The Multistep Beveridge-Nelson Decomposition

Proietti, Tommaso (2009): The Multistep Beveridge-Nelson Decomposition.

Abstract

The Beveridge-Nelson decomposition defines the trend component in terms of the eventual forecast function, as the value the series would take if it were on its long-run path. The paper introduces the multistep Beveridge-Nelson decomposition, which arises when the forecast function is obtained by the direct autoregressive approach, which optimizes the predictive ability of the AR model at forecast horizons greater than one. We compare our proposal with the standard Beveridge-Nelson decomposition, for which the forecast function is obtained by iterating the one-step-ahead predictions via the chain rule. We illustrate that the multistep Beveridge-Nelson trend is more efficient than the standard one in the presence of model misspecification and we subsequently assess the predictive validity of the extracted transitory component with respect to future growth.

• Item Type: MPRA Paper
• Language: English
• Keywords: Trend and Cycle; Forecasting; Filtering.
• Subjects: E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E32 - Business Fluctuations ; Cycles; E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E31 - Price Level ; Inflation ; Deflation; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Buhlmann, P. (1997). Sieve bootstrap for time series. Bernoulli, 3, 123–148.

B¨uhlmann, P. (2002). Bootstraps for time series. Statistical Science, 17, 52–72.

Bloomfield, P. (1976). Fourier Analysis of Time Series: An Introduction. New York: Wiley.

Bondon, P. (2001). Recursive relations for multistep prediction of a stationary time series. Journal of Time Series Analysis, 22, 309–410.

Brockwell, P.J. and Dahlhaus, P. (2003). Generalized Levinson-Durbin and Burg algorithms. Journal of Econometrics, 118, 129-149.

Brockwell, P.J., Dahlhaus, P. and Trinidade, A. (2005). Modified Burg algorithms for multivariate subset autoregression. Statistica Sinica, 15, 197-213.

Clements, M.P. and Hendry, D.F. (1996). Multi-step estimation for forecasting. Oxford Bulletin of Economics and Statistics, 58, 657–684.

Cogley, T. (2002), A simple adaptive measure of core inflation. Journal of Money Credit and Banking, 34, 94-113.

Chevillon, G. (2007). Direct multi-step estimation and forecasting. Journal of Economic Surveys, 21, 746-785.

Cox, D.R. (1961). Prediction by exponentially weighted moving averages and related methods. Journal of the Royal Statistical Society, Series B, 23, 414–422.

Dahlhaus, R. (1988). Small sample effects in time series analysis: a new asymptotic theory and a new estimate. Annals of Statistics, 16, 808-841.

Doornik, J.A. (2006), Ox. An Object-Oriented Matrix Programming Language, Timberlake Consultants Press, London.

Findley, D.F. (1983). On the use of multiple models for multi-period forecasting. Proceedings of Business and Economic Statistics, American Statistical Association, 528-531.

Granger, C. W. J. and Jeon, Y. (2006). Dynamics of model overfitting measured in terms of autoregressive roots. Journal of Time Series Analysis, 27, 347-365.

Granger, C.W. J. and Newbold, P. (1986). Forecasting Economic Time Series. Academic Press.

Harvey, A.C. (1989), Forecasting, Structural Time Series and the Kalman Filter, Cambridge University Press, Cambridge, UK.

Haywood, G. and Tunnicliffe Wilson, G. (1997), Fitting Time Series Models by Minimizing Multistep-ahead Errors: a Frequency Domain Approach, Journal of the Royal Statistical Society, Series B, 59, 237–254.

Hodrick R.J., and Prescott, E.C. (1997). Postwar U.S. Business Cycles: an Empirical Investigation. Journal of Money, Credit and Banking, 29, 1-16.

Hurvich, C.M. and Tsai C.-L. (1997). Selection of a multistep linear predictor for short time series. Statistica Sinica, 7, 395–406.

Ing, C.-K. (2003). Multistep prediction in autoregressive processes. Econometric Theory, 19, 254-279.

Ing, C.-K. (2004). Selecting optimal multistep predictors for autoregressive process of unknown order. Annals of Statistics, 32, 693-722.

Kang, H. (1987). The Tapering Estimation of the First-Order Autoregressive Parameters. Biometrika, 74, 643–645.

Marcellino, M., Stock, J.H. and Watson, M. (2006). A comparison of direct and iterated multistep AR methods for forecasting microeconomic time series. Journal of Econometrics, 135, 499-526.

Morley, J., (2002), A state-space approach to calculating the Beveridge-Nelson decomposition, Economics Letters 75, 123-127.

Morley, J., (2009), The Two Interpretations of the Beveridge-Nelson Decomposition. Working paper, Department of Economics, Washington University.

Morley, J.C., Nelson, C.R. and Zivot E. (2003), Why are the Beveridge-Nelson and unobserved components decompositions of GDP so different? The Review of Economics and Statistics, 85, 235-243.

Nelson, C.R. (2008), The BeveridgeNelson decomposition in retrospect and prospect. Journal of Econometrics, 146, 202–206.

Oh, K.H., E. Zivot, and Creal, D. (2008) The Relationship between the Beveridge-Nelson decomposition and Unobserved Components Models with Correlated Shocks. Journal of Econometrics, 146, 207–219.

Percival D., Walden A. (1993). Spectral Analysis for Physical Applications. Cambridge University Press.

Priestley, M.B. (1981), Spectral Analysis and Time Series. London, Academic Press.

Proietti, T. (1995). The Beveridge-Nelson decomposition: Properties and extensions, Statistical Methods and Applications, 4, 101-124.

Proietti, T. (2006). Trend-cycle decompositions with correlated components, Econometric Reviews, 25, 61-84.

Proietti, T., and Harvey, A.C. (2000). A Beveridge Nelson Smoother. Economics Letters, 67, 139–146.

Shaman, P., and Stine, R.A. (1988). The Bias of Autoregressive Coefficient Estimators. Journal of the American Statistical Association, 83, 842–848.

Shibata, R. (1980). Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. Annals of Statistics, 8, 147-164.

Stock, J.H. and Watson, M. (2007). Why Has U.S. Inflation Become Harder to Forecast? Journal of Money, Credit and Banking, Supplement to Vol. 39, No. 1, 13–33.

Tiao, G. C., and Xu, D. (1993). Robustness of Maximum Likelihood Estimates for Multi-Step Predictions: The Exponential Smoothing Case. Biometrika, 80, 623–641.

Tiao, G. C., and Tsay, R. S. (1994). Some advances in non-linear and adaptive modelling in time-series, Journal of Forecasting, 13, 109–131.

Tjostheim, D. Paulsen J. (1983). Bias of some Commonly-Used Time Series Estimates, Biometrika, 70, 389–399.

Watson, M.W. (1986). Univariate detrending methods with stochastic trends. Journal of Monetary Economics, 18, 49-75.

Weiss, A.A. (1991). Multi-step estimation and forecasting in dynamic models. Journal of Econometrics, 48, 135-149.