Simultaneous Self-Normalized Inference for Time-Varying Linear IV Models
Published:
This working paper, by Jiajing Sun, Haiqi Li, Yongmiao Hong, and Jin Zhou, develops simultaneous inference for smooth coefficient paths in locally stationary linear instrumental-variable models estimated by local generalized method of moments (GMM). The inferential targets are confidence bands over a prespecified finite grid of dates and joint tests of whether an entire coefficient path is constant or linear.
Pointwise confidence intervals do not control these path-level statements, while repeatedly estimating local covariance matrices introduces date-specific nuisance parameters and must account for dependence across overlapping windows. The paper addresses these challenges through self-normalization and multiplier calibration.
Summary and Contribution:
The proposed procedure begins with the feasible influence path of a one-step identity-weight local-quadratic GMM estimator. A kernel-weighted quartic projection removes the leading estimation component from the fitted residual score, while cumulative squared equivalent-kernel loadings describe how variance accumulates within each local window.
The primary statistic uses the adjusted range of the projected, tied-down influence path as its self-normalizer. A matched quadratic root-mean-square self-normalizer provides a benchmark. Common calendar-time multipliers calibrate the maximum statistic across dates and coefficient coordinates, preserving the dependence created by overlapping local windows and jointly observed outcomes.
The theory separates the unconditional orthogonality condition identifying the IV model from the dependence assumptions required for inference. It establishes validity for iid Gaussian multipliers under martingale-difference instrumented scores and for dependent multipliers under serially dependent scores. The framework delivers simultaneous finite-grid confidence bands together with tests of constant and linear coefficient paths.
Evidence:
Monte Carlo experiments show that simultaneous coverage is close to its nominal level in the martingale-score designs. Adjusted-range bands are narrower than matched quadratic bands in every reported design, although relative coverage and power vary across settings.
The empirical application studies time-varying factor exposures for ten Fama-French industry portfolios. Both iid and dependent-multiplier specifications reject system-wide constancy of the factor-exposure paths, while conclusions about whether the paths are linear depend on the dependence calibration.
Availability:
The August 2026 working-paper version is available here:
Recommended citation: Sun, J., Li, H., Hong, Y., & Zhou, J. (2026). Simultaneous Self-Normalized Inference for Time-Varying Linear IV Models. Working paper.