TGARCH

Question 2: TGARCH Consider the following non-linear GARCH model Yt = E[Yt|Ft−1] + εt, (1) εt = zt√qσ2 t , (2) with zt an i.i.d. standard normally distributed random variable, and σ2 t = ω + αε2 t−1 + γε2 t−1I I {εt−1<0} + βσ2 t−1. (3) where α ≥ 0, β ≥ 0,ω > 0 and Ft is the information set up to time t. a) Is the variance model from Equation (3) able to capture leverage effects? If yes, which sign do you expect for γ? Derive Cov(σ2 t+1,εt|Ft−1). Also comment on the sign of Cov(σ2 t+1,εt|Ft−1). Hint: Note that ∫0−∞z3φ(z)dz = −√2/ π and ∫0∞z3φ(z)dz =√2/ π with φ(·) be the pdf of the standard normal distribution Yes, the variance Is model from Equation (3) able to capture leverage effects. Sign of γ is + Cov(σ2 t+1,εt|Ft−1)= E[σ2 t+1 · εt|Ft−1] = E[ξt · (σt · εs )] = Eξt · E[σt · εs ] = 0. Sign of Cov - 0