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First my assumption was that ARMA models take only the autocorrelation of the time series into consideration but not of the error terms (wrong!). But this assumption is wrong!

As the within ARMA models the autocorrelation of the implicit error terms (approx trough the residuals) disappear. An ARMA model with autocorrelated error terms would be miss-specified.

E.g. within ARMA-ARCH or ARMA-GARCH models the observed variables have ARMA white noise errors, but they are ARCH. Therefore, they are dependent, within higher moments.

I do not really understand the last part of it. Tried to explain myself it mathematically but do not really find a solution. Can someone explain me how that is possible?enter image description here


Update:

Thank you for your answer @Richard.

I guess I somehow forgot what the MA term means again.

So just to be clear on the MA part again:

As the MA is a linear combination of the past white noise terms and is useful to explain the serial correlation in residuals in isolation (but only if they are white noise). Often using the MA model we can explain serial correlation in short lags but the problem is that volatility clustering and long-memory effects stay.

So now in (G)ARCH models, the variance is allowed to be serially correlated and also be dependent on past lags. Apart from that all stays the same.

So in the ARMA model they residuals can be dependent but they need to be white noise. In ARCH/GARCH the residuals can also be dependent but they do not need to be white noise (heteroskedastic, volatility clustering). Is that right?

Side question about definitions:

  • variance/error terms: if I talk about the theoretical model
  • residuals: after fitting a specific model

--> if we test a hypothesis we test on the error terms/variance not on the residuals?

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  • $\begingroup$ Well-specified ARMA models must have zero-mean, white noise residuals. White noise means constant mean, constant variance and zero covariance at all lags except lag number 0. There is a difference between white noise and i.i.d. ARCH or GARCH processes are an example that are white noise but not i.i.d. They are dependent, but they satisfy the conditions of white noise. Given all that, what is unclear to you? $\endgroup$ Sep 29, 2023 at 12:18
  • $\begingroup$ Thank you @Richard. Unfortunately, I guess I am still confused about the MA part. Is the MA part not random white noise shocks? e.g. a MA(1) x_t = w_t + beta_1*w_t-1 where w_t is white noise, with mean = 0 and constant variance $\endgroup$ Oct 4, 2023 at 18:14
  • $\begingroup$ The shocks $w_t$ are white noise, but the MA process $x_t$ is not. Perhaps I am just confused by your phrasing? $\endgroup$ Oct 4, 2023 at 19:01
  • $\begingroup$ What do you mean with 𝑥_t is not white noise? You mean it is not white noise in an ARMA-(G)ARCH but white noise in only ARMA right? If I think about an MA(1)-ARCH process: x_t = e_t + beta_1*e_t-1 where x_t is the time series, e_t is white noise with mean=0 and constant variance (for now). So verbally, we take the first lag of the past white noise term and multiply this with a coefficient. But now by adding the ARCH process: e_t=w_t*sigma_t where w_t is white noise with 0 mean and constant variance. However, the conditional variance sigma_t^2 may change over time. $\endgroup$ Oct 7, 2023 at 13:04
  • $\begingroup$ So what we see, in the MA(1)-ARCH model is: that the e_t is still by definition serially uncorrelated, however the conditional variance in the variance of e_t is not constant but rather can change over time. This means to have a ARMA-GARCH model richt? Referring to 'e_t is by definition uncorrelated' because we should not see any correlation in the residuals (by e.g. plotting the ACF) but we might see some correlation when plotting residuals^2 $\endgroup$ Oct 7, 2023 at 13:10

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Well-specified ARMA models must have zero-mean, white noise residuals. White noise means constant mean, constant variance and zero covariance at all lags except lag number 0. There is a difference between white noise and i.i.d. ARCH or GARCH processes are an example that are white noise but not i.i.d. They are dependent, but they satisfy the conditions of white noise.

Regarding your update, everything looks good except for the MA is a linear combination of the past white noise terms and is useful to explain the serial correlation in residuals in isolation (but only if they are white noise). You cannot have both MA and white noise simultaneously.

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