| Title: | Long-Run Variance Estimation in Time Series Regression |
|---|---|
| Description: | Plug-in and difference-based long-run covariance matrix estimation for time series regression. Two applications of hypothesis testing are also provided. The first one is for testing for structural stability in coefficient functions. The second one is aimed at detecting long memory in time series regression. Lujia Bai and Weichi Wu (2024)<doi:10.3150/23-BEJ1680> Zhou Zhou and Wei Biao Wu(2010)<doi:10.1111/j.1467-9868.2010.00743.x> Jianqing Fan and Wenyang Zhang<doi:10.1214/aos/1017939139> Lujia Bai and Weichi Wu(2024)<doi:10.1093/biomet/asae013> Dimitris N. Politis, Joseph P. Romano, Michael Wolf(1999)<doi:10.1007/978-1-4612-1554-7> Weichi Wu and Zhou Zhou(2018)<doi:10.1214/17-AOS1582>. |
| Authors: | Lujia Bai [aut, cre], Weichi Wu [ctb] |
| Maintainer: | Lujia Bai <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 0.1.2 |
| Built: | 2024-11-29 08:55:12 UTC |
| Source: | CRAN |
Simulate data from time-varying time series regression model with change points
bregress2(nn, cp = 0, delta = 0, type = "norm")bregress2(nn, cp = 0, delta = 0, type = "norm")
nn |
sample size |
cp |
number of change points. If cp is between 0 and 1, it specifies the location of the single change point |
delta |
double, magnitude of the jump |
type |
type of distributions of the innovations, default normal. It can also be "t4", "t5" and "t6". |
a list of data, x covariates, y response and e error. n = 300 data = bregress2(n, 2, 1) # time series regression model with 2 changes points
Given a bandwidth, compute its corresponding GCV value
gcv_cov(bw, t, y, X, verbose = 1L)gcv_cov(bw, t, y, X, verbose = 1L)
bw |
double, bandwidth |
t |
vector, scaled time \([0,1]\) |
y |
vector, response |
X |
matrix, covariates matrix |
verbose |
bool, whether to print the numerator and denominator in GCV value |
Generalized cross validation value is defined as
\[n^{-1}| Y-\hat{Y}|^2/[1- \mathrm{tr}(Q(b)) / n]^2\]
When computing \(\mathrm{tr}(Q(b))\),
we use the fact that the first derivative of coefficient function is zero at central point
The ith diagonal value of \(Q(b)\) is actually \(x^T(t_i)S^{-1}_{n}x(t_i)\)
where \(S^{-1}_{n}\) means the top left p-dimension square matrix of
\(S_{n}(t_i) = X^T W(t_i) X\), \(W(t_i)\) is the kernel weighted matrix. Details on
the computation of \(S_n\) could be found in LocLinear and its reference
GCV value
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) value <- gcv_cov(0.2, (1:1000)/1000, data$y, data$x)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) value <- gcv_cov(0.2, (1:1000)/1000, data$y, data$x)
Test for long memory of \(e_i\) in the time series regression
y_i = x_i β_i + e_i, 1≤ i ≤ n
where \(x_i\) is the multivariate covariate process with first component 1, \(\beta_i\) is the functional coefficient, \(e_i\) is the error term which can be long memory. In particular,covariates and the error term are allowed to be dependent.
heter_covariate( data, param = list(B = 2000, lrvmethod = 1, gcv = 1, neighbour = 1, lb = 3, ub = 11, tau_n = 0.3, type = "KPSS"), mvselect = -1, bw = 0.2, shift = 1, verbose_dist = FALSE, hyper = FALSE )heter_covariate( data, param = list(B = 2000, lrvmethod = 1, gcv = 1, neighbour = 1, lb = 3, ub = 11, tau_n = 0.3, type = "KPSS"), mvselect = -1, bw = 0.2, shift = 1, verbose_dist = FALSE, hyper = FALSE )
data |
a list with the vector y and the matrix x, 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 \(m\). In addition, mvselect=-1 provides data-driven smoothing parameters via Minimum Volatility of the long-run covariance estimator as proposed in Chapter 9 of Politis et al.(1999), while mvselect = -2 provides data-driven smoothing parameters via Minimum Volatility of the bootstrap statistics, see Bai and Wu (2024a). |
bw |
the bandwidth parameter in the local linear regression, default 0.2. |
shift |
modify bw by a factor, default 1. |
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,\(m\) and \(\tau_n\), default FALSE. |
param
B, the number of bootstrap simulation, say 2000 *lrvmethod, 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 (2024b), lrvmethod = 2 provides the plug-in estimator using the \(\breve{\beta}\), the pilot estimator proposed in Bai and Wu (2024b)
gcv, 1 or 0, whether to use Generalized Cross Validation for the selection of \(b\), the bandwidth parameter in the local linear regression
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. if not presented, a rule of thumb method will be used for the data-driven range
tau_n, the value of \(\tau\) when no data-driven selection is used. if \(\tau\) is set to \(0\), the rule of thumb \(n^{-2/15}\) will be used
type, c( "KPSS","RS","VS","KS") type of tests, see Bai and Wu (2024a).
ind, types of kernels
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 \(70/81(1 - |u|^{3})^{3}\), \(u \le 1\)
p-value of the long memory test
Heter_LRV, heter_covariate, heter_gradient, gcv_cov, MV_critical
Bai, L., & Wu, W. (2024a). Detecting long-range dependence for time-varying linear models. Bernoulli, 30(3), 2450-2474.
Bai, L., & Wu, W. (2024b). Difference-based covariance matrix estimation in time series nonparametric regression with application to specification tests. Biometrika, asae013.
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.
Politis, D. N., Romano, J. P., and Wolf, M. (1999). Subsampling. Springer Science & Business Media.
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) ### KPSS test B heter_covariate(data, list(B=20, lrvmethod = 1, gcv = 1, neighbour = 1, lb = 3, ub = 11, type = "KPSS"), mvselect = -2, verbose_dist = TRUE)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) ### KPSS test B heter_covariate(data, list(B=20, lrvmethod = 1, gcv = 1, neighbour = 1, lb = 3, ub = 11, type = "KPSS"), mvselect = -2, verbose_dist = TRUE)
Test for long memory of \(e_i\) in the time series regression \[y_i = x_i \beta_i + e_i, 1\leq i \leq n\] where \(x_i\) is the multivariate covariate process with first component 1, \(\beta_i\) is the coefficient, \(e_i\) is the error term which can be long memory. The goal is to test whether the null hypothesis \[\beta_1 = \ldots = \beta_n = \beta\] holds. The alternative hypothesis is that the coefficient function \(\beta_i\) is time-varying. Covariates and the error term are allowed to be dependent.
heter_gradient(data, param, mvselect = -1, verbose_dist = FALSE, hyper = FALSE)heter_gradient(data, param, mvselect = -1, verbose_dist = FALSE, hyper = FALSE)
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 \(m\). 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. |
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,\(m\) and \(\tau_n\), 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 \(\breve{\beta}\), the pilot estimator proposed in Bai and Wu (2024)
gcv, 1 or 0, whether to use Generalized Cross Validation for the selection of \(b\), 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 \(\tau\) 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 \(1-|u|\), \(u \le 1\)
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 \(70/81(1 - |u|^{3})^{3}\), \(u \le 1\)
p-value of the structural stability test
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)# 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)
The function provides a wide range of estimators for the long-run covariance matrix estimation in non-stationary time series with covariates.
Heter_LRV( e, X, m, tau_n = 0, lrv_method = 1L, ind = 2L, print_deg = 0L, rescale = 0L, ncp = 0L )Heter_LRV( e, X, m, tau_n = 0, lrv_method = 1L, ind = 2L, print_deg = 0L, rescale = 0L, ncp = 0L )
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 (2024). |
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 (2024), lrvmethod = 2 provides the plug-in estimator using the \(\breve{\beta}\), the pilot estimator proposed in Bai and Wu (2024) |
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
|
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.
Bai, L., & Wu, W. (2024). Difference-based covariance matrix estimation in time series nonparametric regression with application to specification tests. Biometrika, asae013.
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.
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)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)
This is data to be included in my package
T. S. Lau
Fan, J., and Zhang, W. (1999). Statistical estimation in varying coefficient models. The annals of Statistics, 27(5), 1491-1518.
Nonparametric smoothing
loc_constant(bw, x, y, db_kernel = 0L)loc_constant(bw, x, y, db_kernel = 0L)
bw |
double, bandwidth, between 0 and 1. |
x |
vector, covariates |
y |
matrix, response variables |
db_kernel |
bool, whether to use jackknife kernel, default 0 |
a matrix of smoothed values
n <- 800 p <- 3 t <- (1:n)/n V <- matrix(rnorm(n * p), nrow = p) V3 <- loc_constant(0.2, t, V,1)n <- 800 p <- 3 t <- (1:n)/n V <- matrix(rnorm(n * p), nrow = p) V3 <- loc_constant(0.2, t, V,1)
Local linear estimates for time varying coefficients
LocLinear(bw, t, y, X, db_kernel = 0L, deriv2 = 0L, scb = 0L)LocLinear(bw, t, y, X, db_kernel = 0L, deriv2 = 0L, scb = 0L)
bw |
double, bandwidth |
t |
vector, time, 1:n/n |
y |
vector, response series to be tested for long memory in the next step |
X |
matrix, covariates matrix |
db_kernel |
bool, whether to use jackknife kernel, default 0 |
deriv2 |
bool,whether to return second-order derivative, default 0 |
scb |
bool, whether to use the result for further calculation of simultaneous confidence bands. |
The time varying coefficients are estimated by \[(\hat{\boldsymbol{\beta}}_{b_{n}}(t), \hat{\boldsymbol{\beta}}_{b_{n}}^{\prime}(t)) = \mathbf{arg min}_{\eta_{0},\eta_{1}}[\sum_{i=1}^{n}{y_{i}-\mathbf{x}_{i}^{\mathrm{T}}\eta_{0}-\mathbf{x}_{i}^{\mathrm{T}} \eta_{1}(t_{i}-t)}^{2} \boldsymbol{K}_{b_{n}}(t_{i}-t)]\] where beta0 is \(\hat{\boldsymbol{\beta}}_{b_{n}}(t)\), mu is \(X^T \hat{\boldsymbol{\beta}}_{b_{n}}(t)\)
a list of results
mu: the estimated trend
beta0: time varying coefficient
X_reg: a matrix whose j'th row is \(x_j^T\hat{M}(t_j)\)
t: 1:n/n
bw: bandwidth used
X: covariates matrix
y: response
n: sample size
p: dimension of covariates including the intercept
invM: inversion of M matrix, when scb = 1
Zhou, Z., & Wu, W. B. (2010). Simultaneous inference of linear models with time varying coefficients. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(4), 513-531.
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) n = 500 t = (1:n)/n data = Qct_reg(n, param) result = LocLinear(0.2, t, data$y, data$x)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) n = 500 t = (1:n)/n data = Qct_reg(n, param) result = LocLinear(0.2, t, data$y, data$x)
Comparing bias or mse of lrv estimators based on numerical methods
lrv_measure( data, param, lrvmethod, mvselect = -1, tau = 0, verbose_dist = FALSE, mode = "mse" )lrv_measure( data, param, lrvmethod, mvselect = -1, tau = 0, verbose_dist = FALSE, mode = "mse" )
data |
a list of data |
param |
a list of parameters |
lrvmethod |
int, method of long-run variance estimation |
mvselect |
int, method of MV selection |
tau |
double, value of tau. If tau is 0, a rule-of-thunk value will be applied |
verbose_dist |
bool, whether to output distributional information |
mode |
default "mse", It can be set as "bias". |
empirical MSE of the estimator.
n = 300 param = list(gcv = 1, neighbour = 1,lb = 6, ub = 13, ind = 2) # covariates heterskadecity data = bregress2(n, 2, 1) # with 2 change pointa lrv_measure(data, param, lrvmethod = -1, mvselect = -2) #ols plug-in #debiased difference-based lrv_measure(data, param, lrvmethod = 1, mvselect = -2)n = 300 param = list(gcv = 1, neighbour = 1,lb = 6, ub = 13, ind = 2) # covariates heterskadecity data = bregress2(n, 2, 1) # with 2 change pointa lrv_measure(data, param, lrvmethod = -1, mvselect = -2) #ols plug-in #debiased difference-based lrv_measure(data, param, lrvmethod = 1, mvselect = -2)
Calculation of the variance of the bootstrap statistics for the extended minimum volatility selection.
MV_critical( y, data, R, gridm, gridtau, type = 1L, cvalue = 0.1, B = 100L, lrvmethod = 1L, ind = 2L, rescale = 0L )MV_critical( y, data, R, gridm, gridtau, type = 1L, cvalue = 0.1, B = 100L, lrvmethod = 1L, ind = 2L, rescale = 0L )
y |
vector, as used in the Heter_LRV |
data |
list, a list of data |
R |
a cube of standard.normal random variables. |
gridm |
vector, a grid of candidate m's. |
gridtau |
vector, a grid of candidate tau's. |
type |
integer, 1 KPSS 2 RS 3 VS 4 KS |
cvalue |
double, 1-quantile for the calculation of bootstrap variance, default 0.1. |
B |
integer, number of iterations for the calculation of bootstrap variance |
lrvmethod |
integer, see also Heter_LRV |
ind |
integer, the type of kernel, see also Heter_LRV |
rescale |
bool, whether to rescale when positiveness of the matrix is not obtained. default 0 |
a matrix of critical values
Bai, L., & Wu, W. (2024). Difference-based covariance matrix estimation in time series nonparametric regression with application to specification tests. Biometrika, asae013.
Heter_LRV
###with Long memory parameter 0.2 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) n = 1000 data = Qct_reg(n, param) p = ncol(data$x) t = (1:n)/n B_c = 100 ##small value for testing Rc = array(rnorm(n*p*B_c),dim = c(p,B_c,n)) result1 = LocLinear(0.2, t, data$y, data$x) critical <- MV_critical(data$y, result1, Rc, c(3,4,5), c(0.2, 0.25, 0.3))###with Long memory parameter 0.2 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) n = 1000 data = Qct_reg(n, param) p = ncol(data$x) t = (1:n)/n B_c = 100 ##small value for testing Rc = array(rnorm(n*p*B_c),dim = c(p,B_c,n)) result1 = LocLinear(0.2, t, data$y, data$x) critical <- MV_critical(data$y, result1, Rc, c(3,4,5), c(0.2, 0.25, 0.3))
Smoothing parameter selection for bootstrap tests for change point tests
MV_critical_cp( y, X, t, gridm, gridtau, cvalue = 0.1, B = 100L, lrvmethod = 1L, ind = 2L, rescale = 0L )MV_critical_cp( y, X, t, gridm, gridtau, cvalue = 0.1, B = 100L, lrvmethod = 1L, ind = 2L, rescale = 0L )
y |
vector, as used in the Heter_LRV |
X |
matrix, covariates |
t |
vector, time points. |
gridm |
vector, a grid of candidate m's. |
gridtau |
vector, a grid of candidate tau's. |
cvalue |
double, 1-quantile for the calculation of bootstrap variance, default 0.1. |
B |
integer, number of iterations for the calculation of bootstrap variance |
lrvmethod |
integer, see also Heter_LRV |
ind |
integer, the type of kernel, see also Heter_LRV |
rescale |
bool, whether to rescale when positiveness of the matrix is not obtained. default 0 |
a matrix of critical values
Bai, L., & Wu, W. (2024). Difference-based covariance matrix estimation in time series nonparametric regression with application to specification tests. Biometrika, asae013.
n = 300 t = (1:n)/n data = bregress2(n, 2, 1) # time series regression model with 2 changes points critical = MV_critical_cp(data$y, data$x,t, c(3,4,5), c(0.2,0.25, 0.3))n = 300 t = (1:n)/n data = bregress2(n, 2, 1) # time series regression model with 2 changes points critical = MV_critical_cp(data$y, data$x,t, c(3,4,5), c(0.2,0.25, 0.3))
Selection of smoothing parameters for bootstrap tests by choosing the index minimizing the volatility of bootstrap statistics or long-run variance estimators in the neighborhood computed before.
MV_ise_heter_critical(critical, neighbour)MV_ise_heter_critical(critical, neighbour)
critical |
a matrix of critical values |
neighbour |
integer, number of neighbours |
a list of results,
minp: optimal row number
minq: optimal column number
min_ise: optimal value
Bai, L., & Wu, W. (2024). Difference-based covariance matrix estimation in time series nonparametric regression with application to specification tests. Biometrika, asae013.
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) n = 1000 data = Qct_reg(n, param) p = ncol(data$x) t = (1:n)/n B_c = 100 ##small value for testing Rc = array(rnorm(n*p*B_c),dim = c(p,B_c,n)) result1 = LocLinear(0.2, t, data$y, data$x) gridm = c(3,4,5) gridtau = c(0.2, 0.25, 0.3) critical <- MV_critical(data$y, result1, Rc, gridm, gridtau) mv_result = MV_ise_heter_critical(critical, 1) m = gridm[mv_result$minp + 1] tau_n = gridtau[mv_result$minq + 1]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) n = 1000 data = Qct_reg(n, param) p = ncol(data$x) t = (1:n)/n B_c = 100 ##small value for testing Rc = array(rnorm(n*p*B_c),dim = c(p,B_c,n)) result1 = LocLinear(0.2, t, data$y, data$x) gridm = c(3,4,5) gridtau = c(0.2, 0.25, 0.3) critical <- MV_critical(data$y, result1, Rc, gridm, gridtau) mv_result = MV_ise_heter_critical(critical, 1) m = gridm[mv_result$minp + 1] tau_n = gridtau[mv_result$minq + 1]
Simulate data from time-varying time series regression model
Qct_reg(T_n, param, type = 1)Qct_reg(T_n, param, type = 1)
T_n |
int, sample size |
param |
list, a list of parameters |
type |
type = 1 means the long memory expansion begins from its infinite past, type = 2 means the long memory expansion begins from t = 0 |
list, a list of data, covariates, response and errors.(before and after fractional difference)
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) n = 500 data = Qct_reg(n, param)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) n = 500 data = Qct_reg(n, param)
Simulate data from time-varying trend model
Qt_data(T_n, param)Qt_data(T_n, param)
T_n |
integer, sample size |
param |
a list of parameters
|
a vector of non-stationary time series
param = list(d = -0.2, tvd = 0, tw = 0.8, rate = 0.1, center = 0.3, ma_rate = 0, cur = 1) data = Qt_data(300, param)param = list(d = -0.2, tvd = 0, tw = 0.8, rate = 0.1, center = 0.3, ma_rate = 0, cur = 1) data = Qt_data(300, param)
The function will compute a data-driven interval for the Generalized Cross Validation performed later, see also Bai and Wu (2024) .
rule_of_thumb(y, x)rule_of_thumb(y, x)
y |
a vector, the response variable. |
x |
a matrix of covariates. If the intercept should be includes, the elements of the first column should be 1. |
c(left, right), the vector with the left and right points of the interval
Bai, L., & Wu, W. (2024). Detecting long-range dependence for time-varying linear models. Bernoulli, 30(3), 2450-2474.
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) rule_of_thumb(data$y, data$x)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) rule_of_thumb(data$y, data$x)
bootstrap distribution of the gradient based structural stability test
sim_T(X, t, sigma, m, B, type = 0L)sim_T(X, t, sigma, m, B, type = 0L)
X |
matrix of covariates |
t |
vector of time points |
sigma |
a cube of long-run covariance function. |
m |
int value of window size |
B |
int, number of iteration |
type |
type of tests, residual-based or coefficient-based |
a vector of bootstrap statistics
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 sigma = Heter_LRV(data$y, data$x, 3, 0.3, lrv_method = 1) bootstrap = sim_T(data$x, (1:n)/n, sigma, 3, 20) ### 20 iterationsparam = list(B = 50, bw_set = c(0.15, 0.25), gcv =1, neighbour = 1, lb = 10, ub = 20, type = 0) n = 300 data = bregress2(n, 2, 1) # time series regression model with 2 changes points sigma = Heter_LRV(data$y, data$x, 3, 0.3, lrv_method = 1) bootstrap = sim_T(data$x, (1:n)/n, sigma, 3, 20) ### 20 iterations