estimate_lrv: Computes estimator of the long-run variance of the error...

In marina-khi/multiscale: Multiscale Inference for Nonparametric Time Trend(s)

Computes estimator of the long-run variance of the error terms.

Description

A difference based estimator for the coefficients and long-run variance in case of a nonparametric regression model are AR(p).

Specifically, we assume that we observe Y(t) that satisfy the following equation:

Y(t) = m(t/T) + \epsilon_t.

Here, m(\cdot) is an unknown function, and the errors \epsilon_t are AR(p) with p known. Specifically, we ler \{\epsilon_t\} be a process of the form

\epsilon_t = \sum_{j=1}^p a_j \epsilon_{t-j} + \eta_t,

where a_1,a_2,\ldots, a_p are unknown coefficients and \eta_t are i.i.d.\ with E[\eta_t] = 0 and E[\eta_t^2] = \nu^2.

This function produces an estimator \widehat{\sigma}^2 of the long-run variance

\sigma^2 = \sum_{l=-\infty}^{\infty} cov(\epsilon_0,\epsilon_{l})

of the error terms, as well as estimators \widehat{a}_1, \ldots, \widehat{a}_p of the coefficients a_1,a_2,\ldots, a_p and an estimator \widehat{\nu}^2 of the innovation variance \nu^2.

The exact estimation procedure as well as description of the tuning parameters needed for this estimation can be found in Khismatullina and Vogt (2020).

Usage

estimate_lrv(data, q, r_bar, p)

Arguments

data	A vector of Y(1), Y(2), \ldots, Y(T).
q, r_bar	Tuning parameters.
p	AR order of the error terms.

Value

A list with the following elements:

lrv	Estimator of the long run variance of the error terms \sigma^2.
ahat	Vector of length p of estimated AR coefficients a_1,a_2,\ldots, a_p.
vareta	Estimator of the variance of the innovation term \nu^2.

References

Khismatullina M., Vogt M. Multiscale inference and long-run variance estimation in non-parametric regression with time series errors //Journal of the Royal Statistical Society: Series B (Statistical Methodology). - 2020.

marina-khi/multiscale documentation built on Jan. 15, 2025, 7:28 a.m.