mlrv: 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 Zhangdoi: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.

Author(s)

Maintainer: Lujia Bai bailujia98@gmail.com

Other contributors:

• Weichi Wu wuweichi@mail.tsinghua.edu.cn [contributor]

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

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. (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.

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)

Local linear Regression

Description

Local linear estimates for time varying coefficients

Usage

LocLinear(bw, t, y, X, db_kernel = 0L, deriv2 = 0L, scb = 0L)

Arguments

Details

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 \(X^T \hat{\boldsymbol{\beta}}_{b_{n}}(t)\)

Value

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

References

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.

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)
n = 500
t = (1:n)/n
data = Qct_reg(n, param)
result = LocLinear(0.2, t, data$y, data$x)

Statistics-adapted values for extended minimum volatility selection.

Description

Calculation of the variance of the bootstrap statistics for the extended minimum volatility selection.

Usage

MV_critical(
  y,
  data,
  R,
  gridm,
  gridtau,
  type = 1L,
  cvalue = 0.1,
  B = 100L,
  lrvmethod = 1L,
  ind = 2L,
  rescale = 0L
)

Arguments

Value

a matrix of critical values

References

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

See Also

Heter_LRV

Examples

###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))

Statistics-adapted values for extended minimum volatility selection.

Description

Smoothing parameter selection for bootstrap tests for change point tests

Usage

MV_critical_cp(
  y,
  X,
  t,
  gridm,
  gridtau,
  cvalue = 0.1,
  B = 100L,
  lrvmethod = 1L,
  ind = 2L,
  rescale = 0L
)

Arguments

Value

a matrix of critical values

References

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

Examples

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))

MV method

Description

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.

Usage

MV_ise_heter_critical(critical, neighbour)

Arguments

Value

a list of results,

• minp: optimal row number

• minq: optimal column number

• min_ise: optimal value

References

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

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)
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

Description

Simulate data from time-varying time series regression model

Usage

Qct_reg(T_n, param, type = 1)

Arguments

Value

list, a list of data, covariates, response and errors.(before and after fractional difference)

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)
n = 500
data = Qct_reg(n, param)

Simulate data from time-varying trend model

Description

Simulate data from time-varying trend model

Usage

Qt_data(T_n, param)

Arguments

Value

a vector of non-stationary time series

Examples

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)

Simulate data from time-varying time series regression model with change points

Description

Simulate data from time-varying time series regression model with change points

Usage

bregress2(nn, cp = 0, delta = 0, type = "norm")

Arguments

Value

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

Generalized Cross Validation

Description

Given a bandwidth, compute its corresponding GCV value

Usage

gcv_cov(bw, t, y, X, verbose = 1L)

Arguments

Details

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

Value

GCV value

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)
value <- gcv_cov(0.2, (1:1000)/1000, data$y, data$x)

Long memory tests for non-stationary time series regression

Description

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.

Usage

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
)

Arguments

Details

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

Value

p-value of the long memory test

mlrv functions

Heter_LRV, heter_covariate, heter_gradient, gcv_cov, MV_critical

References

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.

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)
### 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)

Structural stability tests for non-stationary time series regression

Description

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.

Usage

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

Arguments

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 \(\breve{\beta}\), 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 \(70/81(1 - |u|^{3})^{3}\), \(u \le 1\)

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)

This is data to be included in my package

Description

This is data to be included in my package

Author(s)

T. S. Lau

References

Fan, J., and Zhang, W. (1999). Statistical estimation in varying coefficient models. The annals of Statistics, 27(5), 1491-1518.

Nonparametric smoothing

Description

Nonparametric smoothing

Usage

loc_constant(bw, x, y, db_kernel = 0L)

Arguments

Value

a matrix of smoothed values

Examples

n <- 800
p <- 3
t <- (1:n)/n
V <-  matrix(rnorm(n * p), nrow = p)
V3 <- loc_constant(0.2, t, V,1)

Comparing bias or mse of lrv estimators based on numerical methods

Description

Comparing bias or mse of lrv estimators based on numerical methods

Usage

lrv_measure(
  data,
  param,
  lrvmethod,
  mvselect = -1,
  tau = 0,
  verbose_dist = FALSE,
  mode = "mse"
)

Arguments

Value

empirical MSE of the estimator.

Examples

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)

rule of thumb interval for the selection of smoothing parameter b

Description

The function will compute a data-driven interval for the Generalized Cross Validation performed later, see also Bai and Wu (2024) .

Usage

rule_of_thumb(y, x)

Arguments

Value

c(left, right), the vector with the left and right points of the interval

References

Bai, L., & Wu, W. (2024). Detecting long-range dependence for time-varying linear models. Bernoulli, 30(3), 2450-2474.

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)
rule_of_thumb(data$y, data$x)

bootstrap distribution

Description

bootstrap distribution of the gradient based structural stability test

Usage

sim_T(X, t, sigma, m, B, type = 0L)

Arguments

Value

a vector of bootstrap statistics

Examples

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 iterations