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When we are trying to feed time-series data to a GARCH or ARCH model, what kind of data should we give the model?

  • A: Absolute difference between daily prices over-time
  • B: % of the difference between daily prices over-time
  • C: B but squared (to take out the negative values)
  • D: module of B (to take out the negative values)
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It depends on the specific GARCH model you want to estimate. Traditional GARCH takes in squared % returns (C) assuming no mean equation. But there are other GARCH formulations that take in different values, such as the absolute value of % returns (like EGARCH).

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  • $\begingroup$ This is incorrect. Squared returns are not used as inputs in GARCH or ARCH models. $\endgroup$
    – Richard Hardy
    Commented Feb 10, 2022 at 10:43
  • $\begingroup$ Assuming there is no mean equation, the residual/error is the return. And in standard arch/garch models, the squared error is the input (which again is equal to the squared return when there is no mean equation). Every standard text on arch/garch shows this. It’s also available on wikipedia. $\endgroup$
    – RA334
    Commented Feb 10, 2022 at 22:34
  • $\begingroup$ Sorry, but this is wrong. To see that in practice, consider simulating data from a GARCH model and then estimating the model's parameters on the simulated data. If your understanding were correct, you would get parameter estimates that are close to the true values. Since it is not correct, you will likely get something else. Then try what I am suggesting, and you will get something close to the truth (unless your estimation sample is quite short). That should be at least somewhat convincing. $\endgroup$
    – Richard Hardy
    Commented Feb 11, 2022 at 6:42
  • $\begingroup$ I understand what you’re saying but that’s technically not correct. Your exercise simulates returns from a process that specifies a GARCH variance (and can specify an equation for the mean too, like AR or ARMA or not). The GARCH variance equation has squared residuals - taken from the first stage - enter as inputs. Refer to any of the open access online resources on this stuff. Also you can go straight to the source Bollerslev (1986), just look at Eq. 2. Good luck. $\endgroup$
    – RA334
    Commented Feb 12, 2022 at 0:33
  • $\begingroup$ What I am saying is exactly correct. Moreover, the OP writes we are trying to feed time-series data to a GARCH or ARCH model, not to the conditional variance equation of a GARCH or ARCH model, and they are right in their choice. GARCH or ARCH model an entire conditional distribution of the random variable of interest, not just its conditional variance. You never feed squared returns, you feed raw returns. You cannot fit a GARCH or ARCH model for a time series $x_t$ using $x_t^2$ as inputs; rather, you use $x_t$ as inputs. $\endgroup$
    – Richard Hardy
    Commented Feb 12, 2022 at 7:07
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B: % of the difference between daily prices over-time.

This, or rather the closely related logarithmic returns, is what is typically used as inputs to GARCH and ARCH models. GARCH and ARCH model the conditional distribution of the inputs using a particular form of the conditional variance equation. Thus even if this answer alternative (B) is not perfect, it is very close to the truth. None of the other alternatives may work due to technical reasons (see definitions of GARCH and ARCH models).

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