Detecting level shifts in ARMA-GARCH (1,1) Models

Abstract

The purpose of this article is to present a new method to detect level shifts in the context of conditional heteroscedastic models. First, we define precisely what type of outlier we are referring to, a concept that has been scarcely touched in the field of GARCH (1,1) models, and then we go on to present our methodology based on the nature of the Lagrange multiplier tests. The validity and efficiency of the proposed procedure are demonstrated through different simulation experiments. To conclude, we present a practical application of the method to the time series of returns of US short-term interest rates.

Keywords:

• level outliers
• volatility outliers
• level shifts
• GARCH models
• LM tests

JEL Classification :

• C22
• C52

Mathematics Subject Classification :

• 62M10
• 91B28
• 91B84

Acknowledgements

The authors gratefully acknowledge the helpful observations and suggestions of the anonymous referees. This research has been supported by the Spanish Ministry of Education and Science and FEDR under Project SEJ2006-02328 of the Plan Nacional de Investigación Científıca, desarrollo e Innovación Technológica.

Notes

At present, the periodicity of the data is minimal, even at the level of carrying out a transaction 15.

There are other works that study the problems of outliers in ARCH models but, they are only centred on additive outliers. We can cite some of them 3 4 7,11–13,18–22,24 25 27 29 30 32,36–38.

See Catalán and Trívez 10, who present the bases of the methodology proposed and they are developed for the detection and treatment of additive and innovational outliers.

See 25 for the case of tests of additive outliers.

We saw in Table 2 that the effect of an LS-VO on a GARCH\! (1,1) was the following: , ∀ j>0; however, on distributing the variable , we can simplify the previous expression, as in Equation (8).

See 16, which analyses the validity and limitations of the asymptotic theory in interpreting the properties of the tests in finite samples.

The exact distribution of the statistical test may be complicated, but, in practice, asymptotic approximation is used 17.

In the case of level outlier tests, this process requires a quite complex computation, and, moreover on occasions, the definition of the test can create adverse statistical results. To minimize these effects, the following simplification was carried out: the value of the test is only calculated for the observation in which the corresponding VO test reaches the maximum value. . The experiments carried out demonstrate that the tests tend to identify with a high degree the point at which the LS occurs, both for VO and LO; therefore, a distinction will be made between VOs or LOs once the LS has been identified at time k. Note that this is one of the possible simplifications that can be adopted.

On this occasion, we applied the tests only once, we do not correct the series and we re-applied the method iteratively as this is enough for the ends of the experiment.

See the pioneering work of Sargent 31 and Modigliani and Shiller 28 on the modelling of the time structure of interest rates.

Some recent proposals on the ARCH modelling of the heteroscedastic variance of short-term interest rates are Gray 23, Andersen and Lund 1 and Bali 4.

This was one of the processes that provided greater adjustment.