Test for long memory of e_i in the time series regression y_i = x_i _i + e_i, 1 i n where x_i is the multivariate covariate process with first component 1, _i is the coefficient, e_i is the error term which can be long memory. The goal is to test whether the null hypothesis _1 = ... = _n = holds. The alternative hypothesis is that the coefficient function _i is time-varying. Covariates and the error term are allowed to be dependent.
heter_gradient(data, param, mvselect = -1, verbose_dist = FALSE, hyper = FALSE)p-value of the structural stability test
a list with the vector y (response) and the matrix x (covariates), for example, list(x=...,y=...).
a list of parameters, list(B =..., lrvmethod =...,gcv = ..., neighbour =..., lb = ..., ub = ..., tau_n = ..., type = ..., ind = ...)
the value of moving window parameter m. 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.
whether to print intermediate results, i.e., the bootstrap distribution and statistics, default FALSE.
whether to only print the selected values of the smoothing parameters,m and _n, default FALSE.
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 , the pilot estimator proposed in Bai and Wu (2024)
gcv, 1 or 0, whether to use Generalized Cross Validation for the selection of b, 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 m in the extended minimum volatility Selection
ub, the upper bound of the range of m 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 when no data-driven selection is used. if tau is set to 0, the rule of thumb n^-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 1-|u|, u 1
2 Epanechnikov kernel 3/4(1 - u^2), u 1
3 Quartic 15/16(1 - u^2)^2, u 1
4 Triweight 35/32(1 - u^2)^3, u 1
5 Tricube 70/81(1 - |u|^3)^3, u 1
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.
# 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)
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