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arxiv: 2606.26142 · v1 · pith:CQHMMUOOnew · submitted 2026-06-20 · 💰 econ.EM

Determining the Structure of Dynamic Factor Models

Sangmyung Ha This is my paper

Pith reviewed 2026-06-26 10:48 UTC · model grok-4.3

classification 💰 econ.EM
keywords dynamic factor modelsnumber of factorsalternating least squaresmacroeconomic time seriesconsistencyinformation criteriaeigenvalue ratiosfilter length
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The pith

Two procedures determine the number of dynamic factors consistently under weaker conditions than earlier methods.

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

This paper develops two procedures to select the number of dynamic factors that drive large panels of time series. The procedures extend prior criteria to settings where lagged factors can directly affect current observations. An alternating least squares algorithm estimates the dynamic factors without first mapping them to static representations. This step supports choosing the factor count and the filter length together. The methods are shown to remain consistent with milder restrictions on the data process than required before, and they are applied to count primitive shocks in US macroeconomic series.

Core claim

The paper establishes that its two procedures for selecting the number of dynamic factors are consistent even when lagged factors directly influence the observed variables, under conditions weaker than those required by Bai and Ng (2007) and Amengual and Watson (2007). The alternating least squares algorithm developed as an intermediate step estimates the dynamic factors directly, which in turn supports the joint determination of the factor count and the filter length.

What carries the argument

The alternating least squares algorithm that estimates dynamic factors directly rather than through static representations, enabling joint selection of the factor count and the filter length.

If this is right

  • The number of primitive shocks in large macroeconomic panels can be estimated reliably.
  • Factor count and lag length can be chosen jointly instead of in separate stages.
  • The procedures apply to models where lagged factors affect observables directly.
  • Consistency holds under milder assumptions on the data generating process than in prior work.

Where Pith is reading between the lines

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

  • The direct estimation step may reduce errors that accumulate when converting between dynamic and static factor representations.
  • The joint selection feature could be tested in forecasting exercises to check whether it improves out-of-sample accuracy.
  • Similar alternating least squares steps might be adapted to other high-dimensional time series settings with lagged latent variables.

Load-bearing premise

The data are generated by a dynamic factor model in which lagged factors can directly influence the observed variables, and the required rank or moment conditions hold.

What would settle it

A Monte Carlo simulation in which the lagged factor influence is removed or a known incorrect factor count is imposed would show whether the procedures recover the wrong number of factors.

Figures

Figures reproduced from arXiv: 2606.26142 by Sangmyung Ha.

Figure 1
Figure 1. Figure 1: Estimated q using PC2, DC2, and IC2. The DC2 criterion tends to select a larger number of factors, whereas the IC2 cri￾terion typically yields a more parsimonious specification relative to the PC2 criterion. Based on the PC2 results, we find evidence in favor of four dynamic factors prior to 22 [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Estimated m Using PC2, DC2, and IC2. The PC2 criterion consistently selects m = 2 across the sample. The DC2 criterion likewise predominantly indicates m = 2 in the pre-COVID-19 period, but points to an increase to m = 3 in the post-COVID-19 period. In contrast, the IC2 criterion generally favors a more parsimonious specification, selecting shorter filter lengths [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Explained variation of the common component. [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Estimated impulse responses of log output, log CPI, the federal funds rate, [PITH_FULL_IMAGE:figures/full_fig_p059_4.png] view at source ↗
read the original abstract

We propose two procedures for determining the number of dynamic factors, extending Bai and Ng (2002) and Ahn and Horenstein (2013) to dynamic factor models where lagged factors may directly influence the observed variables. As an intermediate step, we develop a simple and computationally efficient alternating least squares algorithm that directly estimates the dynamic factors, rather than their static representations. By working with these direct estimates, our approach enables joint determination of the number of factors and the filter length. Our test is shown to be consistent under weaker conditions than those in Bai and Ng (2007) and Amengual and Watson (2007). We apply our procedures to estimate the number of primitive shocks in a large panel of US macroeconomic time series.

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 two procedures for determining the number of dynamic factors in DFMs where lagged factors may directly affect observables, extending the information-criterion approach of Bai and Ng (2002) and the eigenvalue-ratio method of Ahn and Horenstein (2013). It develops an alternating least squares algorithm for direct estimation of dynamic factors (rather than static representations), enabling joint selection of the number of factors and filter length. The procedures are claimed to be consistent under weaker conditions than Bai and Ng (2007) and Amengual and Watson (2007), with an application to a large US macroeconomic panel to estimate the number of primitive shocks.

Significance. If the consistency results hold, the contribution would be significant for applied macroeconometrics by accommodating more flexible DFM structures with direct lagged effects and providing a computationally efficient ALS procedure for joint determination of factors and lag length. The explicit development of the ALS algorithm for direct dynamic factor estimation is a clear strength that could facilitate reproducible implementations.

minor comments (2)
  1. [Abstract] The abstract refers to 'our test' while the body describes two procedures; standardize terminology for clarity.
  2. [Section 5] The application section would benefit from reporting the selected number of factors and filter length alongside the number of primitive shocks for direct comparison with prior studies.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper and for recommending minor revision. We appreciate the recognition of the ALS algorithm and the extension of existing methods to more flexible DFM structures.

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external priors with independent extensions

full rationale

The paper extends Bai and Ng (2002) and Ahn and Horenstein (2013) information-criterion and eigenvalue-ratio methods to dynamic factor models permitting direct lagged-factor effects, introduces a new alternating least squares algorithm for direct (non-static) factor estimation, and establishes consistency under weaker rank/moment conditions than Bai and Ng (2007) or Amengual and Watson (2007). No step reduces by construction to a self-definition, fitted input renamed as prediction, or load-bearing self-citation; all central claims rest on external literature plus novel algorithmic and proof content that does not presuppose the target results. The derivation chain is therefore self-contained against the cited benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the work relies on standard domain assumptions from the cited econometric literature on factor models.

pith-pipeline@v0.9.1-grok · 5636 in / 1078 out tokens · 38543 ms · 2026-06-26T10:48:16.313393+00:00 · methodology

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Reference graph

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