Since we can not fit ARIMA model when the constant variance assumption is violated, what model can be used to fit univariate time series?
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).
You can fit ARIMA model, but first you need to stabilize the variance by applying suitable transformation. You can also use Box-Cox transformation. This has been done in the book Time Series Analysis: With Applications in R, page 99, and then they use Box-Cox transformation. Check this link Box-Jenkins modelling Another reference is page 169, Introduction to Time Series and Forecasting, Brockwell and Davis, “Once the data have been transformed (e.g., by some combination of Box–Cox and differencing transformations or by removal of trend and seasonal components) to the point where the transformed series X_t can potentially be fitted by a zero-mean ARMA model, we are faced with the problem of selecting appropriate values for the orders p and q.” Therefore, you need to stabilize the variance prior to fit the ARIMA model.
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.