confBounds: Asymptotically Unbiased Confidence Bounds

In smoots: Nonparametric Estimation of the Trend and Its Derivatives in TS

Asymptotically Unbiased Confidence Bounds

Description

Asymptotically Unbiased Confidence Bounds

Usage

confBounds(
  obj,
  alpha = 0.95,
  p = c(0, 1, 2, 3),
  plot = TRUE,
  showPar = TRUE,
  rescale = TRUE,
  ...
)

Arguments

obj	an object returned by either msmooth, tsmooth or dsmooth.
alpha	the confidence level; a single numeric value between 0 and 1; 0.95 is the default.
p	the order of polynomial used for the parametric polynomial regression that is conducted as a benchmark for the trend function; must satisfy 0 \leq p \leq 3; set to 1 by default; is irrelevant, if a derivative of the trend of order greater than zero is being analyzed.
plot	a logical value; for plot = TRUE, the default, a plot is created.
showPar	set to TRUE, if the parametric fitted values are to be shown against the unbiased estimates and the confidence bounds for plot = TRUE; the default is TRUE.
rescale	a single logical value; is set to TRUE by default; if the output of a derivative estimation process is passed to obj and if rescale = TRUE, the estimates and confidence bounds will be rescaled according to x for the plot (see also the details on the parameter ...); the numerical output stays unchanged.
...	further arguments that can be passed to the plot function; if an argument x with time points is not given by the user, x = 1:length(obj$ye) is used per default for the observation time points.

Details

This function is part of the smoots package and was implemented under version 1.1.0. The underlying theory is based on the additive nonparametric regression function

y_t = m(x_t) + \epsilon_t,

where y_t is the observed time series, x_t is the rescaled time on the interval [0, 1], m(x_t) is a smooth trend function and \epsilon_t are stationary errors with E(\epsilon_t) = 0 and short-range dependence.

The purpose of this function is the estimation of reasonable confidence intervals for the nonparametric trend function and its derivatives. The optimal bandwidth minimizes the Asymptotic Mean Integrated Squared Error (AMISE) criterion, however, local polynomial estimates are (usually) biased. The bias is then (approximately)

\frac{h^{k - v} m^{(k)}(x) \beta_{(\nu, k)}}{k!},

where p is the order of the local polynomials, k = p + 1 is the order of the asymptotically equivalent kernel, \nu is the order of the of the trend function's derivative, m^(v) is the \nu-th order derivative of the trend function and \beta_{(\nu, k)} = \int_{-1}^{1} u^k K_{(\nu, k)}(u) du. K_{(\nu, k)}(u) is the k-th order asymptotically equivalent kernel function for estimating m^{(\nu)}. A renewed estimation with an adjusted bandwidth h_{ub} = o(n^{-1 / (2k + 1)}), i.e., a bandwidth with a smaller order than the optimal bandwidth, is conducted. h = h_{A}^{(2k + 1) / (2k)}, where h_{A} is the optimal bandwidth, is implemented.

Following this idea, we have that

\sqrt{nh}[m^{(\nu)}(x) - \hat{m}^{(\nu)}(x)]

converges to

N(0,2\pi c_f R(x))

in distribution, where 2\pi c_f is the sum of autocovariances. Consequently, the trend (or derivative) estimates are asymptotically unbiased and normally distributed.

To make use of this function, an object of class smoots can be given as input that was created by either msmooth, tsmooth or dsmooth. Based on the optimal bandwidth saved within obj, an adjustment to the bandwidth is made so that the estimates following the adjusted bandwidth are (relatively) unbiased.

Based on the input argument alpha, the level of confidence between 0 and 1, the respective confidence bounds are calculated for each observation point.

From the input argument obj, the order of derivative is automatically obtained. By means of the argument p, an order of polynomial is selected for a parametric regression of the trend function. This is only meaningful, if the trend (and not its derivatives) is analyzed. Otherwise, the argument is automatically dropped by the function. Furthermore, if plot = TRUE, a plot of the unbiased trend (or derivative) estimates alongside the confidence bounds is created. If also showPar = TRUE, the estimated parametric trend (or parametric constant value for the derivatives) is added to the confidence bound plot for comparison.

NOTE:

The values that are returned by the function are obtained with respect to the rescaled time points on the interval [0, 1]. While the plot can be adjusted and rescaled by means of a given vector with the actual time points, the numeric output is not rescaled. For this purpose we refer the user to the rescale function of the smoots package.

This function implements C++ code by means of the Rcpp and RcppArmadillo packages for better performance.

Value

A plot is created in the plot window and a list with different components is returned.

alpha

a numeric vector of length 1; the level of confidence; input argument.

b.ub

a numeric vector with one element that represents the adjusted bandwidth for the unbiased trend estimation.

p.estim

a numeric vector with the estimates following the parametric regression defined by p that is conducted as a benchmark for the trend function; for the trend's derivatives or for p = 0, a constant value is the benchmark; the values are obtained with respect to the rescaled time points on the interval [0, 1].

n

the number of observations.

np.estim

a data frame with the three (numeric) columns ye.ub, lower and upper; in ye.ub the unbiased trend estimates, in lower the lower confidence bound and in upper the upper confidence bound can be found; the values are obtained with respect to the rescaled time points on the interval [0, 1].

v

the order of the trend's derivative considered for the test.

Author(s)

• Yuanhua Feng (Department of Economics, Paderborn University),
  Author of the Algorithms
  Website: https://wiwi.uni-paderborn.de/en/dep4/feng/

• Dominik Schulz (Research Assistant) (Department of Economics, Paderborn University),
  Package Creator and Maintainer

References

Beran, J. and Feng, Y. (2002). Local polynomial fitting with long-memory, short-memory and antipersistent errors. Annals of the Institute of Statistical Mathematics, 54(2), 291-311.

Feng, Y., Gries, T. and Fritz, M. (2020). Data-driven local polynomial for the trend and its derivatives in economic time series. Journal of Nonparametric Statistics, 32:2, 510-533.

Feng, Y., Gries, T., Letmathe, S. and Schulz, D. (2019). The smoots package in R for semiparametric modeling of trend stationary time series. Discussion Paper. Paderborn University. Unpublished.

Feng, Y., Gries, T., Fritz, M., Letmathe, S. and Schulz, D. (2020). Diagnosing the trend and bootstrapping the forecasting intervals using a semiparametric ARMA. Discussion Paper. Paderborn University. Unpublished.

Examples

log_gdp <- log(smoots::gdpUS$GDP)
est <- msmooth(log_gdp)
confBounds(est)

smoots documentation built on Sept. 11, 2023, 9:07 a.m.