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The answers to this question on forecast assessment suggest that if the sequence of residuals from the forecast are not properly independent, then the model is missing something and further changes should be made to remove the correlation.

That does make sense to me and it suggests that we should be able to do better than a simple GARCH(1,1) model.

However, in almost all the literature on the subject, this issue is never discussed, and the fact that forecasts produced residuals that are serially correlated is taken as a fact of life. Indeed, people have produced methods for accounting for both serial and contemporaneous correlations when comparing different forecast models.

So, why is this the case? If the GARCH(1,1) model does have such problems, why is it still considered a standard approach for modeling volatility?

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  • $\begingroup$ I guess this is because the suggested autocorrelation in residuals, which are mentioned in the original question, usually is not a problem when using GARCH, which should be obvious, since the volalatility equation of the GARCH model is an ARMA-model of the residuals, which will usually be able to filter any autocorrelation of the residuals. $\endgroup$ – Owe Jessen Feb 25 '11 at 15:07
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    $\begingroup$ It has been my experience that forecast residuals exhibit strong serial correlation. Quite a few papers discuss this issue as a very common problem with forecast assessment. See papers.ssrn.com/sol3/papers.cfm?abstract_id=331800 $\endgroup$ – kaybenleroll Feb 26 '11 at 10:13
  • $\begingroup$ I had a query, I will be thankful if someone could help me: I am currently working on NSE Nifty 50 Futures and was working to forecast volatility by using Garch models..which model and error distribution should i use in order to have good future forecasts? $\endgroup$ – user5378 May 21 '13 at 5:11
  • $\begingroup$ I think the terminology used in the question could be improved. I would suggest clearly separating model residuals (from in-sample fit) from forecast errors (out of sample). Forecast residuals is an uncommon term in my experience. $\endgroup$ – Richard Hardy Oct 26 '16 at 7:37
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One of the reasons the ARCH family of models is used is that you only need price data to generate the model. These data exist back to the 1800s, so ARCH is great for looking at volatility over very long periods. I don't know that I'd say that the ARCH model has a lot of problems -- it solved the problem of not allowing volatility in time or in the level of the underlying process. It allowed Robert Engle to put to rest Milton Friedman's idea that inflation uncertainty varied with inflation level and won Engle the Nobel Prize.

However, I agree with Brian that there are probably better volatility forecasts. ARCH models are necessarily based in the past, not in expectations. I think I would look first at implied volatility from options. Using implied volatility from options will also allow to forecast higher moments like skew and kurtosis.

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    $\begingroup$ Why should I be confident that implied volatility makes a better predictor than volatility forecast from a GARCH-Model? $\endgroup$ – Owe Jessen Feb 25 '11 at 23:36
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    $\begingroup$ The implied volatility should incorporate all market information and investor expectations. An ARCH model needs to be calibrated after a regime change. An extreme example would be if WWIII broke out right now, the price on options would rise quickly and we could back out the implied volatilities. The ARCH models would need to be fed new information at this new higher volatility to calibrate. Really, I would look at both, but as far as a forecast I would feel a little better that the implied vol captures the market's current expectations. $\endgroup$ – Richard Herron Feb 26 '11 at 0:11
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    $\begingroup$ But there's also the caveat that the implied volatility will be a little higher than the realized volatility because it includes an option premium. $\endgroup$ – Richard Herron Feb 26 '11 at 0:12
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    $\begingroup$ But with implied volatility you rely on efficient market hypothesis and all of its assumptions. The GARCH too would react swiftly to rising volatility, as one could see in the financial crisis. $\endgroup$ – Owe Jessen Feb 26 '11 at 11:21
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    $\begingroup$ I can understand the aversion to EMH. But the problem with EMH isn't that it's completely wrong, just that it's not completely right. I don't think the real chinks in EMH are applicable for option-able stocks. For options with volume (i.e., 20-50 days to maturity and volume in the underlying) I think you can feel pretty good that the option price reflects a lot of information. But for illiquid options and stocks, you're right, you should take the info with a grain of salt. But if there's no volume in the underlying, ARCH models will fail you, too. $\endgroup$ – Richard Herron Feb 28 '11 at 3:56
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GARCH(1,1) is a "standard approach for modeling volatility" mainly in academic literature. Most of us in the real world don't use it. Volatility forecasting tends to come more from looking at more-liquid comparables for future market volatility than from fitting fancy retrospective models.

As for ignoring the dependence of residuals, well, folks are probably considering the problem to be less nettlesome than problems introduced by trying to fit complex models with too many parameters.

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    $\begingroup$ As a vol trader I second Brian's opinion - GARCH is popular in academia, not in trading. $\endgroup$ – onlyvix.blogspot.com Feb 26 '11 at 19:51
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If you calculate a var swap using SPX term structure implied vol versus a GARCH(1,1) estimated on 2yrs of past prices, you may see the first 4-5 (non-weekly) expiries offer roughly constant premiums over realised (negative varswap value) which would suggest at least someone is pricing using GARCH in the market.

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  • $\begingroup$ Agree, GARCH and variants are commonly used in industry. I don't see how that is in dispute. $\endgroup$ – Kevin Schmit Apr 8 '12 at 1:10
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You might be conflating two different things:

  1. autocorrelation in model residuals in a fixed sample (window) and
  2. autocorrelation in forecast errors across samples (rolling or expanding windows).

(1) is undesirable as it indicates the model misses a pattern which it should ideally capture. This can be remedied, for example, by changing the model. One may add an ARMA structure to the model's error term (to get ARMA-GARCH from pure GARCH, for example), change the model's autoregressive order, or do some other changes.

(2) can happen by construction and need not indicate any problem with the forecasts or the modelling scheme that is generating them. Indeed, forecast errors of $h$ steps ahead will necessarily be MA($h-1$) processes; see e.g. Diebold "Forecasting in Economics, Business, Finance and Beyond" Chapter 10 "Point Forecast Evaluation", section 10.1 "Absolute Standards for Point Forecasts" (version of 14 December 2015; the linked version might change over time).

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GARCH models were developed by Robert Engle precisely to deal with the problem of auto-correlated residuals (which occurs when you have volatility clustering for example) in time-series regression. To ask "Why are GARCH models used to forecast volatility if residuals are often correlated?" misses this point.

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  • $\begingroup$ GARCH models worked quite nicely for me. Just an ad hoc comment. They weren't terribly complicated or difficult to work with, and as I recall, weren't supposed to be afflicted with auto-correlated residuals! +1 to you... $\endgroup$ – Ellie Kesselman Feb 5 '12 at 19:06

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