gsmooth function

Estimation of Trends and their Derivatives via Local Polynomial Regression

This function is an R function for estimating the trend function and its derivatives in an equidistant time series with local polynomial regression and a fixed bandwidth given beforehand.

gsmooth(y, v = 0, p = v + 1, mu = 1, b = 0.15, bb = c(0, 1))

Arguments

• y: a numeric vector that contains the time series data ordered from past to present.

• v: an integer 0, 1, ... that represents the order of derivative that will be estimated; is set to v = 0 by default.

  Number (‘v’)	Degree of derivative
  0	The function f(x) itself
  1	The first derivative f'(x)
  2	The second derivative f''(x)
  ...	...

• p: an integer $>= ($ v $+ 1)$ that represents the order of polynomial; p - v must be an odd number; is set to v + 1

  by default.

  Exemplary for v = 0:

  Number (‘p’)	Polynomial	‘p - v’	‘p - v’ odd?	‘p’ usable?
  1	Linear	1	Yes	Yes
  2	Quadratic	2	No	No
  3	Cubic	3	Yes	Yes
  ...	...	...	...	...

• mu: an integer 0, 1, 2, ... that represents the smoothness parameter of the kernel weighting function that will be used; is set to 1 by default.

  Number (‘mu’)	Kernel
  0	Uniform Kernel
  1	Epanechnikov Kernel
  2	Bisquare Kernel
  3	Triweight Kernel
  ...	...

• b: a real number $0 <$ b $< 0.5$; represents the relative bandwidth that will be used for the smoothing process; is set to 0.15 by default.

• bb: can be set to 0 or 1; the parameter controlling the bandwidth used at the boundary; is set to 1 by default.

  Number (‘bb’)	Estimation procedure at boundary points
  0	Fixed bandwidth on one side with possible large bandwidth on the other side at the boundary
  1	The k-nearest neighbor method will be used

Returns

The output object is a list with different components:

• b: the chosen (relative) bandwidth; input argument.

• bb: the chosen bandwidth option at the boundaries; input argument.

• mu: the chosen smoothness parameter for the second order kernel; input argument.

• n: the number of observations.

• orig: the original input series; input argument.

• p: the chosen order of polynomial; input argument.

• res: a vector with the estimated residual series; is set to NULL

   for `v > 0`.
  

• v: the order of derivative; input argument.

• ws: the weighting system matrix used within the local polynomial regression; this matrix is a condensed version of a complete weighting system matrix; in each row of ws, the weights for conducting the smoothing procedure at a specific observation time point can be found; the first $[nb + 0.5]$ rows, where $n$ corresponds to the number of observations, $b$ is the bandwidth considered for smoothing and $[.]$ denotes the integer part, contain the weights at the $[nb + 0.5]$ left-hand boundary points; the weights in row $[nb + 0.5] + 1$ are representative for the estimation at all interior points and the remaining rows contain the weights for the right-hand boundary points; each row has exactly $2[nb + 0.5] + 1$ elements, more specifically the weights for observations of the nearest $2[nb + 0.5] + 1$ time points; moreover, the weights are normalized, i.e. the weights are obtained under consideration of the time points $x_t = t/n$, where $t = 1, 2, ..., n$.

• ye: a vector with the estimates of the selected nonparametric order of derivative on the rescaled time interval $[0, 1]$.

Details

The trend or its derivatives are estimated based on the additive nonparametric regression model for an equidistant time series

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 and deterministic trend function and $\epsilon_t$ are stationary errors with $E(\epsilon_t) = 0$ (see also Beran and Feng, 2002).

This function is part of the package smoots and is used in the field of analyzing equidistant time series data. It applies the local polynomial regression method to the input data with an arbitrarily selectable bandwidth. By these means, the trend as well as its derivatives can be estimated nonparametrically, even though the result will strongly depend on the bandwidth given beforehand as an input.

NOTE:

The estimates are obtained with regard to the rescaled time points on the interval $[0, 1]$. Thus, if $\nu > 0$, the estimates might not reflect the values for the actual time points. To rescale the estimates, we refer the user to the rescale function of the smoots

package.

With package version 1.1.0, this function implements C++ code by means of the Rcpp and RcppArmadillo packages for better performance.

Examples

# Logarithm of test data
y <- log(esemifar::gdpG7$gdp)

# Applied gsmooth function for the trend with two different bandwidths
results1 <- gsmooth(y, v = 0, p = 1, mu = 1, b = 0.28, bb = 1)
results2 <- gsmooth(y, v = 0, p = 1, mu = 1, b = 0.11, bb = 1)
trend1 <- results1$ye
trend2 <- results2$ye

# Plot of the results
t <- seq(from = 1962, to = 2019.75, by = 0.25)
plot(t, y, type = "l", xlab = "Year", ylab = "log(US-GDP)", bty = "n",
 lwd = 2,
 main = "Estimated trend for log-quarterly US-GDP, Q1 1947 - Q2 2019")
points(t, trend1, type = "l", col = "red", lwd = 1)
points(t, trend2, type = "l", col = "blue", lwd = 1)
legend("bottomright", legend = c("Trend (b = 0.28)", "Trend (b = 0.11)"),
 fill = c("red", "blue"), cex = 0.6)
title(sub = expression(italic("Figure 1")), col.sub = "gray47",
 cex.sub = 0.6, adj = 0)

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.

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