knsmooth function

Estimation of Nonparametric Trend Functions via Kernel Regression

This function estimates the nonparametric trend function in an equidistant time series with Nadaraya-Watson kernel regression.

knsmooth(y, 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.

• mu: an integer 0, 1, 2, ... that represents the smoothness parameter of the second order kernel 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 0 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.
• res: a vector with the estimated residual series.
• ye: a vector with the estimates of the nonparametric trend.

Details

The trend is 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 $E(\epsilon_t) = 0$.

This function is part of the package smoots and is used for the estimation of trends in equidistant time series. The applied method is a kernel regression with arbitrarily selectable second order kernel, relative bandwidth and boundary method. Especially the chosen bandwidth has a strong impact on the final result and has thus to be selected carefully. This approach is not recommended by the authors of this package.

Examples

# Logarithm of test data
test_data <- gdpUS
y <- log(test_data$GDP)

#Applied knmooth function for the trend with two different bandwidths
trend1 <- knsmooth(y, mu = 1, b = 0.28, bb = 1)$ye
trend2 <- knsmooth(y, mu = 1, b = 0.05, bb = 1)$ye

# Plot of the results
t <- seq(from = 1947, to = 2019.25, 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 = 0.05)"),
 fill = c("red", "blue"), cex = 0.6)
title(sub = expression(italic("Figure 1")), col.sub = "gray47",
 cex.sub = 0.6, adj = 0)

References

Feng, Y. (2009). Kernel and Locally Weighted Regression. Verlag für Wissenschaft und Forschung, Berlin.

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