CUSUM Real-Time Monitoring for Explosive Bubbles

monitor_cusum implements the CUSUM real-time monitoring procedure of Homm & Breitung (2012). You fix a training window [1, T*] that is assumed free of exuberance. The function then compares the standardized cumulative sum of the post-training first differences, S_t = (y_t - y_{T*}) / sigma_hat_t, with the closed-form boundary c_t * sqrt(t), where c_t = sqrt(b_alpha + log(t / T*)), and flags the first date at which the boundary is breached.

Usage

monitor_cusum(
  data,
  r_star = 0.5,
  b_alpha = 4.6,
  boundary = c("asymptotic", "finite"),
  sig_lvl = 95,
  type = c("standard", "kernel"),
  h = 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 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.
b_alpha: The boundary constant (eq. 29 of HB). The default 4.6 is the one-sided asymptotic calibration of HB for a 5\ Section 3). It is an asymptotic upper bound on the false-alarm probability (Chu, Stinchcombe & White 1996) and not an exact size, so it is typically conservative in finite samples. It is ignored when boundary = "finite".
boundary: "asymptotic" (default) uses b_alpha directly. "finite" instead looks up the finite-sample boundary constant of HB (their Table 8) from sig_lvl and the realized ratio of training length to monitoring horizon. sig_lvl must then be one of 90, 95 or 99.
sig_lvl: Significance level on the 0 to 100 scale used throughout the package when boundary = "finite" (default 95). It is ignored when boundary = "asymptotic".
type: "standard" (default) for the original CUSUM statistic of Homm & Breitung (2012), or "kernel" for the volatility-robust "CUSUMV" variant of Astill, Harvey, Leybourne, Taylor & Zu (2023).
h: Bandwidth (window length, N in AHLTZ) of the one-sided kernel spot-variance estimator when type = "kernel". The default 20 is the value that the authors recommend from their experiments (their Section 3: "setting H = 20 delivered a procedure with the best trade-off" between robustness of the false-alarm rate and power). It is ignored when type = "standard".
kernel: Kernel for the spot-variance estimator when type = "kernel", "gaussian" (default) or "uniform". It is ignored when type = "standard".

Value

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

Note

The boundary is closed-form throughout. It is either a fixed asymptotic constant (boundary = "asymptotic", b_alpha = 4.6) or a lookup in the published finite-sample table (boundary = "finite", Table 8 of Homm & Breitung 2012). No simulation and no separate cv function is needed.

monitor (Family A) uses a recursive ADF-family statistic whose boundary needs a wild bootstrap for calibration. This function uses a structurally different statistic, a standardized running sum and not a recursive regression. Its boundary is an asymptotic closed form (the inequality of Chu, Stinchcombe & White 1996, eq. 28 of HB), so it needs no bootstrap and no simulation and does not depend on the data beyond the running variance estimate itself.

type = "kernel" instead uses the volatility-robust modification ("CUSUMV") of Astill, Harvey, Leybourne, Taylor & Zu (2023). Before cumulating, it standardizes each first difference by its own one-sided kernel spot-variance estimate (their eq. 6-7) and not by a single running variance. Their Corollary 1 establishes that the same boundary function gives a controlled asymptotic false-alarm rate even under time-varying volatility. The standard CUSUM statistic does not have this property, because its size-control result requires homoskedasticity.

The function returns its own class and not radf_obj, so it does not work with summary(), \link{datestamp} and tidy. It has its own print() and autoplot() methods instead. print() shows the boundary and the alarm. See vignette("naming-and-analysis", package = "exuber") for which functions fit the shared pipeline and which do not.

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{
# 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_cusum(y, r_star = 0.5)
print(res) # alarm should fire soon after t = 150
#> 
#> ── monitor_cusum (T* = 100 / 200, b_alpha = 4.6) ───────────────────────────────
#> 
#>    series  alarm  alarm_date
#>   series1    160         160
#> 
autoplot(res)
#> Warning: Removed 2 rows containing missing values or values outside the scale range
#> (`geom_segment()`).


# Volatility-robust "CUSUMV" variant (Astill, Harvey, Leybourne, Taylor & Zu 2023):
# the same bubble, but volatility triples at t = 120, after the training window
y_vol <- sim_psy1(n = 200, te = 150, tf = 200, seed = 7,
  e = sim_vol_break(199, tau = 0.6))
res_kernel <- monitor_cusum(y_vol, r_star = 0.5, type = "kernel")
autoplot(res_kernel)
#> Warning: Removed 2 rows containing missing values or values outside the scale range
#> (`geom_segment()`).

# }