Goodness-of-fit tests for Log-GARCH and EGARCH models


This paper studies goodness-of-fit tests and specification tests for an extension of the Log-GARCH model, which is both asymmetric and stable by scaling. A Lagrange-multiplier test is derived for testing the extended Log-GARCH against more general formulations taking the form of combinations of Log-GARCH and exponential GARCH (EGARCH). The null assumption of an EGARCH is also tested. Portmanteau goodness-of-fit tests are developed for the extended Log-GARCH. An application to real financial data is proposed.

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Fig. 1


  1. 1.

    This effect, typically observed on most stock returns series, means that negative returns have more impact on the volatility than positive returns of the same magnitude.

  2. 2.

    Indeed, as remarked by a referee, a practitioner is essentially faced by three choices: (a) leave returns untransformed, i.e., set \(c = 1\), (b) express returns in terms of percentages, i.e., set \(c = 100\), or (c) express returns in terms of basis points, i.e., set \(c = 10,000\). Clearly, it is desirable that the dynamics of the volatility model be not affected by the choice of c.


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The authors would like to thank the referees for their helpful comments. Christian Francq and Jean-Michel Zakoïan also gratefully acknowledge financial support from the Ecodec Labex.

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Correspondence to Christian Francq.

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Francq, C., Wintenberger, O. & Zakoïan, JM. Goodness-of-fit tests for Log-GARCH and EGARCH models. TEST 27, 27–51 (2018).

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  • LM tests
  • Invertibility of time series models
  • Log-GARCH
  • Portmanteau tests
  • Quasi-maximum likelihood

Mathematics Subject Classification

  • 62M10
  • 62P20