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The empirical distribution of sequential LS factors in Multi-level Dynamic Factor Models

This paper demonstrates through Monte Carlo experiments that the asymptotic distribution of Principal Components factors derived by Bai (2003) effectively approximates the finite-sample distribution of Sequential Least Squares factors in multi-level dynamic factor models, while also identifying the most robust estimator for their asymptotic mean squared error.

Original authors: Gian Pietro Bellocca, Ignacio Garrón, Vladimir Rodríguez-Caballero, Esther Ruiz

Published 2026-02-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gian Pietro Bellocca, Ignacio Garrón, Vladimir Rodríguez-Caballero, Esther Ruiz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the weather patterns of a massive country. You have thousands of weather stations (data points) reporting temperature, humidity, and wind speed every hour.

If you just look at the raw data, it's a chaotic mess. But you suspect there are a few "hidden drivers" causing the changes:

The Global Driver: A massive high-pressure system affecting the whole country (like a heatwave).
The Local Drivers: Specific conditions affecting only certain regions (like a foggy morning in the valley or a storm in the mountains).

In the world of economics and finance, this is called a Multi-Level Dynamic Factor Model (ML-DFM). The "Global Driver" is a global economic factor (like a recession), and the "Local Drivers" are regional factors (like a housing bubble in one state).

The Problem: How Do We Measure the Uncertainty?

The authors of this paper, Bellocca, Garrón, Rodríguez-Caballero, and Ruiz, are asking a very specific question: How confident can we be in our estimates of these hidden drivers?

If you estimate that a recession is happening, you need to know: Is this a real recession, or just a fluke in the data? To answer this, statisticians need to calculate the "Mean Squared Error" (MSE)—a fancy way of saying "how far off are we likely to be?"

For a long time, there was a famous rulebook (by a researcher named Bai in 2003) that told us how to calculate this uncertainty for simple models where every station is affected by every factor. But in the real world, factors are often "group-specific" (affecting only some stations).

The big question was: Does the old rulebook still work for these complex, multi-level models?

The Solution: The "Sequential" Approach

To find these hidden drivers, economists use a method called Sequential Least Squares (SLS). Think of this as a detective solving a mystery in two steps:

Step 1: Find the big, country-wide clues (Global Factors).
Step 2: Once the big clues are removed, look at the remaining data to find the local, regional clues (Group-Specific Factors).

This method is popular because it's fast and easy to use. However, until now, no one had a formal mathematical proof of how to calculate the "uncertainty" (the error bars) for the results of this two-step detective work.

The Experiment: A Massive Simulation

The authors decided to test if the old rulebook (Bai's distribution) could be used as a shortcut for the new, complex method (SLS).

They built a virtual world (a Monte Carlo simulation) with:

1000 different scenarios: They created thousands of fake economies with different sizes and different types of noise.
Different types of "Noise": Sometimes the weather stations were independent (uncorrelated). Sometimes, if one station had a glitch, its neighbors did too (cross-sectional correlation). Sometimes the noise was wild and unpredictable (heteroscedasticity).

They ran the SLS detective method on all these fake worlds and compared the results against the "True" hidden factors they had planted in the simulation.

The Big Discoveries

Here is what they found, translated into plain English:

1. The Old Rulebook Works (Surprisingly Well!)
Even though the SLS method is a two-step process and the old rulebook was designed for a one-step process, the math holds up.

Analogy: It's like using a map designed for a straight highway to navigate a winding mountain road. You'd expect it to fail, but the authors found that for most practical purposes, the old map is accurate enough to tell you where you are and how far off you might be.
Why this matters: Economists can now use the simple, well-understood formulas to calculate confidence intervals for complex, multi-level models without needing to invent new, complicated math.

2. The "Noise" Matters
The accuracy of the map depends on the terrain.

If the "noise" (idiosyncratic errors) in the data is random and independent, the old formulas work perfectly.
However, if the noise is correlated (e.g., if a shock in one sector ripples to another), the standard formulas start to lie to you. They might make you think you are more certain than you actually are.

3. The Best Tool for the Job
The paper tested different ways to measure this uncertainty. They found that the best tool is one that:

Acknowledges the ripple effects: It accounts for the fact that errors in one variable can affect others (cross-sectional correlation).
Uses "Subsampling": This is a statistical technique that acts like a "reality check." It simulates the process of estimating the model over and over again on smaller chunks of data to see how much the results wiggle. This accounts for the fact that we don't know the exact "loadings" (how strongly a factor affects a variable) and adds a safety margin for that uncertainty.

The Takeaway

This paper is a green light for economists and data scientists.

For the Practitioner: You can use the popular, easy-to-compute "Sequential Least Squares" method to find global and local economic factors.
For the Statistician: You can trust the standard "Bai (2003)" formulas to tell you how accurate your results are, provided you use the right version of the error calculator (one that handles correlated noise and uses subsampling).

In short: The complex, multi-level models we need for the real world are mathematically sound, and we have the right tools to measure our confidence in them.

1. Problem Statement

The paper addresses a critical gap in the econometric analysis of Multi-level Dynamic Factor Models (ML-DFMs). In many empirical applications (e.g., macroeconomics and finance), latent factors exhibit a hierarchical structure consisting of:

Global factors: Pervasive factors loading on all variables in the system.
Group-specific factors: Non-pervasive factors loading only on specific subsets (groups) of variables.

While the Sequential Least Squares (SLS) estimator (proposed by Breitung and Eickmeier, 2016) is widely used to extract these factors due to its computational simplicity, formal inference remains underdeveloped. Specifically:

There are no explicit expressions for the asymptotic distribution of SLS-estimated factors.
Without a known distribution, it is impossible to construct valid confidence intervals or perform uncertainty quantification (crucial for stress testing and scenario analysis).
It is unclear whether the asymptotic distribution derived by Bai (2003) for standard Principal Component (PC) factors in single-level DFMs can approximate the finite-sample distribution of SLS factors in ML-DFMs.
Existing estimators for the asymptotic Mean Squared Error (MSE) often rely on unrealistic assumptions (e.g., no idiosyncratic cross-sectional correlation) or fail to account for the uncertainty introduced by estimating factor loadings.

2. Methodology

The authors employ a rigorous Monte Carlo simulation framework to investigate the finite-sample properties of SLS factors and the validity of asymptotic approximations.

Model Framework

ML-DFM Structure: The data generating process (DGP) follows a block-structured model where variables are partitioned into $S$ groups. The model includes global factors ( $G_t$ ) and group-specific factors ( $L_{s,t}$ ).
Estimation: Factors are extracted using the SLS algorithm, which iterates between:
1. Initialization: Using Canonical Correlation Analysis (CCA) or PC to get initial global factors.
2. Loading Update: Estimating loadings via cross-sectional Least Squares (LS) with zero restrictions imposed on group-specific loadings.
3. Factor Update: Re-estimating factors via time-series LS.
4. Iteration: Repeating until convergence.

Simulation Design

Dimensions: Various combinations of cross-sectional ( $N$ ) and temporal ( $T$ ) dimensions, including small samples ( $N=50, T=50$ ) and large samples ( $N=600, T=500$ ).
Idiosyncratic Components: The authors test three distinct covariance structures for the error terms ( $\epsilon_{it}$ $ϵ_{i t}$ ):
1. Homoscedastic and cross-sectionally uncorrelated.
2. Heteroscedastic and cross-sectionally uncorrelated.
3. Heteroscedastic and cross-sectionally correlated (using a Toeplitz structure).
Comparison: The study compares SLS results in ML-DFMs against standard PC results in single-level DFMs to isolate the effect of the hierarchical structure.

Estimators Analyzed

The paper evaluates the performance of different estimators for the asymptotic MSE matrix ( $\Gamma_t$ ):

Heteroscedasticity-Robust (HR): Based on Bai and Ng (2006); assumes no cross-sectional correlation.
FPR (Fresoli, Poncela, Ruiz): Allows for cross-sectional correlation via a thresholding estimator.
Subsampling Correction: Based on Maldonado and Ruiz (2021), applied to both HR and FPR to account for the uncertainty in estimating factor loadings in finite samples.

3. Key Contributions

Validation of Asymptotic Approximation: The paper provides the first simulation-based evidence confirming that the Bai (2003) asymptotic distribution for PC factors is a valid approximation for the finite-sample distribution of SLS factors in ML-DFMs. This allows researchers to use established PC inference tools for hierarchical models.
MSE Estimator Performance: The authors demonstrate that the FPR estimator (which accounts for idiosyncratic cross-correlation) significantly outperforms the standard HR estimator when cross-sectional correlation exists. Crucially, FPR does not suffer efficiency losses when correlation is absent.
Importance of Loading Uncertainty: The study highlights that ignoring the uncertainty in factor loading estimation leads to biased MSE estimates. The subsampling correction is shown to be essential for accurate inference, particularly in small-to-moderate samples.
Bias Analysis: The paper identifies that biases in SLS estimates are most pronounced for group-specific factors when their loadings are weak relative to the idiosyncratic variance, and that these biases diminish only with large $N$ and $T$ .

4. Key Results

Distributional Approximation: Histograms of SLS factor estimates closely match the asymptotic normal density derived by Bai (2003), even in moderate sample sizes ( $N=100, T=100$ ). This holds true for both global and group-specific factors.
Impact of Cross-Sectional Correlation:
- When idiosyncratic errors are uncorrelated, HR and FPR estimators perform similarly.
- When errors are cross-sectionally correlated, the HR estimator becomes inconsistent, leading to severely underestimated variances and invalid confidence intervals. The FPR estimator remains robust.
Subsampling Necessity: In all scenarios, the subsampling correction significantly reduces the negative bias in MSE estimation caused by loading uncertainty. Without it, confidence regions are often too narrow.
Bias vs. Variance: In small samples with weak loadings (common for group-specific factors), the bias component of the MSE is non-negligible. Consequently, the asymptotic MSE matrix approximates the empirical covariance matrix better than the empirical MSE matrix in these specific small-sample cases.

5. Significance and Implications

Practical Inference: This paper bridges the gap between theoretical asymptotics and practical application for ML-DFMs. It empowers researchers to construct valid confidence intervals for global and group-specific factors, which is vital for scenario analysis, stress testing, and policy evaluation.
Robustness: By advocating for the FPR estimator with subsampling, the paper provides a robust toolkit for handling the complex dependency structures (both cross-sectional and hierarchical) found in real-world economic and financial data.
Theoretical Extension: It extends the applicability of Bai's (2003) seminal results to hierarchical structures, confirming that the sequential extraction of factors does not fundamentally alter the asymptotic distributional properties, provided the identification restrictions are satisfied.

In summary, the paper establishes that while SLS is a computationally efficient method for ML-DFMs, valid statistical inference requires using the Bai (2003) asymptotic distribution, the FPR estimator for MSE (to handle cross-correlation), and a subsampling correction (to handle loading uncertainty).