# Timeseries¶

 AR1(k, tau_e, *args, **kwargs) Autoregressive process with 1 lag. AR(rho[, sd, tau, constant, init]) Autoregressive process with p lags. GaussianRandomWalk([tau, init, sd, mu]) Random Walk with Normal innovations GARCH11(omega, alpha_1, beta_1, initial_vol, …) GARCH(1,1) with Normal innovations. EulerMaruyama(dt, sde_fn, sde_pars, *args, …) Stochastic differential equation discretized with the Euler-Maruyama method. MvGaussianRandomWalk([mu, cov, tau, chol, …]) Multivariate Random Walk with Normal innovations MvStudentTRandomWalk(nu, *args, **kwargs) Multivariate Random Walk with StudentT innovations
class pymc3.distributions.timeseries.AR1(k, tau_e, *args, **kwargs)

Autoregressive process with 1 lag.

Parameters: k : tensor effect of lagged value on current value tau_e : tensor precision for innovations
class pymc3.distributions.timeseries.AR(rho, sd=None, tau=None, constant=False, init=<pymc3.distributions.continuous.Flat object>, *args, **kwargs)

Autoregressive process with p lags.

$x_t = \rho_0 + \rho_1 x_{t-1} + \ldots + \rho_p x_{t-p} + \epsilon_t, \epsilon_t \sim N(0,\sigma^2)$

The innovation can be parameterized either in terms of precision or standard deviation. The link between the two parametrizations is given by

$\tau = \dfrac{1}{\sigma^2}$
Parameters: rho : tensor Tensor of autoregressive coefficients. The first dimension is the p lag. sd : float Standard deviation of innovation (sd > 0). (only required if tau is not specified) tau : float Precision of innovation (tau > 0). (only required if sd is not specified) constant: bool (optional, default = False) Whether to include a constant. init : distribution distribution for initial values (Defaults to Flat())
class pymc3.distributions.timeseries.GaussianRandomWalk(tau=None, init=<pymc3.distributions.continuous.Flat object>, sd=None, mu=0.0, *args, **kwargs)

Random Walk with Normal innovations

Parameters: mu: tensor innovation drift, defaults to 0.0 sd : tensor sd > 0, innovation standard deviation (only required if tau is not specified) tau : tensor tau > 0, innovation precision (only required if sd is not specified) init : distribution distribution for initial value (Defaults to Flat())
class pymc3.distributions.timeseries.GARCH11(omega, alpha_1, beta_1, initial_vol, *args, **kwargs)

GARCH(1,1) with Normal innovations. The model is specified by

$y_t = \sigma_t * z_t$
$\sigma_t^2 = \omega + \alpha_1 * y_{t-1}^2 + \beta_1 * \sigma_{t-1}^2$

with z_t iid and Normal with mean zero and unit standard deviation.

Parameters: omega : tensor omega > 0, mean variance alpha_1 : tensor alpha_1 >= 0, autoregressive term coefficient beta_1 : tensor beta_1 >= 0, alpha_1 + beta_1 < 1, moving average term coefficient initial_vol : tensor initial_vol >= 0, initial volatility, sigma_0
class pymc3.distributions.timeseries.EulerMaruyama(dt, sde_fn, sde_pars, *args, **kwds)

Stochastic differential equation discretized with the Euler-Maruyama method.

Parameters: dt : float time step of discretization sde_fn : callable function returning the drift and diffusion coefficients of SDE sde_pars : tuple parameters of the SDE, passed as *args to sde_fn
class pymc3.distributions.timeseries.MvGaussianRandomWalk(mu=0.0, cov=None, tau=None, chol=None, lower=True, init=<pymc3.distributions.continuous.Flat object>, *args, **kwargs)

Multivariate Random Walk with Normal innovations

Parameters: mu : tensor innovation drift, defaults to 0.0 cov : tensor pos def matrix, innovation covariance matrix tau : tensor pos def matrix, inverse covariance matrix chol : tensor Cholesky decomposition of covariance matrix init : distribution distribution for initial value (Defaults to Flat())

Notes

Only one of cov, tau or chol is required.

class pymc3.distributions.timeseries.MvStudentTRandomWalk(nu, *args, **kwargs)

Multivariate Random Walk with StudentT innovations

Parameters: nu : degrees of freedom mu : tensor innovation drift, defaults to 0.0 cov : tensor pos def matrix, innovation covariance matrix tau : tensor pos def matrix, inverse covariance matrix chol : tensor Cholesky decomposition of covariance matrix init : distribution distribution for initial value (Defaults to Flat())