Mean Models¶

All ARCH models start by specifying a mean model.

Moving Average Models¶

MAMean is a moving average model of order \(q\) with an optional constant, which can be used with any volatility process and distribution. The model is

where \(\sigma^2_t\) is the conditional variance from the volatility process and \(e_t\) has the distribution of the model. The order is set using lags, and the constant \(\mu\) is omitted when constant=False. The parameters are ordered \([\mu, \theta_1, \ldots, \theta_q]\).

The innovations \(\epsilon_t\) are not observed. They are computed recursively from the data, with the innovations before the first observation set to 0, so that the parameters are estimated by maximizing the conditional likelihood. The effect of the initial values vanishes when the model is invertible. Invertibility is imposed when \(q\) is 1 or 2, but not for larger values of \(q\), and the starting values are always invertible. Exogenous regressors are not supported, and ARX or HARX should be used when they are needed.

In [1]: import arch.data.sp500

In [2]: from arch.univariate import GARCH, MAMean, StudentsT

In [3]: sp500 = arch.data.sp500.load()

In [4]: returns = 100 * sp500["Adj Close"].pct_change().dropna()

In [5]: ma = MAMean(returns, lags=2, volatility=GARCH(1, 0, 1), distribution=StudentsT())

In [6]: res = ma.fit(disp="off")

In [7]: res.params
Out[7]: 
mu          0.067441
theta[1]   -0.060069
theta[2]   -0.031440
omega       0.008391
alpha[1]    0.098476
beta[1]     0.901311
nu          6.483428
Name: params, dtype: float64

The forecast of the mean depends on the most recent innovations for the first \(q\) steps, and is the constant \(\mu\) after that.

In [8]: res.forecast(horizon=4, reindex=False).mean
Out[8]: 
                 h.1       h.2       h.3       h.4
Date                                              
2018-12-31  0.018189  0.041874  0.067441  0.067441

MAMean is a class only. It is not available through arch_model(), so that arch_model(y, mean="ma") raises a ValueError.

Adding Observations to a Model¶

Once a model has been estimated, observations that arrive later can be added using append(), which is available for all mean models. It modifies the model in-place and does not change its parameters, so that forecasts and results that are computed using parameters estimated using earlier data include the new observations without estimating the model again. See Forecasting After New Data Arrive for details and an example.

(G)ARCH-in-mean Models¶

(G)ARCH-in-mean models allow the conditional variance (or a transformation of it) to enter the conditional mean.

Special Requirements¶

A volatility process can only be used in AIM modeling if it supports updating. Specifically, the property updateable must be True. All of the volatility processes in the package support updating. A process that is written by the user must provide a volatility updater, see Volatility Processes.

In [9]: from arch.univariate import GARCH, EGARCH

In [10]: GARCH().updateable
Out[10]: True

In [11]: EGARCH().updateable
Out[11]: True

Writing New Mean Models¶

All mean models must inherit from ARCHModel and provide all public methods. There are two optional private methods that should be provided if applicable.