Recursively Demeaned Sign-Based Bubble Test (s-bar-PWY / s-bar-PSY)

radf_sign_dm computes Harvey, Leybourne & Zu (2020)'s second sign-based analogue of the recursive right-tailed unit root test, denoted \(\bar{s}PWY\)/\(\bar{s}PSY\) in the paper: the same construction as radf_sign, but built on a recursively (expanding-window) demeaned cumulated-sign series, Ctilde_t = sum_{i=2}^{t} (sign(diff(y)_i) - mean(sign(diff(y)_{2:i}))), rather than the raw cumulated sign radf_sign uses.

Usage

radf_sign_dm(data, minw = 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).

Details

Harvey, Leybourne, Tatlow & Zu (2025) show this statistic shares radf_sign's asymptotic level-shift robustness (see that function's Level-shift robustness section) without requiring Assumption 2 of the underlying HLZ (2020) theory (that the innovations' median is zero) – a strictly weaker requirement than radf_sign needs for its own invariance result. Their finite -sample simulations also find the recursive demeaning tends to further reduce size distortion under level shifts relative to radf_sign, though both are asymptotically level-shift robust under the same condition.

Note

Needs radf_sign_dm_cv for critical values (not radf_sign_cv, which is calibrated to the non-demeaned radf_sign statistic instead) – pivotal like radf_sign, so no per-dataset bootstrap is needed.

Carries the radf_obj class and, as of 2026-08-18, its full summary()/datestamp/tidy/autoplot pipeline works, the same fix as radf_sign – radf_sign_dm_cv() now computes badf_cv/bsadf_cv too, not just the three scalar critical values. See vignette("naming-and-analysis", package = "exuber").

Status

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

Examples

# \donttest{
res <- radf_sign_dm(sim_data, minw = 20)
print(res)
#> 
#> ── radf_sign_dm (minw = 20) ────────────────────────────────────────────────────
#> 
#>   series       adf     sadf  gsadf
#>     psy1  -0.07152   2.1729  2.976
#>     psy2   1.47812   3.0005  3.433
#>    evans  -2.14492  -0.9121  1.149
#>      div  -1.05486   2.3305  2.369
#>     blan  -0.17252   1.3508  1.489
#> 

cv <- radf_sign_dm_cv(n = 100, minw = 20)
summary(res, cv = cv)
#> 
#> ── Summary (minw = 20, lag = 0) ───── Sign-Based MC (demeaned) (nboot = 2000) ──
#> 
#> psy1 :
#> # A tibble: 3 × 5
#>   stat    tstat  `90`  `95`  `99`
#>   <fct>   <dbl> <dbl> <dbl> <dbl>
#> 1 adf   -0.0715 0.871  1.27  2.07
#> 2 sadf   2.17   2.36   2.74  3.58
#> 3 gsadf  2.98   2.85   3.20  4.17
#> 
#> psy2 :
#> # A tibble: 3 × 5
#>   stat  tstat  `90`  `95`  `99`
#>   <fct> <dbl> <dbl> <dbl> <dbl>
#> 1 adf    1.48 0.871  1.27  2.07
#> 2 sadf   3.00 2.36   2.74  3.58
#> 3 gsadf  3.43 2.85   3.20  4.17
#> 
#> evans :
#> # A tibble: 3 × 5
#>   stat   tstat  `90`  `95`  `99`
#>   <fct>  <dbl> <dbl> <dbl> <dbl>
#> 1 adf   -2.14  0.871  1.27  2.07
#> 2 sadf  -0.912 2.36   2.74  3.58
#> 3 gsadf  1.15  2.85   3.20  4.17
#> 
#> div :
#> # A tibble: 3 × 5
#>   stat  tstat  `90`  `95`  `99`
#>   <fct> <dbl> <dbl> <dbl> <dbl>
#> 1 adf   -1.05 0.871  1.27  2.07
#> 2 sadf   2.33 2.36   2.74  3.58
#> 3 gsadf  2.37 2.85   3.20  4.17
#> 
#> blan :
#> # A tibble: 3 × 5
#>   stat   tstat  `90`  `95`  `99`
#>   <fct>  <dbl> <dbl> <dbl> <dbl>
#> 1 adf   -0.173 0.871  1.27  2.07
#> 2 sadf   1.35  2.36   2.74  3.58
#> 3 gsadf  1.49  2.85   3.20  4.17
#> 
tidy(res, cv = cv)
#> # A tibble: 5 × 4
#>   id        adf   sadf gsadf
#>   <fct>   <dbl>  <dbl> <dbl>
#> 1 psy1  -0.0715  2.17   2.98
#> 2 psy2   1.48    3.00   3.43
#> 3 evans -2.14   -0.912  1.15
#> 4 div   -1.05    2.33   2.37
#> 5 blan  -0.173   1.35   1.49
datestamp(res, cv = cv)
#> 
#> ── Datestamp (min_duration = 0) ──────────────────── Sign-Based MC (demeaned) ──
#> 
#> psy2 :
#>   Start Peak End Duration   Signal Ongoing
#> 1    39   40  41        2 positive   FALSE
#> 2    84   84  85        1 positive   FALSE
#> 3    86   95 100       14 negative   FALSE
#> 
autoplot(res, cv = cv)

# }