Based on the GJR-GARCH(1,1) model estimates for SPY daily return from 2018 to 2026, a 3 percent loss today will increase the conditional volatility tomorrow more than a 3 percent positive return.

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BlackTom AI · Accepted Answer

The GJR-GARCH(1,1) specification includes an asymmetry (leverage) term, so negative shocks raise next-period variance by alpha + gamma times the squared shock while positive shocks raise it by alpha times the squared shock. 1) Relevant knowledge points and concepts - GJR-GARCH(1,1) conditional variance equation: h_{t+1} = ω + α ε_t^2 + γ I_{ε_t<0} ε_t^2 + β h_t. - ε_t is the return innovation; I_{ε_t<0} is 1 for negative shocks and 0 for positive shocks. - If γ > 0, negative returns increase variance more than positive returns of the same magnitude (the leverage/asymmetry effect). 2) Step-by-step reasoning - Consider a ±3% return (ε_t = ±0.03). The squared shock is ε_t^2 = 0.03^2 = 0.0009 in both cases. - For a +3% return (ε_t > 0): additional contribution to h_{t+1} from yesterday’s shock = α × 0.0009. - For a −3% return (ε_t < 0): contribution = (α + γ) × 0.0009. 3) Intermediate calculations (symbolic and numeric illustration) - Difference = [(α + γ) − α] × 0.0009 = γ × 0.0009. - If γ > 0 (as the GJR-GARCH estimates for SPY indicate), this difference is positive. Example: if α = 0.05 and γ = 0.02 then positive shock adds 0.05×0.0009 = 0.000045 while negative shock adds 0.07×0.0009 = 0.000063, i.e. the negative shock raises variance more. 4) Why the correct answer is correct - The model’s γ parameter captures asymmetry; empirical GJR-GARCH estimates for SPY (2018–2026) show γ > 0, so a 3% loss increases conditional volatility tomorrow by γ×0.0009 more than a 3% gain does. Therefore the statement is true. 5) Why the other option is incorrect - “False” would be correct only if γ ≤ 0 (no asymmetry or reverse asymmetry). Given the reported GJR-GARCH estimates indicate a positive γ, that scenario does not apply. 6) Conclusion - Because the GJR-GARCH model has a positive γ for SPY in the given period, a 3% loss increases conditional volatility tomorrow more than a 3% positive return. Final answer: True.

Based on the GJR-GARCH(1,1) model estimates for SPY daily return from 2018 to 2026, a 3 percent loss today will increase the conditional volatility tomorrow more than a 3 percent positive return.判断题

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