the same variable is used to extract common trends and/or cycles. The third possibility considers a combination of many units of interest and many variables to extract common trends and/or cycles. In this paper, I adopt the third possibility. 2.1 Many Variables and a Single Economic Unit With respect to the first approach, in which many variables of the same unit of interest provide information about a common cycle, Crone and Clayton-Matthews (2005) describe how to use mixed frequency data fromtheU.S. states to estimate state-level monthly indexes ofeconomicactivityforeachstate. ThesetupistheUCmodeldiscussed inStockandWatson (1988, 1989). Observable variables for each state are the first difference of the following: (the log of) nonagricultural employment, the unemployment rate, (the log of) average hours worked in manufacturing, and (the log of) real wage and salary disbursements. A scalar latent stationary series is common to the state-level observable variables and is interpreted as the state’s cycle. The observable series load on the state’s cycle with leads and/or lags. In this analysis, only the cross-sectional structure of the many variables is exploited, not the variability along the state-level dimension. Basistha and Startz (2008) and Fleischman and Roberts (2011) also use several variables, but at the aggregate level, to estimate the U.S. NAIRU in the first case and to estimate the U.S. potential output and its associated business cycle in the second. In both cases, the authors emphasize the advantages of using a multivariate approach in the estimation of unobserved components models. Such advantages include gaining precision of the estimates and the ability to assess the trade-offs among competing signals in a coherent way. 3

2.2 A Single Variable and Many Economic Units Within the second approach —many units of interest and one variable—Del Negro and Otrok(2008)extendafactormodeltoincorporatetime-varyingfactorloadingsandstochastic volatility in order to extract the international business cycle using a panel of 19 countries. The model is estimated with Bayesian methods and allows one to obtain the common and country-specific cycles for the growth rates of GDP, but it does not consider the common GDP trends. Mitra and Sinclair (2012) also use many cross sections and one variable. They propose a multivariate UC model to simultaneously decompose real GDP for each of the G-7 countries into its respective trend and cycle components. The setup considers real GDP as the only observablevariableandassumesthateachcountry’sGDPisdrivenbyspecifictrendandcycle components. The setup allows for possible correlations between any of the contemporaneous shocks to the unobserved (trend and cycle) components. Stock and Watson (2016) propose a multivariate model to estimate trend inflation. They use 17 components of the U.S. personal consumption expenditure inflation to construct an index akin to core inflation, but with time-varying distributed lags of weights, where each sectoral weight depends on the time-varying volatility and persistence of the sectoral inflation series, and on the co-movement among sectors. The modeling framework is a dynamic factor model with time-varying coefficients and stochastic volatility estimated with Bayesian methods. This work is the most similar to the present paper, although it only considers one variable in the analysis—the inflation rate—while this paper considers two, GDP and the unemployment rate. 2.3 Many Variables and Many Economic Units Regarding the third approach—many variables and many cross sections—Gregory, Head and Raynauld (1997) use a dynamic factor model estimated with classical methods to decompose aggregate output, consumption, and investment for the G-7 countries into factors that are common across all countries and aggregates, common across aggregates within a country, and specific to each individual aggregate. Data have to be de-trended to use the dynamic factor models approach because the underlying factors are assumed to be stationary. Similarly, Kose, Otrok and Whiteman (2003) estimate a dynamic latent factor model, but with Bayesian methods, to extract common components frommacroeconomic aggregates (output, consumption, and investment) in a 60-country sample covering seven regions of the world. They allow factors common to the world, the regions, and the countries. Here, too, data are de-trended. For the United States, Owyang, Rapach and Wall (2009) use state-level income and payroll employment data to estimate a dynamic factor model of the 48 contiguous states and the District of Columbia in order to extract business cycle factors. The estimation of the model identifies three common factors underlying the fluctuations in state-level income and employment growth, with the first common factor resembling aggregate fluctuations in real activity at the national level. The factors explain a large proportion of the total variability in state-level variables, although there is still a substantial amount of cross-state heterogeneity. 4

3 Unobserved Components Models for Trend-cycle Decomposition of GDP There is a relatively long history of formulations of UC models to decompose GDP into trend and cycle components. Clark (1987) proposes a univariate estimation in which the log of aggregate real GDP is the sum of orthogonal trend and cycle components and where the cyclefollowsastationaryAR(2)process. Clark(1989)addstheaggregateunemployment rate as an additional variable to inform the trend-cycle decomposition and sets up the following model: y = τy +c (1) t t t u = τu +θ c +θ c (2) t t 1 t 2 t−1 τy = µ+τy +ηy (3) t t−1 t τu = τu +ηu, (4) t t−1 t c = φ c +φ c +ε , (5) t 1 t−1 2 t−2 t with ε σ2 0 0 t ε ηy F ∼ iid N 0 , 0 σ2 0 , (6)  t(cid:12) t−1  3×1  ηy  ηu (cid:12) 0 0 σ2 t (cid:12) ηu  (cid:12)    and F t−1 is the sigma-field co(cid:12)ntaining the information up to period t−1. In equations (1)-(6), y i (cid:12) s the log of aggregate real GDP, τy is its unobserved trend, t t assumed to be a random walk with mean growth rate µ, and c is the unobserved stationary t cycle that follows an AR(2) process with autoregressive coefficients φ and φ . Additionally, 1 2 u istheaggregateunemployment rateandτu isitsunobserved trend, whichfollowsarandom t t walk process, whereas the cyclical component is given by a linear combination of current and lagged values of the cycle of GDP with coefficients θ and θ , in a way resembling Okun’s 1 2 law. Finally, ε is a disturbance to the cycle, while ηy and ηu are trend disturbances, all of t t t them independent of each other. Estimation of this model has been typically made with classical methods using the Kalman filter (see Clark, 1989; Gonzalez-Astudillo and Roberts, 2016, for example). Also, the inflation rate can be introduced in a way resembling a Phillips curve to inform the estimation of the cycle (see Basistha and Nelson, 2007, for example). 3.1 The Unobserved Components Model With Many Cross Sections In this paper, I introduce a cross-sectional dimension to the UC model in equations (1)- (6). Specifically, the cross-sectional information comes from state-level data. The model specifies (the log of) each state’s real GDP, y , and unemployment rate, u , for states it it i = 1,2,...,n in period t = 1,2,...,T as follows: y = τy +c (7) it it it 5

u = τu +θ c +θ c (8) it it 1i it 2i i,t−1 τy = δyτy +ξy (9) it i t it τu = δuτu +ξu (10) it i t it ξy = µ +ξy +ηy (11) it i i,t−1 it ξu = ξu +ηu (12) it i,t−1 it τy = µ+τy +ηy (13) t t−1 t τu = τu +ηu (14) t t−1 t c = α c +υ (15) it i t it υ = ρ υ +ζ (16) it i i,t−1 it c = φ c +φ c +ε , (17) t 1 t−1 2 t−2 t where the error terms are assumed to be white noise, uncorrelated with each other and normally distributed, as follows: ε σ2 0 0 0 0 0 t ε  η η t t u y (cid:12) (cid:12) (cid:12)F ∼ N  0,  0 0 σ 0 η 2 y σ 0 η 2 u 0 0 0 0 0 0  , (18)   η i y t   (cid:12) (cid:12) t−1     0 0 0 σ η 2 i y 0 0     ηu(cid:12)  0 0 0 0 σ2 0   it(cid:12)   η i u    ζ it   (cid:12) (cid:12)     0 0 0 0 0 σ ζ 2 i      (cid:12)    (cid:12) and F is the sigma-fi(cid:12)eld containing the information up to period t − 1. Appendix A t−1 presents the state-space model in matrix form. In the model characterized by equations (7)-(18), each state’s GDP and unemployment rate are the sum of GDP and unemployment rate trends and cycles. The state’s GDP and unemployment ratetrends, τy and τu, respectively, are the sum of two components. The first it it component is a function of the common trends, τy or τu, with loadings δy or δu, respectively. t t i i The common trends, in turn, are unit-root processes with drift µ in the case of GDP and without drift in the case of the unemployment rate. The second component is a state-specific trend, ξy or ξu, which are unit-root processes with drift µ in the case of GDP and without it it i drift in the case of the unemployment rate. The state-specific GDP cycle is c , whereas in the case of the unemployment rate, the it cycle is a linear combination of the cycle of GDP with coefficients θ and θ in a way 1i 2i resembling an Okun’s law, as in Clark (1989). Each state’s GDP cycle loads on the common cycle, c , with coefficient α and is affected by a state-specific disturbance, υ , which in turn t i it follows a stationary AR(1) process with persistence coefficient ρ . Lastly, the common cycle i follows a stationary AR(2) process with persistence coefficients φ and φ . 1 2 The model outlined in equations (7)-(18) differs from the existing UC models for GDP trend-cycle decompositions in the literature because it incorporates cross-sectional information in the analysis. The common trend and cycle of aggregate GDP are informed by the cross-sectional structure of the data. Although the idiosyncratic disturbances are uncorrelated across states, the common trend and cycle allow for correlated movements in the state-level GDPs and unemployment rates. The model differs from the conventional dy- 6

namic factor models approach because it explicitly allows for trends, which are generally removed in the factor analysis from each of the variables to deal with stationary processes. The exception to this de-trending treatment is the setup in Stock and Watson (2016). Their model is similar to the model presented here, except that I do not allow for stochastic volatilities or time-varying loading coefficients, in particular because of the short longitude of the data and the large cross-section. What is allowed in the model considered here, however, is the possibility that the cyclical components, c and υ , follow autocorrelated dynamics, t it whereas Stock and Watson assume that they are martingale processes. Additionally, the model presented here incorporates two variables for each cross section, namely real GDP and the unemployment rate, to inform the trend and cycle of GDP, whereas in Stock and Watson the only variable of interest was inflation to inform its own trend. 4 Estimation Strategy This section lays out the identification restrictions and the Bayesian estimation strategy to obtain the parameters that characterize the model in equations (7)-(18), as well as the common and state-specific trends and cycles of output. 4.1 Identification Regarding the identification of the parameters and the unobserved components, the variance of the error term of the common cycle, σ2, is set to one to be able to identify the scale of ε theloadingcoefficients, α . Thesameisdonewiththevariancesoftheerrortermsofthecomi mon trends, σ2 and σ2 , in order to identify the loading coefficients δy and δu, respectively. ηy ηu i i The drift of the common output trend, µ, is set to zero to allow the unrestricted estimation of the state-specific drifts, µ . To avoid state-level GDP trends with negative slopes, the i loading coefficients, δy, and the state-specific drifts, µ , are restricted to be positive for all i i the states. The variances of the idiosyncratic components of the cycle, σ2, are identified ζi from the covariance and autocovariance structure, including the cross autocovariances, of the state-level GDPs and unemployment rates. Because the common cycle, c , is identified only up to its sign, I restrict α ≥ 0 for t i i = CALIFORNIA. I also restrict the loading coefficients of each state’s unemployment rate on the common unemployment trend, δu, to be nonnegative for all the states in order to i prevent unemployment rate trends at the state level that may move opposite to the common trend permanently. 4.2 Bayesian Estimation The estimation of the UC model with many cross sections in equations (7)-(18) is carried out with Bayesian methods and involves 512 coefficients when the identification restrictions are imposed. I set up the following Metropolis-within-Gibbs procedure.2 Let z be a 2×1 it vector containing the rate of real GDP growth and the first difference of the unemployment rate for i = 1,...,n and t = 1,...,T. Also, let x be a vector including the latent variables it 2Appendix B describes in detail the sampling procedure. 7

for i = 1,...,n and t = 1,...,T. Additionally, denote as Θ the parameters of the statemi space model’s measurement equation for state i and Θ the parameters of the state-space si n n model’s transition equation for state i. For Θ = Θ and Θ = Θ , the steps are m mi s si i=1 i=1 the following: S S 1. Start with initial values for the model’s parameters, Θ = Θ Θ . m s 2. Conditional on the data, Z = [˜z ,˜z ,...,˜z ], and Θ, generaSte a draw of the latent T 1 2 T variables, X = [x˜ ,x˜ ,...,x˜ ], using the Durbin and Koopman (2002) simulation T 1 2 T smoother. Here, ˜z = [z ,z ,...,z ] and x˜ = [x ,x ,...,x ]. t 1t 2t nt t 1t 2t nt 3. Conditional on X ,Z , and Θ , generate a draw of Θ using the Normal-Inverse- T T m s Gamma posterior distribution and the Metropolis-Hastings algorithm. 4. Conditional on X ,Z , and Θ , generate a draw of Θ using the Metropolis-Hastings T T s m algorithm imposing the identification restrictions. 5. Return to step 2. 5 Data I consider real GDPin chained 2009 dollars for the 50 states and the District of Columbia at a quarterly frequency from the first quarter of 2005 to the first quarter of 2016. The information is obtained from the U.S. Bureau of Economic Analysis Regional Economic Accounts. The unemployment rate data at the state level are obtained from the Local Area Unemployment Statistics of the U.S. Bureau of Labor Statistics. Because the aim of the paper is to obtain and characterize the national business cycle from state-level data, this section briefly describes the characteristics of the state economies. I pay special attention to the features of the state economies during the Great Recession. I show the industry contribution to each state’s GDP, the states that were most strongly affected by the recession, and the industries that have contributed the most to the growth of GDP in each state between 2005 and 2015. 5.1 Economic Activities by State This section describes the most relevant industries by state. Using annual information by industries at the state level from 2005 to 2015, Figure 2 presents two maps of the United States with the two most important industries in terms of GDP participation, on average, for each state.3 The industries that contribute the most to the states GDPs, using the North American Industry Classification System, are the following: (i) Manufacturing, (ii) Real Estate and Rental and Leasing, (iii) Mining, (iv) Professional, Scientific, and Technical services, (v) Health Care and Social Assistance, (vi) Finance and Insurance, (vii) Accommodation and 3This analysis excludes the importance of the government sector in each state, which can be large for some of the state economies. 8

Figure 2: Most Important Industries by State 2005-2015 (a) First Industry 14% 13% 12% 14% 14% 25% 13% 16% 20% 19% 14%15% 32% 19% 17% 14% 12% 13% 15%16% 15% 13% 14% 15% 13% 13% 29% 18% 13% 16%2 1 8 7 % % 18% 14% 16% 15% 16% 21% 15% 12% 16% 17% 16% 17% 12% 15% 23% 17% 31% Finance Manufacturing 19% Mining Professional Real Estate (b) Second Industry 14% 9% 10% 12% 11% 13% 12% 9% 12% 12% 11%11% 9% 9% 11% 12% 14% 14 1 % 1% 10% 14% 12% 9% 10% 11% 13% 10% 11% 10% 10%15 9 % % 10% 12% 12% 10% 11% 11% 9% 11% 10% 12% 10% 10% 11% 12% 9% 8% 9% Accommodation Finance 8% Health Manufacturing Mining Professional Real Estate Transportation Note: Percentagesaretheaveragecontributionoftheindustrytothestate’sGDPbetween2005and2015. Source: U.S.BureauofEconomicAnalysis(BEA),RegionalEconomicAccounts. 9

Figure 3: States with Negative Average Real GDP Annual Growth in 2007- 2009 MN ME ID WI MI VTNH IA CTRI PA NV IL IN OH DE NJ MO KY CA TN NC AZ AR SC MS AL GA LA FL -1% to 0% -2% to -1% HI -3% to -2% < -3% Note: Statesinlightgrayexperiencedpositiveaveragegrowthrates. Source: U.S.BureauofEconomicAnalysis(BEA),RegionalEconomicAccounts. Food Services, and (viii) Transportation and Warehousing. By a large margin, the two most important industries for the vast majority of states are manufacturing and real estate. However, some other states have other sectors that contribute importantly to their GDP. For example, mining is an important industry in Alaska, New Mexico, North Dakota, Oklahoma, Texas, West Virginia, and Wyoming. Colorado, the District of Columbia, Maryland, Massachusetts, andVirginiahaveanimportantcontributionfromtheprofessional services sector. Financial services contribute importantly to the GDP of Connecticut, Delaware, Iowa, New York and South Dakota. Health-related services are an important sector in Florida, Maine, Montana, and Rhode Island, while Hawaii and Nevada both have large contributions from accommodation and food services. 5.2 States Most Strongly Affected by the Great Recession The states that were most strongly affected by the Great Recession in terms of real GDP growth rates in the 2007–09 period appear in Figure 3. The states that experienced the lowest average growth rates of real GDP were Florida, Michigan, and Nevada, with rates lower than -3%. Following this group are Arizona and Ohio, with growth rates between -3% and -2%. Having real estate as the first or the second most important economic activity was most likely associated with GDP growth declines across states. This situation is obvious, given the nature of the Great Recession. By contrast, having mining as one of the two most important economic activities—like in Alaska, New Mexico, North Dakota, Oklahoma, Texas, West Virginia, and Wyoming—was associated with positive average growth rates in that period. All told, 30 of the 48 states experienced zero or negative average growth rates in the 2007–09 period. 10

Figure 4: Most Important ContributingIndustries to GDP Growth 2005-2015 6%/23% 5%/19% 124%/91% 3%/10% 2%/-1% 26%/31% 4%/14% 5%/24% 2%/8% 1%/-2% 3%/4% 5%/21% 4%/12% 4%/6% 6%/23% 4%/15% 4%/16% 2%/7% 2%/7% 7%/15% 2%/3%2%/-3% 2%/-6% 6%/28% 4%/20% 4%/17% 4%/6% 6%/11% 7%/11%3%/13% 3%/8% 4%/9% 6%/16% 19%/32% 4%/12% 4%/12% 3%/4% 7%/9% 9%/9% 5%/12% 3%/5% 3%/4%2%/5% 15%/44% 3%/-3% 2%/3% 12%/21% Agriculture Finance 4%/10% Health Information Manufacturing Mining Professional Real Estate Note: ThefirstpercentageistheaveragecontributionofthemostimportantindustrytoGDPgrowth. Thesecond percentageistherateofrealGDPgrowthbetween2005and2015. Source: U.S.BureauofEconomicAnalysis(BEA),RegionalEconomicAccounts. 5.3 Most Important Contributing Industries to the States’ Real GDP Growth Now that we have seen the states that were affected the most by the Great Recession, Figure 4 presents the real GDP growth rates between 2005 and 2015 by state and the industries that contributed the most to the growth of real GDP in each state. The five stateswiththehighest growthratesareNorthDakota(91%), Texas(44%), Oklahoma(32%), Oregon (31%), and Utah (28%). The five states with the lowest growth rates are Nevada (- 6%), Louisiana (-3%), Connecticut (-3%), Michigan (-2%), and Rhode Island (-1%). Mining has been the most important contributor to growth in the three fastest growing states. In Oregon, manufacturing contributed the most, while the real estate sector hasaffected growth the most in Utah. In the majority of states, the construction and manufacturing industries have been the biggest drags with respect to GDP growth. In fact, construction is the sector with the largest negative contribution to the growth of real GDP in Nevada, Michigan, and Rhode Island, whereas the manufacturing sector has the largest negative contribution in Louisiana and Connecticut. In the next section, I present the results of the estimation and link these results to the narrative of this section. 6 Estimation Results This section presents the results of the estimation of the UC model of equations (7)-(18). It describes the results of the estimation focusing on the aggregate GDP trend and cycle. It 11

Table 1: Prior Distributions of the Parameters - State Level Data Parameter Distribution Mean Standard Deviation φ Truncated Normal 1.5 1 1 φ Truncated Normal -0.6 1 2 µ Truncated Normal 0.8 0.8 i α Normal 1 1 i δy Truncated Normal 1 1 i δu Truncated Normal 1 1 i θ Normal -0.25 0.25 1i θ Normal -0.25 0.25 2i ρ Truncated Normal 0 1 i σ2 Inverse Gamma — — ηy i σ2 Inverse Gamma — — ηu i σ2 Inverse Gamma — — ζi also discusses the features of each state’s estimated trend and cycle and the magnitudes of the Okun’s law coefficients. 6.1 Prior Distributions The prior distributions of the parameters of the UC model with many cross sections appear in Table 1. The prior means of the parameters of the common cycle, φ and φ , are similar to 1 2 those found in the literature of trend-cycle decompositions of output. For example, Morley, Nelson and Zivot (2003) find that the estimation of an unobserved components model with aggregate data on real GDP in absence of correlation between trend and cycle innovations ˆ ˆ yields estimated coefficients φ = 1.53 and φ = −0.61. Also with aggregate data, but 1 2 including the unemployment rate along with real GDP, Gonzalez-Astudillo and Roberts (2016) find estimates around 1.6 for φ and -0.65 for φ either with or without correlation 1 2 betweenoutputinnovations. Thejointpriordistributionofthesetwoparametersistruncated to satisfy the weak stationarity feature of the common cycle. The mean growth rate of each state’s GDP has a truncated normal prior distribution with mean 0.8, which yields an annualized growth rate of real GDP around 3%, roughly the historical average. The distributions of the parameters that load on the common cycle, α , are normal with means i and standard deviations equal to one.4 The coefficients that load on the common trends, δy, i and δu, have truncated normal distributions with means and standard deviations equal to i one. The Okun’s lawcoefficients, θ andθ , arenormallydistributed withpriormeans equal 1i 2i to -0.25 each, such that the long-run Okun’s law coefficient for each state is centered at -0.5 under the prior distribution. This is the usual Okun’s law coefficient used in the literature (see Abel, Bernanke and Croushore, 2013). The prior distribution of the parameter of the idiosyncratic cycle, ρ , is a truncated standard normal. In general, the standard deviations i 4In the case of California, the prior distribution of αi is truncated normal with mean and standard deviation equal to one. 12

Table 2: Parameter Estimates of the Common Trend and Cycle Parameter Posterior Mean 90% Confidence Set φ 1.97 [1.95, 1.98] 1 φ -0.98 [-0.99, -0.96] 2 σ 1 — ε σ 1 — ηy σ 1 — ηu of the parameters imply that the distributions are neither too tight nor too narrow. In the current setup, because of the lack of previous estimates of these coefficients in the literature, it may be difficult to characterize the prior distributions of the variances of the idiosyncratic components of the trend and the cycle, σ2 , σ2 , and σ2. Hence, the estimation η i y η i u ζi of these parameters relies exclusively on the likelihood function. 6.2 Estimate of the U.S. Output Gap The estimation produced 60,000 draws from the posterior distribution. The last 30,000 drawswerekept. TheacceptancerateforthecoefficientsdrawnwiththeMetropolis-Hastings algorithm was between 30% and 40%, on average, across states. Results of the estimation for the parameters that characterize the common cycle appear in Table 2. The posterior mean of the parameters φ and φ imply a very persistent hump-shaped response of the cycle to 1 2 a shock, in line with the findings of Fleischman and Roberts (2011). Under the posterior distribution of these coefficients, the lag polynomial has complex roots that imply a 90% confidence set for the period of the cycle between approximately 131⁄ 2 and 19 years, with a mean of approximately 153⁄ 4 years. Using the derivation detailed in Appendix C to obtain the cycle of aggregate GDP from the estimation with state-level data, I obtain the following: Aggregate GDP cycle ≈ α¯c +υ¯ , (19) t t t with α¯ = n w α and υ¯ = n w υ , where the smoothed estimates of c and υ are i=1 i i t i=1 i it t|T it|T used to obtain the estimated cycle and w is the average contribution of state i’s GDP to i P P aggregate GDP in the sample period. The smoothed estimate of the cycle of aggregate GDP for the United States, along with the implied cycle from the CBO estimate of potential output, appears in Figure 5. After reaching a peak of about 0.14% in the fourth quarter of 2005, the estimated cycle starts to decline and becomes negative in the second quarter of 2006. The trough occurs in the fourth quarter of 2009, when the cycle reaches about -7.9%. The estimated cycle differs with respect to the CBO-implied cycle in that during the recession the latter reached -7.3% in the second quarter of 2009, about 1⁄ 2 percentage point higher than and two quarters before the former. However, the trough of the estimated cycle coincides with that of the BOG estimate, which reached -7.6% in the fourth quarter of 2009. The estimations results regarding the output gap at the trough of the recessions can 13

Figure 5: Smoothed Estimate of the Aggregate GDP Cycle from State-level Data 8 8 68% confidence set 6 Smoothed cycle 6 CBO-implied cycle 4 4 2 2 0 0 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 2006 2008 2010 2012 2014 2016 laitnetop/dnert morf noitaived % Source: CongressionalBudgetOffice(CBO)andauthor’scalculations. also illustrate those states that experienced the largest deviations from trend. Figure 6 shows these results. In concordance with the results on the average real GDP annual growth rates in Figure 3, Nevada, Michigan, and Florida are the most affected states. Arizona, California, Delaware, Rhode Island, and Ohio also have negative output gaps larger than 10%. Eleven states—including, for example, Illinois and Indiana—experienced output gaps between 7% and 10%. Eighteen states—including, for example, Massachusetts and Texas— reached output gaps between -5% and -7%. The least affected states that still had negative output gaps were the District of Columbia and states like South Dakota, West Virginia, and Louisiana. Finally, there is a group of states that, according to the estimation, did not go through episodes of negative output gaps during the 2006:Q2-2010:Q1 period. These states are Alaska, New Mexico, Oregon, and Wyoming, three of which are states where the oil and gas sectors, as well as mining, are important contributors. Figure 5 also shows that the aggregate GDP cycle has almost constantly increased after the recession to end at about 0.6% in the first quarter of 2016, well above the CBO-implied estimate of -2.1%. In any case, the 68% confidence set of the estimated cycle contains zero, indicating that it is not unreasonable to predict that the output gap is closed by the first quarter of 2016. At the state level, the results in Figure 7 classify the states according to the estimated output gap at the end of the sample period. Most of the states lie in the region of a positive estimated output gap. The exceptions are Alaska, Louisiana, New Mexico, Oregon, West Virginia, and Wyoming, several of which have relevant oil, gas, and mining industries. In fact, there are reports that West Virginia and Wyoming have recently been suffering from a recession that is most likely due to lower commodity prices (see Matthews, 2016). On the other side of the spectrum, with the largest 14

Figure 6: Largest Negative Output Gaps between 2006:Q2 and 2010:Q1 WA MT ND MN ME OR ID WY SD WI MI NY VTNH IA MA NE PA CTRI NV UT CO KS MO IL IN OH WV MDDE NJ KY VA CA OK TN NC AZ NM AR SC MS AL GA TX LA FL AK Output Gap > 0 -2 < Output Gap < 0 HI -5 < Output Gap < -2 -7 < Ouput Gap < -5 -10 < Output Gap < -7 Output Gap < -10 Figure 7: State Output Gaps in 2016:Q1 WA MT ND MN ME OR ID WY SD WI MI NY VTNH IA MA NE PA CTRI NV UT CO KS MO IL IN OH WV MDDE NJ KY VA CA OK TN NC AZ NM AR SC MS AL GA TX LA FL AK Output Gap > 1 HI 0.5 < Output Gap < 1 0 < Output Gap < 0.5 Output Gap < 0 15

Figure 8: Estimate of the aggregate GDP trend Log of GDP Estimated log of GDP trend CBO log of GDP trend 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Note: TheshadedareaindicatesanNBERrecession. Source: CongressionalBudgetOffice(CBO),FederalReserveEconomicData(FRED),andauthor’scalculations positive output gaps, are California, Colorado, Florida, Nevada, and Michigan. It might be counterintuitive that the states with the most-negative output gaps during the recession, as illustrated in Figure 6, are also the states with the highest output gaps at the end of the sample. One possible explanation is that the GDP trends of these states have been significantlyaffectedbytherecession, hencethelevelofoutputnecessarytoachieveapositive output gap is relatively lower than for other states that did not see their trends adjusted downward. As stated previously, there are differences between the estimate of the aggregate output gap and the CBOs, both in terms of magnitude and timing. This situation has implications for the estimate of the trend. Specifically, between the third quarter of 2008—when the estimated trend peaks during the recession—and the first quarter of 2009, the estimated decline of the aggregate trend in the model used in this paper is about 1.9%. Figure 8 shows the sudden drop in the estimated trend of GDP during the recession, which occurs only gradually in the CBO counterpart. The figure also shows that the estimated trend has experienced a marked deceleration in recent years compared with the years before the recession. In fact, between 2006 and 2008 the trend is estimated to have grown at an average annual rate of about 31⁄ 4 percent, whereas the average growth rate has been about 3⁄ 4 percent since 2012. Additionally, there is a substantial difference compared with the growth rate of the CBO potential in recent years. The CBO estimates that potential has been growing at an annual rate of about 11⁄ 2 percent since 2012, twice the estimate in this paper. At thestatelevel, Figure9 shows selected states whose trends have increased significantly above or below the estimated aggregate trend. The first panel illustrates how the states that benefited the most from high commodity prices (oil, in particular)—North Dakota, Texas, Utah, and Oklahoma—significantly increased their trend output during the past decade. The middle panel shows the states in the rust belt that suffered during the Great Recession. 16

Michigan was not only affected in terms of its cyclical situation, but its trend GDP has also declined after the recession. As mentioned previously, this decline in the trend slope for Michigan can be associated with its positive estimated output gap at the end of the sample. The other states—namely, Indiana, Kentucky, Illinois, and Ohio—have recovered, but at a slower pace than the aggregate. Finally, the third panel shows states that are unusual in their characteristics such that they cannot be grouped with the states in the first two panels. Oregon, for example, has seen its manufacturing industry grow significantly in the past decade. The important increase in trend output for Oregonand a deceleration of output toward the end of the sample can explain the negative output gap discussed previously for this state. South Dakota’s trend may have been driven by industries related mainly to agriculture and financial services. States whose trend output growth has been among the lowest of the country are Louisiana, Maine, and Nevada, according to the model used in this paper. 6.3 State Trend Intercepts and Cycle Loading Coefficients To illustrate the features of the state economies, in Figure 10 I present the results related to the average output trend growth rates, µ , and the common cycle loadings, α . Figure i i 10a shows that the states that experienced higher average growth rates of trend output are North and South Dakota, followed by Alaska, Nebraska, Oregon, Utah, Texas, Washington, and Wyoming. In most of these states, the mining sector strongly influences output growth, as Figure 4 shows. Some of these states with the highest average output trend growth rates also appear in Figure 9 as the states with the fastest growing trends. Those states that do not appear in the state trends figure despite having high estimated values of µ may i have experienced negative idiosyncratic trend shocks. On the other side of the spectrum, with average trend annualized growth rates lower than 0.4%, are Maine and Michigan. As indicated in Section 5.3, the construction industry was the most negative contributor to growth in Michigan, and the same is true for Maine. With regard to the loading coefficients on the common cycle, Figure 10b shows the estimation results for the states ordered by quartiles. Among the states with the highest loadings are Arizona, Florida, Michigan, and Nevada, which are some of the states that experienced the most negative output gaps during the recession years. One can interpret this result as these states being the most sensitive to the national business cycle conditions during the sample period. With respect to the states with the lowest loadings, it turns out that some ofthem actually have negative posterior mean estimates. This set includes Alaska, Louisiana, New Mexico, Oregon, and Wyoming, although in all of the cases, the confidence sets of the loading coefficients include zero. These states seem to be among the less sensitive to the national business cycle conditions. 6.4 Variance Decomposition of States’ Cycles Another way to look at the sensitivity of the states economies cycles to the common cycle is to compute the variance decomposition of the state-specific cycle. From equation (16), the cycle for each state i = 1,2,...,n is specified as c = α c +υ . Hence, one can obtain it i t it the fraction of the variance of the cycle that is due to the common cycle, α2var(c )/var(c ), i t it 17

Figure 9: Estimate of the GDP Trend for Selected States 70 60 50 40 30 20 10 0 -10 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 1Q:5002 morf noitaived % Common GDP Trend North Dakota GDP Trend Texas GDP Trend Utah GDP Trend Oklahoma GDP Trend 20 15 10 5 0 -5 -10 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 1Q:5002 morf noitaived % Common GDP Trend Kentucky GDP Trend Ohio GDP Trend Indiana GDP Trend Illinois GDP Trend Michigan GDP Trend 30 20 10 0 -10 -20 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 1Q:5002 morf noitaived % Common GDP Trend South Dakota GDP Trend Louisiana GDP Trend Oregon GDP Trend Maine GDP Trend Nevada GDP Trend 18

Figure 10: Intercept and Loading Coefficients Estimates WA MT ND (a) µi MN ME OR ID WY SD WI MI NY VTNH IA MA NE PA CTRI NV UT CO KS MO IL IN OH WV MDDE NJ KY VA CA OK TN NC AZ NM AR SC MS AL GA TX LA FL AK μ < 0.1 0.1 < μ < 0.2 HI 0.2 < μ < 0.3 0.3 < μ < 0.5 0.5 < μ < 0.8 μ > 0.8 WA MT ND (b) αi MN ME OR* ID WY* SD WI MI NY VTNH IA MA NE PA CTRI NV UT CO KS MO IL IN OH WV MDDE NJ KY VA CA OK TN NC AZ NM* AR SC MS AL GA TX LA* FL AK* Quartile (α) 1st HI 2nd 3rd 4th Note: Stateswithanasterisk(*)haveanegativeestimatedposteriormeancoefficientαi. 19

Figure 11: Variance Decomposition of the State’s Cycles (cid:58)(cid:36) (cid:48)(cid:55) (cid:49)(cid:39) (cid:48)(cid:49) (cid:48)(cid:40) (cid:50)(cid:53) (cid:44)(cid:39) (cid:58)(cid:60) (cid:54)(cid:39) (cid:58)(cid:44) (cid:48)(cid:44) (cid:49)(cid:60) (cid:57)(cid:55)(cid:49)(cid:43) (cid:44)(cid:36) (cid:48)(cid:36) (cid:49)(cid:40) (cid:51)(cid:36) (cid:38)(cid:55)(cid:53)(cid:44) (cid:49)(cid:57) (cid:56)(cid:55) (cid:38)(cid:50) (cid:46)(cid:54) (cid:48)(cid:50) (cid:44)(cid:47) (cid:44)(cid:49) (cid:50)(cid:43) (cid:58)(cid:57) (cid:48)(cid:39)(cid:39)(cid:40) (cid:49)(cid:45) (cid:46)(cid:60) (cid:57)(cid:36) (cid:38)(cid:36) (cid:50)(cid:46) (cid:55)(cid:49) (cid:49)(cid:38) (cid:36)(cid:61) (cid:49)(cid:48) (cid:36)(cid:53) (cid:54)(cid:38) (cid:48)(cid:54) (cid:36)(cid:47) (cid:42)(cid:36) (cid:55)(cid:59) (cid:47)(cid:36) (cid:41)(cid:47) (cid:36)(cid:46) (cid:43)(cid:44) (cid:91)(cid:3)(cid:31)(cid:3)(cid:20)(cid:19)(cid:8) (cid:20)(cid:19)(cid:8)(cid:3)(cid:31)(cid:3)(cid:91)(cid:3)(cid:31)(cid:3)(cid:24)(cid:19)(cid:8) (cid:24)(cid:19)(cid:8)(cid:3)(cid:31)(cid:3)(cid:91)(cid:3)(cid:31)(cid:3)(cid:27)(cid:19)(cid:8) (cid:27)(cid:19)(cid:8)(cid:3)(cid:31)(cid:3)(cid:91)(cid:3)(cid:31)(cid:3)(cid:28)(cid:19)(cid:8) (cid:28)(cid:19)(cid:8)(cid:3)(cid:31)(cid:3)(cid:91)(cid:3)(cid:31)(cid:3)(cid:28)(cid:24)(cid:8) (cid:91)(cid:3)(cid:33)(cid:3)(cid:28)(cid:24)(cid:8) Note: x=100× α v 2 i a v r( a c r i ( t c ) t) istheproportionofthevarianceofthestate’scyclethatisexplainedbythevariance ofthecommoncycle. and the fraction that corresponds to the idiosyncratic component, var(υ )/var(c ). Figure it it 11 shows the percent of the variance of the cycle of each of the states that is due to the common cycle. Connecticut, California, Georgia, and Nevada are among the states with cycle variability that is explained the most by the variance of the common cycle, whereas Louisiana, New Mexico, SouthDakota, andWestVirginiahavethesmallest variationoftheircycleattributed tothecommoncycle. Inother words, most ofthevariabilityofthecycleinthesestatescomes primarilyfromfeaturesinherenttothosestates. Forexample, NewMexico andWestVirginia are states where mining can have very particular cyclical effects which are not necessarily related to the aggregate economic cycle. 6.5 Variance Decomposition of Aggregate GDP As described in Section 3.1, the common trend and cycle components imply co-movement between the state economies that, in turn, influence the variability of the aggregate GDP. Hence, one can compute the variance decomposition of aggregate GDP growth, as follows:5 var(∆% Aggregate GDP) = var(∆ Aggregate GDP trend)+var(∆ Aggregate GDP cycle). 5The full derivation appears in Appendix D. 20

Figure 12: Okun’s Law Coefficients (θ +θ ) 1i 2i WA MT ND MN ME OR ID WY SD WI MI NY VTNH IA MA NE PA CTRI NV UT CO KS MO IL IN OH WV MDDE NJ KY VA CA OK TN NC AZ NM AR SC MS AL GA TX LA FL AK θ > -0.1 -0.2 < θ < -0.1 HI -0.3 < θ < -0.2 -0.4 < θ < -0.3 θ < -0.4 Note:θisthesumθ1+θ2 foreachstate. Therefore, the percent of the variance of GDP growth that is due to the variance of the change in the cycle is given by the following: var(∆ Aggregate GDP cycle) = 47%. var(∆ Aggregate GDP trend)+var(∆ Aggregate GDP cycle) The contribution of the variability of the cycle to the variability of GDP growth of 47% is lower than estimates obtained with longer samples. For example, Gonzalez-Astudillo and Roberts (2016) find that the contribution is around 60% or 65%, depending on the assumption about the correlation between trend and cycle components. One possible explanation for the lower contribution obtained in this paper is that the sample exclusively contains the Great Recession, which the results indicate affected the trend of GDP more strongly than what estimations with longer samples would obtain. 6.6 State Okun’s Law Coefficients Another dimension along which the results can be analyzed are the cyclical features of the labor markets, in particular the sensitivity of the unemployment gapto the output gapin each state. This feature can be measured by the sum of the Okuns law coefficients, θ +θ . 1i 1i The higher this sum in absolute value, the more responsive is the unemployment rate to the cyclical fluctuations of output. Figure 12 presents the results grouped by values of the posterior mean estimates. The states with more-cyclical labor markets would be California, Colorado, Missouri, Nevada, New Jersey, and Virginia. The states with less cyclical labor markets would be Alaska, New Hampshire, North Dakota, and South Dakota. Three of these states are mining states, a sector that does not require significant labor force inputs. The average of the posterior mean of the Okuns law coefficients across states is about -0.2, which is smaller in 21

Table 3: Prior Distributions of the Parameters - Aggregate Data Parameter Distribution Mean Standard Deviation φ Truncated Normal 1.5 1 1 φ Truncated Normal -0.6 1 2 µ Truncated Normal 0.8 0.8 θ Normal -0.25 0.25 1 θ Normal -0.25 0.25 2 σ2 Inverse Gamma — — ε σ2 Inverse Gamma — — ηy σ2 Inverse Gamma — — ηu absolute value than the usual coefficient of -0.5. 7 Evaluation of Results This section evaluates the performance of the proposed model in comparison to a UC model of aggregate real GDP and unemployment rate. The aim is to offer insights about the gain of including cross-sectional information to estimate the cycle of output. First, I present the results of the estimation of such an aggregate model and compare them to the results of the model with state-level data, in particular with regard to the efficiency of the estimator of the GDP cycle. Then, I evaluate the forecast performance of the UC model with many cross sections proposed in this paper. Finally, I discuss extensions to the model and the limitations of the sample size. 7.1 Unobserved Components Model with Aggregate Data In this section, I compare the results obtained from the estimation of the UC model with many cross sections to the results obtained from the estimation of the model with aggregate data given in equations (1)-(6) over the same sample period. The prior distributions of the coefficients appear in Table 3 and are similar to those in the specification with data at the state level; in this case, however, it is possible to estimate the variances of the aggregate cycle and trend shocks, σ2, σ2 , and σ2 . I use quarterly GDP ε ηy ηu and unemployment rate data from the St. Louis FRED database. Results appear in Table 4. The mean annual growth rate of real GDP is estimated to be about 1.4%, very similar to the estimate with state-level data obtained from weight-averaging the estimated statelevel mean growth rates. With aggregate data, the estimation yields complex roots in the lag polynomial 88% of the times, and the 90% confidence set for the period of the cycle is between approximately 53⁄ 4 and 261⁄ 4 years, a set substantially wider than with state-level data. The estimated mean of the period is approximately 121⁄ 2 years, 31⁄ 4 years shorter than with state-level data. Finally, the variance decomposition of real GDP growth indicates that about 46% of its variation is due to variations in the cycle, 1% below the result with state-level data. 22

Table 4: Parameter Estimates with Aggregate GDP and Unemployment Rate Data Parameter Posterior Mean 90% Confidence Set µ 0.34 [0.18,0.50] φ 1.81 [1.63,1.95] 1 φ -0.85 [-0.98,-0.67] 2 σ 0.27 [0.15,0.40] ε σ 0.56 [0.47,0.67] ηy θ 0.07 [-0.27,0.48] 1 θ -0.70 [-1.21,-0.35] 2 σ 0.20 [0.19,0.30] ηu With respect to the Okun’s law coefficients, the long-runsensitivity of the unemployment gap to the output gap is about -0.64. Owyang, Vermann and Sekhposyan (2013) investigate the Okun’s law coefficients across states, although using the growth rates version of the law, and find that there can be discrepancies between the estimates with state-level data (like those in Figure 12) and the estimate obtained from aggregate data. The smoothed cycle of this estimation along with the CBO-implied output gap appear in Figure 13. The trough of the cycle reaches about -5.1% in the second quarter of 2009; this value is about 23⁄ 4 percentage points higher than the estimate with state-level data. Also, the estimated cycle recovers faster than the estimate with state-level data, reaching about 1.2% in the first quarter of 2016, compared with 0.6% with state-level data. These results are consistent with an estimated output trend growing, at the end of the sample, at an average annual rate of a little above 1 percent with aggregate data, compared with 3⁄ 4 percent with state-level data. Stock and Watson (2016) argue that the width of the confidence set of the cycle, in this case, reflects two sources of uncertainty: the signal extraction uncertainty conditional on the values of the parameters of the model and the parameter uncertainty. The model with state-level data contains more information than the model with aggregate data, and hence the signal extraction uncertainty is smaller in the former. But the model with aggregate data contains many fewer parameters than the model with state-level data, which implies greater parameter uncertainty in the latter. Therefore, there is no a priori ranking of the width of the posterior intervals between the two models. However, theresultsclearlyindicatethattheincorporationofmanycrosssectionsreduces greatly the signal extraction uncertainty despite the large number of parameters that could increase the parameter uncertainty. When one compares the width of the 68% confidence sets between Figure 5, which is the estimate of the cycle with state-level data, and Figure 13, with aggregate data, it is evident the much narrower confidence set is in the case with many cross sections. 23

Figure 13: Smoothed Estimate of the GDP Cycle from Aggregate Data 8 8 68% confidence set 6 Smoothed cycle 6 CBO-implied cycle 4 4 2 2 0 0 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 2006 2008 2010 2012 2014 2016 laitnetop/dnert morf noitaived % Source: CongressionalBudgetOffice(CBO)andauthor’scalculations. 7.2 Forecasting Performance In this section, I perform an out-of-sample forecast evaluation of the proposed model against two models of aggregate real GDP. The first model is a simple AR(1) process for the growth rate of real GDP. The second is the UC model of aggregate real GDP and the unemployment rate, whose estimation is described in the previous section. Given the short span of the sample, the out-of-sample forecast evaluation starts with the sample covering the 2005:Q2 - 2012:Q3 period—that is, 30 quarters—and then predicts real GDP growth one and four quarters ahead. The forecasts are constructed using the prediction step of the Kalman filter directly for the model with aggregate data, and, in the caseofthemodel withstate-level data, obtaining thegrowthrateofaggregaterealGDPfrom the weighted average of the states’ real GDP growth forecasts obtained from the prediction step of the Kalman filter for each of them. The forecast of the AR(1) process is trivial. The forecasts are obtained from 10,000 draws each period. Table 5 presents the average and standard deviation of the mean squared forecast errors of the one- and four-quarter-ahead forecast errors for the three models. A few results stand out from this exercise. First, incorporating richer autoregressive dynamics improves the four-quarter-ahead forecast compared with the simple AR(1) model. Second, incorporating state-level data to the UC model to decompose GDP into trend and cycle more accurately forecasts real GDP growth out of sample. Third, the model with state-level data has a lower variability within the mean square error than the model with aggregate data or the AR(1) model. 24

Table 5: Mean Squared Forecast Errors for Three Competing Models Model One quarter ahead Four quarters ahead AR(1) 0.53 (0.04) 0.57 (0.09) Aggregate 0.57 (0.05) 0.56 (0.08) State-level 0.49 (0.02) 0.48 (0.06) Note: Inparenthesisappearsthestandarddeviationofthemeansquareforecasterror. 7.3 Extensions and Sample Size Limitation Given the importance of inflation in defining potential output, it is natural to incorporate this variable in the system I have considered in this paper. Examples of UC models that include inflation to estimate potential output in the literature include Kuttner (1994); Roberts (2001); Basistha and Nelson (2007); Basistha (2007). Unfortunately, there is not information on price inflation at the state level. However, it is possible to introduce an aggregate measure of inflation in the system in equations (7)-(18) to incorporate the information this variable provides on potential output. Hence, I augment the aforementioned system with the following Phillips curve equation: π = βπ +(1−β)πe +κ×Aggregate GDP cycle +ηπ, t t−1 t t t withβ ∈ [0,1], whereπ denotespriceinflationinperiodt, πe denotesexpected priceinflation t t in period t, Aggregate GDP cycle is the aggregate cycle as defined in equation (19), and t ηπ is an i.i.d. disturbance (see Basistha and Nelson, 2007, for more details). To estimate t the model, I incorporate information on the core personal consumption expenditures price deflator inflation and the corresponding mean of the four-quarter ahead predictions from the survey of professional forecasters. Results adding inflation look very similar to the results without inflation. In particular, the Phillips curve coefficient, κ, results in a value too close to zero for inflation to provide informationabouttheaggregatecycle. Blanchard, CeruttiandSummers(2015),forexample, find that the slope of the Phillips curve has substantially declined in recent years for the United States. Hence, it does not seem that inflation could make a significant difference to the results obtained with real GDP and unemployment rate data at the state level. Regarding the short sample size of the state-level data along the time dimension, it could be a concern that the trend of real GDP is estimated with just about 50 quarters, or about 12 years. In particular, the estimates are probably not characterizing the full time series properties of the output trend or cycle because of the particular features of the Great Recession, which is fully included in the sample. Nevertheless, the aim of the paper is to illustrate a novel way to decompose output into trend and cycle exploiting the cross-sectional variation of the data. In the previous two sections, I have shown the advantages of such an approach compared with estimates obtained using aggregate data only and the same time span. Hence, estimating the model proposed with a longer sample could potentially provide more precise estimates, but it would still keep the appeal compared with models that use aggregate data. 25

8 Conclusions In this paper, I propose a model to obtain a trend-cycle decomposition of GDP using state-level real GDP and unemployment rate data covering the 2005:Q1–2016:Q1 period. The estimation shows that the economy reached a trough in the fourth quarter of 2009 in whichtheoutputgapwasabout-7.9%, whereastheoutputgapisestimatedtobeabout0.6% at the end of the sample in the first quarter of 2016. Across states, the recession hit states in the rust belt especially hard, as well as Arizona, California, Florida, and Nevada. In the first quarter of 2016, states such as Alaska, West Virginia, and Wyoming are experiencing negative output gaps likely because of low commodity prices. An evaluation of the results shows that incorporating cross-sectional information into the estimation of a UC model of trend-cycle decomposition of GDP improves the precision of the estimator of the cycle. The results also show that the model with state-level data performs better in terms of forecasting compared with a model that only uses aggregate data. Althoughintroducing cross-sectional variationfrommany economic units canhelp reduce the uncertainty around the output gap estimate, having a longer sample along the time dimension would better help identify the output trend and, hence, the cycle. Adopting a mixed-frequency approach can help achieve that goal using yearly state-level data. However, thecomputationalcost canbehighasaresult oftheincrease oflatent statestoaccommodate yearly along with quarterly frequencies. One could modify the approach described in this paper to include a time-varying drift in the trend component of GDP to a different cross-sectional system with a longer time span, such as data on GDP at the industry level, to obtain another measure of the output gap. One could also use information on inflation at the country level, along with real GDP and the unemployment rate, to obtain a measure of the common cycle across those countries. These are topics for future research. 26

Appendix A State-space Model in Matrix Form The model in Equations (7)-(17) can be written in matrix form as the following: z = C(Θ )+H(Θ )x +w , w |F ∼ iid N (0,R(Θ )) (20) it mi mi it it it t−1 mi x = F(Θ )x +Gv , v |F ∼ iid N (0,Q(Θ )), (21) it si i,t−1 it it t−1 si where c t c t−1   c ε t−2 t ∆y  ηy  ηy ηy z = it , x =  t , w = it , v =  t, it ∆u it  ηu  it ηu it ηu (cid:20) it (cid:21)  t  (cid:20) it(cid:21) t  υ  ζ   it   it υ     i,t−1 υ   i,t−2   µ +δyµ C(Θ ) = i i , mi 0 (cid:20) (cid:21) α −α 0 δy 0 1 −1 0 H(Θ ) = i i i , mi α θ α (θ −θ ) −α θ 0 δu θ θ −θ −θ (cid:20) i 1i i 2i 1i i 2i i 1i 2i 1i 2i (cid:21) σ2 0 R(Θ )) = η i y , vi 0 σ2 " ηu# i φ φ 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0   0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F(Θ ) =  , si 0 0 0 0 0 0 0 0   0 0 0 0 0 ρ 0 0  i  0 0 0 0 0 1 0 0   0 0 0 0 0 0 1 0     1 0 0 0 0 0 0 0   0 0 0 0 0 1 0 0 G =  , 0 0 1 0   0 0 0 1   0 0 0 0   0 0 0 0     27

σ2 0 0 0 ε 0 σ2 0 0 Q(Θ ) =  ηy , vi 0 0 σ2 0 ηu  0 0 0 σ2  ζi   and Θ = {µ,µ ,α ,θ ,θ ,δy,δu,σ2 ,σ2 } mi i i 1i 2i i i ηy ηu i i Θ = {φ ,φ ,ρ ,σ2,σ2 ,σ2 ,σ2}, si 1 2 i e ηy ηu ζi for i = 1,...,n. B Details on the Metropolis-within-Gibbs Algorithm Let z , x , Θ , and Θ for i = 1,2,...,n, be defined as in Appendix A. Let Z = it it mi si T {˜z ,˜z ,...,˜z } denote the observed data and let X = {x˜ ,x˜ ,...,x˜ }. Here, ˜z = 1 2 T T 1 2 T t n n {z ,z ,...,z } and x˜ = {x ,x ,...,x }. Denote Θ = Θ and Θ = Θ . 1t 2t nt t 1t 2t nt m mi s si i=1 i=1 Partition Θ = Θ1 Θ2 Θ3, where S S si si si si S S Θ1 = {φ ,φ }, si 1 2 Θ2 = {ρ ,σ2}, si i ζi Θ3 = {σ2,σ2 ,σ2 }. si e ηy ηu Notice that the identification conditions imply that Θ3 is not random. si Also, partition Θ = Θ1 Θ2 , where mi mi mi SΘ1 = {µ ,α ,θ ,θ ,δy,δu}, mi i i 1i 2i i i Θ2 = {σ2 ,σ2 }, mi ηy ηu i i where µ has been excluded because it is fixed under the identification conditions. The Metropolis-within-Gibbs procedure is as follows: 1. Draw X from p(X |Z ,Θ ,Θ ) using the Durbin and Koopman (2002) simulation T T T m s smoother. 2. Draw Θ1 from p(Θ1|Z ,X ,Θ ,Θ2,Θ3) using the normal-inverse-gamma distribusi si T T m si si tion. 3. DrawΘ1 fromp(Θ1 |Z ,X ,Θ2 ,Θ ,Θ )fori = 1,2,...,n, usingtheMetropolismi mi T T mi m(/i) s Hastings algorithm, where the proposal density is given by the likelihood function of the following nonlinear bivariate model multiplied by the prior distribution of Θ1 : mi z = C(Θ1 )+H(Θ1 )x +w , w |F ∼ iid N (0,R(Θ2 )), t = 1,2,...,T. it mi mi it it it t−1 mi 28

4. Draw Θ2 from p(Θ2 |Z ,X ,Θ1 ,Θ ,Θ ) for i = 1,2,...,n using the inverse mi mi T T mi m(/i) s gamma distribution. 5. Draw Θ2 from p(Θ2|Z ,X ,Θ ,Θ1,Θ3) for i = 1,2,...,n, using the Metropolissi si T T mi si si Hastings algorithm, where the proposal density is given by the likelihood function of the following nonlinear bivariate model multiplied by the prior distribution of Θ2: si z = C(Θ1 )+H−(Θ1 )x− +e , it mi mi it it where α −α 0 δy 0 H−(Θ1 ) = i i i , mi α θ α (θ −θ ) −α θ 0 δu (cid:20) i 1i i 2i 1i i 2i i (cid:21) c t c t−1   x− = c , it t−2  ηy   t   ηu   t    ∆υ +ηy ey e = it it ≡ it , it θ ∆υ +θ ∆υ +ηu eu (cid:20) 1i it 2i i,t−1 it(cid:21) (cid:20) it(cid:21) and var(ey) = σ2 +σ2 , it ∆υit η i y 1 cov ey,ey = − ρj−1(1−ρ )σ2 , ∀j 6= 0, it i,t−j 2 i i ∆υit (cid:0) var(eu(cid:1)) = θ2 σ2 +σ2 , it 1i ∆υit η i u σ2 θ θ − 1−ρi (θ2 +ρ θ θ +θ2 ) if j = 1 cov eu,eu = ∆υit 1i 2i 2 1i i 1i 2i 2i , it i,t−j (− 2 1ρ i j− (cid:0) 2(1−ρ i )σ ∆ 2 υit (ρ i (θ 1 2 i +θ 2 2 i )+θ 1i θ (cid:1)2i (1+ρ2 i )) if j ≥ 2 (cid:0) (cid:1) 1 1−ρ cov(ey,eu) = σ2 2θ − i θ , it it 2 ∆υit 1i 2 2i (cid:18) (cid:19) 1 cov ey,eu = − ρj−1(1−ρ )σ2 (θ +ρ θ ), ∀j 6= 0, it i,t−j 2 i i ∆υit 1i i 2i (cid:0) (cid:1) σ2 θ − 1−ρiθ if j = 1 cov ey ,eu = ∆υit 2i 2 1i i,t−j it (− 2 1ρ i j− (cid:0) 2(1−ρ i )σ ∆ 2 υ(cid:1) it (θ 2i +ρ i θ 1i ) if j ≥ 2, (cid:0) (cid:1) σ2 with σ2 = 2 ζi . ∆υit 1+ρi C Obtaining the Aggregate Trend and Cycle The objective of the estimation is to obtain the trend-cycle decomposition of aggregate GDP,exploiting thecross-sectional variabilityofstate-level data. Becauseofthenonlinearity implicit in the aggregation of the variables and the fact that state-level GDP appears in logs 29

in the specification (7)-(18), an approximation is needed. The quarterly growth rate of aggregate real GDP is given by the following: n ∆%Y = w ∆%Y t it it i=1 X n ≈ w ∆y it it i=1 X n = w (∆τy +∆c ) it it it i=1 X n = w (δy(µ+ηy)+µ +ηy +α ∆c +∆υ ) it i t i it i t it i=1 X ≈ δ ¯y(µ+ηy)+µ¯ +η¯y + α¯∆c +υ¯y , t t t t ∆ GDP Trend ∆ GDP Cycle | {z } | {z } where Y is real GDP of state i, Y is aggregate real GDP in period t, and the contribution it t of of state’s i GDP to aggregate GDP is denoted by w . it Hence, I can express the trend and cycle components of the aggregate GDP as GDP Trend ≈ δ ¯yτy +ξ ¯y, t t GDP Cycle ≈ α¯c +υ¯y, t t where n µ¯ = w µ , t i i i=1 X n δ ¯y = w δy, i i i=1 X n η¯y = w ηy, t i it i=1 X n ξ ¯y = w ξy, t i it i=1 X n α¯ = w α , i i i=1 X n υ¯y = w υ , t i it i=1 X where w = T w /T is the sample average of each state’s contributions to aggregate i t=1 it GDP. I use the smoothed estimates c and υ to obtain the estimate of the cycle and, by t|T it|T P residual, the trend. 30

D Derivation of the Variance Decomposition of Aggregate GDP Given that (the log of) aggregate GDP can be specified as the sum of its aggregate trend and cycle, one can write the following: var(∆%Y ) = var(∆ Aggregate GDP trend)+var(∆ Aggregate GDP cycle) t ≈ var(δ ¯y∆τ +∆ξ ¯y)+var(α¯∆c +∆υ¯y) t t t t = δ ¯y 2 var(ηy +η¯y )+α¯2var(∆c +∆υ¯y) t t t t n n ≈ δ ¯y 2 1+ w2σ2 +α¯2 2(γ −γ )+ w2σ2 , i η i y ! 0 1 i ∆υit ! i=1 i=1 X X where 1−φ2 γ = 1+φ2 , 0 (1−φ2)−φ2 2 1 φ 1 γ = γ . 1 0 1−φ 2 31

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