Volatility-Robust Alternatives to radf()

library(exuber)

The shared problem

Plain radf() assumes a constant innovation variance. Real series rarely have one, and when volatility varies over time the standard critical values no longer control the size of the test. exuber offers several fixes, and each takes a structurally different approach. Four of the five covered below (radf_sign(), radf_sign_dm(), radf_kp() and radf_sbz()) keep the radf_obj class and support summary(), datestamp(), tidy() and autoplot() in the same way as plain radf(). vignette("naming-and-analysis") shows how each plugs into the pipeline and how we validated it. radf_sbz_union() is the exception. It bundles the statistic of radf_sbz() and the classic supDF into one union-of-rejections call and has its own class, not radf_obj, so only its own print() and autoplot() apply. The time-deformation approach of radf_tt() has its own vignette, vignette("radf-tt"), and this one covers the rest.

All of these functions target non-stationary volatility, meaning a permanent shift or trend in the unconditional innovation variance. They do not target stationary GARCH-type conditional heteroskedasticity, whose variance profile is asymptotically flat and leaves plain radf() with the correct size. The running example is therefore the sim_psy1() bubble driven by sim_vol_break() innovations, whose standard deviation triples half-way through the sample. This is the case in which plain radf() over-rejects the most:

y <- sim_psy1(n = 200, seed = 1, e = sim_vol_break(199))

Function	Paper	Approach
`radf_sign()` / `radf_sign_dm()`	Harvey, Leybourne & Zu (2020)	Transforms the series to the cumulated sign of its first differences, which is exactly invariant to any heteroskedasticity pattern and needs no bootstrap. The `_dm` version demeans first, which makes it robust to level shifts (Harvey, Leybourne, Tatlow & Zu 2025).
`radf_kp()`	Harvey, Leybourne, Taylor & Zu (2024)	Purges volatility. It divides each first difference by a kernel spot-volatility estimate, cumulates the result and runs the ordinary PSY test on the purged series, whose null distribution is identical to the standard homoskedastic one.
`radf_sbz()` / `radf_sbz_cv()`	Harvey, Leybourne & Zu (2019)	A WLS-weighted recursive Dickey-Fuller statistic (`supBZ`). It weights the observations and does not purge or transform the series. It uses the same kernel spot-volatility estimator as `radf_kp()`, but as regression weights.
`radf_sbz_union()`	Harvey, Leybourne & Zu (2019)	Combines `supDF` (the classic statistic) and `supBZ` (the statistic of `radf_sbz()`) as a union of rejections, using a wild bootstrap sized jointly for both. It catches whichever of the two has power on a given series.

Sign-based: `radf_sign()`

radf_sign() has full pipeline support. radf_sign_cv() computes the time-varying badf_cv and bsadf_cv boundary and not only the scalar critical values that summary() needs (see vignette("naming-and-analysis") for the validation):

res <- radf_sign(y, minw = 20)
cv <- radf_sign_cv(n = 200, minw = 20)
summary(res, cv = cv)
#> 
#> ── Summary (minw = 20, lag = 0) ──────────────── Sign-Based MC (nboot = 2000) ──
#> 
#> series1 :
#> # A tibble: 3 × 5
#>   stat   tstat  `90`  `95`  `99`
#>   <fct>  <dbl> <dbl> <dbl> <dbl>
#> 1 adf   -0.293 0.855  1.32  2.06
#> 2 sadf   4.47  2.34   2.70  3.43
#> 3 gsadf  8.88  3.51   3.91  4.92
datestamp(res, cv = cv)
#> 
#> ── Datestamp (min_duration = 0) ─────────────────────────────── Sign-Based MC ──
#> 
#> series1 :
#>   Start Peak End Duration   Signal Ongoing
#> 1    84   84  85        1 positive   FALSE
#> 2    87  103 122       35 positive   FALSE
#> 3   123  123 124        1 positive   FALSE

Kernel-purged: `radf_kp()`

radf_kp() purges volatility and then calls radf() unmodified, so it has full pipeline support in the simplest possible way. It needs no new critical-value code, and radf_mc_cv() applies as it is:

res_kp <- radf_kp(y, minw = 20)
cv_kp <- radf_mc_cv(n = attr(res_kp, "n"), minw = 20)
summary(res_kp, cv = cv_kp)
#> 
#> ── Summary (minw = 20, lag = 0) ────────────────── Monte Carlo (nboot = 1000) ──
#> 
#> series1 :
#> # A tibble: 3 × 5
#>   stat  tstat   `90`   `95`  `99`
#>   <fct> <dbl>  <dbl>  <dbl> <dbl>
#> 1 adf   -1.72 -0.470 -0.140 0.535
#> 2 sadf   1.50  1.07   1.37  1.90 
#> 3 gsadf  2.63  1.98   2.25  2.80

WLS + kernel volatility: `radf_sbz()`

radf_sbz() is a statistic function of its own, separate from the union test. radf_sbz_cv() computes the time-varying badf_cv and bsadf_cv boundary in the same way as radf_tt_cv() and radf_sign_cv() (see vignette("naming-and-analysis") for the validation), so it has full pipeline support:

res_sbz <- radf_sbz(y, minw = 20)
cv_sbz <- radf_sbz_cv(y, minw = 20, nboot = 200, seed = 1)
summary(res_sbz, cv = cv_sbz)
#> 
#> ── Summary (minw = 20, lag = 0) ────────── Wild Bootstrap (SBZ) (nboot = 200) ──
#> 
#> series1 :
#> # A tibble: 3 × 5
#>   stat  tstat  `90`  `95`  `99`
#>   <fct> <dbl> <dbl> <dbl> <dbl>
#> 1 adf   -1.39 0.825  1.13  1.61
#> 2 sadf   1.67 1.58   2.11  3.36
#> 3 gsadf  1.91 3.37   5.06  6.21

The kernel-volatility weighting that makes supBZ robust to heteroskedasticity costs some power relative to the other tests. The default bubble of sim_psy1() is mild (30 periods at rho = 1 + 200^-0.6, then a collapse). It does not clear the 95% critical value of supBZ here, although every other test above rejects on the same series. A stronger bubble that does not collapse (rho = 1.03 from t = 120 to the end of the sample, on the same volatility break) does clear it:

y_strong <- sim_psy1(n = 200, te = 120, tf = 200, c = 0.03, alpha = 0, seed = 1,
                     e = sim_vol_break(199))
res_sbz2 <- radf_sbz(y_strong, minw = 20)
cv_sbz2 <- radf_sbz_cv(y_strong, minw = 20, nboot = 200, seed = 1)
summary(res_sbz2, cv = cv_sbz2)
#> 
#> ── Summary (minw = 20, lag = 0) ────────── Wild Bootstrap (SBZ) (nboot = 200) ──
#> 
#> series1 :
#> # A tibble: 3 × 5
#>   stat  tstat  `90`  `95`  `99`
#>   <fct> <dbl> <dbl> <dbl> <dbl>
#> 1 adf    4.83 0.948  1.65  2.65
#> 2 sadf   4.83 2.24   2.49  3.26
#> 3 gsadf  5.29 2.77   3.00  3.58
datestamp(res_sbz2, cv = cv_sbz2)
#> 
#> ── Datestamp (min_duration = 0) ──────────────────────── Wild Bootstrap (SBZ) ──
#> 
#> series1 :
#>   Start Peak End Duration   Signal Ongoing
#> 1   129  129 130        1 positive   FALSE
#> 2   132  132 133        1 positive   FALSE
#> 3   134  134 135        1 positive   FALSE
#> 4   172  200 200       29 positive    TRUE

This is the same trade-off that radf_sbz_union(), below, hedges against by combining supBZ with the classic supDF.

Union-of-rejections: `radf_sbz_union()`

radf_sbz_union(y, nboot = 200, seed = 1)
#> 
#> ── radf_sbz_union (minw = 27, nboot = 200) ─────────────────────────────────────
#> 
#>    series  supDF  supBZ      U  p_supDF  p_supBZ    p_U
#>   series1  9.329   1.67  9.329        0    0.085  0.005

supDF is the classic PWY statistic, and supBZ is the WLS-weighted version that radf_sbz() also returns on its own. U is their union. Each statistic has its own bootstrap p-value, so one can flag a series without the other. Here supDF rejects, supBZ does not, and the union U follows supDF. The value of U is defined with a bootstrap-derived scaling ratio between supDF and supBZ, and its size guarantee requires that the supDF and supBZ bootstrap draws come from the same resampled series in each replicate. For both reasons radf_sbz_union() cannot be reconstructed by calling radf_sbz_cv() and plain radf_wb_cv() separately. It stays a single bundled call with its own class and is not a radf_obj.

Which to reach for

If you want exact invariance to any heteroskedasticity pattern with no bootstrap, use radf_sign(). Use radf_sign_dm() if a level shift, and not only volatility, is a concern. Both have full pipeline support.
If you want to stay closest to plain radf() with no new critical-value code, use radf_kp().
If you want the efficiency gain from WLS together with full datestamp() and autoplot() support, use radf_sbz() with radf_sbz_cv().
If you want to hedge between the classic and the WLS-weighted statistics on the same series and do not need datestamp() or autoplot(), use radf_sbz_union(). It is the only function in this group without pipeline support, because U bundles both statistics and their scalar joint critical value in one call and does not return a radf_obj.
If volatility is the main concern and you prefer a time-deformation approach without a bootstrap, use radf_tt() (see vignette("radf-tt")).
If the volatility is unknown or complex and a bootstrap is acceptable, radf_wb_cv() remains the general-purpose choice.

The shared problem

Sign-based: radf_sign()

Kernel-purged: radf_kp()

WLS + kernel volatility: radf_sbz()

Union-of-rejections: radf_sbz_union()

Which to reach for

Sign-based: `radf_sign()`

Kernel-purged: `radf_kp()`

WLS + kernel volatility: `radf_sbz()`

Union-of-rejections: `radf_sbz_union()`