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I applied Ljung-Box test for squared residuals for different model specifications (interecept only; ar(1); ma(1); arma(1,1)...).

I always get that the first lag is statistically not significant with p-values of 0.5 and higher. At all other lags p-values are 0.000.

What could this statistical insignificance of the first lag mean? If it wasnt for the first lag it would be clear conditional HTSK but the first lag confuses me...

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If the test rejects the null hypothesis for all lags except one it means that the model is not capable of describing adequately the volatility process. One non-significant lag is poor evidence to exclude the presence of ARCH/GARCH effects, so my advice is: fit an ARCH/GARCH model and repeat the test, if this new specification improves the results therefore the ARCH/GARCH specification is needed to model your data.

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