# 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?

• 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).

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 don't see how variance stabilizing can be done first. You need to see the residuals from the model first to see if the residual variance is changing with time. Then looking at the residuals might suggest how to change the model or stabilize the variance. – Michael R. Chernick Aug 15 '12 at 14:37
• By simply plotting the time series, you can find out if the variance stabilizing should be used or not. This has been done in the book "Time Series Analysis with Applications in R", page 99, and then they use Box-Cox transformation. You can check it by yourself. If you fit without stabilizing the variance, then it will be shown in the residual's plot. The thing is that we should try fixing any violations in the assumption of the ARIMA model before fitting them. I strongly suggest you to be more careful when giving negative points to an answer! Good luck. – Stat Aug 15 '12 at 19:58
• Yes I was the one who downvoted your answer. I agree that you can get a sense of variance inhomogeneity from a plot of the series. But I still do not think it is a good idea to apply a variance stabilizing transformation before trying out models. The models are all tentative. You fit, look at the residuals and modify as necessary. That is the three step Box-Jenkins approach. Initial model identification, followed by fitting and then diagnostic checking with the cycle repeated if the model does not appear to be adequate. – Michael R. Chernick Aug 15 '12 at 20:13
• This means that you did not read Box-Jenkins carefully. Check this link robjhyndman.com/papers/BoxJenkins.pdf Another reference, 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.” You can simply admit that you made a mistake. – Stat Aug 15 '12 at 20:55
• Stat and @Michael, You both have valid points: Stat because often an initial Box-Cox transformation is clearly indicated--so why not begin the iterative modeling process by tentatively applying that transformation?--yet Michael is also right to point out that the focus should be on the model residuals rather than the raw dependent values (a distinction frequently misunderstood in questions here). Neither downvotes nor accusations of making mistakes are needed to carry out this discussion. If you're going to argue, do it about something about which you both truly disagree! – whuber Aug 17 '12 at 12:39

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.