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Published online by Cambridge University Press:  08 March 2016

James Morley
University of New South Wales
Irina B. Panovska
Lehigh University
Tara M. Sinclair*
The George Washington University
Address correspondence to: Tara M. Sinclair, Department of Economics, The George Washington University, Monroe Hall # 340, 2115 G Street NW, Washington, DC 20052, USA; e-mail:


In the aftermath of the global financial crisis, competing measures of the trend in macroeconomic variables such as U.S. real GDP have featured prominently in policy debates. A key question is whether large shocks to macroeconomic variables will have permanent effects—i.e., in econometric terms, do the data contain stochastic trends? Unobserved-components models provide a convenient way to estimate stochastic trends for time series data, with their existence typically motivated by stationarity tests that allow at most a deterministic trend under the null hypothesis. However, given the small sample sizes available for most macroeconomic variables, standard Lagrange multiplier tests of stationarity will perform poorly when the data are highly persistent. To address this problem, we propose the use of a likelihood ratio test of stationarity based directly on the unobserved-components models used in estimation of stochastic trends. We demonstrate that a bootstrap version of this test has far better small-sample properties for empirically relevant data-generating processes than bootstrap versions of the standard Lagrange multiplier tests. An application to U.S. real GDP produces stronger support for the presence of large permanent shocks using the likelihood ratio test than using the standard tests.

Copyright © Cambridge University Press 2016 

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Andrews, D.W.K. (2001) Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica 69, 683734.CrossRefGoogle Scholar
Bailey, R.W. and Taylor, A.M.R. (2002) An optimal test against a random walk component in a non-orthogonal unobserved components model. Econometrics Journal 5, 520532.CrossRefGoogle Scholar
Basistha, A. (2007) Trend–cycle correlation, drift break and the estimation of trend and cycle in Canadian GDP. Canadian Journal of Economics 40, 584606.CrossRefGoogle Scholar
Basistha, A. (2009) Hours per capita and productivity: Evidence from correlated unobserved components model. Journal of Applied Econometrics 24, 187206.CrossRefGoogle Scholar
Berger, T. and Everaert, G. (2010) Labour taxes and unemployment evidence from a panel unobserved component model. Journal of Economic Dynamics and Control 34, 354364.CrossRefGoogle Scholar
Bradley, M., Jansen, D., and Sinclair, T.M. (in press) How well does “core” inflation capture permanent price changes? Macroeconomic Dynamics.Google Scholar
Caner, M. and Kilian, L. (2001) Size distortions of tests of the null hypothesis of stationarity: Evidence and implications for the PPP debate. Journal of International Money and Finance 20, 639657.CrossRefGoogle Scholar
Chernoff, H. (1954) On the distribution of the likelihood ratio. Annals of Mathematical Statistics 25, 573578.CrossRefGoogle Scholar
Clark, P.K. (1987) The cyclical component of U.S. economic activity. Quarterly Journal of Economics 102, 797814.CrossRefGoogle Scholar
Cogley, T. and Nason, J.M. (1995) Effects of the Hodrick–Prescott filter on trend and difference stationary time series: Implications for business cycle research. Journal of Economic Dynamics and Control 19, 253278.CrossRefGoogle Scholar
Davidson, R. and MacKinnon, J.G. (2000) Bootstrap tests: How many bootstraps? Econometric Reviews 19, 5568.CrossRefGoogle Scholar
Davis, R.A., Chen, M., and Dunsmuir, W.T.M. (1995) Inference for MA(1) processes with a root on or near the unit root circle. Probability and Mathematical Statistics 5, 227242.Google Scholar
Davis, R.A., Chen, M. and Dunsmuir, W.T.M. (1996) Inference for seasonal moving average models with a unit root. In Robinson, P. M. and Rosenbaltt, M. (eds.), Athens Conference on Applied Probability and Time Series Analysis: Volume II. Time Series Analysis in Memory of E. J. Hannan, Lecture Notes in Statistics No. 115, pp. 160176. New York: Springer-Verlag.CrossRefGoogle Scholar
Davis, R.A. and Dunsmuir, W.T.M. (1996) Maximum likelihood estimation for MA(1) processes with a root on or near the unit circle. Econometric Theory 12, 129.CrossRefGoogle Scholar
Dickey, D.A. and Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427431.Google Scholar
Elliott, G., Rothenberg, T.J., and Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813836.CrossRefGoogle Scholar
Fischer, S. (2014) The Great Recession: Moving Ahead. Speech at “The Great Recession—Moving Ahead,” a Conference Sponsored by the Swedish Ministry of Finance, Stockholm, Sweden, August 11, 2014. Scholar
Gospodinov, N. (2002) Bootstrap-based inference in models with a nearly noninvertible moving average component. Journal of Business and Economic Statistics 20, 254268.CrossRefGoogle Scholar
Gouriéroux, C., Holly, A., and Monfort, A. (1982) Likelihood ratio test, Wald test, and Kuhn–Tucker test in linear models with inequality constraints on the regression parameters. Econometrica 50, 6380.CrossRefGoogle Scholar
Harvey, A.C. (1989) Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge, UK: Cambridge University Press.Google Scholar
Harvey, A.C. and Jaeger, A. (1993) Detrending, stylized facts and the business cycle. Journal of Applied Econometrics 8, 231247.CrossRefGoogle Scholar
Hobijn, B., Franses, P.H., and Ooms, M. (2004) Generalizations of the KPSS-test for stationarity. Statistica Neerlandica 58, 483502.CrossRefGoogle Scholar
Jansson, M. (2004) Stationarity testing with covariates. Econometric Theory 20, 5694.CrossRefGoogle Scholar
Jansson, M. and Orregaard Nielsen, M. (2012) Nearly efficient likelihood ratio tests of the unit root hypothesis. Econometrica 80, 23212332.Google Scholar
Kim, C.-J. and Nelson, C.R. (1999) Has the U.S. economy become more stable? A Bayesian approach based on a Markov-switching model of the business cycle. Review of Economic and Statistics 81, 608616.CrossRefGoogle Scholar
Kuttner, K.N. (1994) Estimating potential output as a latent variable. Journal of Business and Economic Statistics 12, 361368.Google Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., and Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54, 159178.CrossRefGoogle Scholar
Leybourne, S.J. and McCabe, B.P.M. (1994) A consistent test for a unit root. Journal of Business and Economic Statistics 12, 157166.Google Scholar
Ma, J. and Nelson, C.R. (in press) The superiority of the LM test in a class of econometrics models where standard Wald test performs poorly. In Koopman, Siem Jan and Shephard, Neil (eds.), Unobserved Components and Time Series Econometrics. Oxford University Press, forthcoming.Google Scholar
Ma, J. and Wohar, M.E. (2013) An unobserved components model that yields business and medium-run cycles. Journal of Money, Credit and Banking 45, 13511373.CrossRefGoogle Scholar
MacKinnon, J. (2002) Bootstrap inference in econometrics. Canadian Journal of Economics 35, 615645.CrossRefGoogle Scholar
McCabe, B.P.M. and Leybourne, S.J. (1998) On estimating an ARMA model with an MA unit root. Econometric Theory 14, 326338.CrossRefGoogle Scholar
McConnell, M.M. and Perez-Quiros, G. (2000) Output fluctuations in the United States: What has changed since the early 1980s? American Economic Review 90, 14641476.CrossRefGoogle Scholar
Mitra, S. and Sinclair, T.M. (2012) Output fluctuations in the G-7: An unobserved components approach. Macroeconomic Dynamics 16, 396422.CrossRefGoogle Scholar
Morley, J.C. (2007) The slow adjustment of aggregate consumption to permanent income. Journal of Money, Credit and Banking 39, 615638.CrossRefGoogle Scholar
Morley, J.C. (2011) The two interpretations of the Beveridge–Nelson decomposition. Macroeconomic Dynamics 15, 419439.CrossRefGoogle Scholar
Morley, J.C., Nelson, C.R., and Zivot, E. (2003) Why are the Beveridge–Nelson and unobserved-components decompositions of GDP so different? Review of Economics and Statistics 85, 235243.CrossRefGoogle Scholar
Müller, U.K. (2005) Size and power of tests of stationarity in highly autocorrelated time series. Journal of Econometrics 128, 195213.CrossRefGoogle Scholar
Murray, C.J. (2003) Cyclical properties of Baxter–King filtered time series. Review of Economics and Statistics 85, 472476.CrossRefGoogle Scholar
Nabeya, S. and Tanaka, K. (1988) Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative. Annals of Statistics 16, 218–35.CrossRefGoogle Scholar
Nelson, C.R. and Kang, H. (1981) Spurious periodicity in inappropriately detrended time series. Econometrica 49, 741751.CrossRefGoogle Scholar
Newey, W.K. and West, K.D. (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703708.CrossRefGoogle Scholar
Newey, W.K. and West, K.D. (1994) Automatic lag selection in covariance matrix estimation. Review of Economic Studies 61, 631653.CrossRefGoogle Scholar
Nyblom, J. (1986) Testing for deterministic linear trend in time series. Journal of the American Statistical Association 81, 545549.CrossRefGoogle Scholar
Nyblom, J. and Mäkeläinen, T. (1983) Comparison of tests for the presence of random walk coefficients in a simple linear model. Journal of the American Statistical Association 78, 856864.CrossRefGoogle Scholar
Oh, K.H., Zivot, E.W., and Creal, D.D. (2008) The relationship between the Beveridge–Nelson decomposition and other permanent–transitory decompositions that are popular in economics. Journal of Econometrics 146, 207219.CrossRefGoogle Scholar
Perron, P. (1989) The great crash, the oil price shock and the unit root hypothesis. Econometrica 57, 13611401.CrossRefGoogle Scholar
Perron, P. and Wada, T. (2009) Let's take a break: Trends and cycles in US real GDP. Journal of Monetary Economics 56, 749765.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277301.CrossRefGoogle Scholar
Pötscher, B.M. (1991) Noninvertibility and pseudo-maximum likelihood estimation of misspecified ARMA models. Econometric Theory 7, 435449.CrossRefGoogle Scholar
Proietti, T. (2002) Forecasting with structural time series models. In Clements, M. P. and Hendry, D. F. (eds.), A Companion to Economic Forecasting, 105132. Oxford, UK: Blackwell.Google Scholar
Proietti, T. (2006) Trend–cycle decompositions with correlated components. Econometric Reviews 25, 6184.CrossRefGoogle Scholar
Rothenberg, T. (2000) Testing for unit roots in AR and MA models. In Marriott, P. and Salmon, M. (eds.), Applications of Differential Geometry to Econometrics, 281293. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Rothman, P. (1997) More uncertainty about the unit root in U.S. real GNP. Journal of Macroeconomics 19, 771780.CrossRefGoogle Scholar
Senyuz, Z. (2011) Factor analysis of permanent and transitory dynamics of the US economy and the stock market. Journal of Applied Econometrics 26, 975998.CrossRefGoogle Scholar
Sinclair, T.M. (2009) The relationships between permanent and transitory movements in U.S. output and the unemployment rate. Journal of Money, Credit and Banking 41, 529542.CrossRefGoogle Scholar
Teräsvirta, T. (1977) The invertibility of sums of discrete MA and ARMA processes. Scandinavian Journal of Statistics 4, 165170.Google Scholar
Wada, T. (2012) On the correlations of trend–cycle errors. Economics Letters 116, 396400.CrossRefGoogle Scholar
Watson, M.W. (1986) Univariate detrending methods with stochastic trends. Journal of Monetary Economics 18, 127.CrossRefGoogle Scholar
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