python - 如何在 Python 的 arch_model 中使用 StudentT() 分布时强制 nu=4.0（或某个值）？

Question

我如何在这里提供 nu=4.0 ？

model = arch_model(data)
from arch.univariate import StudentsT
model.distribution = StudentsT()

Answer 1

您可以通过_ord_t_partial_moment将 nu 自由度传递给 StudentsT ：

StudentsT._ord_t_partial_moment(n, z, nu)

来自： arch单变量分布

@staticmethod
    def _ord_t_partial_moment(n: int, z: float, nu: float) -> float:
        r"""
        Partial moments for ordinary parameterization of Students t df=nu

        Parameters
        ----------
        n : int
            Order of partial moment
        z : float
            Upper bound for partial moment integral
        nu : float
            Degrees of freedom

        Returns
        -------
        float
            Calculated moment

        References
        ----------
        .. [1] Winkler et al. (1972) "The Determination of Partial Moments"
               *Management Science* Vol. 19 No. 3

        Notes
        -----
        The order n lower partial moment to z is

        .. math::

            \int_{-\infty}^{z}x^{n}f(x)dx

        See [1]_ for more details.
        """
        if n < 0 or n >= nu:
            return nan
        elif n == 0:
            moment = stats.t.cdf(z, nu)
        elif n == 1:
            c = gamma(0.5 * (nu + 1)) / (sqrt(nu * pi) * gamma(0.5 * nu))
            e = 0.5 * (nu + 1)
            moment = (0.5 * (c * nu) / (1 - e)) * ((1 + (z ** 2) / nu) ** (1 - e))
        else:
            t1 = (z ** (n - 1)) * (nu + z ** 2) * stats.t.pdf(z, nu)
            t2 = (n - 1) * nu * StudentsT._ord_t_partial_moment(n - 2, z, nu)
            moment = (1 / (n - nu)) * (t1 - t2)
        return moment