arch.univariate.MAMean¶

Moving average (MA) mean model, with optional constant, model estimation, simulation and forecasting

Parameters:¶

y: ndarray | DataFrame | Series | None = None¶: nobs element vector containing the dependent variable
lags: int = 1¶: The order of the moving average, q. Must be at least 1.
constant: bool = True¶: Flag whether the model should include a constant
hold_back: int | None = None¶: Number of observations at the start of the sample to exclude when estimating model parameters.
volatility: VolatilityProcess | None = None¶: Volatility process to use in the model
distribution: Distribution | None = None¶: Error distribution to use in the model
rescale: bool | None = None¶: 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, then y is rescaled and the new scale is reported in the estimation results.

lags¶

The order of the moving average

Type:¶: int

constant¶

Whether the model includes a constant

Type:¶: bool

Notes

The MA(q) model is

\[y_t = \mu + \epsilon_t + \theta_1 \epsilon_{t-1} + \ldots + \theta_q \epsilon_{t-q}\]

where the innovations \(\epsilon_t=\sigma_t e_t\) have the conditional variance \(\sigma^2_t\) of the volatility process and \(e_t\) has the distribution of the model. The parameters are ordered \([\mu, \theta_1, \ldots, \theta_q]\), and \(\mu\) is not included when constant is False.

The innovations are not observed. They are computed recursively using

\[\epsilon_t = y_t - \mu - \theta_1 \epsilon_{t-1} - \ldots - \theta_q \epsilon_{t-q}\]

where the innovations before the first observation in the estimation sample are 0. This is the conditional likelihood, and its effect vanishes when the process is invertible, that is when all roots of \(1+\theta_1 z+\ldots+\theta_q z^q\) are outside of the unit circle. Starting values are invertible. Invertibility is imposed when q is 1 or 2 (by bounds and linear constraints, respectively), but is not imposed for larger q since the restrictions are nonlinear. The sample used to estimate the parameters is also the sample used to compute the innovations, and so observations excluded using hold_back or when fitting are not used.

The mean forecast of \(y_{T+h}\) is \(\mu + \sum_{j=h}^{q}\theta_j\epsilon_{T+h-j}\), which uses the last \(q-h+1\) innovations if \(h \leq q\), and is \(\mu\) after that. The forecast variance is \(\sum_{j=0}^{h-1}\psi_j^2\sigma^2_{T+h-j}\) with \(\psi_0=1\), \(\psi_j=\theta_j\) for \(j\leq q\) and 0 otherwise.

Exogenous regressors are not supported.

Examples

Simulate an MA(2) with GARCH errors and estimate the parameters

>>> import numpy as np
>>> from arch.univariate import GARCH, MAMean
>>> model = MAMean(lags=2, volatility=GARCH())
>>> params = np.array([0.1, 0.6, 0.2, 0.05, 0.1, 0.85])
>>> sim_data = model.simulate(params, 1000)
>>> model = MAMean(sim_data["data"], lags=2, volatility=GARCH())
>>> res = model.fit(disp="off")

Forecasts from an MA(q) are not constant for the first q steps

>>> forecasts = res.forecast(horizon=4, reindex=False)

Methods

`append`(y[, x])	Append observations to the model in-place
`bounds`()	Construct bounds for parameters to use in non-linear optimization
`compute_param_cov`(params[, backcast, robust])	Computes parameter covariances using numerical derivatives.
`constraints`()	Construct linear constraint arrays for use in non-linear optimization
`fit`([update_freq, disp, starting_values, ...])	Estimate model parameters
`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
`parameter_names`()	List of parameters names
`resids`(params[, y, regressors])	Compute model residuals
`simulate`(params, nobs[, burn, ...])	Simulates data from a moving average model
`starting_values`()	Returns starting values for the mean model, often the same as the values returned from fit

Properties

`distribution`	Set or gets the error distribution
`name`	The name of the model.
`num_params`	Returns the number of parameters
`volatility`	Set or gets the volatility process
`y`	Returns the dependent variable