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I have read that the presence of autocorrelation detected in the log return can be removed by fitting the simplest plausible ARMA (p, q) model to the data. This autocorrelation was detected using a Ljung-Box test. Furthermore the autocorrelation detected in the squared log returns (using Ljung-Box), indicate that there exists conditional heteroskedasticity of the exchange rate returns series which could be removed by fitting the simplest plausible GARCH model to the ARMA filtered data.

Can someone tell me if this is the correct way how to go about testing the data whether we should fit an ARMA model to the data?

Furthermore can someone direct me to a good reference on why is it more efficient to fit an ARMA-GARCH model at once rather than first fitting an ARMA model, and then fitting a GARCH model on the residuals?

Anna
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  • I have answered a few related questions earlier, but do not have time to look them up right now. Just a brief idea about your last paragraph: think of fitting a regression model $y=\beta_0+\beta_1 x_1+\beta_2x_2+\varepsilon$ sequentially, i.e. by fitting $y=\gamma_0+\gamma_1 x_1+u$ first and then $u=\gamma_2 x_2+v$. Generally you will not get $\gamma_1=\beta_1$ nor $\gamma_2=\beta_2$. Fitting an ARMA-GARCH model sequentially is similar. – Richard Hardy Mar 02 '18 at 08:09
  • @RichardHardy Is the reasoning of the first paragraph correct? – Anna Mar 02 '18 at 08:19
  • Precisely because of what I said in the first comment, the sequential strategy you imply in your first paragraph is not the best way to go. Other than that the first paragraph looks OK. You may keep in mind that Ljung-Box test for autocorrelation does not work really well if there are ARCH effects in addition, and the ARCH tests do not like there being autocorrelation in the data. This is an unfortunate complication. See also the "Related" questions in the right column, e.g. [this](https://stats.stackexchange.com/questions/77925/procedure-for-fitting-an-arma-garch-model?rq=1). – Richard Hardy Mar 02 '18 at 09:16
  • Thank-you because I was reading that the Ljung-Box test shouldn't be applied to the actual raw data but only to the residuals. – Anna Mar 02 '18 at 10:01
  • Really? See [this](https://stats.stackexchange.com/questions/148004/testing-for-autocorrelation-ljung-box-versus-breusch-godfrey) and [this](https://stats.stackexchange.com/questions/6455/how-many-lags-to-use-in-the-ljung-box-test-of-a-time-series/205262#205262). – Richard Hardy Mar 02 '18 at 11:04

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