Asymptotic properties of QMLE for seasonal threshold GARCH model with periodic coefficients
Tóm tắt
Periodic models for volatility process constitute an alternative representation for the seasonal patterns observed in data exhibits a strong seasonal volatility driven by periodic coefficients of high and law variation. Moreover, these varying-parameters can arise also when seasonality is incorporated into the theory of economic decision-making So, in this paper, we propose an extension of time-invariant coefficients threshold GARCH (TGARCH) processes to periodically time-varying coefficients (PTGARCH) one. This parametrization allows us to describe the dynamic volatility through a TGARCH model within each regime (or season), and therefore a new stylized fact that characterize the volatility by seasonal patterns. Hence, theoretical probabilistic properties of this model are derived. The necessary and sufficient conditions which ensure the strict stationarity and ergodicity (in periodic sense) solution of PTGARCH are given. We extend the standard results of the popular quasi-maximum likelihood estimator (QMLE) for estimating the unknown parameters in model. More precisely, the strong consistency and the asymptotic normality of QMLE are studied for the cases when the innovation process is an i.i.d (Strong case) or is not (Semi-strong case). A Monte Carlo study is further conducted to examine the finite sample properties of the QMLE. The simulation results reveal that the QMLE is approximately unbiased and consistent for modest sample sizes when the stationarity conditions hold. Empirical work on the exchange rates of the Algerian Dinar against the single European currency (Euro) shows that our approach also outperforms and fits the data well.
Tài liệu tham khảo
Aknouche A, Bibi A (2008) Quasi-maximum likelihood estimation of periodic \(GARCH\) and periodic \(ARMA-GARCH\) processes. J Time Ser Anal 29(1):19–45
Billingsley P (1961) The Lindebergh–Lévy theorem for martingales. Proc Am Math Soc 12:788–792
Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econom 31(3):307–327
Bollerslev T (2008) Glossary to ARCH (GARCH). In: Bollerslev T, Russell JR, Watson M (eds) Volatility and time series econometrics: essays in honour of Robert F. Angel. Oxford University Press, Oxford, p 2008
Boyles RA, Gardner WA (1983) Cycloergodic properties of discrete-parameter nonstationary stochastic processes. IEEE Trans Inf Theory 29:105–114
Bougerol P, Picard N (1992a) Strict stationarity of generalized autoregressive processes. Ann Probab 20(4):1714–1730
Bougerol P, Picard N (1992b) Stationarity of \(GARCH\) processes and of some nonnegative time series. J Econom 52:115–127
Chan NH (2009) Statistical inference for non-stationary \(GARCH(p, q)\) models. Electron J Stat 3:956–992
Engle RF (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica 50(4):987–1008
Escanciano JC (2009) Quasi-maximum likelihood estimation of semi-strong \(GARCH\) models. Econom Theory 25:561–570
Francq C, Zakoïan JM (2009) Testing the nullity of GARCH coefficients: correction of the standard tests and relative efficiency comparisons. JASA 104:313–324
Francq C, Zakoïan JM (2004) Maximum likelihood estimation of pure \(GARCH\) and \(ARMA-GARCH\) processes. Bernoulli 10(4):605–637
Francq C, Zakoïan JM (2013) Inference in nonstationary asymmetric \(GARCH\) models. Ann Stat 41(4):1970–1998
Francq C, Zakoîan J-M (2010) GARCH models: structure, statistical inference and financial applications. Wiley, New York
Gladyshev EG (1961) Periodically correlated random sequences. Soviet Math Dokl 2:385–388
Glosten LR, Jagannathan R, Runkle D (1993) On the relation between the expected value and the volatilityof the nominal excess return on stocks. J Finance 48(5):1779–1801
Gonzalez-Rivera G, Drosi FC (1999) Efficiency comparisons of maximum likelihood-based estimators in \(GARCH\) models. J Econom 93(1):93–111
Hamdi F, Souam S (2013) Mixture periodic \(GARCH\) models: applications to exchange rate modeling. In: Proceeding of the 5-th international conference on modeling, simulation and optimization (ICMSAO), Hammamet, 28-30 Apr. 2013, pp 1–6
Hamadeh T, Zakoïan JM (2011) Asymptotic properties of \(LS\) and \(QML\) estimators for a class of nonlinear \(GARCH\) processes. J Stat Plann Inference 141:488–507
Jensen ST, Rahbak A (2004a) Asymptotic normality of the \(QMLE\) estimator of \(ARCH\) in the nonstationary case. Econometrica 72(2):641–646
Jensen ST, Rahbak A (2004b) Asymptotic inference for nonstationary \(GARCH\). Econometric Theory 20(6):1203–1226
Kingman JFC (1973) Subadditive ergodic theory. Ann Probab 1(6):883–909
Lee Sw, Hansen BE (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econom Theory 10(1):29–52
Franses PH, Paap R (2000) Modelling day-of-the-week seasonality in the S&P 500 index. Appl Financ Econ 10(5):483–488
Pan J, Wang H, Tong H (2008) Estimation and tests for power-transformed and threshold GARCH models. J Econom 142(1):352–378
Stavros S (2016) Value-at-Risk and backtesting with the \(APARCH\) model and the standardized Pearson type \(IV\) distribution. arXiv:1602.05749
Wang H, Pan J (2014) Normal mixture quasi maximum likelihood estimation for non-stationary \(TGARCH(1,1)\) models. Stat Probab Lett 91:117–123