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I'm looking at a time series that appears to be white noise. The ACF/PACF are in the test bounds. Applying the Ljung-Box test for various (maximum) lags gives me high p-values (i. e. I cannot reject the (joint) null hypothesis of zero autocorrelation). Using AIC favors a MA(1) model but the improvement in fit vs. white noise seems marginal.

However, looking at the squared series (since the mean is approx. zero the residuals are the series itself) gives me a significant autocorrelation for lag one and the Ljung-Box test yields small p-values. This points to volatility clustering, right?

So I tried to fit an "ARMA(0,0,0) + GARCH(1,1)" model (i. e. a GARCH(1,1 model). The gives me the following output:

Error Analysis:
    Estimate  Std. Error  t value Pr(>|t|)    
mu     1.665e-04   1.347e-03    0.124   0.9016    
omega  1.085e-04   6.421e-05    1.690   0.0910 .  
alpha1 1.301e-01   5.235e-02    2.486   0.0129 *  
beta1  7.550e-01   1.055e-01    7.155 8.39e-13 ***

As you can see the only the alpha1 and beta1 coefficients are significant at the 5% level.

My question would be how to proceed. Just using white noise (no ARCH/GARCH) to model the series seems inappropriate given the ACF/PACF of the squared residuals and the Ljung-Box test results. However, the GARCH(1,1) model doesn't seem to provide a very good fit either.


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