GARCH models without positivity constraints: Exponential or Log GARCH - PowerPoint PPT Presentation

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations GARCH models without positivity constraints: Exponential or Log GARCH ? ∗ C. Francq, O. Wintenberger and J-M. Zakoïan CREST and Lille 3 University, France CREST and Dauphine Univ., France CREST and Lille 3 University, France MSDM 2013, March 14-15 ∗ Supported by the project ECONOM&RISK (ANR 2010 blanc 1804 03) Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Objectives Log-GARCH and EGARCH are two models for the log-volatility. Probabilistic properties and estimation of asymmetric Log-GARCH models. Differences and similarities between the log-GARCH and EGARCH models. Testing log-GARCH against EGARCH, or the reverse. Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations The standard GARCH model Engle (1982), Bollerslev (1986) Standard GARCH models: � ǫ t = σ t η t , ( η t ) t ∈ Z iid (0,1) t = ω + � q t − i + � p σ 2 i = 1 α i ǫ 2 j = 1 β j σ 2 t − j with positivity constraints ω > 0 , α i , β j ≥ 0 . Under relevant conditions on the parameter, the model is able to mimic some properties of the financial returns: this is a conditionally heterosckedastic white noise; the squares are positively autocorrelated; the model generates volatility clustering; the marginal distribution can be leptokurtic. Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Two drawbacks of the standard GARCH Do not allows for asymmetries in volatility (leverage 1 effects): decreases of prices have an higher impact on the future volatility than increases of the same magnitude. Leverage effects The positivity constraints on the volatility coefficients entail 2 numerical and statistical difficulties ( e.g. non standard asymptotic distribution of constrained estimators at the boundary of the parameter space). ⇒ numerous extensions (see Bollerslev "Glossary to ARCH (GARCH)", 2009) Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Two log-volatility models ǫ t = σ t η t , η t iid (0,1) Exponential-GARCH model: Nelson (1991) ω + � p t − j + � ℓ log σ 2 j = 1 β j log σ 2 i = 1 γ i + η + t − i + γ i − η − = t t − i Asymmetric log-GARCH model: ω + � p log σ 2 j = 1 β j log σ 2 = t t − j � � + � q log ǫ 2 α i + 1 { ǫ t − i > 0} + α i − 1 { ǫ t − i < 0} t − i i = 1 The log-GARCH model has been introduced by Geweke (1986), Pantula (1986) and Milhøj (1987) (see Sucarrat and Escribano (2010) for the symmetric case). Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Basic features of the log-GARCH model Symmetric log-GARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + α log ǫ 2 = t t − 1 ω + ( α + β )log σ 2 t − 1 + α log η 2 = t − 1 . Symmetric EGARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + γ | η t − 1 | = t � � � � ǫ t − 1 ω + β log σ 2 � � t − 1 + γ � . = � σ t − 1 No positivity constraint on the parameters, but | η t | > 0 . Easily invertible, contrary to the EGARCH. The volatility is not bounded below by a strictly positive constant. Small values have persistent effects on volatility. Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Basic features of the log-GARCH model Symmetric log-GARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + α log ǫ 2 = t t − 1 ω + ( α + β )log σ 2 t − 1 + α log η 2 = t − 1 . Symmetric EGARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + γ | η t − 1 | = t � � � � ǫ t − 1 ω + β log σ 2 � � t − 1 + γ � . = � σ t − 1 No positivity constraint on the parameters, but | η t | > 0 . Easily invertible, contrary to the EGARCH. The volatility is not bounded below by a strictly positive constant. Small values have persistent effects on volatility. Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Basic features of the log-GARCH model Symmetric log-GARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + α log ǫ 2 = t t − 1 ω + ( α + β )log σ 2 t − 1 + α log η 2 = t − 1 . Symmetric EGARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + γ | η t − 1 | = t � � � � ǫ t − 1 ω + β log σ 2 � � t − 1 + γ � . = � σ t − 1 No positivity constraint on the parameters, but | η t | > 0 . Easily invertible, contrary to the EGARCH. The volatility is not bounded below by a strictly positive constant. Small values have persistent effects on volatility. Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Basic features of the log-GARCH model Symmetric log-GARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + α log ǫ 2 = t t − 1 ω + ( α + β )log σ 2 t − 1 + α log η 2 = t − 1 . Symmetric EGARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + γ | η t − 1 | = t � � � � ǫ t − 1 ω + β log σ 2 � � t − 1 + γ � . = � σ t − 1 No positivity constraint on the parameters, but | η t | > 0 . Easily invertible, contrary to the EGARCH. The volatility is not bounded below by a strictly positive constant. Small values have persistent effects on volatility. Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Basic features of the log-GARCH model Symmetric log-GARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + α log ǫ 2 = t t − 1 ω + ( α + β )log σ 2 t − 1 + α log η 2 = t − 1 . Symmetric EGARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + γ | η t − 1 | = t � � � � ǫ t − 1 ω + β log σ 2 � � t − 1 + γ � . = � σ t − 1 No positivity constraint on the parameters, but | η t | > 0 . Easily invertible, contrary to the EGARCH. The volatility is not bounded below by a strictly positive constant. Small values have persistent effects on volatility. Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Basic features of the asymmetric log-GARCH model Asymmetric log-GARCH(1,1): � � log σ 2 ω + β log σ 2 log ǫ 2 = t − 1 + α + 1 { ǫ t − 1 > 0} + α − 1 { ǫ t − 1 < 0} t t − 1 � � log σ 2 = ω + α + 1 { η t − 1 > 0} + α − 1 { η t − 1 < 0} + β t − 1 � � log η 2 α + 1 { η t − 1 > 0} + α − 1 { η t − 1 < 0} t − 1 . + Asymmetric EGARCH(1,1): log σ 2 ω + β log σ 2 t − 1 + γ + η + t − 1 + γ − η − t − 1 . = t Asymmetric random persistence parameter. Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Effect of a small value 5 4 3 2 σ t 2 GARCH EGARCH 1 Log−GARCH 0 η 50 = 1 η 150 = 1 η 201 ≈ 0 η 251 = 1 η 351 = 1 0 GARCH: σ 2 t = 0.06 + 0.09 ǫ 2 t − 1 + 0.89 σ 2 t − 1 log-GARCH: log σ 2 t = 0.033 + 0.03log ǫ 2 t − 1 + 0.93log σ 2 t − 1 EGARCH: log σ 2 t = 0.044 + 0.3 | η t − 1 |+ 0.9log σ 2 t − 1 Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Effect of a sequence of small values 5 4 GARCH EGARCH 3 2 Log−GARCH σ t 2 1 0 η 50 ≈ 0 η 150 ≈ 0 η 251 = 1 η 351 = 1 0 GARCH: σ 2 t = 0.06 + 0.09 ǫ 2 t − 1 + 0.89 σ 2 t − 1 log-GARCH: log σ 2 t = 0.033 + 0.03log ǫ 2 t − 1 + 0.93log σ 2 t − 1 EGARCH: log σ 2 t = 0.044 + 0.3 | η t − 1 |+ 0.9log σ 2 t − 1 Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Estimating and testing the Log-GARCH Numerical illustrations Effect of a large value 5.5 GARCH 4.5 EGARCH Log−GARCH 2 σ t 3.5 2.5 η t = 1 η t = 1 η t = 3 η t = 1 η t = 1 0 GARCH: σ 2 t = 0.06 + 0.09 ǫ 2 t − 1 + 0.89 σ 2 t − 1 log-GARCH: log σ 2 t = 0.033 + 0.03log ǫ 2 t − 1 + 0.93log σ 2 t − 1 EGARCH: log σ 2 t = 0.044 + 0.3 | η t − 1 |+ 0.9log σ 2 t − 1 Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?

Probabilistic properties of the log-GARCH Stationarity conditions Estimating and testing the Log-GARCH Existence of log-moments Numerical illustrations Existence of moments Probabilistic properties of the log-GARCH 1 Stationarity conditions Existence of log-moments Existence of moments Estimating and testing the Log-GARCH 2 Numerical illustrations 3 Francq, Wintenberger and Zakoïan Exponential or Log GARCH ?