Sparse Tree-Based Aggregation for Time Series Regressions

Technical Summary: Sparse Tree-Based Aggregation for Time Series Regressions (StarTime)

Problem Statement
High-dimensional time series regressions, particularly those involving high-order autoregressions (AR) and mixed-frequency data (e.g., daily predictors for quarterly responses), face the "curse of dimensionality." In these settings, predictive information is often distributed across many lags that are highly correlated. While penalized estimators like the Lasso are effective for sparse models, they struggle when the true lag structure is dense and smooth, as is common in financial volatility and macroeconomic dynamics. Standard Lasso approaches may overshrink lagged effects or select unstable subsets of lags, failing to capture the underlying temporal structure. Conversely, existing aggregation methods (e.g., HAR models, MIDAS) often require the aggregation scheme to be specified a priori, lacking flexibility to adapt to unknown temporal resolutions.

Methodology: StarTime
The authors propose StarTime (Sparse Tree-based Aggregation for Time Series), a convex penalization method that addresses dimensionality through data-driven temporal aggregation. The core innovation lies in combining sparsity with flexible aggregation guided by a "temporal tree."

1. Tree-Based Parameterization:

   • The method constructs a temporal tree for each predictor, arranging lags hierarchically from high-frequency (leaf nodes) to low-frequency (root nodes).
   • Original coefficients ($\beta$) are reparametrized as sums of auxiliary parameters ($\gamma$) along the paths from leaves to the root. Specifically, $\beta = A\gamma$, where $A$ is a binary matrix encoding the tree structure.
   • This reparametrization allows the model to induce aggregation: if the auxiliary parameters ($\gamma$) for a specific branch are zeroed out, the descendant lags are fused, effectively estimating a single coefficient for that aggregated frequency.

2. Penalized Estimation:

   • StarTime solves a convex optimization problem minimizing the sum of squared residuals subject to the tree constraint ($\beta = A\gamma$) and an $\ell_1$ penalty on both the auxiliary parameters ($\gamma$) and the original coefficients ($\beta$).
   • The penalty $\lambda_1 \|\gamma\|_1$ encourages temporal aggregation (fusing adjacent lags), while $\lambda_2 \|\beta\|_1$ encourages direct sparsity (setting coefficients to zero).
   • The estimator can produce sparse coefficients, temporally aggregated coefficients at varying frequencies, or a combination of both.

3. Implementation and Tuning:

   • The optimization is solved using an Alternating Direction Method of Multipliers (ADMM) algorithm.
   • Tuning parameters ($\lambda_1, \lambda_2$) are selected via a grid search minimizing a Bayesian Information Criterion (BIC). The authors also propose a "Post StarTime" variant, which refits the model via OLS on the selected and aggregated variables to reduce bias.

Key Contributions

• Novel Framework: The paper introduces a unified framework that simultaneously handles sparsity and temporal aggregation without requiring pre-specified aggregation schemes (unlike HAR or restricted MIDAS).
• Theoretical Guarantees: The authors derive new error bounds for StarTime under Near-Epoch Dependence (NED) assumptions, extending previous results from i.i.d. data to time series settings characterized by non-Gaussianity, serial dependence, and heteroskedasticity. These bounds guarantee consistency for prediction and estimation even when the number of parameters grows faster than the sample size.
• Flexibility: Unlike the HAR model, which imposes a fixed daily-weekly-monthly structure, StarTime determines the appropriate temporal resolution and aggregation structure entirely from the data.

Results

• Simulations:

  • In autoregressive settings, StarTime (particularly the Post variant) consistently outperforms OLS, Ridge, and Lasso in terms of Mean Squared Error (MSE), especially when the true data generating process (DGP) involves smooth, aggregated structures.
  • In mixed-frequency settings, StarTime demonstrates robust performance. It outperforms benchmarks when the true structure involves clear aggregation (DGP 3) and remains competitive even when the tree structure is misspecified (DGP 1 and 2), often outperforming MIDAS-ML in high-dimensional scenarios.
  • StarTime shows superior ability to recover both the correct grouping structure (measured by Adjusted Rand Index) and sparsity patterns (measured by F1 score) compared to benchmarks.

• Empirical Applications:

  • Financial Volatility: Applied to realized variance forecasting for 30 major U.S. assets, StarTime tracks the performance of the established HAR model closely. While HAR remains the top performer, StarTime provides data-driven evidence that the HAR's specific aggregation structure is frequently recovered by the algorithm, validating the economic intuition behind HAR without imposing it a priori.
  • Macroeconomic Nowcasting: The method is applied to GDP growth forecasting using mixed-frequency data (daily, weekly, monthly, quarterly). StarTime effectively handles the high dimensionality arising from frequency mismatches, demonstrating relevance for macroeconomic applications.

Significance and Claims
The paper claims that StarTime offers a powerful, flexible alternative to traditional regularization in time series. Its primary significance lies in its ability to automatically discover the appropriate temporal resolution and aggregation structure from the data, bridging the gap between rigid aggregation models (like HAR) and purely sparse models (like Lasso). The authors emphasize that this data-driven approach is particularly valuable in mixed-frequency settings where the optimal aggregation scheme is unknown. Theoretical results ensure the method's validity in realistic econometric settings with dependent and non-Gaussian data, while empirical results confirm its practical utility in both financial and macroeconomic forecasting.