Test for Co-explosive Behaviour Between Two Series

cobubble_test tests whether two series that each contain an explosive episode are co-explosive. That is, it tests whether the linear combination y_t - alpha - beta * x_{t-lag} is stationary, so that the explosive dynamics in y and x are the same underlying phenomenon, possibly migrating from one series to the other with a lead or lag, and not independent explosive episodes.

Usage

cobubble_test(
  y,
  x,
  lag = NULL,
  lag_grid = -6:6,
  nboot = 499L,
  sig_lvl = 95,
  seed = NULL
)

Arguments

y, x: Numeric vectors of equal length, or objects that as.numeric() can coerce to one. x is the candidate regressor with the explosive episode, and y is tested for co-explosivity with x_{t-lag}.
lag: The lead or lag i in x_{t-lag}. If NULL (default), it is estimated from lag_grid by minimizing the residual variance (i_hat in Section VI).
lag_grid: Candidate lag values searched when lag = NULL. The default -6:6 follows the simulation design of the paper.
nboot: Number of wild bootstrap replications.
sig_lvl: Significance level, on the same 0 to 100 scale as the sig_lvl of datestamp (default 95, which gives a 5\ upper-tail rejection region).
seed: Optional seed for the bootstrap draws.

Value

An object of class cobubble_test_obj: a list with the observed statistic S, the (given or estimated) lag, the bootstrap critical value cv at sig_lvl, the bootstrap p-value p_value and reject, which is TRUE if S exceeds cv, that is, if co-explosivity is rejected.

Details

radf is a right-tailed ADF-family test for the presence of explosiveness. This function is a stationarity (KPSS-type) test instead. The null hypothesis is co-explosivity, that is, that the residuals of y regressed on a constant and x_{t-lag} are I(0). The null limiting distribution of the statistic depends on the pattern of heteroskedasticity in the errors (Evripidou, Harvey, Leybourne & Sollis 2022, Theorem 1). The critical values therefore come from a wild bootstrap that reproduces this pattern of heteroskedasticity in the bootstrap samples (Theorem 2).

Note

The critical value comes from a wild bootstrap of the residuals (Theorem 2), which is computed internally in each call. There is no separate, reusable cv function for this test.

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 statistic, the critical value and the p-value. See vignette("naming-and-analysis", package = "exuber") for which functions fit the shared pipeline and which do not.

Status

References

Evripidou, A. C., Harvey, D. I., Leybourne, S. J., & Sollis, R. (2022). Testing for co-explosive behaviour in financial time series. Oxford Bulletin of Economics and Statistics, 84(3), 624-650.

Examples

# \donttest{
# A co-explosive pair (the process of Evripidou et al.), which is not rejected
xy <- sim_coexplosive(n = 100, seed = 123)
res <- cobubble_test(xy$y, xy$x, nboot = 199L, seed = 1)
print(res)
#> 
#> ── cobubble_test (lag = 0, nboot = 199) ────────────────────────────────────────
#> 
#> S = 0.2364, cv(95%) = 0.4048, p-value = 0.1508
#> Co-explosivity not rejected at the 5% level.
#> 

# Force a specific lead or lag instead of estimating it
res_lag0 <- cobubble_test(xy$y, xy$x, lag = 0L, nboot = 199L, seed = 1)
print(res_lag0)
#> 
#> ── cobubble_test (lag = 0, nboot = 199) ────────────────────────────────────────
#> 
#> S = 0.2364, cv(95%) = 0.4048, p-value = 0.1508
#> Co-explosivity not rejected at the 5% level.
#> 

# Two independent bubbles: co-explosivity correctly rejected
cobubble_test(sim_data$psy1, sim_data$psy2, nboot = 199L, seed = 1)
#> 
#> ── cobubble_test (lag = -2, nboot = 199) ───────────────────────────────────────
#> 
#> S = 1.533, cv(95%) = 0.2998, p-value = 0
#> Co-explosivity rejected at the 5% level.
#> 

# Plot the two series that are tested for co-explosivity
autoplot(res)

# }