Explanation

To understand why GARCH (1,1) will forecast a lower volatility than EWMA under these conditions, let's examine the model equations:

EWMA Model: $\sigma_n^2 = \lambda \sigma_{n-1}^2 + (1-\lambda)r_{n-1}^2$

GARCH (1,1) Model: $\sigma_n^2 = \omega + \alpha r_{n-1}^2 + \beta \sigma_{n-1}^2$

Given the conditions:

• Both models have the same parameter attached to $\sigma_{n-1}^2$ (meaning $\beta = \lambda$)
• $\sigma_{n-1}^2 = r_{n-1}^2$
• Current variance is above long-run variance

For GARCH (1,1), the long-run variance is: $V_L = \frac{\omega}{1-\alpha-\beta}$

Since current variance is above long-run variance, the GARCH model will forecast a variance that reverts toward the long-run mean. This mean-reverting property causes GARCH to forecast lower volatility than EWMA when current volatility is high.

EWMA has no mean-reversion property - it simply weights past observations. When current volatility is above average, EWMA will forecast volatility that remains relatively high, while GARCH will forecast volatility that reverts toward the long-run mean.

Therefore, GARCH (1,1) will forecast lower day n volatility than EWMA under these conditions.