Modeling One-Minute Bitcoin Volatility Beyond Standard GARCH
Article Quant Q&A · Author: Jan Sila
Summary
The document questions whether standard GARCH models adequately describe one-minute Bitcoin returns. The sample is characterized by frequent zero returns, low return magnitudes, and apparent microstructure noise; the author also reports that squared residuals retain patterns similar to the original series after fitting an eGARCH model. A long-sample fit has significant parameters and passes several residual autocorrelation and ARCH tests, but its stability test and goodness-of-fit results raise concerns.
The response argues that conventional GARCH models are generally designed for daily volatility and may fail to capture intraday patterns such as time-of-day effects. It suggests a multiplicative component GARCH approach for intraday returns, or aggregating high-frequency observations into daily realized volatility and modeling that with realized GARCH. These are proposed directions rather than demonstrated solutions, and the discussion does not compare their empirical performance on the Bitcoin data.
Key ideas
- One-minute Bitcoin returns can contain many zero observations and microstructure noise.
- A standard GARCH fit may pass residual dependence tests while still fitting the data poorly.
- Parameter stability and goodness-of-fit diagnostics can reveal problems beyond coefficient significance.
- Intraday component volatility models may accommodate intraday patterns better than conventional GARCH.
- Daily realized volatility from high-frequency data can be modeled with realized GARCH.
Tags
Full text
# model high frequency bitcoin volatility
# model high frequency bitcoin volatility
I am trying to model volatility of 1-minute returns of BTC, but it seems to me that the data do not behave traditionally. I tried fitting GARCH, eGARCH with ARMA (1,1) or (2,0), but I am not confident that either actually fit well. The data exhibit often zero-return and when I compare it to garch-simulated data ours exhibit clearly microstructure noise and behave differently and the returns are rather low in general. The period with highest return in the sample
The ACF and PACF seems little unusual[]
After fitting a garch, the squared residuals of the fit look the same as in the original series -> no arch effect seems to be modelled.
So my question is whether GARCH approach is even valid in this case? The output of one of garch models on full sample (73 k observations) is as follows (smaller sample has insignificant coefficients):
```
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------
GARCH Model : eGARCH(1,1)
Mean Model : ARFIMA(2,0,0)
Distribution : std
Optimal Parameters
------------------------------------
Estimate Std. Error t value Pr(>|t|)
mu 0.000000 0.000000 -1.9930e-03 0.99841
ar1 0.050520 0.003378 1.4956e+01 0.00000
ar2 -0.038044 0.003214 -1.1837e+01 0.00000
omega -0.419248 0.000746 -5.6196e+02 0.00000
alpha1 0.048651 0.003691 1.3182e+01 0.00000
beta1 0.975087 0.000058 1.6759e+04 0.00000
gamma1 0.574946 0.003953 1.4545e+02 0.00000
shape 2.508783 0.005877 4.2687e+02 0.00000
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 0.000000 0.000007 -0.000016 0.999987
ar1 0.050520 0.008469 5.965290 0.000000
ar2 -0.038044 0.004451 -8.547390 0.000000
omega -0.419248 0.034581 -12.123488 0.000000
alpha1 0.048651 0.012999 3.742817 0.000182
beta1 0.975087 0.002306 422.906412 0.000000
gamma1 0.574946 0.016985 33.849762 0.000000
shape 2.508783 0.018145 138.261378 0.000000
LogLikelihood : 692467.3
Information Criteria
------------------------------------
Akaike -14.156
Bayes -14.155
Shibata -14.156
Hannan-Quinn -14.156
Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
statistic p-value
Lag[1] 0.008437 0.9268
Lag[2*(p+q)+(p+q)-1][5] 0.033719 1.0000
Lag[4*(p+q)+(p+q)-1][9] 0.037504 1.0000
d.o.f=2
H0 : No serial correlation
Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
statistic p-value
Lag[1] 1.606e-05 0.9968
Lag[2*(p+q)+(p+q)-1][5] 4.819e-05 1.0000
Lag[4*(p+q)+(p+q)-1][9] 8.032e-05 1.0000
d.o.f=2
Weighted ARCH LM Tests
------------------------------------
Statistic Shape Scale P-Value
ARCH Lag[3] 1.606e-05 0.500 2.000 0.9968
ARCH Lag[5] 3.855e-05 1.440 1.667 1.0000
ARCH Lag[7] 5.737e-05 2.315 1.543 1.0000
Nyblom stability test
------------------------------------
Joint Statistic: 2124.693
Individual Statistics:
mu 11.44
ar1 161.81
ar2 43.00
omega 559.78
alpha1 44.98
beta1 869.78
gamma1 832.78
shape 440.51
Asymptotic Critical Values (10% 5% 1%)
Joint Statistic: 1.89 2.11 2.59
Individual Statistic: 0.35 0.47 0.75
Sign Bias Test
------------------------------------
t-value prob sig
Sign Bias 1.0466011 0.2953
Negative Sign Bias 0.0003836 0.9997
Positive Sign Bias 0.6256527 0.5315
Joint Effect 1.3371473 0.7203
Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
group statistic p-value(g-1)
1 20 85598 0
2 30 101805 0
3 40 112434 0
4 50 120823 0
Elapsed time : 21.09595 2
```
Thank you very much for all suggestions how to continue!
## Answer by Evgenii (score 1, accepted)
https://quant.stackexchange.com/a/48679
In general for stock data, classical GARCH models are designed to model daily volatilities, but not the intraday ones, because, for instance, they do not capture diurnal patterns. So, I would say that models you are estimating are not valid for the 1-minutes returns. And of course, the presence of the microstructure noise makes them even less valid.
As a possible solution, you might want to look at the Multiplicative Component GARCH for intraday returns. Alternatively, you can calculate daily realized volatilities based on high-frequency data you have and employ, for instance, the realized GARCH to model daily volatility dynamics.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.