Time-Transformed Test for Explosive Bubbles under Non-stationary Volatility

radf_tt computes the STADF/GSTADF test statistics of Kurozumi, Skrobotov & Tsarev, a heteroskedasticity-robust alternative to radf that requires no bootstrap: the series is time-deformed using a nonparametric estimate of its variance profile, after which the usual (asymptotic, homoskedastic) recursive sup-ADF critical values apply.

Usage

radf_tt(data, minw = NULL, kernel = c("uniform", "gaussian"), h = NULL)

Arguments

data: A univariate or multivariate numeric time series object, a numeric vector or matrix, or a data.frame. A column may have leading and/or trailing NA values (an uneven/unbalanced panel where series enter or exit the sample at different times) – those periods are filled with NA in badf/bsadf and excluded from that series' adf/sadf/ gsadf. Interior NA values (a gap in the middle of a series) are not supported. When any series is padded this way, the panel statistic (bsadf_panel/gsadf_panel) is not available and is returned as NA, with a warning.
minw: A positive integer. The minimum window size (default = \((0.01 + 1.8/\sqrt{T})T\), where T denotes the sample size).
kernel: Kernel used in the local variance-profile regression, "uniform" (default, as in the paper's simulations) or "gaussian".
h: Bandwidth for the variance-profile kernel regression. Default T^(-2/5), the midpoint (on the log scale) of the paper's cross-validation search range \([T^{-0.5}, T^{-0.3}]\).

Value

An object of class radf_tt_obj/radf_obj: the same adf/badf/sadf/bsadf/gsadf list as radf, computed on the time-transformed series, so it plugs into summary()/datestamp()/tidy()/autoplot() paired with radf_tt_cv.

Details

For critical values, use radf_tt_cv as the primary recommendation: it is pivotal (asymptotically free of the volatility process), so it does not need to be recomputed per dataset, unlike a bootstrap. radf_wb_cv (Harvey, Leybourne, Sollis & Taylor's wild bootstrap) is a bootstrap-based alternative, worth considering if non-pivotality/finite-sample bootstrap robustness is a specific concern.

Note

Carries the radf_obj class and, as of 2026-08-18, its full summary()/datestamp/tidy/autoplot pipeline works – radf_tt_cv() now computes the time-varying badf_cv/bsadf_cv boundary those last two need, not just the three scalar critical values summary() uses. See vignette("naming-and-analysis", package = "exuber").

Status

References

Kurozumi, E., Skrobotov, A., & Tsarev, A. (2024). Time-Transformed Test for Bubbles under Non-stationary Volatility. Journal of Financial Econometrics. doi:10.1093/jjfinec/nbae026

Examples

# \donttest{
# Volatility triples half-way through the sample: the non-stationary-volatility
# case this test is built for (plain radf() over-rejects here)
y <- sim_psy1(n = 200, seed = 1, e = sim_vol_break(199))
res <- radf_tt(y, minw = 20)
print(res)
#> 
#> ── radf_tt (minw = 20, kernel = uniform) ───────────────────────────────────────
#> 
#>    series      adf   sadf  gsadf
#>   series1  -0.9972  3.275  4.229
#> 

cv <- radf_tt_cv(n = 200, minw = 20)
summary(res, cv = cv)
#> 
#> ── Summary (minw = 20, lag = 0) ────────── Time-Transformed MC (nboot = 2000) ──
#> 
#> series1 :
#> # A tibble: 3 × 5
#>   stat   tstat  `90`  `95`  `99`
#>   <fct>  <dbl> <dbl> <dbl> <dbl>
#> 1 adf   -0.997 0.833  1.21  1.97
#> 2 sadf   3.27  2.31   2.65  3.35
#> 3 gsadf  4.23  3.27   3.65  4.42
#> 
tidy(res, cv = cv)
#> # A tibble: 1 × 4
#>   id         adf  sadf gsadf
#>   <fct>    <dbl> <dbl> <dbl>
#> 1 series1 -0.997  3.27  4.23
datestamp(res, cv = cv)
#> 
#> ── Datestamp (min_duration = 0) ───────────────────────── Time-Transformed MC ──
#> 
#> series1 :
#>   Start Peak End Duration   Signal Ongoing
#> 1    21   38  89       68 negative   FALSE
#> 2   148  148 149        1 positive   FALSE
#> 
autoplot(res, cv = cv)

# }