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Bivariate Bubble Relationships: cobubble_test() and contagion_reg()

Source: vignettes/co-explosivity.Rmd
co-explosivity.Rmd

Two different questions about two series

Both functions here look at a pair of series that each contain (or might contain) an explosive episode, but they ask different questions and answer them differently:

  • cobubble_test() (Evripidou, Harvey, Leybourne & Sollis 2022) is a formal hypothesis test: are y and x‘s explosive episodes the same episode, i.e. is y_t - alpha - beta * x_{t-i} stationary for some lag/lead i? This is a KPSS-type test – the null hypothesis is co-explosivity (stationary residual), so rejecting means the two series’ bubbles are not the same underlying process.
  • contagion_reg() (Greenaway-McGrevy & Phillips 2016) performs no formal inference at all. It estimates a time-varying contagion coefficient delta_2(r): at each point r in the sample, how strongly a “peripheral” series y‘s (fixed-window) AR(1) coefficient co-moves with a “core” series’ own coefficient, via a Nadaraya-Watson kernel regression.

Neither carries the radf_obj class – see vignette("naming-and-analysis").

cobubble_test(): are these the same bubble?

sim_coexplosive() is Evripidou et al.’s own DGP: y is a linear function of an explosive x (plus noise), so the pair is co-explosive and 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 independently simulated explosive episodes, sharing no bubble process by construction:

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 comfortably exceeds its (wild-bootstrap, heteroskedasticity-robust) critical value and co-explosivity is rejected – correctly.

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

On the co-explosive pair, y’s AR(1) coefficient tracks x’s 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$delta2 is the full estimated path over cr$r_grid, not just the range print() shows:

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 sitting 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

  • Want a yes/no answer, with a critical value, to “do these two series share the same explosive episode”: cobubble_test().
  • Want to see how the strength of comovement between two series’ AR coefficients evolves over the sample (e.g. to visualize contagion building up before a joint collapse), with no formal test attached: contagion_reg().