Stochastic Unit Root Bubble Test (Kurozumi & Nishi 2025)

ssu_test implements the SSU statistic of Kurozumi & Nishi (2025): a sup-type test for a bubble based on testing for a stochastic (rather than deterministic) unit root in the squared first differences, (Delta y_t)^2 = mu2 + omega*y_{t-1}^2 + eta_t, bias-corrected against its dependence on the correlation between this regression's and the plain ADF regression's innovations.

Usage

ssu_test(data, minw = NULL, sig_lvl = 95)

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).
sig_lvl: Significance level on the package-wide 0-100 scale, one of 90, 95, 99 (the levels Kurozumi & Nishi's Table I tabulates).

Value

An object of class ssu_test_obj: a list with the statistic path (stat, one value per candidate end point from minw to n), the constant crit from Table I, and sadf (the maximum, compared against crit) and detected.

Details

A different generalization from the rest of exuber's volatility -robustness tests: it doesn't touch the innovation variance at all, but instead allows the explosive AR coefficient itself to vary stochastically over time, 1 + c1/T + a*u_t/sqrt(T), rather than the deterministic 1 + c/T^alpha every recursive-ADF-family statistic in this package assumes.

Only the single-recursion SSU statistic (sup over the end point, start fixed at the beginning of the sample) is implemented – not GSSU (the double-recursion generalization), the paper's separate CUSUM/CUSUM-SQ statistics, or the union-of-rejections procedure combining SSU/GSSU with SADF/GSADF.

Note

The critical value is a published closed-table constant (Kurozumi & Nishi (2025)'s Table I, via the internal ssu_q() helper) – no simulation needed.

Returns its own class (not radf_obj), so it does not plug into summary()/\link{datestamp}/tidy; it has its own print() and autoplot() methods instead. Prints its own statistic/critical-value summary – see vignette("naming-and-analysis", package = "exuber") for the full picture of which functions do and don't fit that pipeline.

Status

References

Kurozumi, E., & Nishi, M. (2025). Bubble testing with stochastically varying explosive coefficient. Journal of Time Series Analysis, 46(5), 945-965.

Examples

# \donttest{
# A stochastically varying explosive root, rho_t = 1 + 3/n + 4 * u_t / sqrt(n):
# the alternative ssu_test() is built for (a fixed-root DGP is lbi_test()'s)
y <- sim_psy1(n = 150, te = 75, tf = 150, c = 3, alpha = 1, seed = 2001,
  coef_noise = rnorm(149), coef_a = 4)
res <- ssu_test(y, sig_lvl = 95)
print(res)
#> 
#> ── ssu_test (n = 150, minw = 23, sig_lvl = 95%, crit = 3.3) ────────────────────
#> 
#>    series   sadf  detected
#>   series1  14.53      TRUE
#> 

# Plot the recursive SSU statistic path against its critical value
autoplot(res)

# }