Bloechl, Andreas (2014): Penalized Splines, Mixed Models and the Wiener-Kolmogorov Filter. Discussion Papers in Economics 2014-44 |

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### Abstract

Penalized splines are widespread tools for the estimation of trend and cycle, since they allow a data driven estimation of the penalization parameter by the incorporation into a linear mixed model. Based on the equivalence of penalized splines and the Hodrick-Prescott filter, this paper connects the mixed model framework of penalized splines to the Wiener- Kolmogorov filter. In the case that trend and cycle are described by ARIMA-processes, this filter yields the mean squarred error minimizing estimations of both components. It is shown that for certain settings of the parameters, a penalized spline within the mixed model framework is equal to the Wiener-Kolmogorov filter for a second fold integrated random walk as the trend and a stationary ARMA-process as the cyclical component.

Item Type: | Paper (Discussion Paper) |
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Keywords: | Hodrick-Prescott filter, mixed models, penalized splines, trend estimation, Wiener-Kolmogorov filter |

Faculties: | Economics > Munich Discussion Papers in Economics |

Subjects: | 300 Social sciences > 330 Economics |

JEL Classification: | C220, C520 |

URN: | urn:nbn:de:bvb:19-epub-21406-9 |

Language: | English |

ID Code: | 21406 |

Deposited On: | 08. Sep 2014 08:23 |

Last Modified: | 07. Nov 2020 14:37 |

References: | Bell, W. (1984): "Signal extraction for nonstationary time series," Annals of Statistics, 12, 646-664. Brumback, B. A., Ruppert, D., Wand, M. P. (1999): Comment on "Variable selection and function estimation in additive nonparametric regression using a data-based prior," Journal of the American Statistical Association, 94, 794-797. Claeskens, G., Krivobokova, T., Opsomer, J. (2009): "Asymptotic properties of penalized spline estimators," Biometrika, 96, 529-544. Danthine, J., Girardin, M. (1989); "Business Cycles in Switzerland. A Comparative Study," European Economic Review, 33(1), 31-50. Eilers, P., Marx, B. (1996): "Flexible smoothing with B-splines and penalties," Statistical Science, 11, 89-121. Fahrmeir, L., Kneib, T., Lang, S. (2009): "Regression - Modelle, Methoden und Anwendungen," Springer Verlag, Berlin. Flaig, G. (2005): "Time Series Properties of the German Production Index," Allgemeines Statistisches Archiv, 89, 419-434. Flaig, G. (2012): "Why we should use high values for the smoothing parameter of the Hodrick-Prescott filter," CESifo Working Paper Series No. 3816. Hamilton, J. D. (1994): "Time Series Analysis," Princeton University Press, Princeton/ New Jersey. Harvey, A. C. (1989): "Forecasting, Structural Time Series Models and the Kallman Filter," Cambridge University Press, Cambridge. Hastie, T., Tibshirani, R. (1990): "Generalized Additive Models," Chapman and Hall, London. Hayes, K., Haslett, J. (1999): "Simplifying general least squares," American Statistician, 53, 376-381. Henderson, R. (1924): "On a new method of graduation," Transactions of the Actuarial Society of America, 25, 29-40. Hodrick, R. J., Prescott, E. C. (1997); "Post-War U.S. Business Cycles: An Empirical Investigation," Journal of Money, Credit and Banking, 29, 1-16. Kaiser, R., Maravall, A. (2001): "Measuring Business Cycles in Economic Time Series," Springer Verlag, New York. Kauermann, G., Krivobokova, T., Semmler, W. (2011): "Filtering Time Series with Penalized Splines," Studies in Nonlinear Dynamics & Econometrics, 15(2), Article 2. Kauermann, G., Opsomer, J. (2011): :"Data-driven selection of the spline dimension in penalized spline regression," Biometrika, 98(1), 225-230. Kohn, R., Ansley, C., Wong, C. (1992): "Nonparametric spline regression with autoregressive moving average errors," Biometrika, 79, 335-346. Krivobokova, T., Kauermann, G. (2007): "A note on penalized spline smoothing with correlated errors," Journal of the American Statistical Association, 102, 1328-1337. Leser, C. E. V., (1961): "A simple method of trend construction," Journal of the Royal Statistical Society. Series B (Methodological), 23, 91-107. McCulloch, C. E., Searle, S. R. (2001): "Generalized, Linear, and Mixed Models," Wiley, New York. McElroy, T. (2008): "Matrix Formulas for Nonstationary ARIMA Signal Extraction," Econometric Theory, 24, 988-1009. O’Sullivan, F. (1986): "A statistical perspective on ill-posed inverse problems (c/r:P519- 527)," Statistical Science, 1, 502-518. Opsomer, J., Wang, Y., Yang, Y. (2001): "Nonparametric regression with correlated errors," Statistical Science, 16, 134-153. Paige, R. L. (2010): "The Hodrick-Prescott filter: A special case of penalized spline smoothing," Electronic Journal of Statistics, 4, 856-874. Pinheiro, J. C., Bates, D. M. (2000): "Mixed-Effects Models in S and S-PLUS," Springer-Verlag, New York. Proietti, T. (2005): "Forecasting and signal extraction with misspecified models," Journal of Forecasting, 24, 539-556. Proietti, T. (2007): "Signal Extraction and Filtering by Linear Semiparametric Methods," Computational Statistics and Data Analysis, 52, 935-958. Proietti, T., Luati, A. (2007): "Least Squares Regression: Graduation and Filters," Boumans, M. (Ed.), Measurement in Economics: A Handbook, ch. 16, Academic Press. Robinson, G. K. (1991): "That BLUP s a good thing: The estimation of random effects," Statistical Science, 6, 15-51. Ruppert, D. (2002): "Selecting the number of knots for penalized splines," Journal of Computational and Graphical Statistics, 11, 735-757. Ruppert, R., Wand, M., Carroll, R. (2003): "Semiparametric Regression," Cambridge University Press, New York. Schlicht, E. (2005): "Estimating the Smoothing Parameter in the so-called Hodrick-Prescott Filter," Journal of the Japanese Statistical Society, Vol. 35 No. 1, 99-119. Schlittgen, R., Streitenberg, B. (2001): "Zeitreihenanalyse," Oldenburg Wissenschaftsverlag, München, Wien. Searle, S. R., Casella, G., McCulloch, C. E. (1992): "Variance Components," Wiley, New York. Vonesh, E. F., Chinchilli, V. M. (1997): "Linear and Nonlinear Models for the Analysis of Repeated Measures," Marcel Dekker, New York. Wang, Y. (1998): "Mixed effects smoothing spline analysis of variance," Journal of the Royal Statistical Society, Series B, 60, 159-174 Whittaker, E. T. (1923): "On a new method of graduation," Proceedings of the Edinburgh Mathematical Society, 41, 63-75. Whittle, P. (1983): "Prediction and Regulation by Linear Least-Square Methods," University of Minnesota Press, Minneapolis. |