Questions:

(i) In regard to volatility estimate approaches, what is the single key difference between moving average (MA) and exponentially weighted moving average (EWMA)?

(ii) What is the single key difference between exponentially weighted moving average (EWMA) and generalized autoregressive conditional heteroscedasticity (GARCH)?

(iii) What is the significance of lambda in EWMA?

(iv) Empirically, volatility tends to cluster (i.e., is “sticky” to itself). How would this be reflected in the GARCH(1,1) parameters? How would it’s opposite be reflected?

(v) GARCH(1,1) is generally considered superior to EWMA, but when would EWMA be better?

Answers:

Answer:

(i) The moving average (MA) is equally-weighted (or un-weighted); EWMA assigns exponentially decreasing weights to observations (squared returns) as it “looks back” over the historical series. In short, EWMA gives greater weight to more recent observations.

(ii) Like EWMA, GARCH(1,1) assigns exponentially decreasing weights to the historical series. They have this in common. The key difference is that GARCH(1,1) adds a term for mean-reversion.

(iii) Lambda is the decay factor in EWMA. It is the ratio of weight (n-2) to weight (n-1), and generically (n-m-1) to (n-m). (1-lambda) is the weight assigned to the most recent observation. A HIGHER lambda indicates SLOWER decay; a LOWER lambda indicates HIGHER decay.

(iv) Cluster would be captured in HIGHER PERSISTENCE, where persistence is the sum of alpha + beta (gamma is the “leftover” weight, so if alpha + beta is higher, gamma will be lower and that meant the series does not “decay” toward it’s mean-reverting level as quickly). The opposite of persistence, here, would be greater decay. This would be reflected in a higher GAMMA weight.

(v) If alpha + beta > 1, then this implies gamma < 0. In this case, according to Hull, the GARCH(1,1) series is unstable and EWMA is preferred.