Non-parametric estimation of a time varying GARCH model

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Non-parametric estimation of a time varying GARCH model Neelabh Rohan 1 and T. V. Ramanathan 2 Department of Statistics and Centre for Advanced Studies University of Pune, 411 007, INDIA Abstract In this paper, a non-stationary time-varying GARCH (tvGARCH) model has been introduced by allowing the parameters of a stationary GARCH model to vary as functions of time. It is shown that the tvGARCH process is locally stationary in the sense that it can be locally approximated by stationary GARCH processes at fixed time points. We develop a two step local polynomial procedure for the estimation of the parameter functions of the proposed model. Several asymptotic properties of the estimators have been established including the asymptotic optimality. It has been found that the tvGARCH model performs better than many of the standard GARCH models for various real data sets. Mathematical Subject classification: 62M10, 62G05 Keywords: Local polynomial estimation, time-varying GARCH, volatility modelling. 1 Corresponding author Email: neelabh.stats@yahoo.co.in 2 Email: ram@stats.unipune.ac.in 2
1 Introduction The first decade of the 21 st century left the global economies grappling with the conse- quences of the financial crisis followed by an uninvited rash of currency wars. Many of the emerging economies started receiving large capital inflows that have the potential to destabilizing the economy. Perhaps, the most deleterious consequence of capital inflows has been the strengthening of domestic currency, which can lead to a loss in export com- petitiveness. This, in turn led to currency wars-the phenomenon of several emerging and developed countries intervening in currency market simultaneously in order to ensure that their currency will not be the only one that appreciates. Such a phenomenon may induce instability and hence non-stationarity in the bilateral exchange rate volatility process, implying the failure of standard stationary volatility models. In this paper, we address this problem by considering a GARCH model with time varying parameters. Non-stationary volatility models have got considerable attention recently, see for ex- ample Mercurio and Spokoiny (2004), Mikosch and Starica (2004), Starica and Granger (2005), Dahlhaus and Subba Rao (2006), Amado and Terasvirta (2008), Fryzlewicz, Sap- atinas and Subba Rao (2008) and Chen and Hong (2009) and among others. Dahlhaus and Subba Rao (2006) proposed a time-varying ARCH (tvARCH) model for the volatility process by allowing the parameters of a stationary ARCH model to change slowly through time. Fryzlewicz et al. (2008) developed a least-squares estimation procedure for such a tvARCH model. We generalize the tvARCH model introduced by Dahlhaus and Subba Rao (2006) to time varying GARCH (tvGARCH) by allowing the parameters of a stationary GARCH model to vary as functions of time. Dahlhaus and Subba Rao (2006) showed that the tvARCH model can be approximated by stationary ARCH processes locally. We extend their results to the tvGARCH model and show that a non-stationary tvGARCH process can be locally approximated by stationary processes at specific time points. Therefore, the tvGARCH model is asymp- totically locally stationary at every point of observation, but it is globally non-stationary because of time-varying parameters. Such an approximation further helps us in deriving the asymptotic distribution of the estimators. An alternative approach to incorporate non-stationarity in the volatility process is the varying coefficient GARCH model (see Číˇzek and Spkoiny (2009) and references therein). The estimation of a varying coefficient GARCH model requires the search for local time 3
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Page 11 and 12: 4 Asymptotic results Towards provin
Page 13 and 14: Notations. bj = bj(u0) = Bias(ˆαj
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Page 25 and 26: Hence the result can be easily prov
Page 27 and 28: The variance expression given in Th
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Page 31 and 32: 31 Table 1: Summary statistics of t
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Page 37 and 38: volatility volatility 0 10 20 30 0

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