radf_sign computes the sign-based variant of the recursive right-tailed
unit root test of Harvey, Leybourne & Zu (2020). Instead of applying the
(double-)supremum ADF test to the series itself, it applies it to the cumulated
sign of the first differences, C_t = sum(sign(diff(y))). The sign()
function removes all information about magnitudes, so the recursive DF statistic
of C_t is exactly invariant to the pattern of volatility in the
innovations, even when volatility changes over time. Unlike radf,
the test needs no wild bootstrap to control its size under heteroskedasticity.
The critical values in radf_sign_cv are pivotal, so they are
computed once and not for each dataset.
Arguments
- data
A univariate or multivariate numeric time series object, a numeric vector or matrix, or a data.frame. A column may have leading or trailing
NAvalues, which describes an unbalanced panel in which series enter or exit the sample at different times. Those periods are filled withNAinbadfandbsadfand excluded from theadf,sadfandgsadfof that series. InteriorNAvalues (a gap in the middle of a series) are not supported. When any series is padded in this way, the panel statistics (bsadf_panelandgsadf_panel) are not available, and the function returnsNAfor them 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).
Value
An object of class radf_sign_obj/radf_obj. It is the same
adf/badf/sadf/bsadf/gsadf list as for
radf, computed on the sign-transformed series, and it pairs with
radf_sign_cv.
Details
The price of this invariance is power. The paper finds that the sign-based test
outperforms the standard PSY test for many specifications of time-varying
volatility and bubbles, but not for all of them, and the standard test can still
win in some. The strategy that the paper recommends in practice is a
bootstrap-based union of rejections that combines both tests. We have not
implemented it (see the package's enhancement notes for the cost and benefit
considerations), and this function provides the standalone sign-based test only.
sadf is the single-supremum sPWY statistic (r1 = 0 fixed), and
gsadf is the double-supremum sPSY statistic.
Note
The test needs the critical values from radf_sign_cv, and
neither radf_wb_cv nor any other bootstrap applies. The statistic is
pivotal (exactly invariant to heteroskedasticity), so its critical values are
simulated once and not for each dataset.
The result carries the radf_obj class. Since 2026-08-18 the full
summary(), datestamp, tidy and autoplot
pipeline works, because radf_sign_cv() now computes the time-varying
badf_cv and bsadf_cv boundary that the last two need, and not only
the three scalar critical values that summary() uses. See
vignette("naming-and-analysis", package = "exuber").
Level-shift robustness
Harvey, Leybourne, Tatlow & Zu (2025) show that this test keeps its standard
null distribution, the one without level shifts, in the presence of
deterministic level shifts, provided that the number of shifts grows strictly
more slowly than sqrt(T). The size of the shifts does not matter. This
is a materially weaker requirement than the one the standard PSY test needs for
size control, which restricts the number and the magnitude of the shifts
jointly. In their simulations the standard test is never correctly sized once
the number of shifts grows at rate sqrt(T), while this test stays close to
its nominal size.
![[Experimental]](figures/lifecycle-experimental.svg)
References
Harvey, D. I., Leybourne, S. J., & Zu, Y. (2020). Sign-based unit root tests for explosive financial bubbles in the presence of deterministically time-varying volatility. Econometric Theory, 36(1), 122-169.
Harvey, D. I., Leybourne, S. J., Tatlow, D., & Zu, Y. (2025). Unit root tests for explosive financial bubbles in the presence of deterministic level shifts. Oxford Bulletin of Economics and Statistics, 87(5), 879-901. doi:10.1111/obes.12668
See also
radf_sign_cv for critical values,
radf_sign_dm for the recursively demeaned sign-based analogue,
which has the same level-shift robustness, and radf for the
standard test, which is not invariant.
Other volatility-robust tests:
cusum_test(),
radf_kp(),
radf_sbz(),
radf_sbz_union(),
radf_sign_dm(),
radf_tt(),
ssu_test()
Examples
# \donttest{
# Volatility triples half-way through the sample. This is the case of
# non-stationary volatility that this test is built for, and plain radf()
# over-rejects here
y <- sim_psy1(n = 200, seed = 1, e = sim_vol_break(199))
res <- radf_sign(y, minw = 20)
print(res)
#>
#> ── radf_sign (minw = 20) ───────────────────────────────────────────────────────
#>
#> series adf sadf gsadf
#> series1 -0.2933 4.468 8.879
#>
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
#>
tidy(res, cv = cv)
#> # A tibble: 1 × 4
#> id adf sadf gsadf
#> <fct> <dbl> <dbl> <dbl>
#> 1 series1 -0.293 4.47 8.88
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
#>
autoplot(res, cv = cv)
# }