class arch.univariate.HARX(y=None, x=None, lags=None, constant=True, use_rotated=False, hold_back=None, volatility=None, distribution=None, rescale=None)[source]

Heterogeneous Autoregression (HAR), with optional exogenous regressors, model estimation and simulation

  • y ({ndarray, Series}) – nobs element vector containing the dependent variable

  • x ({ndarray, DataFrame}, optional) – nobs by k element array containing exogenous regressors

  • lags ({scalar, ndarray}, optional) – Description of lag structure of the HAR. Scalar included all lags between 1 and the value. A 1-d array includes the HAR lags 1:lags[0], 1:lags[1], … A 2-d array includes the HAR lags of the form lags[0,j]:lags[1,j] for all columns of lags.

  • constant (bool, optional) – Flag whether the model should include a constant

  • use_rotated (bool, optional) – Flag indicating to use the alternative rotated form of the HAR where HAR lags do not overlap

  • hold_back (int) – Number of observations at the start of the sample to exclude when estimating model parameters. Used when comparing models with different lag lengths to estimate on the common sample.

  • volatility (VolatilityProcess, optional) – Volatility process to use in the model

  • distribution (Distribution, optional) – Error distribution to use in the model

  • rescale (bool, optional) – Flag indicating whether to automatically rescale data if the scale of the data is likely to produce convergence issues when estimating model parameters. If False, the model is estimated on the data without transformation. If True, than y is rescaled and the new scale is reported in the estimation results.


>>> import numpy as np
>>> from arch.univariate import HARX
>>> y = np.random.randn(100)
>>> harx = HARX(y, lags=[1, 5, 22])
>>> res =
>>> from pandas import Series, date_range
>>> index = date_range('2000-01-01', freq='M', periods=y.shape[0])
>>> y = Series(y, name='y', index=index)
>>> har = HARX(y, lags=[1, 6], hold_back=10)


The HAR-X model is described by

\[y_t = \mu + \sum_{i=1}^p \phi_{L_{i}} \bar{y}_{t-L_{i,0}:L_{i,1}} + \gamma' x_t + \epsilon_t\]

where \(\bar{y}_{t-L_{i,0}:L_{i,1}}\) is the average value of \(y_t\) between \(t-L_{i,0}\) and \(t - L_{i,1}\).



Construct bounds for parameters to use in non-linear optimization

compute_param_cov(params[, backcast, robust])

Computes parameter covariances using numerical derivatives.


Construct linear constraint arrays for use in non-linear optimization

fit([update_freq, disp, starting_values, …])

Fits the model given a nobs by 1 vector of sigma2 values

fix(params[, first_obs, last_obs])

Allows an ARCHModelFixedResult to be constructed from fixed parameters.

forecast(params[, horizon, start, align, …])

Construct forecasts from estimated model


List of parameters names

resids(params[, y, regressors])

Compute model residuals

simulate(params, nobs[, burn, …])

Simulates data from a linear regression, AR or HAR models


Returns starting values for the mean model, often the same as the values returned from fit



Set or gets the error distribution


The name of the model.


Returns the number of parameters


Set or gets the volatility process


Gets the value of the exogenous regressors in the model


Returns the dependent variable