The Scale-Ordered Contagion methodology

The mechanism

A shock in a source market \(s\) releases information that the receiving market \(r\) absorbs through its own exponential filter. The transmitted response is the convolution of source generation and receiver absorption – a bi-exponential whose power spectrum is the product of two Lorentzians, one with corner at the source rate \(\alpha_s\) and one at the receiver rate \(\alpha_r\):

\[ S(\omega) = A^2\,\frac{\alpha_s^2}{\alpha_s^2+\omega^2}\, \frac{\alpha_r^2}{\alpha_r^2+\omega^2}. \]

The slower market supplies the binding corner: it governs the frequency band in which contagion power concentrates. The spectrum is symmetric in the two rates – the basis of the shape-symmetry prediction below.

w <- seq(0, pi, length.out = 200)
plot(w, soch_spectrum(2.0, 0.2, w), type = "l", lwd = 2,
     xlab = expression(omega), ylab = "S(omega)",
     main = "Product-Lorentzian transmission spectrum (fast source, slow receiver)")

From spectrum to wavelet scales

Projecting the spectrum onto MODWT octave bands gives a closed-form transfer-entropy-by-scale profile, soch_scale_power(). The two mixed directions (fast\(\to\)slow and slow\(\to\)fast) have identical normalised profiles, while fast/fast peaks fine and slow/slow peaks coarse:

P <- list(
  "fast|fast" = soch_scale_power(2.0, 2.0),
  "fast|slow" = soch_scale_power(2.0, 0.2),
  "slow|fast" = soch_scale_power(0.2, 2.0),
  "slow|slow" = soch_scale_power(0.2, 0.2)
)
plot_scale_profiles(P, normalise = TRUE)

The peak frequency obeys \(\omega^\star \in [\alpha_\wedge/\sqrt3,\ \alpha_\wedge]\), and the predicted peak scale is \(k^\star \approx \log_2(\pi/\alpha_\wedge)\):

soch_peak_frequency(2.0, 0.2)   # near the slow rate 0.2
#> [1] 0.1962205
soch_peak_scale(2.0, 0.2)       # a coarse scale
#> [1] 4.000949

Three falsifiable predictions

  • SOCH-A – the peak scale is set by the slower market’s rate, so emerging-inclusive pairs peak at coarser scales.
  • SOCH-B – the shape of the scale profile is the same in both directions (the spectrum is symmetric in the two rates).
  • SOCH-C – the level is directionally asymmetric, scaling with connectivity and source-shock content.

Identification

Fitting an observed profile to the closed form recovers the pair’s rates by nonlinear least squares, concentrating out the level. Because the scale power is symmetric, a single direction identifies only the unordered pair \(\{\alpha_\wedge, \alpha_\vee\}\); pooling across pairs that share a market resolves the ordering and yields market-level rates.

truth <- 3 * soch_scale_power(2.0, 0.2)        # known rates + a level
fit <- soch_fit_pair(truth)
c(amin = round(fit$amin, 3), amax = round(fit$amax, 3), R2 = round(fit$R2, 4))
#> amin amax   R2 
#>  0.2  2.0  1.0

See the Replication vignette to reproduce the headline G20 results.