Simultaneous inference and trend specification testing in ARMA model with trend via innovation distribution function - Zhong, Chen (Statistical papers (Berlin, Germany), 2025)
The innovation distribution function in the ARMA model with trend is estimated by the kernel distribution estimator (KDE). The KDE is shown to converge weakly to a Gaussian process with certain covariance structure, yielding a simultaneous confidence band (SCB) for the innovation distribution function. Additionally, a goodness-of-fit test statistics for testing whether the trend function belongs to some parametric form is proposed through measuring the difference between the nonparametric and parametric residuals via the KDE of innovation distribution function. The asymptotic distribution of the test statistics under the null is developed and the asymptotic power against local alternatives is examined as well. Bootstrap methods are employed to implement the trend specification test and work well in the numerical studies. The proposed theory is illustrated by two data sets including spot price returns and global air temperature data.