The goal of
smoots is to provide an easy way to estimate the
nonparametric trend and its derivatives in trend-stationary, equidistant
time series with short-memory stationary errors. The main functions
allow for data-driven estimates via local polynomial regression with an
automatically selected optimal bandwidth.
You can install the released version of
This is a basic example which shows you how to solve a common problem.
tempNH in the package includes the mean monthly temperature
changes in degrees Celsius of the Northern Hemisphere (NH) from 1880 to
2018. The data was obtained from the Goddard Institute for Space Studies
of the National Aeronautics and Space Administration (NASA). To make use
smoots package, it has to be assumed that the data follows an
additive model consisting of a deterministic, nonparametric trend
function and a zero-mean stationary rest with short-range dependence.
The user-friendly and simply applicable function
msmooth() for the
estimation of trend function in the additive model will be used.
library(smoots) # Call the package
data <- tempNH # Call the 'tempNH' data frame Yt <- data$Change # Store the actual values as a vector # Estimate the trend function via the 'smoots' package results <- msmooth(Yt, p = 1, mu = 1, bStart = 0.15, alg = "A") # Easily access the main estimation results b.opt <- results$b0 # The optimal bandwidth trend <- results$ye # The trend estimates resid <- results$res # The residuals b.opt #>  0.101089
An optimal bandwidth of 0.101089 was selected by the iterative plug-in
algorithm (IPI) within
msmooth(). Moreover, the estimated trend fits
the data suitably and the residuals seem to be stationary. Since the
trend was obtained without any parametric assumptions with respect to
the rest term, the detrended values could now be further analyzed by
means of any suitable parametric approach,
e.g. autoregressive-moving-average (ARMA) models.
With the package version 1.1.0 different functions were newly introduced that allow for forecasting trend-stationary series and other functionalities. Based on the previous example, the following code shows how to obtain point forecasts and 95% forecasting intervals for the mean monthly temperature changes data and directly create a plot of the forecasting results. For simplicity, it is assumed that the rest term of the additive model follows an ARMA(1,1) model with normally distributed innovations. However, via the functions a bootstrap method for non-Gaussian cases can be applied as well. Furthermore, an automatic order selection for the ARMA model is also built-in that is triggered, if no values are passed to the respective arguments that define the orders.
n <- length(Yt) # Create a vector with exact time points (optional for the plot) time <- seq(from = 1880 + 1 / 12, to = 2019, by = 1 / 12) # Application of the forecasting function with automatic creation of a graphic forecast <- modelCast(results, p = 1, q = 1, h = 5, alpha = 0.95, method = "norm", plot = TRUE, x = time, type = "b", col = "deepskyblue4", pch = 20, lty = 2, main = "Title (series is cut-off)", xlab = "Exemplary x-axis label", ylab = "Exemplary y-axis label")
forecast #> k=1 k=2 k=3 k=4 k=5 #> fcast 1.1041137 1.1151230 1.1240264 1.1313490 1.1374849 #> 2.5% 0.7468131 0.7281308 0.7212654 0.7199682 0.7213253 #> 97.5% 1.4614144 1.5021152 1.5267873 1.5427297 1.5536445
Another contribution that was made to package version 1.1.0 is a function for testing the trend graphically for linearity. Based on a previously obtained nonparametric estimate of the trend or its derivatives, an asymptotically unbiased series of estimates with its confidence bounds is obtained and plotted. By choice, different polynomial regression lines can be displayed alongside the nonparametric trend estimates and its confidence bounds. The estimated slope of a simple linear regression model of the trend and the constant with value zero are displayed against the estimates of the first and second derivatives, respectively. If, for a selected confidence level 100s%, clearly more than (1 - s)100% of the estimated parametric line lies outside of the confidence bounds, the null hypothesis can be rejected. The following example is based yet again on the mean monthly temperature changes data and illustrates the linearity test with respect to the nonparametric trend. The derivatives are skipped at this point for simplicity. Moreover, a confidence level of 95% was chosen.
# Calculation of confidence bounds with creation of a graphic bounds <- confBounds(results, alpha = 0.95, p = 1, x = time)
bounds #> ----------------------------------------------- #> | Results of the confidence bounds estimation | #> ----------------------------------------------- #> #> Number of observations: 1668 #> Order of derivative: 0 #> Adjusted bandwidth: 0.0570
For the (asymptotically) unbiased estimation of the trend function an adjusted bandwidth of 0.057 was used. Since more than 5% of the linear regression line (blue) lies outside of the grey confidence bounds, we can reject the null hypothesis that the trend is linear.
The trend estimation functions can also be used for the implementation
of semiparametric generalized autoregressive conditional
heteroskedasticity (Semi-GARCH) models and its various variants in
Financial Econometrics (see also the examples in the documentation of
smoots fourteen functions are available.
Original functions since version 1.0.0:
dsmooth: Data-driven Local Polynomial for the Trend’s Derivatives in Equidistant Time Series
gsmooth: Estimation of Trends and their Derivatives via Local Polynomial Regression
knsmooth: Estimation of Nonparametric Trend Functions via Kernel Regression
msmooth: Data-driven Nonparametric Regression for the Trend in Equidistant Time Series
tsmooth: Advanced Data-driven Nonparametric Regression for the Trend in Equidistant Time Series
Newly introduced with version 1.1.0:
rescale: Rescaling Derivative Estimates
critMatrix: ARMA Order Selection Matrix
optOrd: Optimal Order Selection
normCast: Forecasting Function for ARMA Models under Normally Distributed Innovations
bootCast: Forecasting Function for ARMA Models via Bootstrap
trendCast: Forecasting Function for Nonparametric Trend Functions
modelCast: Forecasting Function for Trend-Stationary Time Series
rollCast: Backtesting Semi-ARMA Models with Rolling Forecasts
confBounds: Asymptotically Unbiased Confidence Bounds
For further information on each of the functions, we refer the user to the manual or the package documentation.
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