Assumption 2.Suppose that Y_t is generated by (1), and let Assumption 1 hold. Then, we assume that there exists an estimator, $\widehat{d}$, of the true fractional integration parameter, d, which satisfies the condition that $E{(\widehat{d}-d)}^2=o(1)$.

Remark 13.Assumption 2 does not specify any particular method for estimating d. Obvious candidate estimators to consider are the LW and ELW estimators discussed in Section 2. For consistency, the bandwidth, m, used for the LW estimator must satisfy the condition that $\frac{1}{m}+\frac{m}{T}\to 0$ as T → ∞, while for ELW the required rate on m is that $\frac{1}{m}+\frac{m{(\log m)}^{\frac{1}{2}}}{T}+\frac{\log T}{m^{\gamma }}\to 0$ as T → ∞, for any γ > 0. For ELW the range of permissible values in the optimisation in (3) is such that $d_2-d_1\le \frac{9}{2}$, although note that this does not restrict the value of d itself. The full set of required conditions for consistency are given in Assumptions A1 to A4 of Robinson (1995) for the LW estimator and in Assumptions 1 to 5 of Phillips and Shimotsu (2005) for the ELW estimator. Assumption 2, however, requires the stronger concept of convergence in mean square. This would need to be verified for any given estimator. Given convergence in distribution, a sufficient condition for Assumption 2 to hold is that ${\widehat{d}}^2$ is uniformly integrable.

假设2。假设Y _t是由(1) 生成的，并让假设1 成立。然后，我们假设存在一个估计量，$\widehat{d}$, 的真分数积分参数d满足条件$乙{(\widehat{d}-d)}^2=○(1)$.

备注 13.假设 2 没有指定估计d 的任何特定方法。要考虑的明显候选估计量是第 2 节中讨论的LW和ELW估计量。为了一致性，用于LW估计量的带宽m必须满足以下条件：$\frac{1}{米}+\frac{米}{吨}\to 0$作为T  → ∞，而对于ELW，m上所需的速率是$\frac{1}{米}+\frac{米{(\mathrm{日志}米)}^{\frac{1}{2}}}{吨}+\frac{\mathrm{日志}吨}{{米}^{\gamma }}\to 0$作为T  → ∞，对于任何γ  > 0。对于ELW，(3) 中优化中的允许值范围是这样的$d_2-d_1\le \frac{9}{2}$，但请注意，这并不限制d本身的值。在 Robinson (1995) 的假设 A1 到 A4 中针对LW估计量以及在 Phillips 和 Shimotsu (2005) 的假设 1 到 5 中针对ELW估计量给出了一致性所需的全套条件。然而，假设 2 需要更强的均方收敛概念。这需要针对任何给定的估算器进行验证。鉴于分布收敛，假设 2 成立的充分条件是${\widehat{d}}^2$ 是一致可积的​​。