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

Source: vignettes/alternative-tests.Rmd
alternative-tests.Rmd

Why not just use radf()

radf()’s GSADF statistic tests one specific alternative: a fixed explosive AR(1) root. lbi_test(), ssu_test() and quantile_test() are standalone hypothesis tests – not built on radf()’s recursive core, and not fit into the summary()/datestamp()/tidy()/autoplot() pipeline (see vignette("naming-and-analysis")) – each targeting a different alternative 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 specific alternative (so it can out-power GSADF exactly there).
ssu_test() Kurozumi & Nishi (2025) A stochastically varying explosive coefficient – the root itself has a random component, not a fixed value.
quantile_test() Wu, Shi & Wu (2025) Explosiveness in the tau-th conditional quantile of y_t on y_{t-1}, not the conditional mean.

Same alternative, lbi_test() detects 

set.seed(1)
n <- 60
y <- 100 * 1.03^(1:n) + cumsum(rnorm(n, sd = 1)) # fixed rho = 1.03
lbi_test(y)
#> 
#> ── lbi_test (n = 60, level = 95%) ──────────────────────────────────────────────
#> 
#>    series   stat   crit  detected
#>   series1  6.885  1.645      TRUE

A different alternative: where lbi_test() misses and ssu_test() doesn’t

ssu_test() is designed for a root that itself varies stochastically over time, not a fixed one:

make_stochastic_bubble <- function(n, te_frac = 0.5, c1 = 3, a = 4) {
  y <- numeric(n)
  y[1] <- rnorm(1)
  Te <- round(te_frac * n)
  for (t in 2:n) {
    if (t <= Te) {
      y[t] <- y[t - 1] + rnorm(1)
    } else {
      rho_t <- 1 + c1 / n + a * rnorm(1) / sqrt(n) # random root, not fixed
      y[t] <- rho_t * y[t - 1] + rnorm(1)
    }
  }
  y
}
set.seed(2001)
y <- make_stochastic_bubble(150)
ssu_test(y, level = 0.95)
#> 
#> ── ssu_test (n = 150, minw = 23, level = 95%, crit = 3.3) ──────────────────────
#> 
#>    series  sadf  detected
#>   series1  12.2      TRUE
lbi_test(y)
#> 
#> ── lbi_test (n = 150, level = 95%) ─────────────────────────────────────────────
#> 
#>    series   stat   crit  detected
#>   series1  0.147  1.645     FALSE

ssu_test() detects it; lbi_test(), built for a fixed root, does not on this same draw. This isn’t a defect in lbi_test() – it’s the whole reason two tests exist: each is the (locally) most powerful test for its own alternative, and neither dominates the other everywhere.

Testing the quantile, not the mean

quantile_test() picks (or is given) a quantile tau and tests for explosiveness there instead of in the conditional mean:

quantile_test(sim_data$psy2, nrep = 100, seed = 1)
#> 
#> ── quantile_test (n = 100, level = 95%) ────────────────────────────────────────
#> 
#>    series   tau  tstat    crit  delta  detected
#>   series1  0.35  5.364  0.7143  0.361      TRUE

tau = "optimal" (the default) searches tau_grid and reports the quantile with the strongest signal, shown here fixed at a specific value for a faster, reproducible example.

Which to reach for

  • A genuinely fixed explosive root, and want a test with power advantages over standard GSADF for exactly that case: lbi_test().
  • Suspect the explosive root itself is noisy/time-varying rather than constant: ssu_test().
  • Suspect explosiveness shows up more in the tails (or a specific quantile) of the distribution than in the mean: quantile_test().
  • Not sure which alternative applies, or want the most widely used benchmark: radf()’s GSADF remains the default first test to run.