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arxiv: 2606.03665 · v1 · pith:GSLB7NKMnew · submitted 2026-06-02 · 💰 econ.EM · stat.ME

Sparse Tree-Based Aggregation for Time Series Regressions

Marie Corillon , Stephan Smeekes , Ines Wilms This is my paper

Pith reviewed 2026-06-28 07:51 UTC · model grok-4.3

classification 💰 econ.EM stat.ME
keywords time series regressiontemporal aggregationsparse estimationmixed-frequency dataautoregressive modelsconvex penalizationhigh-dimensional datatree structure
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The pith

StarTime uses a temporal tree to aggregate lags at varying frequencies and reduce dimensionality in high-order time series regressions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that arranging lags in a hierarchical temporal tree allows a convex penalization method to choose which coefficients to aggregate at different frequencies, set to zero, or both, serving as an alternative to standard sparsity penalties for high-dimensional autoregressions and mixed-frequency models. This produces new error bounds and better recovery of the underlying structure in simulations. A sympathetic reader would care because economic time series often require many lags or mixed sampling rates, which quickly inflate the number of parameters and make estimation unstable. The method therefore offers a structured way to lower dimension while respecting the frequency relationships among lags.

Core claim

StarTime is a convex penalization method that arranges lags hierarchically in a temporal tree from high to low frequency and then selects which coefficients to aggregate at possibly different frequencies, to sparsify, or to combine both operations, delivering new error bounds for estimation accuracy in high-order autoregressions and mixed-frequency regressions.

What carries the argument

The temporal tree that organizes lags from high to low frequency so that the penalization can select aggregation at chosen nodes or sparsity.

If this is right

  • Improved finite-sample estimation accuracy and recovery of both aggregation patterns and sparsity relative to standard benchmarks.
  • New theoretical error bounds that justify the use of tree-structured aggregation in high-dimensional time series settings.
  • Direct applicability to financial and macroeconomic regressions that mix high-frequency and low-frequency variables.
  • Flexible selection of aggregation levels that can vary across different parts of the lag structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same tree logic could be adapted to spatial or network data where observations have a natural hierarchical scale.
  • StarTime might be combined with existing mixed-frequency nowcasting tools to handle real-time data releases without manual lag truncation.
  • If the tree is misspecified in practice, a data-driven way to learn the hierarchy itself would be a natural next step.

Load-bearing premise

The temporal tree correctly reflects the true hierarchical frequency relationships among lags so that aggregation at selected nodes preserves the underlying dynamics.

What would settle it

A Monte Carlo experiment in which the data-generating process has lag relationships that violate the assumed tree hierarchy and StarTime then produces higher mean squared error than ordinary lasso or the unaggregated model.

Figures

Figures reproduced from arXiv: 2606.03665 by Ines Wilms, Marie Corillon, Stephan Smeekes.

Figure 1
Figure 1. Figure 1: Top: Log-transformed realized variance for Apple Inc. (AAPL) from March 20, 2008 to [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Toy example of a temporal tree aggregating months into quarters and a semester. Left [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Temporal tree aggregating 20 days into four weeks and one month. The sparsity pattern [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Tree structures for a mixed-frequency regression with a quarterly response and monthly [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Performance metrics of the estimators across the three AR-based DGPs. [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Performance metrics of the estimators across the three mixed-frequency DGPs. [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparative forecast performance across varying fitting schemes. Each panel displays [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Rolling-window coefficient heatmaps for Post StarTime (horizon 1, 20 lags) applied to [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of squared forecast errors for the Reduced Set (49 predictors) with [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Evolution of the squared forecast errors for the Full Set (398 predictors) with [PITH_FULL_IMAGE:figures/full_fig_p056_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of the squared forecast errors for the Full Set (398 predictors) with [PITH_FULL_IMAGE:figures/full_fig_p057_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Evolution of the squared forecast errors for the Reduced Set (49 predictors) with [PITH_FULL_IMAGE:figures/full_fig_p058_12.png] view at source ↗
read the original abstract

High-dimensional time series regressions are often regularized to produce sparse coefficients. We show that temporal aggregation provides a powerful alternative to reduce dimensionality in high-order autoregressions and mixed-frequency regressions. To this end, we propose StarTime (Sparse Tree-based Aggregation for Time Series), a convex penalization method that uses a temporal tree to arrange lags hierarchically from high to low frequency. StarTime then flexibly selects coefficients to be aggregated at possibly varying frequencies, sparse or a combination thereof. We provide new error bounds for StarTime, demonstrate improved estimation accuracy and recovery of aggregation and sparsity in simulations relative to benchmarks, and illustrate StarTime's relevance for financial and macroeconomic applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proposes StarTime, a convex penalization method for high-dimensional time series regressions that arranges lags in a temporal tree from high to low frequency and selects coefficients for aggregation at varying frequencies (or sparsity, or combinations thereof). It claims this reduces dimensionality in high-order autoregressions and mixed-frequency regressions, provides new error bounds, and yields improved estimation accuracy and recovery of aggregation/sparsity patterns relative to benchmarks in simulations, with relevance to financial and macroeconomic applications.

Significance. If the error bounds hold under the stated conditions and the simulation results are robust, the method offers a structured alternative to standard regularization by exploiting temporal hierarchies for dimensionality reduction. The explicit combination of aggregation and sparsity via tree-based penalization is a clear strength, as is the provision of new theoretical bounds (when verified) and the reproducible simulation benchmarks.

minor comments (2)
  1. [Abstract] The abstract states that 'new error bounds' are provided, but without a named section or equation reference in the provided material it is unclear whether these are derived under the same assumptions as the tree construction; adding an explicit statement of the bound (e.g., in the theory section) would strengthen the central claim.
  2. The simulation section should clarify whether aggregation levels are chosen ex ante or involve any post-hoc selection, as this directly affects the interpretation of the reported accuracy gains over benchmarks.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of StarTime's contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces StarTime as a new convex penalization method based on a temporal tree structure for aggregating lags in high-dimensional time series regressions. It claims new error bounds derived from the method's formulation and validates performance via external simulations and applications. No load-bearing step in the abstract or described claims reduces by construction to fitted parameters, self-citations, or renamed inputs; the derivation chain appears self-contained with independent theoretical and empirical content.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the temporal tree hierarchy and the convexity of the penalization; no explicit free parameters are named in the abstract but tuning parameters for the penalty are implied by any regularization method. No invented entities beyond the method itself.

free parameters (1)
  • penalty tuning parameter
    Standard in convex penalization methods; required to control the trade-off between aggregation, sparsity, and fit.
axioms (1)
  • domain assumption The temporal tree arranges lags hierarchically from high to low frequency without loss of information for the regression problem.
    Invoked when the method selects aggregation nodes; this is the structural premise that allows dimensionality reduction.

pith-pipeline@v0.9.1-grok · 5636 in / 1271 out tokens · 15503 ms · 2026-06-28T07:51:47.572779+00:00 · methodology

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Reference graph

Works this paper leans on

72 extracted references · 16 canonical work pages

  1. [1]

    Journal of Financial Econometrics , volume=

    A simple approximate long-memory model of realized volatility , author=. Journal of Financial Econometrics , volume=

  2. [2]

    Andersen, T. G. and Bollerslev, T. and Diebold, F. X. and Labys, P. , year =. Modeling and forecasting realized volatility , journal =

  3. [3]

    and Santa-Clara, P

    Ghysels, E. and Santa-Clara, P. and Valkanov, R. , year =. The

  4. [4]

    The realized Laplace transform of volatility , volume =

    Todorov, Viktor and Tauchen, George , year =. The realized Laplace transform of volatility , volume =. Econometrica , doi =

  5. [5]

    , title =

    Litvinova, J. , title =. 2003 , note =

    2003

  6. [6]

    Leverage and volatility feedback effects in high-frequency data , volume =

    Bollerslev, Tim and Litvinova, Julia and Tauchen, George , year =. Leverage and volatility feedback effects in high-frequency data , volume =. Journal of Financial Econometrics , doi =

  7. [7]

    2008 , issn =

    Nowcasting: The real-time informational content of macroeconomic data , journal =. 2008 , issn =. doi:https://doi.org/10.1016/j.jmoneco.2008.05.010 , url =

    work page doi:10.1016/j.jmoneco.2008.05.010 2008

  8. [8]

    Temporal aggregation and economic time series , volume =

    Rossana, Robert and Seater, John , year =. Temporal aggregation and economic time series , volume =

  9. [9]

    A comparison of MIDAS and bridge equations , journal =

    Christian Schumacher , keywords =. A comparison of MIDAS and bridge equations , journal =. 2016 , issn =. doi:https://doi.org/10.1016/j.ijforecast.2015.07.004 , url =

    work page doi:10.1016/j.ijforecast.2015.07.004 2016

  10. [10]

    Temporal aggregation of univariate and multivariate time series models: A survey , volume =

    Veredas, David and Silvestrini, Andrea , year =. Temporal aggregation of univariate and multivariate time series models: A survey , volume =. Journal of Economic Surveys , doi =

  11. [11]

    Wu , journal =

    Takeshi Amemiya and Roland Y. Wu , journal =. The effect of aggregation on prediction in the autoregressive model , urldate =

  12. [12]

    1973 , issn =

    Some consequences of temporal aggregation and systematic sampling for ARMA and ARMAX models , journal =. 1973 , issn =. doi:https://doi.org/10.1016/0304-4076(73)90015-8 , url =

    work page doi:10.1016/0304-4076(73)90015-8 1973

  13. [13]

    G. C. Tiao , journal =. Asymptotic behaviour of temporal aggregates of time series , urldate =

  14. [14]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume =

    Zou, Hui and Hastie, Trevor , title = ". Journal of the Royal Statistical Society Series B: Statistical Methodology , volume =. 2005 , month =. doi:10.1111/j.1467-9868.2005.00503.x , url =

    work page doi:10.1111/j.1467-9868.2005.00503.x 2005

  15. [15]

    2016 , issn =

    On the use of high frequency measures of volatility in MIDAS regressions , journal =. 2016 , issn =. doi:https://doi.org/10.1016/j.jeconom.2016.04.012 , url =

    work page doi:10.1016/j.jeconom.2016.04.012 2016

  16. [16]

    Journal of the American Statistical Association , volume =

    Xiaohan Yan and Jacob Bien , title =. Journal of the American Statistical Association , volume =. 2021 , publisher =. doi:10.1080/01621459.2020.1796677 , URL =

    work page doi:10.1080/01621459.2020.1796677 2021

  17. [17]

    Journal of Machine Learning Research , volume=

    Tree-based node aggregation in sparse graphical models , author=. Journal of Machine Learning Research , volume=. 2022 , author+an=

    2022

  18. [18]

    Journal of Business & Economic Statistics , volume=

    Machine learning time series regressions with an application to nowcasting , author=. Journal of Business & Economic Statistics , volume=. 2022 , publisher=

    2022

  19. [19]

    Foundations and Trends in Machine Learning , volume=

    Distributed optimization and statistical learning via the alternating direction method of multipliers , author=. Foundations and Trends in Machine Learning , volume=. 2011 , publisher=

    2011

  20. [20]

    Journal of Classification , volume=

    Comparing partitions , author=. Journal of Classification , volume=. 1985 , publisher=

    1985

  21. [21]

    Journal of Statistical Software , year =

    Regularization paths for generalized linear models via coordinate descent , author =. Journal of Statistical Software , year =

  22. [22]

    Mixed frequency data sampling regression models: The

    Eric Ghysels and Virmantas Kvedaras and Vaidotas Zemlys , journal =. Mixed frequency data sampling regression models: The. 2016 , volume =

    2016

  23. [23]

    2022 , note =

    midasml: Estimation and prediction methods for high-dimensional mixed frequency time series data , author =. 2022 , note =

    2022

  24. [24]

    2011 , publisher =

    Statistics for high-dimensional data: Methods, theory and applications , author =. 2011 , publisher =

    2011

  25. [25]

    and Molstad, A

    Fu, J. and Molstad, A. J. and Zou, H. , year =. A direct approach to tree-guided feature aggregation for high-dimensional regression , howpublished =

  26. [26]

    2025 , url =

    R: A Language and Environment for Statistical Computing , author =. 2025 , url =

    2025

  27. [27]

    Journal of the American Statistical Association , year =

    Data-driven tuning parameter selection for high-dimensional vector autoregressions , author =. Journal of the American Statistical Association , year =

  28. [28]

    Regression shrinkage and selection via the lasso , urldate =

    Robert Tibshirani , journal =. Regression shrinkage and selection via the lasso , urldate =

  29. [29]

    Hoerl, A. E. and Kennard, R. W. , year =. Ridge regression: Biased estimation for nonorthogonal problems , journal =. doi:10.1080/00401706.1970.10488634 , url =

    work page doi:10.1080/00401706.1970.10488634 1970

  30. [30]

    Extended BIC for small- n -large- P sparse GLM , urldate =

    Jiahua Chen and Zehua Chen , journal =. Extended BIC for small- n -large- P sparse GLM , urldate =

  31. [31]

    2026 , pages =

    HARd to beat: The overlooked impact of rolling windows in the era of machine learning , journal =. 2026 , pages =. doi:https://doi.org/10.1016/j.ijforecast.2025.06.003 , url =

    work page doi:10.1016/j.ijforecast.2025.06.003 2026

  32. [32]

    Journal of Financial Econometrics , volume =

    Hecq, Alain and Margaritella, Luca and Smeekes, Stephan , title =. Journal of Financial Econometrics , volume =. 2023 , doi =

    2023

  33. [33]

    Journal of Statistical Software , year =

    Stephan Smeekes and Ines Wilms , title =. Journal of Statistical Software , year =. doi:10.18637/jss.v106.i12 , url =

    work page doi:10.18637/jss.v106.i12

  34. [34]

    Journal of Computational and Graphical Statistics , volume =

    Alain Hecq and Marie Ternes and Ines Wilms , title =. Journal of Computational and Graphical Statistics , volume =. 2022 , publisher =. doi:10.1080/10618600.2022.2058003 , URL =

    work page doi:10.1080/10618600.2022.2058003 2022

  35. [35]

    Journal of Business & Economic Statistics , volume=

    FRED-MD: A Monthly Database for Macroeconomic Research , author=. Journal of Business & Economic Statistics , volume=. 2016 , publisher=. doi:10.1080/07350015.2015.1086655 , url=

    work page doi:10.1080/07350015.2015.1086655 2016

  36. [36]

    The Quarterly Journal of Economics , volume=

    Measuring economic policy uncertainty , author=. The Quarterly Journal of Economics , volume=. 2016 , publisher=

    2016

  37. [37]

    Forecasting returns in the

    Taylor, Nick , year =. Forecasting returns in the. International Journal of Forecasting , volume =. doi:10.1016/j.ijforecast.2019.01.009 , url =

    work page doi:10.1016/j.ijforecast.2019.01.009 2019

  38. [38]

    and McCracken, Michael W

    Amburgey, Aaron J. and McCracken, Michael W. , title =. Journal of Applied Econometrics , volume =. doi:https://doi.org/10.1002/jae.2943 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/jae.2943 , abstract =

    work page doi:10.1002/jae.2943

  39. [39]

    Patton , keywords =

    Andrew J. Patton , keywords =. Volatility forecast comparison using imperfect volatility proxies , journal =. 2011 , note =. doi:https://doi.org/10.1016/j.jeconom.2010.03.034 , url =

    work page doi:10.1016/j.jeconom.2010.03.034 2011

  40. [40]

    2026 , howpublished =

    Economic Policy Uncertainty Index , author =. 2026 , howpublished =

    2026

  41. [41]

    Econometrica , volume =

    The model confidence set , author =. Econometrica , volume =. 2011 , publisher =

    2011

  42. [42]

    Journal of Business & Economic Statistics , volume =

    Comparing predictive accuracy , author =. Journal of Business & Economic Statistics , volume =. 1995 , publisher =

    1995

  43. [43]

    and Watson, Mark W

    Stock, James H. and Watson, Mark W. , journal=. Why has. 2007 , publisher=

    2007

  44. [44]

    and Watson, Mark W

    Marcellino, Massimiliano and Stock, James H. and Watson, Mark W. , journal=. A comparison of direct and iterated multistep. 2006 , publisher=

    2006

  45. [45]

    New housing registrations as a leading indicator of the

    Cheung, Calista and Granovsky, Dmitry , year=. New housing registrations as a leading indicator of the

  46. [46]

    2024 , howpublished =

    Cortes, Gustavo and LaPoint, Cameron , title =. 2024 , howpublished =

    2024

  47. [47]

    Economic Perspectives , volume=

    Monitoring financial stability: A financial conditions index approach , author=. Economic Perspectives , volume=. 2011 , publisher=

    2011

  48. [48]

    NBER Macroeconomics Annual 1989, Volume 4 , pages=

    New indexes of coincident and leading economic indicators , author=. NBER Macroeconomics Annual 1989, Volume 4 , pages=. 1989 , publisher=

    1989

  49. [49]

    2025 , howpublished=

    Business cycle dating procedure: Frequently asked questions , author=. 2025 , howpublished=

    2025

  50. [50]

    Journal of Economic Literature , volume=

    Forecasting volatility in financial markets: A review , author=. Journal of Economic Literature , volume=. 2003 , publisher=

    2003

  51. [51]

    Unrestricted mixed data sampling (

    Foroni, Claudia and Marcellino, Massimiliano and Schumacher, Christian , journal=. Unrestricted mixed data sampling (. 2015 , publisher=

    2015

  52. [52]

    2015 , edition =

    Statistical learning with sparsity: The lasso and generalizations , author=. 2015 , edition =. doi:10.1201/b18401 , url =

    work page doi:10.1201/b18401 2015

  53. [53]

    Journal of Econometrics , volume =

    Lasso inference for high-dimensional time series , author =. Journal of Econometrics , volume =. 2023 , issn =

    2023

  54. [54]

    Journal of Econometrics , volume=

    An automated approach towards sparse single-equation cointegration modelling , author=. Journal of Econometrics , volume=. 2021 , publisher=

    2021

  55. [55]

    Journal of Econometrics , volume=

    Bayesian MIDAS penalized regressions: Estimation, selection, and prediction , author=. Journal of Econometrics , volume=. 2021 , publisher=

    2021

  56. [56]

    Journal of Economic Surveys , volume=

    Machine learning advances for time series forecasting , author=. Journal of Economic Surveys , volume=. 2023 , publisher=

    2023

  57. [57]

    Econometric reviews , volume=

    MIDAS regressions: Further results and new directions , author=. Econometric reviews , volume=. 2007 , publisher=

    2007

  58. [58]

    Econometric Reviews , volume=

    Lassoing the HAR model: A model selection perspective on realized volatility dynamics , author=. Econometric Reviews , volume=. 2016 , publisher=

    2016

  59. [59]

    International Journal of Forecasting , volume=

    The impact of sentiment and attention measures on stock market volatility , author=. International Journal of Forecasting , volume=. 2020 , publisher=

    2020

  60. [60]

    Journal of Financial Econometrics , volume=

    A machine learning approach to volatility forecasting , author=. Journal of Financial Econometrics , volume=. 2023 , publisher=

    2023

  61. [61]

    Journal of Financial Econometrics , volume=

    Volatility forecasting with machine learning and intraday commonality , author=. Journal of Financial Econometrics , volume=. 2024 , publisher=

    2024

  62. [62]

    Journal of Financial Econometrics , volume=

    When MIDAS meets LASSO: The power of low-frequency variables in forecasting value-at-risk and expected shortfall , author=. Journal of Financial Econometrics , volume=. 2025 , publisher=

    2025

  63. [63]

    Journal of Financial Econometrics , volume=

    High-dimensional Granger causality tests with an application to VIX and news , author=. Journal of Financial Econometrics , volume=. 2024 , publisher=

    2024

  64. [64]

    Journal of Machine Learning Research , volume=

    High dimensional forecasting via interpretable vector autoregression , author=. Journal of Machine Learning Research , volume=

  65. [65]

    Statistical Science , number =

    Xiaohan Yan and Jacob Bien , title =. Statistical Science , number =. 2017 , doi =

    2017

  66. [66]

    Journal of Econometrics , volume=

    _1 -regularization of high-dimensional time-series models with non-Gaussian and heteroskedastic errors , author=. Journal of Econometrics , volume=. 2016 , publisher=

    2016

  67. [67]

    1992 , publisher=

    Business cycles: Theory, history, indicators, and forecasting , author=. 1992 , publisher=

    1992

  68. [68]

    Journal of Economic Surveys , volume=

    The resurgence of inventory research: What have we learned? , author=. Journal of Economic Surveys , volume=

  69. [69]

    and Catania, L

    Bernardi, M. and Catania, L. , title =. International Journal of Computational Economics and Econometrics , year =. doi:10.1504/IJCEE.2018.091037 , url =

    work page doi:10.1504/ijcee.2018.091037 2018

  70. [70]

    Automatic time series forecasting: The forecast package for

    Rob J Hyndman and Yeasmin Khandakar , journal =. Automatic time series forecasting: The forecast package for. 2008 , doi =

    2008

  71. [71]

    , year =

    Leamer, Edward E. , year =. Housing is the business cycle , type =

  72. [72]

    and Ng, Serena , title =

    McCracken, Michael W. and Ng, Serena , title =. Federal Reserve Bank of. 2021 , volume =

    2021