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 known (or assumed) to span the entire sample: LBI = (y_T - y_1) / (sigma_tilde * sqrt(T - 1)), with sigma_tilde^2 the sample variance of first differences. Heteroskedasticity-robust by construction (the statistic's invariance property does not depend on the exact form of the innovation variance), with a standard normal null distribution – no bootstrap, no simulation, no published table.

Usage

lbi_test(data, level = 0.95)

# S3 method for class 'lbi_test_obj'
autoplot(object, ...)

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.
level: Nominal confidence level for the (one-sided, right-tailed – positive bubbles only) test (default 0.95).
object: An object of class lbi_test_obj, the output of lbi_test.

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 (single, full-sample window) test is implemented. Breitung & Diegel's own headline contribution is a sequential/ exponentially-weighted extension for monitoring an unknown start date, whose exact weighting scheme and boundary constant are not pinned down here and are not implemented.

Note

The critical value is closed-form: the standard normal (qnorm) quantile at level – no bootstrap, no simulation, no table needed.

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

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, level = 95%) ──────────────────────────────────────────────
#> 
#>    series   stat   crit  detected
#>   series1  4.892  1.645      TRUE
#> 

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

# }