Alternative Tests: lbi_test(), ssu_test(), quantile_test()

library(exuber)

Why not just use radf()

The GSADF statistic in radf() tests against one specific alternative, an explosive AR(1) root that stays fixed. lbi_test(), ssu_test() and quantile_test() are standalone hypothesis tests. They do not use the recursive core of radf() and they do not feed into the summary(), datestamp(), tidy() and autoplot() pipeline (see vignette("naming-and-analysis")). Each targets a different alternative, one where GSADF-style tests can lose power.

Function Paper Alternative it targets
lbi_test() Breitung & Diegel (2025) A fixed explosive root, tested with the locally best invariant statistic for that alternative, so it can have more power than GSADF in exactly that case.
ssu_test() Kurozumi & Nishi (2025) A stochastically varying explosive coefficient: the root has a random component and is not a fixed value.
quantile_test() Wu, Shi & Wu (2025) Explosiveness in the tau-th conditional quantile of y_t given y_{t-1}, instead of in the conditional mean.

A fixed root, which lbi_test() detects

y <- sim_psy1(n = 60, te = 1, tf = 60, c = 0.03, alpha = 0, seed = 1) # fixed rho = 1.03 throughout
lbi_test(y)
#> 
#> ── lbi_test (n = 60, sig_lvl = 95%) ────────────────────────────────────────────
#> 
#>    series  stat   crit  detected
#>   series1  6.12  1.645      TRUE

A varying root, which ssu_test() detects and lbi_test() misses

ssu_test() is designed for a root that varies stochastically over time. The coef_noise and coef_a arguments of sim_psy1() generate this alternative, with rho_t = 1 + c/n + coef_a * u_t / sqrt(n), so the root is random and not fixed:

y <- sim_psy1(n = 150, te = 75, tf = 150, c = 3, alpha = 1, seed = 2001,
              coef_noise = rnorm(149), coef_a = 4)
ssu_test(y, sig_lvl = 95)
#> 
#> ── ssu_test (SSU, n = 150, minw = 23, sig_lvl = 95%, crit = 3.3) ───────────────
#> 
#>    series   sadf  detected
#>   series1  15.02      TRUE
lbi_test(y)
#> 
#> ── lbi_test (n = 150, sig_lvl = 95%) ───────────────────────────────────────────
#> 
#>    series    stat   crit  detected
#>   series1  0.3121  1.645     FALSE

On this draw ssu_test() detects the bubble and lbi_test(), which is built for a fixed root, does not. This is not a defect of lbi_test(). Each test is the (locally) most powerful one against its own alternative, and neither dominates the other everywhere, which is why both exist.

Testing a quantile instead of the mean

quantile_test() picks a quantile tau (or takes one from you) and tests for explosiveness there instead of in the conditional mean. This pays off with heavy-tailed innovations, where the conditional-mean regression is least reliable, so the example below drives the PSY bubble with t(3) shocks:

y_t3 <- sim_psy1(n = 100, seed = 1, e = sim_innov(99, dist = "t", df = 3))
quantile_test(y_t3, nrep = 100, seed = 1)
#> 
#> ── quantile_test (n = 100, sig_lvl = 95%) ──────────────────────────────────────
#> 
#>    series   tau  tstat    crit  delta  detected
#>   series1  0.25  4.684  0.6824  0.379      TRUE

By default (tau = "optimal") the function searches tau_grid and reports the quantile with the strongest signal. The example above fixes the quantile at a specific value so that it runs faster and can be reproduced.

Which to reach for

  • If you believe the explosive root is fixed and want more power than GSADF in that case, use lbi_test().
  • If you suspect the explosive root is noisy or varies over time, use ssu_test().
  • If you suspect explosiveness shows up in the tails of the distribution, or at a specific quantile, more than in the mean, use quantile_test().
  • If you are unsure which alternative applies, or want the most widely used benchmark, start with the GSADF test in radf().