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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 enter image description here and the returns are rather low in general. The period with highest return in the sample enter image description here

The ACF and PACF seems little unusual[sample afc and pacf]

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!

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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.

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  • $\begingroup$ Thanks, I have been studying further about microstructure noise and realised volatility! Thanks a lot for nudging in the right direction $\endgroup$ – Jan Sila Sep 22 at 13:20

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