GARCH and EWMA
- Thread starter ellenlcy
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Hi,
The EWMA can only project the current volatility into future. σ(t)^2 = (1- λ)*r(t-1)^2 + λ*sigma(t-1)^2 where σ(t) the current estimate of volatility based on EWMA should forecast the same volatility estimate σ(t) for the future period also. There is no forecast of volatility possible with EWMA for a point of time t in future we just assume the same estimate σ(t) as the volatility for point t.In a sense
there is no genuine forecast possible with EWMA like we could not find or forecast volatility for a point of time t in future as is possible with the GARCH as:
E(σ(n+t)^2)=VL+(alpha+beta)^t*(σ(n)^2-VL)
please see these:
https://forum.bionicturtle.com/threads/garch-1-1-vs-ewma-for-forecasting-volatility.9231/
thanks
The EWMA can only project the current volatility into future. σ(t)^2 = (1- λ)*r(t-1)^2 + λ*sigma(t-1)^2 where σ(t) the current estimate of volatility based on EWMA should forecast the same volatility estimate σ(t) for the future period also. There is no forecast of volatility possible with EWMA for a point of time t in future we just assume the same estimate σ(t) as the volatility for point t.In a sense
there is no genuine forecast possible with EWMA like we could not find or forecast volatility for a point of time t in future as is possible with the GARCH as:
E(σ(n+t)^2)=VL+(alpha+beta)^t*(σ(n)^2-VL)
please see these:
https://forum.bionicturtle.com/threads/garch-1-1-vs-ewma-for-forecasting-volatility.9231/
thanks
Thanks! @ShaktiRathore That helps a lot.
Hi @ellenlcy You raise a good point, thank you! For reference, here is the question:
@ShaktiRathore 's reference is helpful, because there has been a subtle discrepancy, that I do perceive. I do not like this (old) question #4 above, sorry; I realize I probably wrote it. I did used to say (years ago, I am not sure to what extent the FRM reading informed that view) that the N-day forward forecast of an EWMA volatility, σ(t+N), is the current estimate, σ(t) because EWMA (without simulating future return innovations) doesn't model a long-term variance or otherwise contain any forward-looking mechanism. So, based on the model, σ(t) = σ(t+N) for any (N).
But I've come to think that's an inferior interpretation to saying "EWMA does not forecast." Here's an analogy. If we have a model (eg, DCF) that tells us the intrinsic price of AAPL is $120.00 today, do we say that $120.00*e^(kN) is the N-year forecast of the price if the discount rate is (K)? Maybe we do, but more likely we say "No, it's not a model for N-year forward price." We would add features to the model to claim it could forecast, rather then simply extrapolate into the future. So I would distinguish between forecast and extrapolate.
For that reason, I think question 4 above is flawed. Better is Carol Alexander (from @ShaktiRathore 's link):
@ShaktiRathore 's reference is helpful, because there has been a subtle discrepancy, that I do perceive. I do not like this (old) question #4 above, sorry; I realize I probably wrote it. I did used to say (years ago, I am not sure to what extent the FRM reading informed that view) that the N-day forward forecast of an EWMA volatility, σ(t+N), is the current estimate, σ(t) because EWMA (without simulating future return innovations) doesn't model a long-term variance or otherwise contain any forward-looking mechanism. So, based on the model, σ(t) = σ(t+N) for any (N).
But I've come to think that's an inferior interpretation to saying "EWMA does not forecast." Here's an analogy. If we have a model (eg, DCF) that tells us the intrinsic price of AAPL is $120.00 today, do we say that $120.00*e^(kN) is the N-year forecast of the price if the discount rate is (K)? Maybe we do, but more likely we say "No, it's not a model for N-year forward price." We would add features to the model to claim it could forecast, rather then simply extrapolate into the future. So I would distinguish between forecast and extrapolate.
For that reason, I think question 4 above is flawed. Better is Carol Alexander (from @ShaktiRathore 's link):
Hi David,
I reviewed your answer again. Got a new question mark this time about this sentence:
because EWMA (without simulating future return innovations) doesn't model a long-term variance or otherwise contain any forward-looking mechanism.
Are you saying EWMA is not simulating future return innovation? I remember the innovation is the squared return item, u(n-1)^2, in the GARCH, which also exists in the EWMA. Confused about that.
Or because EWMA doesn't have long-term variance? But why long-term variance becomes innovation here?
Or because two items in EWMA are both about past information, only long-term variance has trend information about future?
Hope you can confirm one of my thoughts......
Thanks!
I reviewed your answer again. Got a new question mark this time about this sentence:
because EWMA (without simulating future return innovations) doesn't model a long-term variance or otherwise contain any forward-looking mechanism.
Are you saying EWMA is not simulating future return innovation? I remember the innovation is the squared return item, u(n-1)^2, in the GARCH, which also exists in the EWMA. Confused about that.
Or because EWMA doesn't have long-term variance? But why long-term variance becomes innovation here?
Or because two items in EWMA are both about past information, only long-term variance has trend information about future?
Hope you can confirm one of my thoughts......
Thanks!
GARCH(1,1) gives us a (conditional) estimate for the the current ("today's") volatility, where (n) is today and (n-1) was yesterday:
GARCH(1,1): σ(n) = α*µ(n-1)^2 + β*σ(n-1)^2 + γL, where α+β+γ=1.0
You are correct that yesterday's return, µ(n-1), can be called the "innovation" and that is what I mean. EWMA can be viewed as the special case of GARCH(1,1) where gamma (the weight not the omega term!) is zero:
EWMA: σ(n) = α*µ(n-1)^2 + β*σ(n-1)^2 + γL, where α+β=1.0
In both cases, the volatility estimate is updated by the latest "innovation," µ(n-1). Both of these have in common that they are estimates of current conditional volatility.
As opposed to a volatility forecast which would given an answer to the question, what is σ(n+T)? The point above (i.e., that EWMA either cannot forecast or "is not a very good forecasting model") is that EWMA by itself can really only say that σ(n+T) = σ(n) whereas GARCH can "forecast" σ(n+T) by assuming the daily volatility estimate eventually converges on the long-run variance or volatility (L). When I mentioned simulation, I just meant that in either case of course we can simulate future returns (i.e., future innovations). In that case, if we simulate up to a future µ(n+T-1), the we can estimate a future σ(n+T) per either or any method. But a simulation of future returns is not a feature of the model; our volatility estimates will become highly dependent on our simulation model and sample.
I hope that's useful. Volatility is deep and tricky. It helps to recall that volatility is not itself observable, it is a manufactured statistic so to speak. It can be historical, conditional, current, forecast, or implied! Thanks,
GARCH(1,1): σ(n) = α*µ(n-1)^2 + β*σ(n-1)^2 + γL, where α+β+γ=1.0
You are correct that yesterday's return, µ(n-1), can be called the "innovation" and that is what I mean. EWMA can be viewed as the special case of GARCH(1,1) where gamma (the weight not the omega term!) is zero:
EWMA: σ(n) = α*µ(n-1)^2 + β*σ(n-1)^2
In both cases, the volatility estimate is updated by the latest "innovation," µ(n-1). Both of these have in common that they are estimates of current conditional volatility.
As opposed to a volatility forecast which would given an answer to the question, what is σ(n+T)? The point above (i.e., that EWMA either cannot forecast or "is not a very good forecasting model") is that EWMA by itself can really only say that σ(n+T) = σ(n) whereas GARCH can "forecast" σ(n+T) by assuming the daily volatility estimate eventually converges on the long-run variance or volatility (L). When I mentioned simulation, I just meant that in either case of course we can simulate future returns (i.e., future innovations). In that case, if we simulate up to a future µ(n+T-1), the we can estimate a future σ(n+T) per either or any method. But a simulation of future returns is not a feature of the model; our volatility estimates will become highly dependent on our simulation model and sample.
I hope that's useful. Volatility is deep and tricky. It helps to recall that volatility is not itself observable, it is a manufactured statistic so to speak. It can be historical, conditional, current, forecast, or implied! Thanks,
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