A computational bootstrap procedure to compare two dependent time series

On testing for the equality of autocovariance in time series

“…In the more recent work, Jin et al (2019) relax the independence assumption of the two univariate series and they build a statistic…”

“…We denote S * L as a version of S L from ( 7) that utilizes the Bartlett estimated covariance, A ′ ŴA. By replacing the estimated covariance matrix in theorem 2.1 of Jin et al (2019) with A ′ ŴA given by Bartlett's formula, Ŝ * L has the same limiting distribution as ŜL . The test rejects equality of autocovariances when S L, or S * L, exceeds the 𝛼th quantile of the distribution of sup l≥0 { ∑ l i=0 𝜒 2 1 − 2(l + 1)}.…”

“…As in Jin and Wang (2016), working with this distribution is nontrivial. The results of Jin and Wang (2016) and Jin et al (2019) are for two univariate time series. We argue that their results can be extended to the multivariate case with relative ease using our exposition from above.…”

“…The penalty term prevents larger lags from being selected simply due to their larger U * L value. For these statistics, like that in Jin et al (2019), we employ a blocks of blocks bootstrap to estimate the null distribution discussed below.…”

“…A data-driven test for both independent and dependent time series was proposed by Jin (2011) who found it to be equivalent to a goodness-of-fit test for a constant model. A test using a blocks of blocks bootstrap to compare the second-order characteristics of two dependent stationary time series was developed and shined with regards to short and nonnormal series in Jin et al (2019). They showed through simulation that their test is more powerful that those of (Decowski & Li, 2015;Li & Lu, 2018;.…”