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I estimated ARIMA model with daily gold time series. The residuals' corelogram is flat but its squared is not flat. Already I tried eVİEWS heterodasticity >> arch effect and ı found prob value 0.00 so there is heterodasticity. Can ı continue with ARIMA?

https://yadi.sk/i/rT6UHvOud338Xw 2009q3-2018 work days

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3 Answers 3

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Depending on the estimator, it is not necessarily a problem.

In the context of OLS the Gauss-Markov theorem will not apply, this means that the standard errors will be biased. However, the OLS estimator itself is consistent in the presence of heteroscedastic errors. A quick-fix for your problem could be to use a "robust" estimator of the standard errors, see e.g. the wikipedia page on the subject.

A better fix would be to actively model the heteroscedasticity in the errors, e.g. using a GARCH model. One often finds that the GARCH(1,1) model is sufficient to remove ARCH effects from financial data, such as your return series for gold prices, see e.g. the following working paper Hansen & Lunde 2001 (or the published version, Hansen & Lunde 2005)

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  • $\begingroup$ Thx for your answer. Finally I can continue with ARIMA ıf ı dont want to make better my model? :) Because this is my master theises and last day I noticed this. I am afraid of my teacher will ask why did you continue with arma. So ı search powerful evidence , reference .It must not that ıt is your choice. $\endgroup$
    – murat tuna
    Aug 5, 2019 at 12:07
  • $\begingroup$ Note that OLS is not applicable to ARIMA or ARMA models unless the MA order is 0. $\endgroup$ Aug 5, 2019 at 12:11
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    $\begingroup$ Thank you for the clarification Richard. Murat, you can proceed with your ARIMA model. As stated above the presence of ARCH effects in your residuals will not affect the estimates themselves. However, if you are interested in conducting inference, you should look into using either the robust standard errors, or use a GARCH-type model to account for the heteroscedasticity, both of which should be relatively straight forward to implement depending on which statistical software you use. $\endgroup$
    – seims
    Aug 5, 2019 at 12:20
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If, after estimating any time-series model including ARIMA, your estimated residuals are not independent (so in this case their squares are autocorrelated), then this likely means that there exists a better specification for the model, i.e. maybe you are not specifying the conditional mean structure appropriately or you are making the wrong assumption about the shape of the conditional covariances, or all of them.. If I understand your situation correctly, in this case since the squared innovations are correlated and the simple innovations are not, I would say the second one is likely.. therefore maybe you are right using an Arima because the true process does follow an Arima in its mean but you need to specify other parts of the model better. If your conditional mean specification of the Arima removes the non-stationarity in the process and the autocorrelation in its first difference, then it means that you are correctly specifying the conditional mean of the process and that the true problem lies elsewhere. Here likely in the conditional variance.

Try with simulated data and check that, when you don’t mispecify the model to be estimated, then retrieving the innovations leads to 1) uncorrelated innovations 2) uncorrelated squared innovations 3) innovations that follow the distribution that generate the data (so for example if you have simulated normal shocks then the estimated innovations/shocks should proxy a normal distribution, look for simplicity at a QQ Plot or test). Notice that here by innovations I mean the standardized residuals of the model (standardized based on the conditional mean and variance structures forecasted for each t)

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  • $\begingroup$ Thx for answer. My estimated residuals are independent but theirs square is not independent $\endgroup$
    – murat tuna
    Aug 5, 2019 at 12:03
  • $\begingroup$ Exactly.. try to see whether using a Garch on the Error term (or any other conditional covariance model) removes the significant correlation in the squared residuals (likely it will happen as long as you correctly specify the model for the conditional covariance, I.e. you choose the right form of the equation, with the right number of lags and the right model..) $\endgroup$
    – Fr1
    Aug 5, 2019 at 12:06
  • $\begingroup$ Yes it is possible to try but ı dont have time for this. I search only a powerful reference to show. I will write or say "according to ...." I continued with ARIMA. Thank you again. $\endgroup$
    – murat tuna
    Aug 5, 2019 at 12:10
  • $\begingroup$ You can do so, because, in the case where your Arima specification was faulty, then you would likely have correlation between non-squared residuals, which is not your case.. $\endgroup$
    – Fr1
    Aug 5, 2019 at 12:13
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    $\begingroup$ Don’t worry, I think you have understood.. maybe the language I am using is too complicated.. but the answer to the question “can I still use arima?” Is yes.. maybe you’d better check also whether the residuals are integrated of order 0, so that if the series is integrated of order 1, your estimated arima removes the non-stationarity in the mean $\endgroup$
    – Fr1
    Aug 5, 2019 at 12:26
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I would not trust a model with evidented heteroscedastic errors ... thus my preferred approach is to .... (AFTER adjusting for pulses,level shifts,seasonal pulses and local time trends i.e. adjusting the conditional mean structure ) pursue this thread .

To treat heteroscedasticity consider employing the GLS approach suggested by TSAY http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html . The weights he generates lead directly to Weighted Least Squares. I have had success with this feature which I incorporated into AUTOBOX, a commercial time series package to treat deterministic change points in model error variance AFTER validating that the model parameters didn't change over time which surprisingly is generally ignored elsewhere !

If doesn't resolve your problem then

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  • $\begingroup$ I use daily gold price (LMBA troy/ons ) so I cant seasonal adjustment.Some sources says daily time series dont have seasonal. Second the series is stationary at 1.difference witl all options (none,trend,intercept.) I try different models but there is not excellent model. Already ıf I am not curious about squared of residuals correlogram there will npt be problem. I think it is enough to look only correlogram of residuals ?On the other hand GLS method isnt give permission by Eviews to estimate ARMA.Thx for your help $\endgroup$
    – murat tuna
    Aug 5, 2019 at 11:59
  • $\begingroup$ post your data and I will try and help further .... $\endgroup$
    – IrishStat
    Aug 5, 2019 at 12:16
  • $\begingroup$ how can ı post my data $\endgroup$
    – murat tuna
    Aug 5, 2019 at 12:18
  • $\begingroup$ I should know BUT I don't so email it to me at my contact email address . By the way time series analysis requires observations for EVERY DAY thus if you are missing days due to holidays ...then fill in estimated values via interpolation. Also specify the beginning date. $\endgroup$
    – IrishStat
    Aug 5, 2019 at 12:26
  • $\begingroup$ I added up , thank you again $\endgroup$
    – murat tuna
    Aug 5, 2019 at 12:39

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