Summary
The paper presents a procedure based on the EM algorithm for the indirect estimation of the parameters of BiLinear GARCH (BL-GARCH) models. BL-GARCH generalize the class of GARCH models by considering interactions of past shocks and volatilities in the conditional variance equation. In this way the response of the conditional variance to past information becomes asymmetric allowing to account for the so called leverage effect, typically characterizing the behaviour of financial time series. The results of an application to a time series of stock market returns are presented.
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Notes
2Note that, for the BL-GARCH(1,1) model, the positive definiteness of R1 is a necessary and sufficient condition for the positivity of \(h_{t}^{2}\)
3As in Storti & Vitale (2003), for the sake of simplicity and for notational convenience we have limited our attention to the case in which r = s. This does not imply any loss of generality since the extension of this result to the more general case in which r≠s is straightforward.
4In the state-space models jargon, filtering is the operation of recursively estimating the current state vector given past and present information. In this sense, the filtering step implies only a forward pass through the data. Differently, smoothing implies the estimation of the current state vector given the whole stretch of data. In Gaussian linear state-space models, this can be achieved by means of the Kaiman smoother combining a filtering step with a back-filtering step performed on data in reversed order. In this sense, the smoothing step implies a forward and a backward pass through the data.
5The results, which have not been reported here for the sake of brevity, are available upon request.
6However, the test is general and can be readily applied to higher order models. For more details see Storti (2002).
7The algorithm took approximately 11 seconds to converge on a laptop PC equipped with a Intel Pentium IV processor at 2.0 Ghz. The code used was written in MATLAB.
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The paper has been partially supported by a grant from MIUR, Italy, within the project Modelli Stocastici e Metodi di Simulazione per l’Anaiisi di Dati Dipendenti, 2000.
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Storti, G., Vitale, C. Likelihood inference in BL-GARCH models. Computational Statistics 18, 387–400 (2003). https://doi.org/10.1007/BF03354605
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DOI: https://doi.org/10.1007/BF03354605