multiarch

Multivariate GARCH for Python. The arch package only fits univariate models, so Python has had no maintained multivariate GARCH library. Courses and research therefore send Python users to R (rmgarch, tsmarch). multiarch fills that gap:

• Models: CCC, DCC and asymmetric DCC, optionally as Gaussian or Student-t copulas, plus GO-GARCH.
• Margins: GARCH(1,1) or GJR-GARCH(1,1) with normal or Student-t errors.
• Inference: two-step standard errors that correct for the first estimation stage.
• Large cross-sections: a composite-likelihood estimator for many assets.
• Outputs: forecasts, simulation, portfolio VaR and ES, rolling backtests and specification tests.
• Validation: it reproduces rmgarch's likelihood exactly. It runs 2–9× faster than rmgarch and 7–30× faster than tsmarch.

pip install multiarch

Requires Python ≥ 3.10, NumPy, SciPy, pandas and numba.

Quick start

import multiarch as ma

res = ma.DCC(returns).fit()          # returns: (T, N) DataFrame, any units
print(res.summary())

res.conditional_correlation          # (T, N, N) R_t
res.conditional_covariance           # (T, N, N) H_t = D_t R_t D_t
res.correlation("JPM", "INTC")       # one rho_t path as a Series
res.forecast(10)                     # mean, variance, correlation, covariance
res.value_at_risk([0.5, 0.5], level=0.01, value=1e6)   # {'VaR': ..., 'ES': ...}

examples/course_workflow.py walks through the standard course sequence: EWMA → CCC → DCC → constant-correlation test → portfolio VaR → rolling VaR backtest.

What's in it

Estimation is scale-free: percent and fractional returns give the same parameters, with ω and μ rescaled.

Validation

All numbers come from validation/ and are reproducible: Rscript rmgarch_fit.R . then python compare.py, and Rscript tsmarch_fit.R . then python compare_tsmarch.py. The data are daily returns of DJI stocks from rmgarch's dji30ret (T = 5,521).

Against rmgarch 1.3 (R 4.6)

ll(ours @ rm) is multiarch's log-likelihood evaluated at rmgarch's own estimates. If both packages implement the same model, it equals rmgarch's reported value.

Timings for multiarch include the two-step standard errors.

• The likelihood functions are identical, and so are the correlation paths at the same parameters.
• The 10-step DCC covariance forecasts agree to 2e-14.
• Estimates differ only through the optimiser, and multiarch reaches higher margin likelihoods. Because estimation is two-step, a better first stage can still leave the joint log-likelihood slightly lower (the t and copula rows). That is a property of the two-step estimator, not a different model.
• Standard errors for the DCC parameters a and b are within 5% of rmgarch's. For the shape ν and the aDCC asymmetry g they are 17–31% larger. A 300-sample Monte Carlo (mc_se.py) shows that multiarch's two-step SEs are calibrated: the median SE is 0.0059 against a sampling SD of 0.0059 for a, and 0.0124 against 0.0139 for b.
• GO-GARCH: given rmgarch's ICA rotation and factor parameters, the likelihood agrees to 0.06, a difference that comes from how the constant mean is estimated. multiarch's own FastICA finds a rotation with a likelihood 277 points higher on this data, in 0.04 s against 0.47 s.

Against tsmarch (the package used in Financial Risk Forecasting notes)

The margins match tsgarch exactly (σ_t to 1e-14). A residual gap of under 0.03 log-likelihood points remains in tsmarch's compiled DCC recursion; it fades to 1e-7 by the end of the sample.

Notes on rmgarch's DCCtest

rmgarch::DCCtest departs from Engle & Sheppard (2001) in two ways. It calls qr.solve(regressand, regressors), which regresses the lags on the target instead of the reverse, and it divides by √σ² instead of σ². validation/compare.py reproduces its statistic exactly (65.34) from its own residuals. On the same residuals, the textbook statistic returned by constant_correlation_test is 41.08.

Scaling

From validation/bench.py (T = 2,500, no SEs):

The full-likelihood DCC estimate of b is biased downward as N grows: 0.930 against a true 0.95 at N = 50. The composite estimator does not have this bias.

Conventions

The default ("arch") follows the arch package for the margins (exponential backcast) and uses Q̄ = Z′Z/T with Q₁ = Q̄. convention="rmgarch" reproduces rmgarch's start-up:

• σ²₁ is the sample mean of ε².
• Q̄ = cov(Z).
• The first Q step uses a pre-sample shock of ones in DCC and of zeros in copula-DCC.
• Copula uniforms are clipped at 0.99999.

convention="tsmarch" reproduces tsgarch's σ²₁ = ω + persistence · mean(ε²) and uses Q₁ = Q̄ = cov(Z). These choices only affect the first few dozen observations.

Limitations

• Margins are GARCH(1,1) or GJR(1,1) with a constant or zero mean. There are no ARMA means or higher orders.
• DCC is of order (1,1).
• Estimation is two-step. The two-step SEs treat Q̄ and N̄ as known, as rmgarch does.
• Composite-likelihood SEs are Hessian-only.
• GO-GARCH SEs ignore estimation error in the rotation.
• Copula models use parametric margins only. Empirical and semi-parametric margins are not implemented.
• BEKK and DCC-MIDAS are not implemented.

References

• Bollerslev (1990), Modelling the coherence in short-run nominal exchange rates, REStat.
• Engle (2002), Dynamic conditional correlation, JBES.
• Engle & Sheppard (2001), Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH, NBER WP 8554.
• Cappiello, Engle & Sheppard (2006), Asymmetric dynamics in the correlations of global equity and bond returns, J. Fin. Econometrics.
• Engle, Shephard & Sheppard (2008), Fitting vast dimensional time-varying covariance models, working paper.
• van der Weide (2002), GO-GARCH: a multivariate generalized orthogonal GARCH model, J. Applied Econometrics.
• Patton (2006), Modelling asymmetric exchange rate dependence, IER.
• Kupiec (1995); Christoffersen (1998); Hosking (1980).

License

MIT