Coverageshortmemory {LSEbootLS}

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

n

(type: numeric) size of the simulated series.

R

(type: numeric) number of realizations of the Monte Carlo experiments.

N

(type: numeric) sample size of each block.

S

(type: numeric) shifting places from block to block. Observe that the number of blocks M is determined by the following formula M = \left\lfloor \frac{T-N}{S} + 1 \right\rfloor, where \left\lfloor . \right\rfloor takes a single numeric argument x and returns a numeric vector containing the integers formed by truncating the values in x toward 0.

mu

(type: numeric) trend coefficient of the regression model.

dist

(type: character) white noise distribution for calculating coverage, it includes the "normal", "exponential" and "uniform" univariate distributions.

method

(type: character) methods are asymptotic ("asym"), bootstrap percentile ("boot"), bootstrap-t ("boott") and bootstrap-SP ("bootSP").

alpha

(type: numeric) numeric vector with values to simulate the time varying autoregressive parameters of model LSAR(1), \phi(u).

beta

(type: numeric) numeric vector with values to simulate the time varying scale factor parameters of model LSAR(1), \sigma(u).

start

(type: numeric) numeric vector, initial values for parameters to run the model.

Subdivisions

(type: numeric) the number of subintervals produced in the subdivision (integration) process; only required in the asymptotic method.

m

(type: numeric) parameter that allows to remove the first m observations when simulating the LSAR process.

NN

(type: numeric) parameter that allows to remove the first NN observations of noise from the LSAR model.

B

(type: numeric) the number of bootstrap replicates, NULL indicates the asymptotic method.

case

(type: character) nonlinear ("no-linear") and linear cases ("linear").

sign

nominal significance level

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")