The package includes two published benchmarks, one for the GARCH(1,1) and one for the APARCH(1,1) model. Below we present the results of the former based on the benchmark of Fiorentini, Calzolari, and Panattoni (1996). Note that this is not a saved benchmark, but rather estimated every time it is called. As can be seen from the values of the log relative error, this achieves a very high rate of accuracy, addressing the questions raised by Hill and McCullough (2019).

| Estimate | Std Error (H) | Std Error (OPG) | Std Error (QML) | ||||
---|---|---|---|---|---|---|---|---|

tsgarch | FCP | tsgarch | FCP | tsgarch | FCP | tsgarch | FCP | |

$\mu$ | -0.0062 | -0.0062 | 0.0085 | 0.0085 | 0.0084 | 0.0084 | 0.0092 | 0.0092 |

| 6.1518 | 6.9758 | 6.4222 | 6.3652 | ||||

$\omega$ | 0.0108 | 0.0108 | 0.0029 | 0.0029 | 0.0013 | 0.0013 | 0.0065 | 0.0065 |

| 5.0391 | 6.1340 | 5.4333 | 6.2619 | ||||

$\alpha_1$ | 0.1531 | 0.1531 | 0.0265 | 0.0265 | 0.0140 | 0.0140 | 0.0535 | 0.0535 |

| 6.3803 | 5.9348 | 5.1801 | 7.4731 | ||||

$\beta_1$ | 0.8060 | 0.8060 | 0.0336 | 0.0336 | 0.0166 | 0.0166 | 0.0725 | 0.0725 |

| 6.3809 | 6.5216 | 6.7323 | 6.1553 | ||||

Notes to table: values in italic are the log relative errors (LRE). The FCP benchmark is from the paper by Fiorentini, G., G. Calzolari and L. Panattoni, Analytic Derivatives and the Computation of GARCH Estimates, Journal of Applied Econometrics 11 (1996), 399--417. |

Fiorentini, Gabriele, Giorgio Calzolari, and Lorenzo Panattoni. 1996.
“Analytic Derivatives and the Computation of GARCH
Estimates.” *Journal of Applied Econometrics* 11 (4):
399–417.

Hill, Chelsey, and BD McCullough. 2019. “On the Accuracy of Garch
Estimation in r Packages.” *Econometric Research in
Finance* 4 (2): 133–56.