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Wavelet Estimation of Gegenbauer Processes: Simulation and Empirical Application

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Abstract

The aim of this paper is to estimate the parameters of a stationary Gegenbauer process using a wavelet methodology where the selection of the orthonormal basis is given by generalized variance portmanteau test. Two other maximum likelihood estimators, including the Whittle and the wavelets—Whitcher (Technometrics 46:225–238, 2004) estimators, are also considered. We have shown by Monte-Carlo experiments that the new selection procedure improves considerably the Whittle and Whitcher estimators. Moreover, to assess the impact of volatility in the estimation methods, we assumed that the innovations \(\varepsilon _{t}\) are generated by univariate GARCH process. Simulation experiments show that the wavelets estimators perform better under most situations than the Whittle estimator. We then applied this new selection method to the consumer price index in monthly frequencies for the United States and find that this is more appropriate for forecasts.

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Notes

  1. This section draws heavily on Boubaker and Boutahar (2011) and Boubaker and Péguin-Feissolle (2013).

  2. The wavelet function (filter) proceeds as a special filter possessing specific properties, such that (i) it integrates to zero, i.e., \( \sum _{l=0}^{L-1}h_{l}=0\), (ii) has unit energy, i.e., \( \sum _{l=0}^{L-1}h_{l}^{2}=1\) and (iii) is orthogonal to its even shifts, i.e., \(\sum _{l=0}^{L-1}h_{l}h_{l+2n}=0,\) for all nonzero integers \(n.\)

  3. For Daubechies wavelets, the number of vanishing moments is half the filter length.

  4. A robust theoretical framework for critically sampled wavelet transformation is Mallat’s Multiresolution Analysis (see Mallat 1989) which is a design method underlying the conception of the DWT and the construction of the wavelet bases.

  5. The portmanteau statistic introduced by Box and Pierce (1970) is \({\hat{Q}_{m}}=N\sum \nolimits _{\varkappa =1}^{m}{\hat{ r}_{\varkappa }^{2}}\), and that of introduced by Ljung and Box (1978) is given by \({\hat{Q}_{m}}=N\left( {N+2}\right) \sum \nolimits _{\varkappa =1}^{m}{{ {\left( {N-\varkappa }\right) }^{-1}}\hat{r}_{\varkappa }^{2}}\).

  6. Whitcher (2001) and Collet and Fadili (2005) propose a method for simulating of Gegenbauer processes using the wavelet packets transform.

  7. We also used others values of \(m\). We observe that the sensitivity of the results to the value of \(m\) is very small. Thus, the choice of \(m\) is arbitrary. The different results are supplied upon request.

  8. We conduct the same simulations using \(D(L)\) and \(LA(L)\) filters and we find that the \(MB(L)\) performs better the simulations than \(D(L)\) and \(LA(L)\) regardless of the method of selection of the best basis. Here, we do not report the results, these are available upon request.

  9. Other simulations with other different parameters yield similar conclusion.

  10. A preliminary analysis to test the stationarity, using the ADF (Dickey and Fuller 1979, 1981), the PP (Phillips and Perron 1988) and the KPSS (Kwiatkowski et al. 1992) tests of the series in level shows evidence of presence of unit roots. So, we consider the log CPI in first difference.

  11. We note that we do not find evidence of time-varying volatility. Therefore, we do not estimate the GARCH-type dynamics for conditional variance.

  12. MSE: the Mean Square Error and MAPE: the Mean Absolute Prediction Error expressed as a percentage.

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  1. IPAG LAB, IPAG Business School, 184 boulevard Saint-Germain, 75006, Paris, France

    Heni Boubaker

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  1. Heni Boubaker
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Boubaker, H. Wavelet Estimation of Gegenbauer Processes: Simulation and Empirical Application. Comput Econ 46, 551–574 (2015). https://doi.org/10.1007/s10614-014-9471-6

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