LocLinear {mlrv}

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

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.

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)