*5.1. Study of ML Estimators*

Through a Monte Carlo (MC) study, we evaluate here the finite sample behavior of the ML estimators of the GBS-ACD model parameters presented in Section 3. The sample sizes considered were *n* = 500, 1000, and 3000. The number of MC replications was *B* = 1000. The data-generating process for each of the realizations is

$$X\_i = \psi\_i \epsilon\_i, \quad \ln \psi\_i = 0.10 + 0.90 \ln \psi\_{i-1} + 0.10 \left[\frac{x\_{i-1}}{\Psi\_{i-1}}\right], \tag{27}$$

where the distribution of *i* is a generalized gamma with density *f*(*x*; *μ*, *σ*, *ν*) = *θθ zθν* exp(−*θz*)/(Γ(*θ*)*x*) with *z* = (*x*/*μ*)*<sup>μ</sup>* and *θ* = 1/*σ*<sup>2</sup>|*ν*|2. Note that stationarity conditions only require |*β*| < 1, and in (27), *β* = 0.9; see Bauwens and Giot (2000).

We estimate the GBS-ACD model parameters through the following two-step algorithm:


The estimation results from the simulation study are presented in Table 2. The following sample statistics for the ML estimates are reported: Mean, coefficients of skewness (CS) and kurtosis (CK), relative bias (the RB, in absolute values, is defined as |E(*τ*) − *<sup>τ</sup>*|/*<sup>τ</sup>*, where *τ* is an estimator of a parameter *τ*), and root mean squared error (√MSE). The sample CS and CK are, respectively, given by

$$\text{CS}(\mathbf{x}) = \frac{\sqrt{n[n-1]}}{[n-2]} \frac{n^{-1} \sum\_{i=1}^{n} [x\_i - \mathfrak{x}]^3}{\left[n^{-1} \sum\_{i=1}^{n} (x\_i - \mathfrak{x})^2\right]^{3/2}} \quad \text{and} \quad \text{CK}(\mathbf{x}) = \frac{n^{-1} \sum\_{i=1}^{n} [x\_i - \mathfrak{x}]^4}{\left[n^{-1} \sum\_{i=1}^{n} (x\_i - \mathfrak{x})^2\right]^{3/2}}$$

where *x* = (*xi*, ... , *xn*) denotes an observation of the sample. This definition of kurtosis is the raw measure, not excess kurtosis, which subtracts three from this quantity. From Table 2, we note that, as the sample size increases, the RBs and √MSE become smaller. We can also note that both *β* and *γ* are persistently skewed and somewhat unstable; nonetheless, they remain close to a normal distribution in terms of their skewness and kurtosis values.


**Table 2.** Results of the Monte Carlo (MC) experiments based on the generalized gamma distribution.
