GARCH Models. Eduardo Rossi University of Pavia. December Rossi GARCH Financial Econometrics / 50

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1 GARCH Models Eduardo Rossi University of Pavia December 013 Rossi GARCH Financial Econometrics / 50

2 Outline 1 Stylized Facts ARCH model: definition 3 GARCH model 4 EGARCH 5 Asymmetric Models 6 GARCH-in-mean model 7 The News Impact Curve Rossi GARCH Financial Econometrics / 50

3 Stylized Facts Stylized Facts GARCH models have been developed to account for empirical regularities in financial data. Many financial time series have a number of characteristics in common. Asset prices are generally non stationary. Returns are usually stationary. Some financial time series are fractionally integrated. Return series usually show no or little autocorrelation. Serial independence between the squared values of the series is often rejected pointing towards the existence of non-linear relationships between subsequent observations. Rossi GARCH Financial Econometrics / 50

4 Stylized Facts Stylized Facts Volatility of the return series appears to be clustered. Normality has to be rejected in favor of some thick-tailed distribution. Some series exhibit so-called leverage effect, that is changes in stock prices tend to be negatively correlated with changes in volatility. A firm with debt and equity outstanding typically becomes more highly leveraged when the value of the firm falls.this raises equity returns volatility if returns are constant. Black, however, argued that the response of stock volatility to the direction of returns is too large to be explained by leverage alone. Volatilities of different securities very often move together. Rossi GARCH Financial Econometrics / 50

5 ARCH model: definition The ARCH Model ARCH process (Bollerslev, Engle and Nelson, 1994) The process {ε t (θ 0 )} follows an ARCH (AutoRegressive Conditional Heteroskedasticity) model if and the conditional variance E t 1 [ε t (θ 0 )] = 0 t = 1,,... σ t (θ 0 ) Var t 1 [ε t (θ 0 )] = E t 1 [ ε t (θ 0 ) ] t = 1,,... depends non trivially on the σ-field generated by the past observations: {ε t 1 (θ 0 ), ε t (θ 0 ),...}. Rossi GARCH Financial Econometrics / 50

6 ARCH model: definition The ARCH Model Let {y t (θ 0 )} denote the stochastic process of interest with conditional mean µ t (θ 0 ) E t 1 (y t ) t = 1,,... µ t (θ 0 ) and σ t (θ 0 ) are measurable with respect to the time t 1 information set. Define the {ε t (θ 0 )} process by ε t (θ 0 ) y t µ t (θ 0 ). Rossi GARCH Financial Econometrics / 50

7 ARCH model: definition The ARCH Model It follows from eq.(5) and (5), that the standardized process with z t (θ 0 ) ε t (θ 0 ) σ t (θ 0 ) 1/ t = 1,,... E t 1 [z t (θ 0 )] = 0 Var t 1 [z t (θ 0 )] = 1 will have conditional mean zero (E t 1 [z t (θ 0 )] = 0) and a time invariant conditional variance of unity. Rossi GARCH Financial Econometrics / 50

8 ARCH model: definition The ARCH Model We can think of ε t (θ 0 ) as generated by ε t (θ 0 ) = z t (θ 0 ) σ t (θ 0 ) 1/ where ε t (θ 0 ) is unbiased estimator of σ t (θ 0 ). Let s suppose z t (θ 0 ) NID (0, 1) and independent of σ t (θ 0 ) because z t Φ t 1 χ (1). E t 1 [ ε t ] = Et 1 [ σ t ] Et 1 [ z t ] = Et 1 [ σ t ] = σ t Rossi GARCH Financial Econometrics / 50

9 ARCH model: definition The ARCH Model If the conditional distribution of z t is time invariant with a finite fourth moment, the fourth moment of ε t is E [ ε 4 ] [ ] [ ] [ ] [ ] t = E z 4 t E σ 4 t E z 4 t E σ [ ] [ ] t = E z 4 t E ε t where the last equality follows from E[σ t ] = E[E t 1 (ε t )] = E[ε t ] E [ ε 4 ] [ ] [ ] t E z 4 t E ε t by Jensen s inequality. The equality holds true for a constant conditional variance only. Rossi GARCH Financial Econometrics / 50

10 ARCH model: definition If z t NID (0, 1), then E [ ] zt 4 = 3, the unconditional distribution for εt is therefore leptokurtic E [ ε 4 ] t E [ ε 4 t ] /E [ ε t ] 3 3E [ ε ] t The kurtosis can be expressed as a function of the variability of the conditional variance. Rossi GARCH Financial Econometrics / 50

11 ARCH model: definition The ARCH Model In fact, if ε t Φ t 1 N ( ) 0, σt [ ] E t 1 ε 4 t E [ ε 4 ] t [ ] = 3E t 1 ε t ( ) = 3E [E t 1 ε ] t 3 { E [ ( )]} E t 1 ε [ ( )] t = 3 E ε t E [ ε 4 ] [ ( )] t 3 E ε [ ] t = 3E {E t 1 ε } t 3 { E [ ( )]} E t 1 ε t E [ ε 4 ] t = 3 [ E ( ε )] [ ] t + 3E {E t 1 ε } t 3 { E [ ( )]} E t 1 ε t E [ ] ε 4 E t [E (ε t )] = {E t 1 [ ε t ] } { E [ E t 1 ( ε t )]} k = [E (ε t )] = Var { [ ]} E t 1 ε t [E (ε t )] = Var { } σ t [E (ε t )]. Rossi GARCH Financial Econometrics / 50

12 ARCH model: definition The ARCH Model Another important property of the ARCH process is that the process is conditionally serially uncorrelated. Given that E t 1 [ε t ] = 0 we have that with the Law of Iterated Expectations: E t h [ε t ] = E t h [E t 1 (ε t )] = E t h [0] = 0. This orthogonality property implies that the {ε t } process is conditionally uncorrelated: Cov t h [ε t, ε t+k ] = E t h [ε t ε t+k ] E t h [ε t ] E t h [ε t+k ] = = E t h [ε t ε t+k ] = E t h [E t+k 1 (ε t ε t+k )] = = E t h [ε t E t+k 1 [ε t+k ]] = 0 The ARCH model has showed to be particularly useful in modeling the temporal dependencies in asset returns. Rossi GARCH Financial Econometrics / 50

13 ARCH model: definition The ARCH(q) Model The ARCH(q) model introduced by Engle (198) is a linear function of past squared disturbances: q σt = ω + α i ε t i In this model to assure a positive conditional variance the parameters have to satisfy the following constraints: ω > 0 e α 1 0, α 0,..., α q 0. Defining i=1 σ t ε t v t where E t 1 (v t ) = 0 we can write (13) as an AR(q) in ε t : where α (L) = α 1 L + α L α q L q. ε t = ω + α (L) ε t + v t Rossi GARCH Financial Econometrics / 50

14 ARCH model: definition The ARCH(q) Model (1 α(l))ε t = ω + v t q The process is weakly stationary if and only if α i < 1; in this case the unconditional variance is given by i=1 E ( ε t ) = ω/ (1 α1... α q ). Rossi GARCH Financial Econometrics / 50

15 ARCH model: definition The ARCH(q) Model The process is characterized by leptokurtosis in excess with respect to the normal distribution. In the case, for example, of ARCH(1) with ε t Φ t 1 N ( 0, σ t ), the kurtosis is equal to: with 3α1 < 1, when 3α 1 = 1 we have E ( ε 4 ) ( ) t /E ε ( ) ( ) t = 3 1 α 1 / 1 3α 1 E ( ε 4 t ) /E ( ε t ) =. In both cases we obtain a kurtosis coefficient greater than 3, characteristic of the normal distribution. Rossi GARCH Financial Econometrics / 50

16 ARCH model: definition The ARCH Regression Model We have an ARCH regression model when the disturbances in a linear regression model follow an ARCH process: y t = x tb + ε t E t 1 (ε t ) = 0 ε t Φ t 1 N ( 0, σt ) E t 1 ( ε t ) σ t = ω + α (L) ε t Rossi GARCH Financial Econometrics / 50

17 GARCH model The GARCH(p,q) model In order to model in a parsimonious way the conditional heteroskedasticity, Bollerslev (1986) proposed the Generalised ARCH model, i.e GARCH(p,q): σ t = ω + α (L) ε t + β (L) σ t. where α (L) = α 1 L α q L q, β (L) = β 1 L β p L p. The GARCH(1,1) is the most popular model in the empirical literature: σ t = ω + α 1 ε t 1 + β 1 σ t 1. Rossi GARCH Financial Econometrics / 50

18 GARCH model The GARCH(p,q) model To ensure that the conditional variance is well defined in a GARCH(p,q) model all the coefficients in the corresponding linear ARCH( ) should be positive: ( ) 1 p q σt = 1 β i L i ω + α j ε t j = ω + i=1 φ k ε t k 1 k=0 σ t 0 if ω 0 and all φ k 0. The non-negativity of ω and φ k is also a necessary condition for the non negativity of σ t. j=1 Rossi GARCH Financial Econometrics / 50

19 GARCH model The GARCH(p,q) model In order to make ω e {φ k } k=0 well defined, assume that : i. the roots of the polynomial β (x) = 1 lie outside the unit circle, and that ω 0, this is a condition for ω to be finite and positive. ii. α (x) e 1 β (x) have no common roots. These conditions are establishing nor that σ t neither that { σ t } t= is strictly stationary. For the simple GARCH(1,1) almost sure positivity of σ t requires, with the conditions (i) and (ii), that (Nelson and Cao, 199), ω 0 β 1 0 α 1 0 Rossi GARCH Financial Econometrics / 50

20 GARCH model The GARCH(p,q) model For the GARCH(1,q) and GARCH(,q) models these constraints can be relaxed, e.g. in the GARCH(1,) model the necessary and sufficient conditions become: ω 0 0 β 1 < 1 β 1 α 1 + α 0 α 1 0 For the GARCH(,1) model the conditions are: ω 0 α 1 0 β 1 0 β 1 + β < 1 β 1 + 4β 0 Rossi GARCH Financial Econometrics / 50

21 GARCH model The GARCH(p,q) model These constraints are less stringent than those proposed by Bollerslev (1986): ω 0 β i 0 i = 1,..., p α j 0 j = 1,..., q These results cannot be adopted in the multivariate case, where the requirement of positivity for { σ t } means the positive definiteness for the conditional variance-covariance matrix. Rossi GARCH Financial Econometrics / 50

22 GARCH model Stationarity The process {ε t } which follows a GARCH(p,q) model is a martingale difference sequence. In order to study second-order stationarity it s sufficient to consider that: Var [ε t ] = Var [E t 1 (ε t )] + E [Var t 1 (ε t )] = E [ σt ] and show that is asymptotically constant in time (it does not depend upon time). A process {ε t } which satisfies a GARCH(p,q) model with positive coefficient ω 0, α i 0 i = 1,..., q, β i 0 i = 1,..., p is covariance stationary if and only if: α (1) + β(1) < 1 This is a sufficient but non necessary conditions for strict stationarity. Because ARCH processes are thick tailed, the conditions for covariance stationarity are often more stringent than the conditions for strict stationarity. Rossi GARCH Financial Econometrics / 50

23 GARCH model Strict Stationarity GARCH(1,1) Nelson shows that when ω > 0, σt < a.s. and { } ε t, σt is strictly stationary if and only if E [ ln ( )] β 1 + α 1 zt < 0 E [ ln ( β 1 + α 1 zt )] [ ( ln E β1 + α 1 zt )] = ln (α1 + β 1 ) when α 1 + β 1 = 1 the model is strictly stationary. E [ ln ( )] β 1 + α 1 zt < 0 is a weaker requirement than α 1 + β 1 < 1. Example ARCH(1), with α 1 = 1, β 1 = 0, z t N (0, 1) E [ ln ( zt )] [ ( )] ln E z t = ln (1) It s strictly but not covariance stationary. The ARCH(q) is covariance stationary if and only if the sum of the positive parameters is less than one. Rossi GARCH Financial Econometrics / 50

24 GARCH model Forecasting volatility Forecasting with a GARCH(p,q) (Engle and Bollerslev 1986): σ t+k = ω + q α i ε t+k i + i=1 p β i σt+k i we can write the process in two parts, before and after time t: σ t+k = ω + i=1 n [ αi ε t+k i + β i σt+k i] m [ + αi ε t+k i + β i σt+k i] i=1 where n = min {m, k 1} and by definition summation from 1 to 0 and from k > m to m both are equal to zero. Thus [ ] n [ ( )] m [ E t σ t+k = ω + (αi + β i ) E t σ t+k i + αi ε t+k i + β i σt+k i ]. i=1 i=k i=k Rossi GARCH Financial Econometrics / 50

25 GARCH model In particular for a GARCH(1,1) and k > : E t [ σ t+k ] = k (α 1 + β 1 ) i ω + (α 1 + β 1 ) k 1 σt+1 i=0 [1 (α 1 + β 1 ) k 1] = ω [1 (α 1 + β 1 )] + (α 1 + β 1 ) k 1 σ t+1 = σ [ 1 (α 1 + β 1 ) k 1] + (α 1 + β 1 ) k 1 σ t+1 = σ + (α 1 + β 1 ) k 1 [ σ t+1 σ ] When the process is covariance stationary, it follows that E t [ σ t+k ] converges to σ as k. Rossi GARCH Financial Econometrics / 50

26 GARCH model The IGARCH(p,q) model The GARCH(p,q) process characterized by the first two conditional moments: E t 1 [ε t ] = 0 σt [ ] q E t 1 ε t = ω + α i ε t i + i=1 p β i σt i where ω 0, α i 0 and β i 0 for all i and the polynomial 1 α (x) β(x) = 0 has d > 0 unit root(s) and max {p, q} d root(s) outside the unit circle is said to be: Integrated in variance of order d if ω = 0 i=1 Integrated in variance of order d with trend if ω > 0. Rossi GARCH Financial Econometrics / 50

27 GARCH model The IGARCH(p,q) model The Integrated GARCH(p,q) models, both with or without trend, are therefore part of a wider class of models with a property called persistent variance in which the current information remains important for the forecasts of the conditional variances for all horizon. So we have the Integrated GARCH(p,q) model when (necessary condition) α (1) + β(1) = 1 Rossi GARCH Financial Econometrics / 50

28 GARCH model The IGARCH(p,q) model To illustrate consider the IGARCH(1,1) which is characterised by α 1 + β 1 = 1 σt = ω + α 1 ε t 1 + (1 α 1 ) σt 1 σt = ω + σt 1 ( + α 1 ε t 1 σt 1 ) 0 < α 1 1 For this particular model the conditional variance k steps in the future is: E t [ σ t+k ] = (k 1) ω + σ t+1 Rossi GARCH Financial Econometrics / 50

29 EGARCH The EGARCH(p,q) Model GARCH models assume that only the magnitude and not the positivity or negativity of unanticipated excess returns determines feature σ t. There exists a negative correlation between stock returns and changes in returns volatility, i.e. volatility tends to rise in response to bad news, (excess returns lower than expected) and to fall in response to good news (excess returns higher than expected). Rossi GARCH Financial Econometrics / 50

30 EGARCH The EGARCH(p,q) Model If we write σ t as a function of lagged σ t and lagged z t, where ε t = z t σ t σ t = ω + q α j zt jσ t j + j=1 p β i σt i it is evident that the conditional variance is invariant to changes in sign of the z ts. Moreover, the innovations z t j σ t j are not i.i.d. The nonnegativity constraints on ω and φ k, which are imposed to ensure that σt remains nonnegative for all t with probability one. These constraints imply that increasing zt in any period increases σt+m for all m 1, ruling out random oscillatory behavior in the σt process. i=1 Rossi GARCH Financial Econometrics / 50

31 EGARCH The EGARCH(p,q) Model The GARCH models are not able to explain the observed covariance between ε t and ε t j. This is possible only if the conditional variance is expressed as an asymmetric function of ε t j. In GARCH(1,1) models, shocks may persist in one norm and die out in another, so the conditional moments of GARCH(1,1) may explode even when the process is strictly stationary and ergodic. GARCH models essentially specify the behavior of the square of the data. In this case a few large observations can dominate the sample. Rossi GARCH Financial Econometrics / 50

32 EGARCH The EGARCH(p,q) Model In the EGARCH(p,q) model (Exponential GARCH(p,q)) put forward by Nelson the σ t depends on both size and the sign of lagged residuals. This is the first example of asymmetric model: ln ( σt ) p = ω + β i ln ( σt i ) q + α i [φz t i + ψ ( z t i E z t i )] i=1 i=1 α 1 1, E z t = (/π) 1/ when z t NID(0, 1), where the parameters ω, β i, α i are not restricted to be nonnegative. Let define g (z t ) φz t + ψ [ z t E z t ] by construction {g (z t )} t= is a zero-mean, i.i.d. random sequence. Rossi GARCH Financial Econometrics / 50

33 EGARCH The EGARCH(p,q) Model The components of g (z t ) are φz t and ψ [ z t E z t ], each with mean zero. If the distribution of z t is symmetric, the components are orthogonal, but not independent. Over the range 0 < z t <, g (z t ) is linear in z t with slope φ + ψ, and over the range < z t 0, g (z t ) is linear with slope φ ψ. The term ψ [ z t E z t ] represents a magnitude effect. Rossi GARCH Financial Econometrics / 50

34 EGARCH The EGARCH(p,q) Model If ψ > 0 and φ = 0, the innovation in ln ( σ t+1) is positive (negative) when the magnitude of z t is larger (smaller) than its expected value. If ψ = 0 and φ < 0, the innovation in conditional variance is now positive (negative) when returns innovations are negative (positive). A negative shock to the returns which would increase the debt to equity ratio and therefore increase uncertainty of future returns could be accounted for when α i > 0 and φ < 0. Rossi GARCH Financial Econometrics / 50

35 EGARCH The EGARCH(p,q) Model Nelson assumes that z t has a GED distribution (exponential power family). The density of a GED random variable normalized is: f (z; υ) = υ exp [ ( ) 1 υ ] z/λ < z <, 0 < υ λ (1+1/υ) Γ (1/υ) Rossi GARCH Financial Econometrics / 50

36 EGARCH The EGARCH(p,q) Model where Γ ( ) is the gamma function, and λ υ is a tail thickness parameter. [ ] 1/ ( /υ) Γ (1/υ) /Γ (3/υ) z s distribution υ = standard normal distribution υ < thicker tails than the normal υ = 1 double exponential distribution υ > thinner tails than the normal υ = uniformly distributed on [ 3 1/, 3 1/] With this density, E z t = λ1/υ Γ (/υ). Γ (1/υ) Rossi GARCH Financial Econometrics / 50

37 Asymmetric Models Non linear ARCH(p,q) Model The Non linear ARCH(p,q) model (Engle - Bollerslev 1986): q p σt γ = ω + α i ε t i γ + β i σ γ t i i=1 i=1 q p σt γ = ω + α i ε t i k γ + β i σ γ t i i=1 i=1 for k 0, the innovations in σt γ will depend on the size as well as the sign of lagged residuals, thereby allowing for the leverage effect in stock return volatility. Rossi GARCH Financial Econometrics / 50

38 Asymmetric Models GJR model The Glosten - Jagannathan - Runkle model (1993): p q σt = ω + β i σt i ( + αi ε t 1 + γ i S t i t i) ε i=1 i=1 where S t = { 1 if εt < 0 0 if ε t 0 Rossi GARCH Financial Econometrics / 50

39 Asymmetric Models Asymmetric GARCH(p,q) The Asymmetric GARCH(p,q) model (Engle, 1990): σ t = ω + q α i (ε t i + γ) + i=1 The QGARCH by Sentana (1995): σ t = σ + Ψ x t q + x t qax t q + p β i σt i i=1 p β i σt i when x t q = (ε t 1,..., ε t q ). The linear term (Ψ x t q ) allows for asymmetry. The off-diagonal elements of A accounts for interaction effects of lagged values of x t on the conditional variance. The QGARCH nests several asymmetric models. i=1 Rossi GARCH Financial Econometrics / 50

40 Asymmetric Models The APARCH model The proliferation of GARCH models has inspired some authors to define families of GARCH models that would accomodate as many individual as models as possible. The Asymmetric Power ARCH (Ding, Engle and Granger, 1993) r t = µ + ɛ t where ɛ t = σ t z t z t N (0, 1) q p σt δ = ω + α i ( ɛ t i γ i ɛ t i ) δ + β j σt j δ i=1 j=1 ω > 0 δ 0 α i 0 i = 1,..., q 1 < γ i < 1 i = 1,..., q β j 0 j = 1,..., p Rossi GARCH Financial Econometrics / 50

41 Asymmetric Models The APARCH model This model imposes a Box-Cox transformation of the conditional standard deviation process and the asymmetric absolute residuals. The Box-Cox transformation for a positive random variable Y t : { Y (λ) t = Y λ t 1 λ λ 0 log Y t λ = 0 The asymmetric response of volatility to positive and negative shocks is the well known leverage effect. This generalized version of ARCH model includes seven other models as special cases. Rossi GARCH Financial Econometrics / 50

42 Asymmetric Models The APARCH model 1 ARCH(q) model, just let δ = and γ i = 0, i = 1,..., q, β j = 0, j = 1,..., p. GARCH(p,q) model just let δ = and γ i = 0, i = 1,..., q. 3 Taylor/Schwert s GARCH in standard deviation model just let δ = 1 and γ i = 0, i = 1,..., q. 4 GJR model just let δ =. Rossi GARCH Financial Econometrics / 50

43 GARCH-in-mean model The GARCH-in-mean Model The GARCH-in-mean (GARCH-M) proposed by Engle, Lilien and Robins (1987) consists of the system: y t = γ 0 + γ 1 x t + γ g ( σ t ) + ɛt σ t = β 0 + q α i ɛ t 1 + i=1 p β i σt 1 i=1 ɛ t Φ t 1 N(0, σ t ) When y t (r t r f ), where (r t r f ) is the risk premium on holding the asset, then the GARCH-M represents a simple way to model the relation between risk premium and its conditional variance. Rossi GARCH Financial Econometrics / 50

44 GARCH-in-mean model This model characterizes the evolution of the mean and the variance of a time series simultaneously. The GARCH-M model therefore allows to analize the possibility of time-varying risk premium. It turns out that: y t Φ t 1 N(γ 0 + γ 1 x t + γ g ( σ t ), σ t ) In applications, g ( σ t ) = σ t, g ( σ t ) = ln ( σ t ) and g ( σ t ) = σ t have been used. Rossi GARCH Financial Econometrics / 50

45 The News Impact Curve The News Impact Curve The news have asymmetric effects on volatility. In the asymmetric volatility models good news and bad news have different predictability for future volatility. The news impact curve characterizes the impact of past return shocks on the return volatility which is implicit in a volatility model. Holding constant the information dated t and earlier, we can examine the implied relation between ε t 1 and σ t, with σ t i = σ i = 1,..., p. This impact curve relates past return shocks (news) to current volatility. This curve measures how new information is incorporated into volatility estimates. Rossi GARCH Financial Econometrics / 50

46 The News Impact Curve For the GARCH model the News Impact Curve (NIC) is centered on ɛ t 1 = 0. GARCH(1,1): σ t = ω + αɛ t 1 + βσ t 1 The news impact curve has the following expression: σ t = A + αɛ t 1 A ω + βσ Rossi GARCH Financial Econometrics / 50

47 The News Impact Curve In the case of EGARCH model the curve has its minimum at ɛ t 1 = 0 and is exponentially increasing in both directions but with different paramters. EGARCH(1,1): σ t = ω + β ln ( σ t 1) + φzt 1 + ψ ( z t 1 E z t 1 ) where z t = ɛ t /σ t. The news impact curve is [ ] φ + ψ A exp σt σ ɛ t 1 = [ ] φ ψ A exp σ ɛ t 1 for ɛ t 1 > 0 for ɛ t 1 < 0 [ A σ β exp ω α ] /π φ < 0 ψ + φ > 0 Rossi GARCH Financial Econometrics / 50

48 The News Impact Curve The EGARCH allows good news and bad news to have different impact on volatility, while the standard GARCH does not. The EGARCH model allows big news to have a greater impact on volatility than GARCH model. EGARCH would have higher variances in both directions because the exponential curve eventually dominates the quadrature. Rossi GARCH Financial Econometrics / 50

49 The News Impact Curve The Asymmetric GARCH(1,1) (Engle, 1990) σt = ω + α (ɛ t 1 + γ) + βσt 1 the NIC is σt = A + α (ɛ t 1 + γ) A ω + βσ ω > 0, 0 β < 1, σ > 0, 0 α < 1. is asymmetric and centered at ɛ t 1 = γ. Rossi GARCH Financial Econometrics / 50

50 The News Impact Curve The Glosten-Jagannathan-Runkle model The NIC is σ t = ω + αɛ t + βσ t 1 + γs t 1 ɛ t 1 { σt = is centered at ɛ t 1 = γ. S t 1 = { 1 if ɛt 1 < 0 0 otherwise A + αɛ t 1 if ɛ t 1 > 0 A + (α + γ) ɛ t 1 if ɛ t 1 < 0 A ω + βσ ω > 0, 0 β < 1, σ > 0, 0 α < 1, α + β < 1 Rossi GARCH Financial Econometrics / 50