Simultaneous Inference Bands for Autocorrelations

Abstract

Sample autocorrelograms typically come with significance (non-rejection) bands for the null hypothesis of no temporal correlation. These bands have three shortcomings. First, they build on pointwise intervals and suffer from joint undercoverage (overrejection) under the null hypothesis. Second, in cases where this null is clearly violated one would rather prefer to see confidence bands to quantify estimation uncertainty. Third, they are invalid under conditional heteroskedasticity. We propose both simultaneous significance and confidence bands for time series and series of regression residuals. Our simultaneous inference bands are as easy to construct as their pointwise counterparts and at the same time provide an intuitive and visual quantification of sampling uncertainty as well as valid statistical inference. For residuals from dynamic regressions, we show how our inference bands from the case of observed time series and static regressions need to be adjusted due to a correction term in the asymptotic variances. For all our bands, we also provide robust versions, allowing for conditional heteroskedasticity. We analyse the finite-sample performance of our inference bands in a simulation study and illustrate their use in applications to monthly US inflation, Fama-French excess returns and residuals from Phillips curve regressions.