Real-Time Monitoring • exuber

library(exuber)

Monitoring vs. testing

radf()/datestamp() and the dating_*() family (see vignette("dating-methods")) all work on a finished sample: they answer “was there a bubble, and when” after every observation is already in hand. The monitor_*() family answers a different, real-time question: fix a training window [1, T*] believed free of exuberance, calibrate a boundary on it, then watch each new observation T*+1, T*+2, ... and raise an alarm the first time the boundary is breached. All four functions share this r_star/alarm/alarm_date shape; they differ in what statistic they monitor and how the boundary is calibrated.

Function	Statistic monitored	Boundary	Static, full-sample counterpart
`monitor()`	`radf()`’s own `badf`/`bsadf` recursion	`"bootstrap"` (Phillips & Shi 2020 wild-bootstrap quantile), `"kurozumi"` (closed-form, Kurozumi 2020), or `"fluc"` (closed-form, Homm & Breitung 2012)	`radf()`/`datestamp()` – not a `_test()`, but the same recursive-ADF core
`monitor_cusum()`	A CUSUM of the training-window-standardized series	Homm & Breitung (2012)’s asymptotic (or finite-sample) constant	none – Homm & Breitung’s CUSUM boundary is inherently a training/monitoring construction, with no full-sample form
`monitor_lbi()`	Breitung & Diegel (2025)’s locally-best-invariant CUSUM (`mCUSUM`/`wCUSUM`, via `c_bar`)	Their Table 1 constant	`lbi_test()`, the static version of the same statistic
`monitor_quantile()`	A recursive quantile regression at `tau`	A simulated first-crossing boundary (Wu, Shi & Wu 2025)	`quantile_test()`, the static version of the same statistic

monitor() reuses badf/bsadf directly (the same recursive-ADF core as the radf_*() family) but is named for what it does, not that internal detail – see vignette("naming-and-analysis") for why. Two of its three siblings (monitor_lbi(), monitor_quantile()) are the sequential extension of an existing static test of the same name minus the monitor_ prefix; monitor_cusum() has no such counterpart, since its source paper (Homm & Breitung 2012) proposed CUSUM as a monitoring detector only.

The same bubble, five monitors

A training window of pure random walk (T* = 100), followed by more random walk, then a genuine explosive regime (rho = 1.04) starting at t = 150:

make_bubble_series <- function(n, T_star, bstart, rho = 1.04) {
  y <- numeric(n)
  y[seq_len(T_star)] <- cumsum(rnorm(T_star))
  for (t in (T_star + 1):n) {
    y[t] <- if (t < bstart) y[t - 1] + rnorm(1) else rho * y[t - 1] + rnorm(1)
  }
  y
}
set.seed(7)
y <- make_bubble_series(200, T_star = 100, bstart = 150)

monitor_lbi(y, r_star = 100)
#> 
#> ── monitor_lbi (T* = 100 / 200, c_bar = 0, b_alpha = 1.95) ─────────────────────
#> 
#>    series  alarm  alarm_date
#>   series1    159         159
monitor_cusum(y, r_star = 0.5)
#> 
#> ── monitor_cusum (T* = 100 / 200, b_alpha = 4.6) ───────────────────────────────
#> 
#>    series  alarm  alarm_date
#>   series1    164         164
monitor_quantile(y, tau = 0.5, nrep = 200, seed = 1)
#> 
#> ── monitor_quantile (n = 200, minw = 27, tau = 0.5, level = 95%) ───────────────
#> 
#>    series  delta  boundary  alarm  alarm_date
#>   series1  0.695     1.757    163         163
monitor(y, r_star = 0.5, nboot = 200, seed = 1)
#> 
#> ── monitor (T* = 100 / 200, minw = 27, level = 95%, boundary = bootstrap) ──────
#> 
#>    series  boundary  alarm  alarm_date
#>   series1     2.025    160         160
monitor(y, r_star = 0.5, boundary = "kurozumi")
#> 
#> ── monitor (T* = 100 / 200, minw = 27, level = 95%, boundary = kurozumi) ───────
#> 
#>    series  boundary  alarm  alarm_date
#>   series1     1.038    161         161

monitor_lbi() and monitor_quantile() each have a static, full-sample sibling that asks the retrospective version of the same question – run on the whole series rather than watching for a first crossing:

lbi_test(y)
#> 
#> ── lbi_test (n = 200, level = 95%) ─────────────────────────────────────────────
#> 
#>    series   stat   crit  detected
#>   series1  6.313  1.645      TRUE
quantile_test(y, tau = 0.5)
#> 
#> ── quantile_test (n = 200, level = 95%) ────────────────────────────────────────
#> 
#>    series  tau  tstat    crit  delta  detected
#>   series1  0.5  13.96  0.4221  0.695      TRUE

Every monitor here alarms within about 15 points of the true bubble start (150), none before it – that “never before T* (or the true start)” property is exactly what each function’s own test suite checks under the null. Alarm timing differs by design: monitor()’s ADF-family statistics tend to detect mid-sample bubbles fastest (the literature’s own finding, e.g. Kurozumi 2020/2021), while CUSUM-type detectors (monitor_cusum(), monitor_lbi()) are typically slower but computationally simpler and don’t need a bootstrap.

Which to reach for

Fastest detection, willing to pay for a wild bootstrap per call: monitor(boundary = "bootstrap") (the default).
Same statistic, no bootstrap, an off-the-shelf published constant instead: monitor(boundary = "kurozumi") or boundary = "fluc".
A simpler CUSUM-based alternative with its own closed-form boundary: monitor_cusum(), or monitor_lbi() for Breitung & Diegel’s locally-best-invariant version (c_bar > 0 trades a little size for power against slow-building bubbles).
Monitoring a specific quantile of the distribution rather than the mean behavior: monitor_quantile().