Volatility-Robust Alternatives to radf()
Source:vignettes/volatility-robust-radf.Rmd
volatility-robust-radf.RmdThe shared problem
Plain radf() assumes constant innovation variance. Real
series rarely have that, and under time-varying volatility its standard
critical values no longer control size. exuber has several fixes for
this, each taking a structurally different approach. Four of the five
below (radf_sign(), radf_sign_dm(),
radf_kp(), radf_sbz()) keep the
radf_obj class and now have full
summary()/datestamp()/tidy()/autoplot()
support, the same as plain radf() (see
vignette("naming-and-analysis") for exactly how each one
plugs into the pipeline and how that was validated);
radf_sbz_union() doesn’t – it bundles
radf_sbz()’s statistic and the classic supDF
into one union-of-rejections call with its own class, not a
radf_obj, so none of those four generics apply to it, only
its own print()/autoplot().
radf_tt()’s time-deformation approach has its own dedicated
vignette, vignette("radf-tt"); this one covers the
rest.
All of them target non-stationary volatility – a permanent
shift or trend in the unconditional innovation variance, not stationary
GARCH-type conditional heteroskedasticity (whose variance profile is
asymptotically flat, leaving plain radf() size-correct). So
the running example is sim_psy1()’s bubble driven by
sim_vol_break() innovations, whose standard deviation
triples half-way through the sample – the case where plain
radf() over-rejects 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) | Transform to the cumulated sign of first differences –
exactly invariant to any heteroskedasticity pattern, no bootstrap
needed. _dm demeans first for level-shift robustness
(Harvey, Leybourne, Tatlow & Zu 2025). |
radf_kp() |
Harvey, Leybourne, Taylor & Zu (2024) | Purge volatility: divide each first difference by a kernel spot-volatility estimate, cumulate, then run ordinary PSY on the purged series – null distribution is identical to the standard homoskedastic one. |
radf_sbz() / radf_sbz_cv()
|
Harvey, Leybourne & Zu (2019) | A WLS-weighted recursive-DF statistic (supBZ),
weighting rather than purging or transforming: same kernel
spot-volatility estimator as radf_kp(), used as regression
weights instead. |
radf_sbz_union() |
Harvey, Leybourne & Zu (2019) |
supDF (classic) and supBZ
(radf_sbz()’s), unioned via a jointly-sized wild bootstrap
– catches whichever of the two has power on a given series. |
Sign-based: radf_sign()
As of 2026-08-18, this also gets full pipeline support –
radf_sign_cv() computes the time-varying
badf_cv/bsadf_cv boundary now, not just the
scalar critical values 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 FALSEKernel-purged: radf_kp()
Because radf_kp() purges volatility and then calls
radf() unmodified, it gets full pipeline support too, the
simplest way of all (no new critical-value machinery at all –
radf_mc_cv() applies unmodified):
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 2.80WLS + kernel volatility: radf_sbz()
As of 2026-08-22, radf_sbz() is split from the union
test into its own statistic function, with radf_sbz_cv()
computing the time-varying badf_cv/bsadf_cv
boundary the same way
radf_tt_cv()/radf_sign_cv() do (see
vignette("naming-and-analysis") for the validation), so it
gets 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 6.21The kernel-volatility weighting that makes supBZ
heteroskedasticity-robust also trades away some power relative to the
other tests: sim_psy1()’s default, mild bubble (30 periods
at rho = 1 + 200^-0.6, then a collapse) doesn’t clear
supBZ’s 95% critical value here, even though every other
test above rejects on the same series. A stronger, uncollapsed episode –
rho = 1.03 from t = 120 to the sample end, on
the same volatility break – does:
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 TRUEThat’s the same trade-off radf_sbz_union() below exists
to hedge against by unioning 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 0.005supDF is the classic PWY statistic, supBZ
the WLS-weighted version (the same one radf_sbz() now
returns on its own), and U their union – each with its own
bootstrap p-value, so a series can be flagged by one without the other,
as here: supDF rejects, supBZ doesn’t, and the
union U follows supDF. U’s value
is defined using a bootstrap-derived
supDF/supBZ scaling ratio, and its size
guarantee requires supDF/supBZ bootstrap draws
paired from the same resampled series per replicate – both reasons
radf_sbz_union() can’t be reconstructed from separately
calling radf_sbz_cv() and plain radf_wb_cv(),
and why it stays a single bundled call with its own class rather than a
radf_obj.
Which to reach for
- Want exact invariance to any heteroskedasticity pattern
with no bootstrap at all:
radf_sign()(orradf_sign_dm()if a level shift, not just volatility, is a concern). Also full pipeline support. - Want to stay closest to plain
radf(), no new critical-value machinery at all:radf_kp(). - Want the WLS efficiency gain with full
datestamp()/autoplot()support:radf_sbz()+radf_sbz_cv(). - Want to hedge between the classic and WLS-weighted statistics on the
same series, and don’t need
datestamp()/autoplot():radf_sbz_union()– the one function in this group with no pipeline support, sinceUbundles both statistics and their (scalar-only) joint critical value in one call rather than returning aradf_obj. - Volatility is the whole story and a bootstrap-free, time-deformation
approach is preferred:
radf_tt(), seevignette("radf-tt"). - Volatility is genuinely unknown/complex and a bootstrap is
acceptable: plain
radf_wb_cv()remains the general-purpose choice.