# Weak stationarity and ARMA-ARCH/GARCH models?

I am slightly irritated about weak stationarity in connection to ARCH/GARCH models. I do not know the answer and I am not sure about it:

The basic question is:

Do we have to test weak stationarity before applying an ARMA-GARCH model?

Further on it can be said:

ADF and others test the mean equation, but this is not for the volatility equation, so what test do we have to use for the autocovariance-stationarity?

My thoughts:

Standard ARMA models assume the unconditional mean and unconditional variance to be constant. For ARMA-GARCH models this is also the case: The unconditional mean and unconditional variance need to be constant, whereas in case of the ARMA-GARCH models the conditional variance does not need to be constant.

It is correct that for the mean equation we may have to think about using a trend-stationary or difference-stationary model. But this is only concerning the mean equation, yes.

Conditional variance can be tested by testing for ARCH effects (Box-Ljung, Lagrange Multiplier).

So for ARMA-GARCH models we still need weak stationarity, since the unconditional mean and unconditional variance need to be constant. So I am not sure, but we have to test for weak stationarity before applying an ARMA-GARCH model? And especially with financial returns, do we also have to test for it? And which test do we use (and which command is implemented in R, so what command can you suggest?)

I know that if the unconditional variance is nonstationary (not finite and then it is also not constant) an integrated GARCH may be appropriate. But just because it is not constant I cannot say I use an integrated GARCH model?

I also know that for ARMA-GARCH processes all the "characteristical" roots lie outside the unit circle. So in case of a ARMA-GARCH(1,1) $\alpha_1+\beta_1<1$ is necessary. But I only see this after estimation? This is not a test for covariance stationarity?

EDIT: It basically pins down to (see the comments): How can I test the unconditional variance to be constant? I mean in order to apply a GARCH model I have to make sure that I have constant unconditional mean (ADF test and so) and I have to test for constant unconditional variance (how?). I know that I have to further make sure that I have nonconstant conditional variance for GARCH processes, otherwise having a constant conditional variance ARMA is sufficient (test for ARCH effects).

• The answer is yes. All the standard GARCH models assume that GARCH process is the stationary process. – mpiktas Nov 30 '13 at 9:41
• @mpiktas Mh, I knew that before. So I know that ARMA-GARCH processes need weak stationarity, but the question is about testing? So do I have to test before applying it? (Let's do not discuss about mean stationarity, since I know this is concerning the ARMA part, so the mean equation and I want to focus on the GARCH topic). – Jen Bohold Nov 30 '13 at 9:42
• @mpiktas As I said, I know that ARMA-GARCH processes are stationary processes, but the question is about testing? So do I have to test for weak stationarity (and what tests?) in order to apply an ARMA-GARCH process? Especially in case of financial returns? – Jen Bohold Nov 30 '13 at 9:44
• Yes you have to make sure that the series is stationary. Concerning your last question, no $\alpha_1+\beta_1<1$ ensures that the theoretical model is a valid GARCH model, but if the sum is larger than one, this might be an indication, that GARCH(1,1) model is not apropriate for the data. – mpiktas Nov 30 '13 at 9:45
• @mpiktas Yeah, but what stationarity? I know this, but how can I test the unconditional variance to be constant? I mean in order to apply a GARCH model I have to make sure that I have constant unconditional mean (ADF test and so) and I have to test for constant unconditional variance (how?) I know that I have to further make sure that I have nonconstant conditional variance for GARCH processes, otherwise having a constant conditional variance ARMA is sufficient. – Jen Bohold Nov 30 '13 at 9:48

Strict stationarity is the strongest form of stationarity. It means that the joint statistical distribution of any collection of the time series variates never depends on time. So, the mean, variance and any moment of any variate is the same whichever variate you choose. However, for day to day use strict stationarity is too strict. Hence, the following weaker definition is often used instead. Stationarity of order 2 which includes a constant mean, a constant variance and an autocovariance that does not depend on time. (second-order stationary or stationary of order 2). A weaker form of stationarity that is first-order stationary which means that the mean is a constant function of time, time-varying means to obtain one which is first-order stationary.

Using traditional stationarity tests such us PP.test (Phillips-Perron Unit Root Test), kpss test or Augmented Dickey-Fuller Tests are not adequate if you are to perform regression via other methods than ARIMA (due that in Arima the orders are fixed and that no other factors that produce non stationarity are included in the model). For non Arima cases stationarity tests in the frequency domain are more adequate.

Tests in the frequency domain : The Priestley-Subba Rao (PSR) test for nonstationarity (fractal package). Based upon examining how homogeneous a set of spectral density function (SDF) estimates are across time, across frequency, or both.

The test you refer to is a test also in the frequency domain (which tests a second order unit root test) where the wavelet looks at a quantity called βj(t) which is closely related to a wavelet-based time-varying spectrum of the time series (it is a linear transform of the evolutionary wavelet spectrum of the locally stationary wavelet processes of Nason, von Sachs and Kroisandt, 2000). So we see if βj(t) function varies over time or is constant by looking at Haar wavelet coefficients of the estimate so is stationary if all haar coefficients are zero (locits package).

There are other concerns about stationarity such us long range dependence, fractional integrated processes (ARFIMA) where the term d (differenciation) refers to long term memory processes.

The effect of higher order non stationarity, long term dependencies is that they are in effect reflected systematically in the errors of a regression, however its impact and thus validity of the regression is difficult to measure

From: "Financial Econometrics: From basics to advanced modeling techniques" Rachev, Mittnik et al. (Wiley 2006). You can use the following statistical test for your issue :

Sanso, A., Arago, V., and Carrion, J. (2004), “Testing for Changes in the Unconditional Variance of Financial Time Series,” Revista de Economia Financiera,4, 32–53.

As a result, you will know if the UNconditional variance of your time series can be considered constant or not (from a given confidence level).

Their test statistic can be easily implemented.

Enjoy!

As a practial "test" you could estimate your GARCH model on several subsamples and simply see if the unconditonal variance c*(1-(a+b))^-1 changes a lot or not.

I think it's imposible to prove if the unconditional variance is constant in his time serie, because you have only one realization of you underlying process. But someone here can to know how to extrapolate the conclusions to all possible realizations (and therefore to unconditional variance) with a single realization of the serie without using assumptions for ergodicity. I am not sure how this assumption works, but I would say that there is the key.

I believe you could use ADF test (unit root test) on the squared series for stationarity check of ARCH/GARCH models. Essentially, ARCH model is about the auto-correlation in squared Yt while ARMA model is about auto-correlation of Yt itself. ARCH model gives the equation below and note that if this holds, the unconditional variance is constant $\frac{w}{1-\alpha}$ . $$E(r_t^2|r_{t-1}, r_{t-2},...) = E(\sigma_t^2e_t^2|r_{t-1}, r_{t-2},...) = \sigma_t^2 = w + \alpha*r_{t-1}^2$$ So this can be thought of as an AR(1) process of $r_{t}^2$. Stationarity requires $\alpha <1$ so the ADF could be applied here while treating $r_{t}^2$ as $s_{t}$.

However here is some research pointing out that the unit root test "should be expected to have very poor size properties and so they attached little significance to the rejection of the unit root null". For detailed discussions, this paper by Wright (1999) compared the standard unit root test on volatility agaist the modified unit root approach proposed by Perron and Ng, which proved to be a better test method.

References

Wright, J. H. (1999). Testing for a unit root in the volatility of asset returns. Journal of Applied Econometrics, 309-318.

Standard ARMA models assume the unconditional mean and unconditional variance to be constant. For ARMA-GARCH models this is also the case

I don't think it is true, they're constant conditional variance models, see e.g. MATLAB's help on ARMA and GARCH. Whether you have to test before or after applying the model is up to you. We usually have a battery of standard tests, including ADF and Engle's ARCH, applied to our series during the data exploration phase. We never jump right into modeling before understanding the series, gathering literature, studying their basic time series properties such trends and cycles etc.

However, for the purposes of exposition, you can always apply the model then test the residuals for unit-root and homoscedasticity assumptions if they were implied, such as in constant conditional variance models.

• Are you claiming (in your first line) that ARMA-GARCH is a constant conditional variance model? That is not correct. – Richard Hardy Feb 2 '16 at 19:46
• @RichardHardy, simple AR(1): $x_t=c+\alpha x_{t-1}+\varepsilon_t$, where $\varepsilon_t\sim\mathcal{N}(0,\sigma^2)$. So, $Var(x_t|I_{t-1})=\sigma^2$, constant – Aksakal Feb 2 '16 at 19:56
• GARCH is explicitly a model for non-constant conditional variance; AR(1) in your example is just a special case which is not sufficient to prove the general case -- and your claim as phrased in the answer is about the general case (or am I reading it incorrectly?). Also, another question: you say that the quoted statement is not true; but unconditional mean and variance in ARMA-GARCH indeed are constant (unless ARMA has a unit root or GARCH is actually IGARCH). Did you mean to stress the latter special cases? – Richard Hardy Feb 2 '16 at 20:16