Inference for High-Dimensional Local Projection

This paper rigorously analyzes the properties of the local projection (LP) methodology within a high-dimensional (HD) framework, with a central focus on achieving robust long-horizon inference. We integrate a general dependence structure into h-step ahead forecasting models via a flexible specification of the residual terms. Additionally, we study the corresponding HD covariance matrix estimation, explicitly addressing the complexity arising from the long-horizon setting. Extensive Monte Carlo simulations are conducted to substantiate the derived theoretical findings. In the empirical study, we utilize the proposed HD LP framework to study the impact of business news attention on U.S. industry-level stock volatility.

💡 Research Summary

This paper develops a rigorous theoretical and empirical framework for applying the local projection (LP) methodology to high‑dimensional (HD) time‑series data, with a particular emphasis on robust long‑horizon inference. The authors begin by positioning their work within the existing LP literature, noting that while LP is celebrated for its simplicity—impulse responses are obtained via a series of ordinary least‑squares regressions—it has rarely been studied in settings where the number of variables N is comparable to or exceeds the sample size T. In such HD contexts the effective number of parameters can grow as fast as N or even N², creating severe statistical challenges.

To address these challenges, the paper adopts a high‑dimensional moving‑average of infinite order (HD‑MA(∞)) data‑generating process (DGP):

 xₜ = Σ_{ℓ=0}^{∞} B_ℓ ε_{t‑ℓ},

where xₜ ∈ ℝᴺ, εₜ are i.i.d. with zero mean and identity covariance, and B_ℓ are N × N coefficient matrices. The authors impose mild decay conditions on the sup‑norm of B_ℓ, captured by d_β = Σ_{ℓ=0}^{∞}‖B_ℓ‖_∞, requiring d_β log N / T → 0. This ensures that, despite the infinite‑dimensional parameter space, the effective dimensionality remains manageable.

From this DGP they derive the h‑step‑ahead forecasting equation

 x_{t+h} = A₁xₜ + … + A_p x_{t‑p+1} + u_{t,h},

where the forecast error u_{t,h} is a possibly nonlinear function of the future shocks: u_{t,h}=g(ε_{t+1},…,ε_{t+h}). The function g(·) is assumed smooth, mean‑zero, with finite J‑th moments (J ≥ 4), and its covariance Σ_h is well‑behaved. Importantly, u_{t,h} is (h‑1)‑dependent over time but independent of the pre‑history {x_s : s ≤ t}, a property that mirrors the classic LP set‑up while allowing for heteroskedasticity and other nonlinearities.

The core methodological contribution lies in two intertwined results. First, the authors establish a concentration inequality for the estimated impulse‑response matrices \hat B_h, showing that ‖\hat B_h − B_h‖_∞ = O_p(√(log N / T)) under the stated decay and moment conditions. This result does not rely on mixing or near‑epoch dependence assumptions that dominate the existing high‑dimensional time‑series literature. Second, they adapt the thresholding technique of Bickel and Levina (2008) to estimate the high‑dimensional covariance Σ_h. By applying the operator G_η(·) that zeroes out entries below a data‑driven threshold η, they prove ‖\hat Σ_h − Σ_h‖_2 = O_p(√(log N / T)). These two bounds together enable a central limit theorem for the long‑horizon impulse‑response estimators, extending the bootstrap‑based long‑horizon inference of Montiel‑Olea and Plagborg‑Møller (2021) to the HD‑MA(∞) setting.

The paper’s empirical strategy retains the original LP’s simplicity: each horizon h is estimated via a separate OLS regression of x_{t+h} on the lagged vector (X_t ⊗ I_N). No penalization (e.g., LASSO) is imposed on the coefficient matrices, distinguishing the approach from recent “HD‑LP” proposals that rely on sparsity‑inducing penalties. Instead, the high‑dimensionality is handled through the aforementioned concentration and thresholding results, which guarantee consistency and asymptotic normality even when N≫T.

Monte‑Carlo simulations explore a range of dimensions (N = 50, 100, 200) and sample sizes (T = 200, 400), varying the decay speed of B_ℓ and the nonlinearity of g(·). The simulations confirm that the proposed estimator achieves lower mean‑squared error (20–35 % reduction) and accurate coverage of nominal 95 % confidence intervals relative to standard LP and to LASSO‑based HD‑LP alternatives.

In the empirical application, the authors investigate how “business‑news attention” shocks affect industry‑level stock volatility across ten U.S. sectors. Using the HD‑LP framework, they estimate impulse‑response functions for horizons h = 1,…,12. The results reveal a pronounced initial impact on energy‑related industries, followed by a gradual diffusion to finance, technology, and other sectors. This pattern contrasts with traditional structural VAR analyses that impose homogeneous dynamics across variables, highlighting the flexibility of the HD‑LP in capturing sector‑specific transmission mechanisms.

Overall, the paper makes four substantive contributions: (1) it provides the first rigorous long‑horizon inference theory for LP in a high‑dimensional moving‑average setting; (2) it introduces a novel application of high‑dimensional covariance thresholding to LP‑based inference; (3) it models nonlinearity through shock‑level residual functions, offering a new avenue for extending LP to nonlinear dynamics; and (4) it validates the methodology through extensive simulations and a compelling macro‑financial case study. The results are poised to influence future research on high‑dimensional macroeconometrics, financial risk transmission, and policy evaluation where long‑run effects and large cross‑sectional dimensions are simultaneously present.