Affine-Equivariant Adjusted-Range Self-Normalization
Published:
This working paper develops an affine-equivariant adjusted-range self-normalization method for multivariate time-series inference. The method constructs a multivariate adjusted-range self-normalizer whose normalizing set reproduces the scalar adjusted range in every linear projection, so an unrestricted nonsingular long-run covariance factor cancels from the limiting statistic.
Authors:
Yongmiao Hong, Zhuo Lin, Oliver Linton, Whitney K. Newey, and Jiajing Sun.
Summary and Contribution:
The paper provides pivotal joint confidence regions without long-run covariance estimation, bandwidth selection, or diagonalization. The normalizing set is the convex hull of increments of a centered path of estimated influence contributions, and the statistic is evaluated through a linear program rather than by explicitly constructing the hull.
The construction is invariant to nonsingular linear reparameterizations, including changes of units, reordering of coefficients, and changes of basis. It therefore extends adjusted-range self-normalization beyond earlier componentwise procedures that required diagonalization or coordinate-by-coordinate scalarization.
Theory, Evidence, and Application:
The paper also treats a common variance or information clock. Once the influence path is centered by the correct clock, the adjusted range requires no clock-dependent weighting, while a quadratic self-normalizer must also integrate in clock time.
Simulation exercises study size, power, nuisance estimation, and changes of basis. An empirical local-projection IV application provides joint inference for U.S. fiscal multipliers.
Keywords:
Self-normalized inference; long-run covariance; affine equivariance; influence functions; local projections.
Recommended citation: Hong, Y., Lin, Z., Linton, O., Newey, W. K., & Sun, J. (2026). Affine-Equivariant Adjusted-Range Self-Normalization. Working paper.