# What model can be used when the constant variance assumption is violated?

Since we can not fit ARIMA model when the constant variance assumption is violated, what model can be used to fit univariate time series?

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Assuming there are no independant regressors in the fitted model, non constant variance is really only a problem when the variance of the error term is time-dependant. Then: arma+garch –  user603 Aug 13 '12 at 15:32

There are a number of modelling options to account for a non-constant variance, for example ARCH (and GARCH, and their many extensions) or stochastic volatility models.

An ARCH model extend ARMA models with an additional time series equation for the square error term. They tend to be pretty easy to estimate (the fGRACH R package for example).

SV models extend ARMA models with an additional time series equation (usually a AR(1)) for the log of the time-dependent variance. I have found these models are best estimated using Bayesian methods (OpenBUGS has worked well for me in the past).

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I would first ask why the residuals from an ARIMA model don't have constant variance before I would abandon the approach. Do the residuals themselve exhibit no correlation structure? If they do maybe some moving average terms need to be incorporated into the model.

But now let us suppose that the residuals do not appear to have any autocorrelation structure. then in what ways is the variance changing with time (increasing, decreasing, or fluctuating up and down)? The way the variance is changing may be a clue to what is wrong with the existing model. Perhaps there are covariates that are crosscorrelated with this time series. In that case the covariates could be added to the model. The residuals may then no longre exhibit nonconstant variance.

You may say that if the series is cross correlated with a covariate that show show up in the autocorrelation of the residuals. But that would not be the case if the correlation is mostly at lag 0.

If neither the addition of moving average terms nor the introduction of covariates helps solve the problem, you could perhaps consider identifying a time varying function for the residual variance based on a few parameters. Then that relationship could be incorporated in the likelihood function in order to modify the model estimates.

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