Local Threshold Tests Ignore Cross Dependence Asymptotically

Detecting a jump in one regression curve is already a boundary problem: the object of interest is the discontinuity at an unknown or specified threshold, not a smooth derivative. This paper asks for the same inference in a harder setting, where there may be hundreds of heterogeneous nonparametric time series, the number of series $N$ may exceed the time length $T$, and the panels may share strong latent factors. The authors target jumps in conditional mean functions of the form $Y_{jt}=h_j(X_{jt},U_{jt})+\tau_j(X_{jt})1\{X_{jt}\geq c_{0j}\}+e_{jt}$, with additional latent variation allowed through $U_{jt}$.

Core Contribution

The contribution is a pair of uniform tests: one for whether any series has a jump, and one for whether jump effects are homogeneous across series. The paper’s main statistical point is sharper than a standard high-dimensional Gaussian approximation. Because each jump estimate is built from local left-versus-right smoothing around the threshold, the leading cross-sectional covariance of the studentized statistics becomes asymptotically negligible under the stated density and smoothness conditions. That lets the critical values come from the maximum of independent Gaussian variables after estimating only per-series variances, even when the underlying data-generating process has strong cross-sectional dependence.

The paper also adds a finite-sample correction for cases where higher-order cross-sectional terms still matter. When the localization does not wash out enough dependence in samples of realistic size, the authors reinsert an estimated correlation matrix into the Gaussian critical-value calculation rather than relying on the diagonal approximation.

Technical Approach

For each series and candidate threshold, the method estimates the left and right limits of the conditional mean using local linear weights and forms a studentized jump estimate. The existence statistic $\hat I$ is the maximum absolute standardized jump across series and threshold locations. The homogeneity statistic $\hat Q$ compares each estimated jump to the cross-sectional average jump, again using a max statistic so that sparse alternatives are not averaged away.

Figure 1 is a useful visual check on why the max-based construction matters: when only a fraction of series has a nonzero jump, the uniform test gains power where pooled threshold regressions lose signal.

Results and Analysis

The simulations cover known and unknown thresholds, serial dependence, cross-sectional factor dependence, homogeneous and heterogeneous jumps, and both level and derivative discontinuities. The pattern is consistent: size is usually close to nominal, and power rises with $T$, $N$, and the fraction of nonzero coefficients. In an unknown-threshold design with 20% nonzero coefficients, Table A4 reports power 0.967 at the 5% level for the uniform test with $N=100,T=800$, versus 0.327 for a pooled parametric threshold test. Table A5 reports a similar comparison against a pooled nonparametric test: 0.971 versus 0.124 under the same $N,T$ and 20% signal setting. That is the empirical case for using a maximum statistic rather than pooling across heterogeneous series.

The dependence-aware refinement is also supported by the simulations. In the higher-order dependence design DGP8, Table A7 shows that the independent-Gaussian version has low size at $N=100,T=800$ and $\alpha=0.05$ (0.031), while the correlation-adjusted version is closer to target size (0.059) and has higher power (0.342 versus 0.241). The Benjamini-Hochberg comparison is also conservative in that setting, with size 0.037 and power 0.253.

The empirical applications make the method concrete. For S&P 500 constituents from January 2010 to May 2024, the paper tests for jumps in the news impact curve, using lagged stock returns as the threshold variable and Garman-Klass volatility as the response. AT&T shows a significant threshold effect at the 1% level, while Akamai does not, illustrating that the procedure can separate individual nonlinearities rather than forcing a common pooled effect.

The election application studies U.S. House incumbency effects across states. The appendix table reports an existence statistic of 5.117, above the 1% critical value 3.392, and a homogeneity statistic of 3.096, above the 5% critical value 2.917 but below the 1% cutoff. California is shown with a significant threshold effect, while Pennsylvania is not.

Caveats

The evidence is a mix of formal asymptotics, simulations, and two observational applications. The theory depends on bandwidth, density, smoothness, and moment assumptions that may be hard to verify in applied panels. The empirical examples show detected discontinuities, but they do not establish causal mechanisms for volatility or incumbency effects.