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I'm trying to capture heteroskedasticity in the returns of a price time series using a GARCH model.

A basic intuition suggests that I should fit the GARCH model on log-returns: indeed, if the price is divided by $2$ at a certain point in time, it'd give a return of $-0.5$. If it is multiplied by $2$, it gives a return of $1$. So we'd have different amplitude of return for a price move that is actually of the same amplitude because prices are in an exponential scale. If we take log-return, "divided by $2$" gives a log-return of $\log(0.5)\approx-0.3$ and "multiplied by $2$" to gives a log-return of $\log(2)\approx0.3$ : we're good, they are the same in absolute value.

However, after trying the GARCH on log-returns (i.e., the log of the gross return), it appears that log-returns remove a lot of the heteroskedasticity from the actual returns, leading the GARCH not to distinguish clearly between periods of high activity and period of low activity.

To sum up, if I use simple returns, the GARCH distinguish clearly periods of high volatility, but the same price move has a different amplitude depending on if it goes up or down, which biases the estimation of the variance in some way.

On the other hand, if I use log-returns, I don't have the "bias" of the exponential scale, but the result has less heteroskedasticity, which is not good for my strategy since I scale positions depending on volatility.

What is usually used in practice to forecast volatility? Is it more appropriate, in general, to fit a GARCH on returns or on log-returns to estimate volatility?

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    $\begingroup$ Hi: I would look at one of the early papers on arch or garch ( mid to late 80's ) and see what is used and hopefully it's discussed. It kind of makes sense that you would reduce volatility by using log returns because the log transformation is supposed to do that in general not just in return settings. $\endgroup$
    – mark leeds
    Jun 28, 2023 at 14:52
  • $\begingroup$ Here's one that I remember being pretty good. Hopefully it discusses what you talked about. kroner.com/attachments/AcademicPapers/Survey%20(English).pdf $\endgroup$
    – mark leeds
    Jun 28, 2023 at 14:58
  • $\begingroup$ One issue with not using log returns is that the GARCH assumes a conditional normal distribution (or similar unbounded distribution) for returns. But relative returns below -100% are impossible and using log-returns is then more consistent. But it might not always matter in applications. $\endgroup$
    – fes
    Jun 28, 2023 at 14:58
  • $\begingroup$ GARCH is not about the type of conditional heteroskedasticity (CH) that can be reduced using a logarithmic transformation. GARCH is about autoregressive CH while the logarithmic transformation can fix the case when variance increases together with the level of the variable. GARCH can be used for both kinds of returns: percentage ones and logarithmic ones. $\endgroup$ Jun 28, 2023 at 15:02
  • $\begingroup$ @markleeds, I searched the document for relevant keywords and did not find anything about log-returns, except for one reference which cannot be generalized. Since log-returns are so popular in financial modelling, I suppose the paper simply does not pay attention to the choice. Otherwise, log-returns would definitely have been mentioned in the discussion. $\endgroup$ Jun 28, 2023 at 15:06

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What is usually used in practice to forecast volatility?

I believe it is log-returns.

Is it more appropriate, in general, to fit a GARCH on returns or on log-returns to estimate volatility?

The general mathematical specification of the model does not restrict the use of the model to a single type of returns. However, the distribution usually assumed for standardized residuals is typically defined on the real line, yielding nonzero density for values under -100% which are impossible for simple returns. Thus, log-returns are more natural and I would start with them. However, values under -100% are so far down in the left tail that I think for all practical purposes they can be ignored.

I would use whichever type of return is easier to interpret and to model (where you get a better fit and especially out-of-sample forecasts). After all, if you have a model for simple returns, you can back out a model for prices, then log-prices and then log-returns, and the other way around. It is enough to model one of these, and then models for the other ones are implied.

it appears that log-returns remove a lot of the heteroskedasticity from the actual returns, leading the GARCH not to distinguish clearly between periods of high activity and period of low activity.

GARCH is not about the type of conditional heteroskedasticity (CH) that can be reduced using a logarithmic transformation. GARCH is about autoregressive CH while the logarithmic transformation can fix the case when variance increases together with the level of the variable.

...which biases the estimation of the variance in some way.

I do not think it biases the estimation of the target quantity, but you may find that the target quantity does not measure what you are interested in measuring. But again, if you have a model for any particular one-to-one transformation of prices, you can back out a model for any other one-to-one transformation, so you can derive all kinds of quantities from a model for one particular transformation such as log-returns or simple returns.

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I would not use anything but log returns in finance. The reasons logs are used so frequently are summarised here. Commonly used textbooks like Ruey Tsay, Analysis of Financial Time Series always define (G)ARCH with log returns. Historical volatility is basically always computed using log returns. I would flip the question around and ask why you would not want to use log returns.

If the results are very different, you need to ask yourself a few questions:

  • How does the forecast perform: Ultimately, a good forecast should fit. if you realize one model fits better for your data, stick to the one that works better.
  • Is the model fit properly? Are the residuals weak white noise after you specified the mean equation? You need to remove the sample mean if it is significantly different from zero. If you use the log returns, you're essentially making the assumption that there is no conditional variation in the mean.

Side remark, one weakness is that the model assumes positive and negative shocks have the same effects on volatility. In reality, financial assets react different to positive and negative shocks. You can use EGARCH (exponential GARCH) to mitigate this.

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    $\begingroup$ EGARCH has all kinds of statistical/econometric faults; log-GARCH is a good alternative. Also, you can use log-returns without making an assumption of constant conditional mean. Just specify the conditional mean equation appropriately for your GARCH model. $\endgroup$ Jul 1, 2023 at 10:15
  • $\begingroup$ Thanks for the comment and correcting my typos @Richard Hardy. $\endgroup$
    – AKdemy
    Jul 1, 2023 at 11:10
  • $\begingroup$ No problem! I hope this is not too intrusive of me. $\endgroup$ Jul 1, 2023 at 11:21
  • $\begingroup$ Not at all! Thanks. $\endgroup$
    – AKdemy
    Jul 1, 2023 at 11:26
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If your strategy is mostly based on volatility, I suggest you to use a different approach; by using a GARCH model you're assuming that the volatility is unknown but you can model it. The Realized Variance is a non-parametric method that allows you to calculate the daily variance knowing the intraday returns (generalizing, the variance in a given timeframe knowing lower timeframe returns). The Realized volatility - square root of the realized variance - is given by: $\displaystyle\text{RV}=\sqrt{\sum_{t=1}^Tr_t^2}$ but this formula is highly biased due to the so-called "microstructure noise". There are multiple valid methods in literature that let you have an unbiased estimation of the RV, like the Realized kernel (a practical and clear approach to this method can be found here). The big difference between RV estimation and GARCH model is that the first one is non-parametric and gives you the possibility to retrieve the volatility directly from your time series, without potential bias given by the optimization and parametrization of the GARCH model. This approach is a starting point to the volatility forecasting, that need a different model - e.g. HAR model of Corsi(2009), Bayesian method by Liu and Maheu (2009), bagging by Hillebrand and Modeiros (2010) and so on.

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    $\begingroup$ You should mention this requires data on a much higher frequency. As such it is not a direct alternative for GARCH. Also, regarding your first paragraph: the relationship between volatility at different frequencies depends on the data generating process. Without further assumptions on that, one cannot infer daily volatility from 5-minute (or some other short interval) volatility or vice versa. From this perspective, the 5-minute RV being a nonparametric method does not help when we are after daily volatility. $\endgroup$ Jun 30, 2023 at 14:22

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