Equal positive and negative return shocks can have unequal volatility effects; asymmetric recursions expose that assumption.

Base-R mechanics; rugarch and stochvol are optional production tools.

sigma_t^2 = omega + alpha e^2 + gamma I(e<0)e^2 + beta sigma_(t-1)^2

1. Step 1Define every variable and unit.
2. Step 2Check assumptions and information timing.
3. Step 3Calculate, verify, and interpret in context.

Variable

Fixed prior variance and book parameters.

Model choice

Under symmetric innovations, a common covariance-stationarity condition is alpha + gamma/2 + beta < 1.

Calculation

Negative-to-positive next-variance ratio = 1.385; illustrative GJR stationarity sum = 0.93.

Interpretation

Holding prior variance fixed isolates the specification's sign-sensitive response.

Reasonableness check

The curve is a model implication, not an observed causal law or investment signal.

Holding prior variance fixed isolates the specification's sign-sensitive response. Limitation: The curve is a model implication, not an observed causal law or investment signal.

Negative and positive shocks of equal magnitude imply different next variance.

Run the self-contained starter code in the protected STATLAB R runner. The code prints an auditable result and avoids network or unrestricted file access.

Compare your result with the definition, units, and model assumptions—not only with a target number.

Complete these without looking at the selected solutions. More than one defensible program may exist.

These are compact solution routes. Confirm dimensions, units, and any changed modelling choices.

Compare sign-sensitive GJR, EGARCH, APARCH, GARCH-M, and stochastic-volatility ideas.

Name the assumption, evidence you would seek, and how the recommendation might change.

Reach 70% to complete the chapter. Explanations appear after submission.