Sequential LBI Monitoring for an Unknown Bubble Start Date (Breitung & Diegel 2025)

monitor_lbi implements the sequential (constant-boundary) extension of lbi_test's locally best invariant statistic, for monitoring a series in real time when the bubble's start date is unknown: after a training window [1, T*] assumed free of exuberance, the (optionally exponentially weighted) partial sum of post-training first differences is compared against a constant boundary, flagging the first monitoring date it is breached.

Usage

monitor_lbi(data, r_star = 0.5, c_bar = 0, level = 0.95)

# S3 method for class 'monitor_lbi_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.
r_star: The end of the training window: a fraction in (0, 1) of the sample (default 0.5), or an integer observation count if >= 1.
c_bar: Exponential up-weighting parameter for later (more bubble-like) monitoring observations (their eq. 12), >= 0. 0 (default) is the flat-weight "mCUSUM" variant, appropriate when a bubble is equally likely to start at any point in the monitoring window; the paper's own suggested value for a moderate power boost when a bubble partway through is more plausible is 2. Critical values (level) are the same for every c_bar.
level: Nominal confidence level, one of 0.90, 0.95, 0.975, 0.99, 0.995 (Breitung & Diegel's Table 1 only tabulates these).
object: An object of class monitor_lbi_obj, the output of monitor_lbi.

Value

An object of class monitor_lbi_obj: a list with the monitoring-region statistic path (stat), the constant boundary, the training window length T_star, and alarm/alarm_date (the first breach, NA if none).

Details

Their eq. 15 shows this partial sum, normalized by the fixed monitoring horizon length (not sqrt(t), unlike monitor_cusum's Chu-Stinchcombe-White-style boundary), converges to a standard Brownian motion on [0, 1] under the null – so a single constant boundary controls size uniformly across the whole monitoring window. The paper shows this constant-boundary detector ("mCUSUM" at c_bar = 0, "wCUSUM" at c_bar > 0) is more powerful than the classical time-varying-boundary CUSUM test it is compared against.

Note

The critical value is a published constant boundary (Breitung & Diegel (2025)'s Table 1) – a table lookup, no simulation.

Returns its own class (not radf_obj), so it does not plug into summary()/\link{datestamp}/tidy/autoplot – prints its own boundary/alarm 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{
# A martingale training window, explosive from t = 150 to the sample end
y <- sim_psy1(n = 200, te = 150, tf = 200, seed = 7)
res <- monitor_lbi(y, r_star = 100)
print(res) # alarm should fire soon after t = 150
#> 
#> ── monitor_lbi (T* = 100 / 200, c_bar = 0, b_alpha = 1.95) ─────────────────────
#> 
#>    series  alarm  alarm_date
#>   series1    155         155
#> 
autoplot(res)
#> Warning: Removed 1 row containing missing values or values outside the scale range
#> (`geom_segment()`).


# wCUSUM: exponentially up-weight later monitoring observations
autoplot(monitor_lbi(y, r_star = 100, c_bar = 2))
#> Warning: Removed 1 row containing missing values or values outside the scale range
#> (`geom_segment()`).

# }