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Time-Transformed Test (STADF/GSTADF)

Source: vignettes/radf-tt.Rmd
radf-tt.Rmd

Why another test

radf() (the classic PSY GSADF test) assumes the innovation variance is constant. Real financial series usually don’t have constant volatility, and Harvey, Leybourne, Sollis & Taylor (2016) show that when volatility is time-varying, radf()’s standard critical values no longer control size – radf_wb_cv() already addresses this in exuber via a wild bootstrap.

radf_tt() implements a different, bootstrap-free fix from Kurozumi, Skrobotov & Tsarev (2024, Journal of Financial Econometrics): instead of resampling, it time-deforms the series using a nonparametric estimate of its variance profile, so that the deformed series behaves like a constant-volatility random walk under the null. The resulting statistic’s null distribution is then the same (pivotal) distribution as under homoskedasticity, so ordinary asymptotic critical values apply – no bootstrap, and no per-dataset resimulation needed.

Basic usage 

set.seed(1)
y <- sim_psy1(100)
res <- radf_tt(y)
res
#> 
#> ── radf_tt (minw = 19, kernel = uniform) ───────────────────────────────────────
#> 
#>    series      adf   sadf  gsadf
#>   series1  -0.7186  2.662  3.083

radf_tt_cv() gives the matching (pivotal) asymptotic critical values; because the null distribution doesn’t depend on the volatility path, one call with a large n approximates the whole family, unlike radf_wb_cv()’s per-dataset bootstrap:

cv <- radf_tt_cv(n = 300, minw = 30, nrep = 1000, seed = 1)
cv$gsadf_cv
#>      90%      95%      99% 
#> 3.248911 3.584415 4.246916

What’s actually estimated

Under the hood, radf_tt():

  1. estimates the time-varying AR(1) coefficient with a local kernel regression, and from its (truncated) residuals builds a monotone variance profile eta_hat(s), s in [0, 1];
  2. inverts it and uses the inverse to resample/time-deform the series;
  3. runs a (GLS-demeaned, no-intercept) recursive sup-ADF statistic on the deformed series – the same statistic family as radf(), but built to need no fitted intercept, matching the paper’s derivation.

kernel ("uniform", the paper’s own choice, or "gaussian") and h (bandwidth; default a fixed plug-in, not the paper’s full cross-validation search – see the package’s enhancement notes for the cost/benefit reasoning) can both be adjusted.

Verifying against the paper

Kurozumi, Skrobotov & Tsarev’s footnote 4 gives an exact published asymptotic critical value triple for minw/n = 0.1: (2.319, 2.626, 3.223) at the (10%, 5%, 1%) levels – for the STADF statistic (the single-sup, r1 = 0 case). exuber’s test suite (tests/testthat/test-tt.R) reproduces this via radf_tt_cv()’s own Monte Carlo and checks it lands within Monte Carlo/finite-sample tolerance of the published numbers.