heter_gradient: Structural stability tests for non-stationary time series...

In mlrv: Long-Run Variance Estimation in Time Series Regression

Structural stability tests for non-stationary time series regression

Description

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Test for long memory of \mjseqne_i in the time series regression \mjsdeqny_i = x_i \beta_i + e_i, 1\leq i \leq n where \mjseqnx_i is the multivariate covariate process with first component 1, \mjseqn\beta_i is the coefficient, \mjseqne_i is the error term which can be long memory. The goal is to test whether the null hypothesis \mjsdeqn\beta_1 = ... = \beta_n = \beta holds. The alternative hypothesis is that the coefficient function \mjseqn\beta_i is time-varying. Covariates and the error term are allowed to be dependent.

Usage

heter_gradient(data, param, mvselect = -1, verbose_dist = FALSE, hyper = FALSE)

Arguments

data	a list with the vector y (response) and the matrix x (covariates), for example, list(x=...,y=...).
param	a list of parameters, list(B =..., lrvmethod =...,gcv = ..., neighbour =..., lb = ..., ub = ..., tau_n = ..., type = ..., ind = ...)
mvselect	the value of moving window parameter \mjseqnm. In addition, mvselect=-1 provides data-driven smoothing parameters via Minimum Volatility of the long-run covariance estimator, while mvselect = -2 provides data-driven smoothing parameters via Minimum Volatility of the bootstrap statistics.
verbose_dist	whether to print intermediate results, i.e., the bootstrap distribution and statistics, default FALSE.
hyper	whether to only print the selected values of the smoothing parameters,\mjseqnm and \mjseqn\tau_n, default FALSE.

Details

param

• B, the number of bootstrap simulation, say 2000

• lrvmethod the method of long-run variance estimation, lrvmethod = -1 uses the ols plug-in estimator as in Wu and Zhou (2018), lrvmethod = 0 uses the plug-in estimator in Zhou (2010), lrvmethod = 1 offers the debias difference-based estimator in Bai and Wu (2024), lrvmethod = 2 provides the plug-in estimator using the \mjseqn\breve\beta, the pilot estimator proposed in Bai and Wu (2024)

• gcv, 1 or 0, whether to use Generalized Cross Validation for the selection of \mjseqnb, the bandwidth parameter in the local linear regression, which will not be used when lrvmethod is -1, 1 or 2.

• neighbour, the number of neighbours in the extended minimum volatility, for example 1,2 or 3

• lb, the lower bound of the range of \mjseqnm in the extended minimum volatility Selection

• ub, the upper bound of the range of \mjseqnm in the extended minimum volatility Selection

• bw_set, the proposed grid of the range of bandwidth selection, which is only useful when lrvmethod = 1. if not presented, a rule of thumb method will be used for the data-driven range.

• tau_n, the value of \mjseqn\tau when no data-driven selection is used. if \mjseqntau is set to \mjseqn0, the rule of thumb \mjseqnn^-1/5 will be used

• type, default 0, uses the residual-based statistic proposed in Wu and Zhou (2018). “type” can also be set to -1, using the coefficient-based statistic in Wu and Zhou (2018).

• ind, types of kernels

• 1 Triangular \mjseqn1-|u|, \mjseqnu \le 1

• 2 Epanechnikov kernel \mjseqn3/4(1 - u^2), \mjseqnu \le 1

• 3 Quartic \mjseqn15/16(1 - u^2)^2, \mjseqnu \le 1

• 4 Triweight \mjseqn35/32(1 - u^2)^3, \mjseqnu \le 1

• 5 Tricube \mjseqn70/81(1 - |u|^3)^3, \mjseqnu \le 1

Value

p-value of the structural stability test

References

Bai, L., & Wu, W. (2024). Difference-based covariance matrix estimation in time series nonparametric regression with application to specification tests. Biometrika, asae013.

Wu, W., and Zhou, Z. (2018). Gradient-based structural change detection for nonstationary time series M-estimation. The Annals of Statistics, 46(3), 1197-1224.

Politis, D. N., Romano, J. P., and Wolf, M. (1999). Subsampling. Springer Science & Business Media.

Examples

# choose a small B for tests
param = list(B = 50, bw_set = c(0.15, 0.25), gcv =1, neighbour = 1, lb = 10, ub = 20, type = 0)
n = 300
data = bregress2(n, 2, 1) # time series regression model with 2 changes points
param$lrvmethod = 0 # plug-in
heter_gradient(data, param, 4, 1)
param$lrvmethod = 1 # difference based
heter_gradient(data, param, 4, 1)

mlrv documentation built on Sept. 11, 2024, 6:57 p.m.