IDEAS home Printed from https://ideas.repec.org/a/eee/intfor/v41y2025i3p982-1002.html
   My bibliography  Save this article

Time-varying parameters as ridge regressions

Author

Listed:
  • Goulet Coulombe, Philippe

Abstract

Time-varying parameter (TVP) models are frequently used in economics to capture structural change. I highlight a rather underutilized fact—that these are actually ridge regressions. Instantly, this makes computations, tuning, and implementation much easier than in the state-space paradigm. Among other things, solving the equivalent dual ridge problem is computationally very fast even in high dimensions, and the crucial ‘amount of time variation’ is tuned by cross-validation. Evolving volatility is dealt with using a two-step ridge regression. I consider extensions that incorporate sparsity (the algorithm selects which parameters vary and which do not) and reduced-rank restrictions (variation is tied to a factor model). To demonstrate the usefulness of the approach, I use it to study the evolution of monetary policy in Canada using large time-varying local projections and TVP-VARs with demanding lag lengths. The applications require the estimation of up to 4600 TVPs, a task within the reach of the new method.

Suggested Citation

  • Goulet Coulombe, Philippe, 2025. "Time-varying parameters as ridge regressions," International Journal of Forecasting, Elsevier, vol. 41(3), pages 982-1002.
    • Handle: RePEc:eee:intfor:v:41:y:2025:i:3:p:982-1002
    DOI: 10.1016/j.ijforecast.2024.08.006
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0169207024000931
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ijforecast.2024.08.006?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Stevanovic Dalibor, 2016. "Common time variation of parameters in reduced-form macroeconomic models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 20(2), pages 159-183, April.
    2. Bryan T. Kelly & Semyon Malamud & Kangying Zhou, 2022. "The Virtue of Complexity in Return Prediction," NBER Working Papers 30217, National Bureau of Economic Research, Inc.
    3. Bergmeir, Christoph & Hyndman, Rob J. & Koo, Bonsoo, 2018. "A note on the validity of cross-validation for evaluating autoregressive time series prediction," Computational Statistics & Data Analysis, Elsevier, vol. 120(C), pages 70-83.
    4. Angelia L. Grant & Joshua C.C. Chan, 2017. "A Bayesian Model Comparison for Trend‐Cycle Decompositions of Output," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 49(2-3), pages 525-552, March.
    5. Hauzenberger Niko & Huber Florian & Koop Gary, 2024. "Dynamic Shrinkage Priors for Large Time-Varying Parameter Regressions Using Scalable Markov Chain Monte Carlo Methods," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 28(2), pages 201-225, April.
    6. Goulet Coulombe, Philippe & Leroux, Maxime & Stevanovic, Dalibor & Surprenant, Stéphane, 2021. "Macroeconomic data transformations matter," International Journal of Forecasting, Elsevier, vol. 37(4), pages 1338-1354.
    7. Michael W. McCracken & Serena Ng, 2021. "FRED-QD: A Quarterly Database for Macroeconomic Research," Review, Federal Reserve Bank of St. Louis, vol. 103(1), pages 1-44, January.
    8. Philippe Goulet Coulombe & Maxime Leroux & Dalibor Stevanovic & Stéphane Surprenant, 2022. "How is machine learning useful for macroeconomic forecasting?," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(5), pages 920-964, August.
    9. Miguel A.G. Belmonte & Gary Koop & Dimitris Korobilis, 2014. "Hierarchical Shrinkage in Time‐Varying Parameter Models," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 33(1), pages 80-94, January.
    10. Christiane Baumeister & Lutz Kilian, 2014. "What Central Bankers Need To Know About Forecasting Oil Prices," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 55(3), pages 869-889, August.
    11. George Kapetanios & Massimiliano Marcellino & Fabrizio Venditti, 2019. "Large time‐varying parameter VARs: A nonparametric approach," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 34(7), pages 1027-1049, November.
    12. Davide Pettenuzzo & Allan Timmermann, 2017. "Forecasting Macroeconomic Variables Under Model Instability," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(2), pages 183-201, April.
    13. Antonello D'Agostino & Luca Gambetti & Domenico Giannone, 2013. "Macroeconomic forecasting and structural change," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 28(1), pages 82-101, January.
    14. Rachidi Kotchoni & Maxime Leroux & Dalibor Stevanovic, 2019. "Macroeconomic forecast accuracy in a data‐rich environment," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 34(7), pages 1050-1072, November.
    15. Champagne, Julien & Sekkel, Rodrigo, 2018. "Changes in monetary regimes and the identification of monetary policy shocks: Narrative evidence from Canada," Journal of Monetary Economics, Elsevier, vol. 99(C), pages 72-87.
    16. Pooyan Amir-Ahmadi & Christian Matthes & Mu-Chun Wang, 2020. "Choosing Prior Hyperparameters: With Applications to Time-Varying Parameter Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 38(1), pages 124-136, January.
    17. repec:rim:rimwps:18-37 is not listed on IDEAS
    18. James Cloyne & Patrick Hürtgen, 2016. "The Macroeconomic Effects of Monetary Policy: A New Measure for the United Kingdom," American Economic Journal: Macroeconomics, American Economic Association, vol. 8(4), pages 75-102, October.
    19. Bin Chen & Yongmiao Hong, 2012. "Testing for Smooth Structural Changes in Time Series Models via Nonparametric Regression," Econometrica, Econometric Society, vol. 80(3), pages 1157-1183, May.
    20. repec:ulb:ulbeco:2013/13388 is not listed on IDEAS
    21. Florian Huber & Gary Koop & Luca Onorante, 2021. "Inducing Sparsity and Shrinkage in Time-Varying Parameter Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(3), pages 669-683, July.
    22. Marta Banbura & Domenico Giannone & Lucrezia Reichlin, 2010. "Large Bayesian vector auto regressions," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 25(1), pages 71-92.
    23. Philippe Goulet Coulombe, 2022. "A Neural Phillips Curve and a Deep Output Gap," Working Papers 22-01, Chair in macroeconomics and forecasting, University of Quebec in Montreal's School of Management.
    24. Goulet Coulombe, Philippe & Marcellino, Massimiliano & Stevanović, Dalibor, 2021. "Can Machine Learning Catch The Covid-19 Recession?," National Institute Economic Review, National Institute of Economic and Social Research, vol. 256, pages 71-109, May.
    25. Philippe Goulet Coulombe, 2024. "The macroeconomy as a random forest," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 39(3), pages 401-421, April.
    26. Mikio Ito & Akihiko Noda & Tatsuma Wada, 2014. "International stock market efficiency: a non-Bayesian time-varying model approach," Applied Economics, Taylor & Francis Journals, vol. 46(23), pages 2744-2754, August.
    27. Florian Frommlet & Grégory Nuel, 2016. "An Adaptive Ridge Procedure for L0 Regularization," PLOS ONE, Public Library of Science, vol. 11(2), pages 1-23, February.
    28. Christina D. Romer & David H. Romer, 2004. "A New Measure of Monetary Shocks: Derivation and Implications," American Economic Review, American Economic Association, vol. 94(4), pages 1055-1084, September.
    29. Blanchard, Oliver & Cerutti, Eugenio & SUmmers, Lawrence, 2015. "Inflation and Activity - Two Explorations and Their Monetary Policy Implications," Working Paper Series 15-070, Harvard University, John F. Kennedy School of Government.
    30. Stock, James H & Watson, Mark W, 1996. "Evidence on Structural Instability in Macroeconomic Time Series Relations," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(1), pages 11-30, January.
    31. Korobilis, Dimitris, 2019. "High-dimensional macroeconomic forecasting using message passing algorithms," MPRA Paper 96079, University Library of Munich, Germany.
    32. Marco Del Negro & Giorgio E. Primiceri, 2015. "Time Varying Structural Vector Autoregressions and Monetary Policy: A Corrigendum," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 82(4), pages 1342-1345.
    33. Koop, Gary & Korobilis, Dimitris, 2013. "Large time-varying parameter VARs," Journal of Econometrics, Elsevier, vol. 177(2), pages 185-198.
    34. Ramey, V.A., 2016. "Macroeconomic Shocks and Their Propagation," Handbook of Macroeconomics, in: J. B. Taylor & Harald Uhlig (ed.), Handbook of Macroeconomics, edition 1, volume 2, chapter 0, pages 71-162, Elsevier.
    35. Joris de Wind & Luca Gambetti, 2014. "Reduced-rank time-varying vector autoregressions," CPB Discussion Paper 270.rdf, CPB Netherlands Bureau for Economic Policy Analysis.
    36. Marco Del Negro & Giorgio E. Primiceri, 2015. "Time Varying Structural Vector Autoregressions and Monetary Policy: A Corrigendum," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 82(4), pages 1342-1345.
    37. Florian Huber & Gary Koop & Michael Pfarrhofer, 2020. "Bayesian Inference in High-Dimensional Time-varying Parameter Models using Integrated Rotated Gaussian Approximations," Papers 2002.10274, arXiv.org.
    38. Mikkel Plagborg‐Møller & Christian K. Wolf, 2021. "Local Projections and VARs Estimate the Same Impulse Responses," Econometrica, Econometric Society, vol. 89(2), pages 955-980, March.
    39. Boivin, Jean, 2006. "Has U.S. Monetary Policy Changed? Evidence from Drifting Coefficients and Real-Time Data," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 38(5), pages 1149-1173, August.
    40. Chan, Joshua C.C. & Eisenstat, Eric, 2018. "Comparing hybrid time-varying parameter VARs," Economics Letters, Elsevier, vol. 171(C), pages 1-5.
    41. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    42. Jennifer L. Castle & Jurgen A. Doornik & David F. Hendry & Felix Pretis, 2015. "Detecting Location Shifts during Model Selection by Step-Indicator Saturation," Econometrics, MDPI, vol. 3(2), pages 1-25, April.
    43. Annalisa Cadonna & Sylvia Frühwirth-Schnatter & Peter Knaus, 2020. "Triple the Gamma—A Unifying Shrinkage Prior for Variance and Variable Selection in Sparse State Space and TVP Models," Econometrics, MDPI, vol. 8(2), pages 1-36, May.
    44. Marta Banbura & Domenico Giannone & Lucrezia Reichlin, 2010. "Large Bayesian vector auto regressions," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 25(1), pages 71-92.
    45. JAMES G. MacKINNON, 2006. "Bootstrap Methods in Econometrics," The Economic Record, The Economic Society of Australia, vol. 82(s1), pages 2-18, September.
    46. Diebold, Francis X & Mariano, Roberto S, 2002. "Comparing Predictive Accuracy," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 134-144, January.
    47. Niko Hauzenberger & Florian Huber & Gary Koop & Luca Onorante, 2022. "Fast and Flexible Bayesian Inference in Time-varying Parameter Regression Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(4), pages 1904-1918, October.
    48. Mikio Ito & Akihiko Noda & Tatsuma Wada, 2017. "An Alternative Estimation Method of a Time-Varying Parameter Model," Papers 1707.06837, arXiv.org, revised Dec 2017.
    49. Joris de Wind & Luca Gambetti, 2014. "Reduced-rank time-varying vector autoregressions," CPB Discussion Paper 270, CPB Netherlands Bureau for Economic Policy Analysis.
    50. Victor Chernozhukov & Kaspar Wüthrich & Yinchu Zhu, 2018. "Exact and robust conformal inference methods for predictive machine learning with dependent data," CeMMAP working papers CWP16/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    51. James H. Stock & Mark W. Watson, 1998. "A Comparison of Linear and Nonlinear Univariate Models for Forecasting Macroeconomic Time Series," NBER Working Papers 6607, National Bureau of Economic Research, Inc.
    52. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
    53. Kilian,Lutz & Lütkepohl,Helmut, 2018. "Structural Vector Autoregressive Analysis," Cambridge Books, Cambridge University Press, number 9781107196575, January.
    54. Dimitris Korobilis, 2021. "High-Dimensional Macroeconomic Forecasting Using Message Passing Algorithms," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(2), pages 493-504, March.
    55. Giraitis, L. & Kapetanios, G. & Yates, T., 2014. "Inference on stochastic time-varying coefficient models," Journal of Econometrics, Elsevier, vol. 179(1), pages 46-65.
    56. Stock, James H & Watson, Mark W, 2002. "Macroeconomic Forecasting Using Diffusion Indexes," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(2), pages 147-162, April.
    57. Barakchian, S. Mahdi & Crowe, Christopher, 2013. "Monetary policy matters: Evidence from new shocks data," Journal of Monetary Economics, Elsevier, vol. 60(8), pages 950-966.
    58. Petrova, Katerina, 2019. "A quasi-Bayesian local likelihood approach to time varying parameter VAR models," Journal of Econometrics, Elsevier, vol. 212(1), pages 286-306.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Cho, Dooyeon & Jung, Jaehun, 2025. "Machine learning goes beyond: Time-varying monetary policy and oil price pass-through to inflation expectations," Journal of Macroeconomics, Elsevier, vol. 85(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Philippe Goulet Coulombe, 2020. "Time-Varying Parameters as Ridge Regressions," Papers 2009.00401, arXiv.org, revised Nov 2024.
    2. Philippe Goulet Coulombe, 2024. "The macroeconomy as a random forest," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 39(3), pages 401-421, April.
    3. Niko Hauzenberger, 2020. "Flexible Mixture Priors for Large Time-varying Parameter Models," Papers 2006.10088, arXiv.org, revised Nov 2020.
    4. Hauzenberger, Niko, 2021. "Flexible Mixture Priors for Large Time-varying Parameter Models," Econometrics and Statistics, Elsevier, vol. 20(C), pages 87-108.
    5. Barbara Rossi, 2021. "Forecasting in the Presence of Instabilities: How We Know Whether Models Predict Well and How to Improve Them," Journal of Economic Literature, American Economic Association, vol. 59(4), pages 1135-1190, December.
    6. Goulet Coulombe, Philippe & Leroux, Maxime & Stevanovic, Dalibor & Surprenant, Stéphane, 2021. "Macroeconomic data transformations matter," International Journal of Forecasting, Elsevier, vol. 37(4), pages 1338-1354.
    7. Hauzenberger, Niko & Huber, Florian & Klieber, Karin, 2023. "Real-time inflation forecasting using non-linear dimension reduction techniques," International Journal of Forecasting, Elsevier, vol. 39(2), pages 901-921.
    8. Marta Banbura & Andries van Vlodrop, 2018. "Forecasting with Bayesian Vector Autoregressions with Time Variation in the Mean," Tinbergen Institute Discussion Papers 18-025/IV, Tinbergen Institute.
    9. Niko Hauzenberger & Florian Huber & Gary Koop & James Mitchell, 2023. "Bayesian Modeling of Time-Varying Parameters Using Regression Trees," Working Papers 23-05, Federal Reserve Bank of Cleveland.
    10. Giacomo Rella, 2021. "The Fed, housing and household debt over time," Department of Economics University of Siena 850, Department of Economics, University of Siena.
    11. Zheng, Tingguo & Ye, Shiqi & Hong, Yongmiao, 2023. "Fast estimation of a large TVP-VAR model with score-driven volatilities," Journal of Economic Dynamics and Control, Elsevier, vol. 157(C).
    12. Yousuf, Kashif & Ng, Serena, 2021. "Boosting high dimensional predictive regressions with time varying parameters," Journal of Econometrics, Elsevier, vol. 224(1), pages 60-87.
    13. Hauzenberger Niko & Huber Florian & Koop Gary, 2024. "Dynamic Shrinkage Priors for Large Time-Varying Parameter Regressions Using Scalable Markov Chain Monte Carlo Methods," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 28(2), pages 201-225, April.
    14. Joshua C.C. Chan & Rodney W. Strachan, 2023. "Bayesian State Space Models In Macroeconometrics," Journal of Economic Surveys, Wiley Blackwell, vol. 37(1), pages 58-75, February.
    15. Nima Nonejad, 2021. "An Overview Of Dynamic Model Averaging Techniques In Time‐Series Econometrics," Journal of Economic Surveys, Wiley Blackwell, vol. 35(2), pages 566-614, April.
    16. Florian Huber & Gary Koop & Michael Pfarrhofer, 2020. "Bayesian Inference in High-Dimensional Time-varying Parameter Models using Integrated Rotated Gaussian Approximations," Papers 2002.10274, arXiv.org.
    17. Andrea Carriero & Francesco Corsello & Massimiliano Marcellino, 2022. "The global component of inflation volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(4), pages 700-721, June.
    18. Niko Hauzenberger & Florian Huber & Gary Koop & James Mitchell, 2020. "Bayesian Modelling of TVP-VARs Using Regression Trees," Working Papers 2308, University of Strathclyde Business School, Department of Economics, revised Aug 2023.
    19. Michael Pfarrhofer, 2024. "Forecasts with Bayesian vector autoregressions under real time conditions," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 43(3), pages 771-801, April.
    20. Koop, Gary & Korobilis, Dimitris & Pettenuzzo, Davide, 2019. "Bayesian compressed vector autoregressions," Journal of Econometrics, Elsevier, vol. 210(1), pages 135-154.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:intfor:v:41:y:2025:i:3:p:982-1002. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/ijforecast .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.