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

monitor_lbi(data, r_star = 0.5, c_bar = 0, sig_lvl = 95)

monitor_lbi implements the sequential, constant-boundary extension of the locally best invariant statistic of lbi_test. It monitors a series in real time when the start date of the bubble is unknown. After a training window [1, T*] that is assumed free of exuberance, it compares the (optionally exponentially weighted) partial sum of the post-training first differences with a constant boundary and flags the first monitoring date at which the boundary is breached.

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.
r_star	The end of the training window: a fraction in (0, 1) of the sample (default 0.5), or an integer number of observations if >= 1.
c_bar	Exponential up-weighting parameter for later monitoring observations, which are more likely to be bubble-like (their eq. 12), >= 0. The default 0 is the flat-weight "mCUSUM" variant, which is appropriate when a bubble is equally likely to start at any point in the monitoring window. For a moderate gain in power when a bubble that starts partway through the window is more plausible, the paper suggests 2. The critical values (sig_lvl) are the same for every c_bar.
sig_lvl	Significance level on the 0 to 100 scale used throughout the package, one of 90, 95, 97.5, 99 or 99.5. Table 1 of Breitung & Diegel tabulates only these.

Value

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

Details

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

# 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()`).