Bootstrap Methods for Regression Models with Locally Stationary Errors

Description

Implements bootstrap methods for linear regression models with errors following a time-varying process, focusing on approximating the distribution of the least-squares estimator for regression models with locally stationary errors. It enables the construction of bootstrap and classical confidence intervals for regression coefficients, leveraging intensive simulation studies and real data analysis. The methodology is based on the approach described in Ferreira et al. (2020), allowing errors to be locally approximated by stationary processes.

Author(s)

Maintainer: Nicolas Loyola nloyola2016@udec.cl

Authors:

• Guillermo Ferreira gferreri@udec.cl

• Joel Muñoz joelmuno@udec.cl

Calculate the coverage of several long-memory models

Description

Generates coverage metrics for a parameter of interest using a specified long-memory model.

Usage

Coveragelongmemory(
  n,
  R,
  N,
  S,
  mu = 0,
  dist,
  method,
  B = NULL,
  nr.cores = 1,
  seed = 123,
  alpha,
  beta,
  start,
  sign = 0.05
)

Arguments

Details

This function estimates the parameters in the linear regression model for t = 1, ..., T,

Y_{t,T} = X_{t,T} \beta + \epsilon_{t,T},

where a locally stationary fractional noise process (LSFN) is described by the equation:

\epsilon_{t,T} = \sum_{j=0}^\infty \psi_j(u) \eta_{t-j}

for u=t/T in [0,1], where \psi_j(u) = \frac{\Gamma[j + d(u)]}{\Gamma[j+1] \Gamma[d(u)]} and d(u) is the smoothly varying long-memory coefficient. This model is referred to as locally stationary fractional noise (LSFN).

In this particular case, d(u) is modeled as a linear polynomial, and \sigma(u) as a quadratic polynomial.

Resampling methods evaluated:

• asym: Asymptotic method that uses the asymptotic variance of the estimator, based on the Central Limit Theorem, to construct confidence intervals under the assumption of normality in large samples.

• boot: Standard bootstrap that generates replicas of the estimator \hat{\beta} by resampling the adjusted residuals \hat{\epsilon}_t. It approximates the distribution of the estimator by the variability observed in the bootstrap replicas of \hat{\beta}.

• boott: Adjusted bootstrap that scales the bootstrap replicas of the estimator \hat{\beta} by its standard error, aiming to refine the precision of the confidence interval and adjust for the variability in the parameter estimation.

For more details, see references.

Value

A data frame containing the following columns:

• n: Size of each simulated series.

• method: Statistical method used for simulation.

• coverage: Proportion of true parameter values within the intervals.

• avg_width: Average width of the intervals.

• sd_width: Standard deviation of the interval widths.

References

Ferreira G., Mateu J., Vilar J.A., Muñoz J. (2020). Bootstrapping regression models with locally stationary disturbances. TEST, 30, 341-363.

Examples

Coveragelongmemory(n=500,R=5,N=60,S=40,mu=0.5,dist="normal",method="asym",
beta=c(0.1,-2),alpha=c(0.15,0.25, 0.1),start = c(0.1,-2,0.15,0.2, 0.1))

Calculate the coverage for several short-memory models

Description

Generates coverage metrics for a parameter of interest using a specified short-memory model.

Usage

Coverageshortmemory(
  n,
  R,
  N,
  S,
  mu,
  dist,
  method,
  alpha,
  beta,
  start,
  Subdivisions = 100,
  m = 500,
  NN = 100,
  B,
  case,
  sign = 0.05
)

Arguments

Details

This function estimates the parameters in the linear regression model for t = 1, ..., T,

Y_{t,T} = X_{t,T} \beta + \epsilon_{t,T},

where a locally stationary autoregressive process of order one (LSAR(1)) is described by the equation:

\epsilon_{t,T} = \phi(u) \epsilon_{t-1,T} + \sigma(u) \eta_t

where u=t/T in [0,1], with \phi(u) is the autoregressive coefficient which is modeled as a linear polynomial, \sigma(u) is modeled as a quadratic polynomial, and \eta_t is a white noise sequence with zero mean and unit variance. This setup is referred to as a locally stationary autoregressive model (LSAR(1)).

Resampling methods evaluated:

• asym: Asymptotic method that uses the asymptotic variance of the estimator, based on the Central Limit Theorem, to construct confidence intervals under the assumption of normality in large samples.

• boot: Standard bootstrap that generates replicas of the estimator \hat{\beta} by resampling the adjusted residuals \hat{\epsilon}_t. It approximates the distribution of the estimator by the variability observed in the bootstrap replicas of \hat{\beta}.

• boott: Adjusted bootstrap that scales the bootstrap replicas of the estimator \hat{\beta} by its standard error, aiming to refine the precision of the confidence interval and adjust for the variability in the parameter estimation.

• bootSP: Symmetrized Percentile-t method, a variation of the boot-t that symmetrizes the bootstrap distribution around zero to handle skewed distributions or outliers more effectively. This method enhances the accuracy of confidence intervals by adjusting for asymmetries in the bootstrap replicas.

For more details, see references.

Value

A data frame containing the following columns:

• n: Size of each simulated series.

• method: Statistical method used for simulation.

• coverage: Proportion of true parameter values within the intervals.

• avg_width: Average width of the intervals.

• sd_width: Standard deviation of the interval widths.

References

Ferreira G., Mateu J., Vilar J.A., Muñoz J. (2020). Bootstrapping regression models with locally stationary disturbances. TEST, 30, 341-363.

Examples

Coverageshortmemory(n=100,R=10,N=60,S=40,mu=0.5,dist="normal",method="asym",alpha=c(0.25,0.2),
beta=c(1,1,-0.5),start=c(0.15,0.15,1,1,-0.5),case="no-linear")

US Monthly Inflation Data

Description

Monthly inflation rates for the United States. The data covers the period from January 1965 to December 2011, totaling 564 observations.

Usage

USinf

Format

A time series object with 564 elements

Monthly inflation rate, expressed as a percentage.

Source

International Financial Statistics (IFS)

Calculate the bootstrap LSE for a long memory model

Description

Bootstrap procedure to approximate the sampling distribution of the LSE for time series linear regression with errors following a Locally Stationary process.

Usage

application(
  formula,
  data,
  start,
  d.order,
  s.order,
  N,
  S,
  B = 1,
  nr.cores = 1,
  seed = 123
)

Arguments

Details

This function estimates the parameters in the linear regression model for t = 1, ..., T,

Y_{t,T} = X_{t,T} \beta + \epsilon_{t,T},

where the error term \epsilon_{t,T} follows a Locally Stationary Autoregressive Fractionally Integrated Moving Average (LS-ARFIMA) structure, given by:

\epsilon_{t,T} =(1 - B)^{-d(u)} \sigma(u)\eta_t,

where u=t/T in [0,1], d(u) represents the long-memory parameter, \sigma(u) is the noise scale factor, and \{\eta_t\} is a white noise sequence with zero mean and unit variance.

Particularly, we model d(u) and \sigma(u) as polynomials of order d.order and s.order respectively.

d(u) = \sum_{i=0}^{d.order} \delta_i u^i,

\sigma(u) = \sum_{j=0}^{s.order} \alpha_j u^j,

For more details, see references.

Value

A list with the following elements:

• coeff: A tibble of estimated model coefficients, including intercepts, regression coefficients (\beta), and coefficients of the \delta and \alpha polynomials. Contains columns for coefficient name, estimate, t-value and p-value.

• estimation: A matrix of bootstrap replicates for regression coefficients (\beta).

• delta: A matrix of bootstrap replicates for the \delta polynomial coefficients.

• alpha: A matrix of bootstrap replicates for the \alpha polynomial coefficients.

References

Ferreira G., Mateu J., Vilar J.A., Muñoz J. (2020). Bootstrapping regression models with locally stationary disturbances. TEST, 30, 341-363.

Examples

n    <- length(USinf)
shift<-201
u1<-c((1:shift)/shift,rep(0, n-shift))
u2<-c(rep(0, shift),(1:(n-shift))/(n-shift))
u<-(1:n)/n
switch <- c(rep(1,shift), rep(0, n-shift))
x1<-switch*u
x2<-(1-switch)*u

test <- data.frame(USinf, x1=x1, x2=x2)

application(formula=USinf~x1+x2,data=test,N=150,S=50,B=10,
start = c(0.16,2.0,-7,8,-3,0.25,-0.25,0.01),
d.order=4,s.order=2,nr.cores=1)