CUSUM Real-Time Monitoring for Explosive Bubbles

monitor_cusum implements Homm & Breitung (2012)'s CUSUM real-time monitoring procedure: fix a training window [1, T*] assumed free of exuberance, then compare the standardized cumulative sum of post-training first differences, S_t = (y_t - y_{T*}) / sigma_hat_t, against a closed-form boundary c_t * sqrt(t), c_t = sqrt(b_alpha + log(t / T*)), flagging the first date it is breached.

Usage

monitor_cusum(
  data,
  r_star = 0.5,
  b_alpha = 4.6,
  boundary = c("asymptotic", "finite"),
  level = 0.95,
  type = c("standard", "kernel"),
  N = 20,
  kernel = c("gaussian", "uniform")
)

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.
b_alpha: The boundary constant (HB's eq. 29). Default 4.6, HB's own one-sided asymptotic calibration for a 5\ (their Section 3); this is an asymptotic upper bound on the false- alarm probability (Chu, Stinchcombe & White 1996), not an exact size, so it is typically conservative in finite samples. Ignored when boundary = "finite".
boundary: "asymptotic" (default) uses b_alpha directly. "finite" instead looks up HB's own finite-sample boundary constant (their Table 8) from level and the realized training length/monitoring-horizon ratio – level must then be one of 0.90, 0.95, 0.99.
level: Nominal confidence level when boundary = "finite" (default 0.95); ignored when boundary = "asymptotic".
type: "standard" (default) for Homm & Breitung (2012)'s original CUSUM statistic, or "kernel" for Astill, Harvey, Leybourne, Taylor & Zu (2023)'s volatility-robust "CUSUMV" variant.
N: Bandwidth/window length for the one-sided kernel spot-variance estimator when type = "kernel". Default 20, the authors' own empirically-recommended value (their Section 3: "setting H = 20 delivered a procedure with the best trade-off" between false-alarm robustness and power). Ignored when type = "standard".
kernel: Kernel for the spot-variance estimator when type = "kernel", "gaussian" (default) or "uniform". Ignored when type = "standard".

Value

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

Note

The boundary is closed-form throughout: a fixed asymptotic constant (boundary = "asymptotic", b_alpha = 4.6) or a published finite-sample table lookup (boundary = "finite", Homm & Breitung (2012)'s Table 8) – no simulation, no separate cv function.

Unlike radf_monitor (Family A, a recursive ADF-family statistic requiring a wild bootstrap to calibrate its boundary), this is a structurally different statistic – a standardized running sum, not a recursive regression – with an asymptotic closed-form boundary (Chu, Stinchcombe & White 1996's inequality, HB's eq. 28): no bootstrap, no simulation, no dependence on the data beyond the running variance estimate itself.

type = "kernel" instead uses Astill, Harvey, Leybourne, Taylor & Zu (2023)'s volatility-robust modification ("CUSUMV"): each first difference is standardized by its own one-sided kernel spot-variance estimate (their eq. 6-7) instead of a single running variance, before cumulating. Their Corollary 1 establishes the same boundary function delivers a controlled asymptotic false-alarm rate even under time-varying volatility, unlike the standard CUSUM statistic, which requires homoskedasticity for its own size-control result to hold.

Status

References

Homm, U., & Breitung, J. (2012). Testing for speculative bubbles in stock markets: A comparison of alternative methods. Journal of Financial Econometrics, 10(1), 198-231.

Chu, C. S. J., Stinchcombe, M., & White, H. (1996). Monitoring structural change. Econometrica, 64(5), 1045-1065.

Astill, S., Harvey, D. I., Leybourne, S. J., Taylor, A. M. R., & Zu, Y. (2023). CUSUM-based monitoring for explosive episodes in financial data in the presence of time-varying volatility. Journal of Financial Econometrics, 21(1), 187-227.

Examples

# \donttest{
res <- monitor_cusum(sim_data$sim_psy1, r_star = 0.5)
#> Warning: Unknown or uninitialised column: `sim_psy1`.
#> Error: unsupported class
print(res)
#> Error: object 'res' not found
# }