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# GARCH(1,1) vs EWMA for Forecasting Volatility

#### jairamjana

##### Member
So I link this video which explains GARCH(1,1) as a measure to forecast future volatility.

Now we know EWMA is a special case of GARCH which sums alpha and beta equal to 1 and therefore ignores any impact on long run variance, implying that variance is not mean reverting.. Again when we substitute in the formula we get E(Variance(n+t)) = Variance(n) since alpha + beta = 1.. So future volatility will always be a constant of current estimated volatility.. Now we don't know the parameters alpha , beta and gamma in practise.. we estimate them with the help of maximum likelihood method and there is a possibility that alpha + beta > 1 which according to common understanding means GARCH measure will be unstable so practitioners should use EWMA instead..
Once we are forced to abandon GARCH as a measure isn't it the case that we can never estimate future volatility henceforth and hence there is no point in forecasting..
Is my interpretation right?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @jairamjana Yes, I although you might arguably overstate the futility, I agree with your interpretation with respect to EWMA: EWMA is not a forecasting model, so if you are forced to use EWMA, that it tantamount to an acknowledgement that you cannot forecast! I copied a few paragraphs from my favorite volatility source, Carol Alexander (www.amazon.com/gp/product/0470998016/ ), emphasis mine:
II.3.8.4 Forecasting with EWMA
The exponentially weighted average provides a methodology for calculating an estimate of the variance at any point in time, and we denote this estimate σ(t)^2, using the subscript t because the estimate changes over time. But the EWMA estimator is based on an i.i.d. returns model. The ‘true’ variance of returns at every point is constant, it does not change over time. That is, EWMA is not a model for the conditional variance σ(t)^2. Without a proper model it is not clear how we should turn our current estimate of variance into a forecast of variance over some future horizon.

However, a EWMA model for the conditional variance could be specified as
σ(t)^2 = (1- λ)*r(t-1)^2 + λ*sigma(t-1)^2, r(t) | I(t-1) ~ N(0, σ^2)

This is a restricted version of the univariate symmetric normal GARCH model (introduced in the next chapter) but the restrictions are such that the forecast conditional volatility must be constant, i.e. σ(t)^2 = σ^2 for all t. So, after all, even if we specify the model (II.3.37) it reduces to the i.i.d. model for returns. Hence, the returns model for EWMA estimator is the same as the model for the equally weighted average variance and covariance estimator, i.e. that the returns are generated by multivariate normal i.i.d. processes.The fact that our estimates are time varying is merely due to a ‘fancy’ exponential weighting of sample data. The underlying model for the dynamics of returns is just the same as in the equally weighted average case!

A EWMA volatility forecast must be a constant, in the sense that it is the same for all time horizons. The EWMA model will forecast the same average volatility, whether the forecast is over the next 10 days or over the next year. The forecast of average volatility, over any forecast horizon, is set equal to the current estimate of volatility. This is not a very good forecasting model. Similar remarks apply to the EWMA covariance. We can regard EWMA as a simplistic version of bivariate GARCH. But then, using the same reasoning as above, we see that the EWMA correlation forecast, over any risk horizon, is simply set equal to the current EWMA correlation estimate. So again we are reduced to a constant correlation model.

The base horizon for the forecast is given by the frequency of the data – daily returns will give the 1-day covariance matrix forecast, weekly returns will give the 1-week covariance matrix forecast and so forth. Then, since the returns are assumed to be i.i.d. the square-rootof-time rule will apply. So we can convert a 1-day covariance matrix forecast into an h-day forecast by multiplying each element of the 1-day EWMA covariance matrix by h. Since the choice of λ itself is ad hoc some users choose different values of λ for forecasting over different horizons. For instance, in the RiskMetricsTM methodology described below a relatively low value of λ is used for short term forecasts and a higher value of λ is used for long term forecasts. However, this is merely an ad hoc rule." --- Source: II.3.8 of Carol Alexander's Market Risk Analysis, Practical Financial Econometrics (Volume II) www.amazon.com/gp/product/0470998016/

#### jairamjana

##### Member
I have heard great things about her books on Market Risk.. I am just a student so its a bit on the higher side to buy the set.. When I actually enter this field I will look to buy her book.. I hear she also has given spreadsheet case studies for her references...I am sure I will find it all useful as I finish my studies...
Personally I think FRM Part 1 speeds through the material without really going into considerable detail.. Understanding volatility, copulas and even regression requires mathematical acumen but of course Objective of FRM is to gain a conceptual understanding rather than working knowledge of a broad set of topics..
And I love learning new things..

Thank you again @David Harper CFA FRM

#### Tania Pereira

##### Member
Subscriber
Hi Harper,I would like to see the demonstration of V = w/ (1-a-b). Do you have this demonstration or paper about it? I would like to know how this formula was developed.
Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Tania Pereira @ShaktiRathore gave it to you! The key is that the weights (a, b and c, in Shakti's formula above) must sum to 1.0, by design. One way to look at GARCH(1,1) is that it generalizes EWMA which is an infinite series reducing to a recursive that needs only (λ) weight and (1-λ); i.e., EWMA has two weights which must sum to 1.0, also. GARCH(1,1) is actually also exponentially declining by β (ie, Shakt's b above) such that EWMA's λ is analogous to GARCH(1,1)'s β. If there were no constant term (Shakti's above), GARCH would be EWMA with a+b = 1.0, just as λ + (1 - λ) = 1.0.

A way to think about it is: if β (or b; the exponential decay weight) is, say, 90% or 0.90, then EWMA would assign all of the other 0.10 to the most recent return, but GARCH(1,1) splits the remaining 0.10 between the most recent return (a) and the long-run volatility (c). I hope that's interesting!

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