Specifying a Nonlinear AR-GARCH-in-Mean Model

Summary

The document explains how to represent a time series with a nonlinear autoregressive mean, a GARCH(1,1) conditional variance, and an ARCH-in-mean effect. The mean equation includes the previous observation, its square, and the conditional variance; the variance equation depends on the previous squared innovation and previous conditional variance.

The answer recommends combining an AR(1)-GARCH(1,1) specification with the variance-in-mean term and a lagged squared observation as an explanatory variable in the mean. It says to use an econometric package that supports both ARCH-in-mean effects and additional regressors, citing a package as an example. The response clarifies the model structure but does not provide software steps, estimation choices, diagnostics, or empirical results. Users would still need to select a suitable error distribution and check whether the fitted specification is statistically and economically appropriate.

Key ideas

• The mean equation combines a lagged observation, its square, and conditional variance.
• The conditional variance follows a GARCH(1,1) process driven by past squared innovations and variance.
• The model can be specified as AR-GARCH with an ARCH-in-mean effect and a lagged squared regressor.
• Estimation requires software that supports both variance in the mean and additional explanatory variables.

Tags

# Fitting a non linear AR + GARCH(1,1)-M model


# Fitting a non linear AR + GARCH(1,1)-M model












I want to fit the following model to a time series:

$$ y_{t}=\alpha_{0}+\alpha_{1}y_{t-1}+\alpha_{2}y_{t-1}^{2}+\lambda h_{t}+\varepsilon_{t} $$

$$ h_{t}=\beta_{0}+\beta_{1}\varepsilon_{t-1}^{2}+\beta_{2}h_{t-1} $$

How can I do this with R or with any other statistical software?

Thanks

## Answer by Malick (score 1)

https://quant.stackexchange.com/a/22495

I would fit an AR(1)-GARCH(1,1) with arch in mean effect and with the square of the serie at lag 1 as explanatory variable in the mean process :

The AR(1)-GARCH(1,1) component for :

$$ y_{t}=\alpha_{0}+\alpha_{1}y_{t-1}+\varepsilon_{t} $$ $$ h_{t}=\beta_{0}+\beta_{1}\varepsilon_{t-1}^{2}+\beta_{2}h_{t-1} $$ + the arch in mean effect ($\lambda h_{t}$ ): $$ y_{t}=\alpha_{0}+\alpha_{1}y_{t-1} +\lambda h_{t}+\varepsilon_{t} $$ $$ h_{t}=\beta_{0}+\beta_{1}\varepsilon_{t-1}^{2}+\beta_{2}h_{t-1} $$ + the explanatory variable in the mean process ($ y_{t-1}^{2} $):

$$ y_{t}=\alpha_{0}+\alpha_{1}y_{t-1}+\alpha_{2}y_{t-1}^{2}+\lambda h_{t}+\varepsilon_{t} $$ $$ h_{t}=\beta_{0}+\beta_{1}\varepsilon_{t-1}^{2}+\beta_{2}h_{t-1} $$ All econometrical packages allowing to add 1)an arch in mean effect and 2) an explanatory variable can be used (the G@rch package (Ox) for example).

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