Locally Best Invariant Test for a Bubble (Breitung & Diegel 2025)

lbi_test implements the static locally best invariant (LBI) test of Breitung & Diegel (2025) for a bubble that is known, or assumed, to span the entire sample: LBI = (y_T - y_1) / (sigma_tilde * sqrt(T - 1)), where sigma_tilde^2 is the sample variance of the first differences. The test is robust to heteroskedasticity by construction, because the invariance property of the statistic does not depend on the exact form of the innovation variance. The null distribution is standard normal, so the test needs no bootstrap, no simulation and no published table.

Usage

lbi_test(data, 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 or trailing NA values, which describes an unbalanced panel in which series enter or exit the sample at different times. Those periods are filled with NA in badf and bsadf and excluded from the adf, sadf and gsadf of that series. Interior NA values (a gap in the middle of a series) are not supported. When any series is padded in this way, the panel statistics (bsadf_panel and gsadf_panel) are not available, and the function returns NA for them with a warning.
sig_lvl: Significance level of the one-sided, right-tailed test (positive bubbles only), on the 0 to 100 scale used throughout the package (default 95). Any value in [50, 100) is accepted, because the critical value is a closed-form normal quantile.

Value

An object of class lbi_test_obj: a list with the test statistic stat, the standard-normal critical value crit and detected (logical, stat > crit).

Details

Only the static test, which uses a single full-sample window, is implemented. The main contribution of Breitung & Diegel is a sequential, exponentially weighted extension for monitoring a series when the start date is unknown. Its exact weighting scheme and boundary constant are not pinned down here, so this function does not implement it.

Note

The critical value is closed-form: it is the standard normal (qnorm) quantile at sig_lvl, so no bootstrap, simulation or table is needed.

The function returns its own class and not radf_obj, so it does not work with summary(), \link{datestamp} and tidy. It has its own print() and autoplot() methods instead. print() shows the statistic, the critical value and whether the bubble was detected. See vignette("naming-and-analysis", package = "exuber") for which functions fit the shared pipeline and which do not.

Status

References

Breitung, J., & Diegel, M. (2025). A locally best invariant sequential test for explosive behavior in the presence of nonstationary volatility. Journal of Time Series Analysis.

Examples

# \donttest{
y <- sim_psy1(n = 60, te = 1, tf = 60, seed = 1) # explosive from the start
res <- lbi_test(y)
print(res)
#> 
#> ── lbi_test (n = 60, sig_lvl = 95%) ────────────────────────────────────────────
#> 
#>    series   stat   crit  detected
#>   series1  4.892  1.645      TRUE
#> 

# Compare the statistic to its critical value
autoplot(res)

# }