Real-time monitoring

CUSUM Real-Time Monitoring for Explosive Bubbles

monitor_cusum Replication record →

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")
)

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.

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\% significance level (their 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).

Status

[Experimental]

Examples

These examples are copied from the package's own documentation and are run by R CMD check on every release. The printed output (after #>) and the plots were produced by running them against the current package source.

# 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()`).

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.