A Distributed Lag Approach to the Generalised Dynamic Factor Model

Referee: [§2] §2 (theoretical insight on finite-lag representation): the assertion that the dynamic common component χ_it admits χ_it = ∑_{k=0}^K λ_{i,k}' f_{t-k} with finite K and contemporaneously pervasive f_t is stated to hold 'under reasonable conditions,' yet no explicit decay rate on the tail ∑_{k>K} ||λ_{i,k}|| or spectral-density condition is supplied to guarantee that the truncation error is o_p(1) uniformly. This assumption is load-bearing for the reduction to static PCA + OLS and for the subsequent consistency claims.

Authors: The finite-lag representation follows directly from the absolute summability of the dynamic loadings that defines the GDFM (as in Forni et al., 2000, 2005). Under the standard condition ∑_k ||λ_{i,k}|| < ∞ together with uniform boundedness of the loadings, the tail can be made smaller than any ε > 0 by a sufficiently large but fixed K, uniformly in i. We will revise §2 to state this summability condition explicitly and note its implication for uniform o_p(1) truncation error. revision: partial

Referee: [§3] §3 (consistency and asymptotic normality): the proofs of consistency for the dynamic common component and asymptotic normality for the weak common component rest on the finite-K representation; if near-unit-root dynamics (common in macro series) force K to grow with T or violate contemporaneous pervasiveness once lags are folded in, the rates derived under fixed K would not apply. The manuscript does not provide a uniform bound or a data-driven rule for K that preserves the o_p(1) property.

Authors: The theory is derived under the maintained stationarity assumption of the GDFM, where summability guarantees that a fixed K delivers the required approximation uniformly. Near-unit-root behavior is typically addressed by separate treatment of stochastic trends outside the standard GDFM. We will add a paragraph in §3 discussing the choice of K and noting that information criteria applied to the OLS step can be used in practice to verify that the truncation error remains negligible, while the fixed-K rates continue to hold under the summability condition. revision: partial

Referee: [Section 5] Empirical application (Section 5): the claim that a 'sizeable weak common component' is uncovered, especially in sentiment indicators, depends on the chosen K and on the precise definition of weak versus strong factors after the lag augmentation; without reported robustness checks on K or explicit data-exclusion rules, it is unclear whether the finding is robust to the truncation that underpins the estimator.

Authors: The selected K is motivated by the decay pattern of the estimated dynamic loadings and the improvement in OLS fit. We will augment Section 5 with a robustness table showing results for K = 2, 4, and 6; the weak common component in sentiment indicators remains sizeable and statistically distinguishable from zero across these values. We will also clarify the post-augmentation definition of weak factors. revision: yes