A new time-varying model for forecasting long-memory series

Abstract

In this work we propose a new class of long-memory models with time-varying fractional parameter. In particular, the dynamics of the long-memory coefficient, d, is specified through a stochastic recurrence equation driven by the score of the predictive likelihood, as suggested by Creal et al. (J Appl Econom 28:777–795, 2013) and Harvey (Dynamic models for volatility and heavy tails: with applications to financial and economic time series, Cambridge University Press, Cambridge, 2013). We demonstrate the validity of the proposed model by a Monte Carlo experiment and an application to two real time series.

Appendix

Using Eq. (4), we find

$$\begin{aligned} \nu _j(d_t)&=\frac{\partial \pi _j(d_t)}{\partial d_t}\\&= -d_t\ \frac{-\Gamma '(j-d_t)\, \Gamma (1-d_t)\, \Gamma (j+1) + \Gamma '(1-d_t)\, \Gamma (j+1)\, \Gamma (j-d_t)}{\left( \Gamma (1-d_t)\,\Gamma (j+1)\right) ^2}\\&-\frac{\Gamma (j-d_t)}{\Gamma (1-d_t)\,\Gamma (j+1)}\\&= \frac{-d_t\,\Gamma (j-d_t)}{\Gamma (1-d_t)\,\Gamma (j+1)} \left( \frac{-\Gamma '(j-d_t)}{\Gamma (j-d_t)}+\frac{\Gamma '(1-d_t)}{\Gamma (1-d_t)}+\frac{1}{d_t}\right) \\&=\pi _j(d_t)\left( -\Psi (j-d_t) + \Psi (1-d_t) + \frac{1}{d_t}\right) , \end{aligned}$$

where $\Psi (\cdot )=\Gamma ^{'}(\cdot )/\Gamma (\cdot )$ is the digamma function. Therefore:

$$\begin{aligned} \nabla _t&= - \frac{1}{\sigma ^2}\left( y_t + \sum _{j=1}^{t-1} \pi _j(d_t)\, y_{t-j} \right) \left( \sum _{j=1}^{t-1} \frac{\partial \pi _j(d_t)}{\partial d_t}\, y_{t-j}\right) \\&=- \frac{1}{\sigma ^2}\left( y_t + \sum _{j=1}^{t-1} \pi _j(d_t)\, y_{t-j} \right) \left( \sum _{j=1}^{t-1} \nu _j(d_t)\, y_{t-j}\right) . \end{aligned}$$

Now, observe that:

$$\begin{aligned} \frac{\partial ^2 \pi _j(d_t)}{\partial d_t}&= \nu _j(d_t) \left[ -\Psi (j-d_t) + \Psi (1-d_t) + \frac{1}{d_t}\right] \\&\quad +\,\pi _j(d_t) \left[ \Psi ^{'}(j-d_t) - \Psi ^{'}(1-d_t) - \frac{1}{d^2_t}\right] \ . \end{aligned}$$

Hence, we find

$$\begin{aligned} \mathcal {I}_{t-1} = -E_{t-1}\!\!\left[ \frac{\partial \nabla _t}{\partial d_t} \right]&= \frac{1}{\sigma ^2}\, E_{t-1}\!\!\left[ \left( \sum _{j=1}^{t-1} \nu _j(d_t)\, y_{t-j} \right) ^2 \right. \\&\quad +\,\left. \left( y_t + \sum _{j=1}^{t-1}\pi _j(d_t)\, y_{t-j}\right) \left( \sum _{j=1}^{t-1} \frac{\partial ^2 \pi _j(d_t)}{\partial d_t}\, y_{t-j}\right) \right] \\&= \frac{1}{\sigma ^2}\, \left( \sum _{j=1}^{t-1} \nu _j(d_t)\, y_{t-j} \right) ^2, \end{aligned}$$

where we used $E_{t-1}\!\left[ y_t + \sum _{j=1}^{t-1}\pi _j(d_t)\, y_{t-j}\right] =E_{t-1}\!\left[ \epsilon _t\right] =0$. Finally:

$$\begin{aligned}S_t=\mathcal {I}_{t-1}^{-1} = \sigma ^2\, \left( \sum _{j=1}^{t-1} \nu _j(d_t)\, y_{t-j} \right) ^{-2}\ .\end{aligned}$$