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Bayesian Nonlinear Ornstein-Uhlenbeck Models with Stochastic Volatility
bayesianOU fits a Bayesian nonlinear Ornstein-Uhlenbeck
(OU) / nonlinear error-correction model with cubic drift, stochastic
volatility (SV), and Student-t innovations. Sector-specific parameters
use a non-centered hierarchical (partial-pooling)
parameterization, and sampling is done with Stan
(reduce_sum for within-chain parallelism). Model comparison
uses PSIS-LOO (Vehtari, Gelman, Gabry 2017).
# install.packages("remotes")
remotes::install_github("IsadoreNabi/bayesianOU")
# Stan backend (cmdstanr recommended):
install.packages("cmdstanr", repos = c("https://mc-stan.org/r-packages/", getOption("repos")))
cmdstanr::install_cmdstan()
# Or rstan:
install.packages("rstan")library(bayesianOU)
Y <- as.matrix(your_prices_data) # market prices by sector
X <- as.matrix(your_production_prices_data) # prices of production
TMG <- your_tmg_series # aggregate profit rate
COM <- as.matrix(your_com_data) # organic composition of capital
K <- as.matrix(your_capital_data) # total capital by sector
results <- fit_ou_nonlinear_tmg(
results_robust = list(),
Y = Y, X = X, TMG = TMG, COM = COM, CAPITAL_TOTAL = K,
fit_window = "train", # honest out-of-sample evaluation
chains = 4, iter = 4000, warmup = 2000,
verbose = TRUE
)
validate_ou_fit(results) # MCMC + LOO (incl. Pareto-k) + OOS
kappa_stability_evidence(results) # dynamic mean-reversion evidence
plot_beta_tmg(results)
plot_drift_curves(results)For each sector s, on training-standardized series, the
one-step (Euler, dt = 1) increment is
dY[t,s] = kappa_s * (theta_s - Y[t-1,s] + a3_s * (Y[t-1,s] - theta_s)^3) # cubic OU drift
+ (beta0_s + beta1 * zTMG[t]) * X[t-1,s] # TMG-modulated pass-through
+ gamma * COM_std[t-1,s] # organic composition in mean
+ eps[t,s], eps ~ Student-t(nu, 0, sigma[t,s])
log sigma^2[t,s] = h[t,s], h ~ stationary AR(1) (alpha_s, rho_s, sigma_eta_s)kappa_s mean-reversion speed, theta_s
equilibrium level, a3_s < 0 cubic term. The implied
long-run equilibrium is Y* = theta_s + (beta/kappa_s) X,
i.e. a nonlinear error-correction toward a linear function of the prices
of production.theta_s, kappa_s, a3_s, beta0_s are built
hierarchically as intercept + slope * COM_s + sd * z,
z ~ N(0,1) (non-centered).nu > 2 (finite variance). sigma[t,s] is
the Student-t scale; the conditional standard deviation
is sigma * sqrt(nu/(nu-2)).Training/test and “out-of-sample”. A Bayesian fit on
all data is perfectly valid for inference and for
PSIS-LOO (which approximates leave-one-out predictive
performance from the full-data posterior — this is exactly what
rstanarm does, and it is not leakage). Leakage only arises
if one labels a block “out-of-sample” while it still entered the
likelihood. This package keeps the two designs coherent via
fit_window:
fit_window = "train" (default): the likelihood and
log_lik are summed only over the training
window 2:T_train; the test block is genuinely held out, so
evaluate_oos is a real out-of-sample evaluation.fit_window = "full": fit on all observations
(full-information). PSIS-LOO is then valid over all points, but
evaluate_oos becomes in-sample.Priors are configurable and neutral by default. The
key hypothesis prior is neutral: beta1 ~ Normal(0, 0.5) (no
sign baked in). The Student-t df uses a weakly-informative
nu_tilde ~ Gamma(2, 0.1) (prior mean nu ~ 22),
and SV persistence rho_s ~ Normal(0.7, 0.2). Override any
of these through the priors argument and run a sensitivity
analysis.
Leakage in preprocessing is avoided by default.
use_train_loadings = TRUE computes the common-factor
loadings from the training window only and then projects the full
series, so the orthogonalized TMG regressor does not see the future.
Stability assumption. a3_s < 0 is
enforced, so mean reversion strengthens with the deviation. This is an
assumption (global stabilization), not an estimated result; it
precludes detecting locally expansive / self-amplifying regimes.
Half-life. With the Euler dt = 1 discretization, the
discrete persistence of the linear part is 1 - kappa;
interpret half-lives as log(0.5)/log(1 - kappa), not
log(2)/kappa (these coincide only as
kappa -> 0).
PSIS-LOO caveat. The model has one latent SV state
per observation, so plain PSIS-LOO tends to be optimistic and to produce
high Pareto-k. validate_ou_fit surfaces a Pareto-k summary
and warns; for genuine forecasting assessment prefer leave-future-out
(LFO-CV).
validate_ou_fit).tests/testthat/test-recovery.R).fit_window = "train") and PSIS-LOO comparison against
baselines.@software{bayesianOU,
author = {Gómez Julián, José Mauricio},
title = {bayesianOU: Bayesian Nonlinear Ornstein-Uhlenbeck Models},
year = {2024},
note = {ORCID: 0009-0000-2412-3150},
url = {https://github.com/IsadoreNabi/bayesianOU}
}MIT License
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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