# Does applying ARMA-GARCH require stationarity?

I am going to use the ARMA-GARCH model for financial time series and was wondering whether the series should be stationary before applying the said model. I know to apply ARMA model the series should be stationary, however I'm not sure for ARMA-GARCH since I'm including GARCH errors which imply volatility clustering and non-constant variance and hence non-stationary series no matter what transformation I do.

Are financial time series usually stationary or non-stationary? I tried applying ADF test to a few volatile series and got p-value<0.01 which seems to indicate stationarity but the principle of volatile series itself tells us that the series isn't stationary.

Can somebody clear that up for me?I'm getting really confused

Copying from the abstract of Engle's original paper:
"These are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances. For such processes, the recent past gives information about the one-period forecast variance".

Continuing with the references, as the author who introduced GARCH shows (Bollerslev, Tim (1986). "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics, 31:307-327) for the GARCH(1,1) process, it suffices that $\alpha_1 + \beta_1 <1$ for 2nd-order stationarity.

Stationarity (the one needed for estimation procedures), is defined relative to the unconditional distribution and moments.

To summarize here discussion in the comments, the GARCH modeling approach is an ingenious way to model suspected heteroskedasticity over time, i.e. of some form of heterogeneity of the process (which would render the process non-stationary) as an observed feature that comes from the existence of memory of the process, in essence inducing stationarity at the unconditional level.

In other words, we took our two "great opponents" in stochastic process analysis (heterogeneity and memory), and used the one to neutralize the other -and this is indeed an inspired strategy.

• I'm not sure how this answers my question?Can you explain?Is it possible for a volatile series to be defined as stationary? – ankc Oct 12 '13 at 17:46
• If a time series exhibit volatility clustering doesn't that mean that the series in non-stationary and GARCH cannot be applied to it(if it's non-stationary)? – ankc Oct 12 '13 at 18:32
• I take it that by "volatility clustering" you mean that it appears that the time series is characterized by different variance in different intervals. First, this is just an indication of possible non-stationarity, not proof. Second, the ARCH model and its extensions attempt to explain this "volatility clustering" by modelling the conditional variance as time-changing, while maintaining the assumption of a constant unconditional variance (and hence, the assumption of 2nd-order stationarity). – Alecos Papadopoulos Oct 12 '13 at 19:13
• Well lets assume that there is indeed volatility clustering. The series itself would be non-stationary so how can I apply a GARCH model to a non-stationary series as mpiktas did say that GARCH should be applied to stationary series. – ankc Oct 12 '13 at 19:33
• No, volatility clustering does not necessarily imply non-stationarity. So if it can be "explained" by GARCH modelling, then you can operate on the assumption of unconditional stationarity. Indeed, this appears a bit circular - but then again, we can almost never be sure that an actual observed stochastic process is, or is not, stationary. – Alecos Papadopoulos Oct 12 '13 at 19:38

Yes the the series should be stationary. GARCH models are actually white noise processes with not trivial dependence structure. Classical GARCH(1,1) model is defined as

$$r_t=\sigma_t\varepsilon_t,$$

with

$$\sigma_t^2=\alpha_0+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2,$$

where $\varepsilon_t$ are independent standard normal variables with unit variance.

Then

$$Er_t=EE(r_t|\varepsilon_{t-1},\varepsilon_{t-2},...)=E\sigma_tE(\varepsilon_t|\varepsilon_{t-1},\varepsilon_{t-2},...)=0$$

and

$$Er_tr_{t-h}=EE(r_tr_{t-h}|\varepsilon_{t-1},\varepsilon_{t-2},...)=Er_{t-h}\sigma_{t}E(\varepsilon_t|\varepsilon_{t-1},\varepsilon_{t-2},...)=0$$

for $h>0$. Hence $r_t$ is a white noise process. However it is possible to show that $r_t^2$ is actually a $ARMA(1,1)$ process. So GARCH(1,1) is stationary process, yet has non-constant conditional variance.

• How can a series be stationary if it exhibits volatility?How do you define stationarity when applying a GARCH model? – ankc Oct 12 '13 at 18:24
• Would it be okay if I include AR and MA terms in my mean equation?If the return series exhibit some autocorrelation at short lags. – ankc Oct 12 '13 at 18:43
• Stationary means constant mean, variance and correlation depending only on lag. AR and MA terms can be included in the mean equation. The key in GARCH processes is conditional volatility. Note that volatility is not variance. The mean volatility is series variance. – mpiktas Oct 12 '13 at 19:28
• As reference take for example the SP500 data in R, the return data seems to be constant in its mean but exhibit blatant conditional heteroskedasticity. So it is possible to apply a GARCH model on it despite having non constant variance? – ankc Oct 12 '13 at 19:40
• usually can I apply the GARCH model to any log return series that exhibits volatility clustering?I am asking this because I saw in a dissertation that the ADF test was applied to test for stationarity, so I thought that stationarity was necessary before applying the GARCH model. – ankc Oct 12 '13 at 20:17

For anyone who is wondering about this question still, i will clarify - Volatility clustering does not at all imply that the series is non-stationary. It would suggest that there is a shifting conditional variance regime - which may still satisfy constancy of the unconditional distribution.

The GARCH(1,1) model of Bollerslev is not weakly stationary when $$\alpha_{1}+\beta>1$$, however it is actually still stricktly stationary for a much larger range, Nelson 1990. Further Rahbek & Jensen 2004 (Asymptotic inference in the non-stationary GARCH), showed that the ML estimator of $$\alpha_{1}$$ and $$\beta$$ is consistent and asymptotically normal for any parameter specification that ensure the model is non-stationary. Combining this with the results of Nelson 1990 (all weak or strict stationary GARCH(1,1) models have MLE estimator as consistent and asymptotically normal), suggests that any parameter combination whatsoever of $$\alpha_{1}$$ and $$\beta>1$$ will have consistent and Asymptotically normal estimators.

It is important to note however that if the GARCH(1,1) model is non stationary, the constant term in the conditional variance is not estimated consistently.

Regardless, this suggests that you do not have to worry about stationarity before estimating the GARCH model. You do however have to wonder whether it seems to have a symmetric distribution, and whether the series has high persistence, as this is not allowed in the classical GARCH(1,1) model. When you have estimated the model it is of interest to test whether $$\alpha_{1}+\beta=1$$ if you are working with financial timeseries, since this would imply a trending conditional variance which is hard to immagine being a behavioral tendency amongst investors. Testing this however can be done with a normal LR test.

Stationarity is fairly misunderstood, and is only partially connected to whether the variance or mean seems to be ocationally changing - as this can still ocour while the process maintains a constant unconditional distribution. The reason you may think that the seeming shifts in variance may cause a departure from stationarity, is because such a thing as permanent levelshift in the variance equation (or the mean equation) would by definition break stationarity. But if the changes are caused by the dynamic specification of the model, it may still be stationary even though the mean is impossible to identify and the volatility constantly changes. Another Beautiful example of this is the DAR(1,1) model introduced by Ling in 2002.

• Good answer! Does DAR(1,1) standard for ARIMA(1,1,0)? If not what is it & why didn't you address the non-stationary ARIMA models? – Michael R. Chernick Nov 25 '19 at 22:19

Stationarity is a theoretical concept which is then modified to other forms like Weak Sense Stationarity which can be tested easily. Most of the tests like adf test as you have mentioned test for linear conditions only. the ARCH effects are made for series which do not have autocorrelation in the first order but there is dependence in the squared series.

The ARMA-GARCH process you talk about, here the second order dependence is removed using the GARCH part and then any dependence in the linear terms is captured by the ARMA process.

The way to go about is to check for the autocorrelation of the squared series, if there is dependence, then apply the GARCH models and check the residuals for any linear time series properties which can then be modelled using ARMA processes.

• I was thinking of fitting the ARMA first, then fitting the residuals to a GARCH model. Is this wrong?How can I "check the residuals for any linear time series properties which can then be modelled using ARMA processes."?Can the ljung-box test be used to detect ARCH effect? – ankc Oct 12 '13 at 18:48
• simplest way is to look for the auto correlation function of the squared series. if it is significant then try out the GARCH model. if the autocorrelation of the square of the residuals gets removed, then the GARCH does help to model the dependence in the squared series. – htrahdis Oct 12 '13 at 18:52
• If I do that my mean return will be 0 right?I want to be able to get a mean that will not be a straight line, like a mean function that will depend on AR and MA terms + the GARCH error. – ankc Oct 12 '13 at 18:59
• there are three things : one is the decision of whether there are GARCH effects present, the other is a justification of using ARMA and GARCH and the third is to actually fit the model when the above two are affirmative. the fitting is not so simple as do it in two different stages. you have to fit both the ARMA and the GARCH parts simultaneously. There are methods available for this. – htrahdis Oct 12 '13 at 19:06
• Would the use of ARMA be justified if there are correlations in the return series?I think there are packages in R that does the fitting. I only need to know when to apply an ARMA-GARCH or simply a GARCH. Can I use ljung-box test to test for GARCH effects? – ankc Oct 12 '13 at 19:29