Bivariate Bubble Relationships: cobubble_test() and contagion_reg()

library(exuber)

Two different questions about two series

Both functions look at a pair of series that each contain, or might contain, an explosive episode. They ask different questions.

  • cobubble_test() (Evripidou, Harvey, Leybourne & Sollis 2022) is a formal hypothesis test of whether the explosive episodes of y and x are the same episode, that is, whether y_t - alpha - beta * x_{t-i} is stationary for some lag or lead i. It is a KPSS-type test whose null hypothesis is co-explosivity (a stationary residual). Rejecting the null therefore means that the bubbles in the two series do not come from the same underlying process.
  • contagion_reg() (Greenaway-McGrevy & Phillips 2016) does no formal inference. It estimates a time-varying contagion coefficient delta_2(r) with a Nadaraya-Watson kernel regression. At each point r in the sample, the coefficient measures how strongly the fixed-window AR(1) coefficient of a “peripheral” series y moves with the coefficient of a “core” series.

Neither function returns the radf_obj class (see vignette("naming-and-analysis")).

cobubble_test(): are these the same bubble?

sim_coexplosive() implements the data generating process of Evripidou et al.: y is a linear function of an explosive x plus noise. The pair is co-explosive, so a correct test should not reject:

xy <- sim_coexplosive(n = 100, seed = 123)
res <- cobubble_test(xy$y, xy$x, nboot = 199, seed = 1)
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.

For contrast, sim_data$psy1 and sim_data$psy2 are simulated independently, so by construction they share no bubble process:

cobubble_test(sim_data$psy1, sim_data$psy2, nboot = 199, 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.

Here S clearly exceeds its wild-bootstrap critical value, which is robust to heteroskedasticity, and co-explosivity is rejected, as it should be.

contagion_reg(): how strongly do they co-move, and when?

For the co-explosive pair, the AR(1) coefficient of y follows that of x almost one for one throughout the sample:

cr <- contagion_reg(xy$y, xy$x, d = 0)
cr
#> 
#> ── contagion_reg (n = 100, S = 33, d = 0, h = 0.6567) ──────────────────────────
#> 
#> delta_2(r) range: [0.948, 0.965]

cr$delta2 holds the whole estimated path over cr$r_grid, while print() shows only its range:

plot(cr$r_grid, cr$delta2, type = "l",
     xlab = "r (fraction of sample)", ylab = expression(delta[2](r)),
     main = "Estimated time-varying contagion coefficient")

For contrast, the two independent sim_data series give a coefficient path that stays far below one:

cr_null <- contagion_reg(sim_data$psy1, sim_data$psy2, d = 0)
range(cr_null$delta2)
#> [1] 0.1628972 0.1823787
range(cr$delta2)
#> [1] 0.9476613 0.9652861

Which to reach for

  • If you want a yes or no answer, with a critical value, to the question of whether two series share the same explosive episode, use cobubble_test().
  • If you want to see how the comovement between the AR coefficients of two series changes over the sample, for example to watch contagion build up before a joint collapse, and you do not need a formal test, use contagion_reg().