Heter_LRV {mlrv}

Long-run covariance matrix estimators

Description

The function provides a wide range of estimators for the long-run covariance matrix estimation in non-stationary time series with covariates.

Usage

Heter_LRV(
  e,
  X,
  m,
  tau_n = 0,
  lrv_method = 1L,
  ind = 2L,
  print_deg = 0L,
  rescale = 0L,
  ncp = 0L
)

Arguments

e

vector, if the plug-in estimator is used, e should be the vector of residuals, OLS or nonparametric ones. If the difference-based debiased method is adopted, e should be the response time series, i.e., \(y\). Specially, e should also be the response time series, i.e., \(y\), if the plug-in estimator using the \(\breve{\beta}\), the pilot estimator proposed in Bai and Wu (2023).

X

a matrix \(n\times p\)

m

integer, the window size.

tau_n

double, the smoothing parameter in the estimator. If tau_n is 0, a rule-of-thumb value will be automatically used.

lrv_method

the method of long-run variance estimation, lrvmethod = 0 uses the plug-in estimator in Zhou (2010), lrvmethod = 1 offers the debias difference-based estimator in Bai and Wu (2023), lrvmethod = 2 provides the plug-in estimator using the \(\breve{\beta}\), the pilot estimator proposed in Bai and Wu (2023)

ind

types of kernels

print_deg

bool, whether to print information of non-positiveness, default 0\(n\times p\)

rescale

bool, whether to use rescaling to correct the negative eigenvalues, default 0

ncp

1 no change points, 0 possible change points

• 1 Triangular \(1-|u|\), \(u \le 1\)

• 2 Epanechnikov kernel \(3/4(1-u^{2})\), \(u \le 1\)

• 3 Quartic \(15/16(1-u^{2})^{2}\), \(u \le 1\)

• 4 Triweight \(35/32(1-u^{2})^{3}\), \(u \le 1\)

• 5 Tricube \(70/81(1-|u|^{3})^{3}\), \(u \le 1\)

Value

a cube. The time-varying long-run covariance matrix \(p \times p \times n\), where p is the dimension of the time series vector, and n is the sample size.

References

Bai, L., & Wu, W. (2023). Difference-based covariance matrix estimate in time series nonparametric regression with applications to specification tests.

Zhou, Z. and Wu, W. B. (2010). Simultaneous inference of linear models with time varying coefficients.J. R. Stat. Soc. Ser. B. Stat. Methodol., 72(4):513–531.

Examples

param = list(d = -0.2, heter = 2, tvd = 0,
tw = 0.8, rate = 0.1, cur = 1, center = 0.3,
ma_rate =  0, cov_tw =  0.2, cov_rate = 0.1,
cov_center = 0.1, all_tw  = 1, cov_trend = 0.7)
data = Qct_reg(1000, param)
sigma = Heter_LRV(data$y, data$x, 3, 0.3, lrv_method = 1)