Point forecasts and the respective forecasting intervals for trend-stationary time series are calculated.
modelCast(
obj,
p = NULL,
q = NULL,
h = 1,
method = c("norm", "boot"),
alpha = 0.95,
it = 10000,
n.start = 1000,
pb = TRUE,
cores = future::availableCores(),
np.fcast = c("lin", "const"),
export.error = FALSE,
plot = FALSE,
...
)The function returns a \(3\) by \(h\) matrix with its columns
representing the future time points and the point forecasts, the lower
bounds of the forecasting intervals and the upper bounds of the
forecasting intervals as the rows. If the argument plot is set to
TRUE, a plot of the forecasting results is created.
#'If export.error = TRUE is selected, a list with the following
elements is returned instead.
the \(3\) by \(h\) forecasting matrix with point forecasts and bounds of the forecasting intervals.
an it by \(h\) matrix, where each column
represents a future time point \(n + 1, n + 2, ..., n + h\); in each column
the respective it simulated forecasting errors are saved.
an object of class smoots; must be the output of a trend
estimation process and not of a first or second derivative estimation
process.
an integer value \(>= 0\) that defines the AR order \(p\) of the
underlying ARMA(\(p,q\)) model within the rest term (see the section
Details for more information); is set to NULL by default; if no
value is passed to p but one is passed to q, p is set to
0; if both p and q are NULL, optimal orders
following the BIC for \(0 \leq p,q \leq 5\) are chosen; is
set to NULL by default; decimal numbers will be rounded off to
integers.
an integer value \(\geq 0\) that defines the MA order
\(q\) of the underlying ARMA(\(p,q\)) model within X; is set to
NULL by default; if no value is passed to q but one is passed
to p, q is set to 0; if both p and q are
NULL, optimal orders following the BIC for
\(0 \leq p,q \leq 5\) are chosen; is set to NULL by
default; decimal numbers will be rounded off to integers.
an integer that represents the forecasting horizon; if \(n\) is
the number of observations, point forecasts and forecasting intervals will be
obtained for the time points \(n + 1\) to \(n + h\); is set to
h = 1 by default; decimal numbers will be rounded off to integers.
a character object; defines the method used for the calculation
of the forecasting intervals; with "norm" the intervals are obtained
under the assumption of normally distributed innovations; with "boot"
the intervals are obtained via a bootstrap; is set to "norm" by
default.
a numeric vector of length 1 with \(0 < \) alpha
\( < 1\); the forecasting intervals will be obtained based on the
confidence level (\(100\)alpha)-percent; is set to
alpha = 0.95 by default, i.e., a \(95\)-percent confidence level.
an integer that represents the total number of iterations, i.e.,
the number of simulated series; is set to 10000 by default; only
necessary, if method = "boot"; decimal
numbers will be rounded off to integers.
an integer that defines the 'burn-in' number
of observations for the simulated ARMA series via bootstrap; is set to
1000 by default; only necessary, if method = "boot";decimal
numbers will be rounded off to integers.
a logical value; for pb = TRUE, a progress bar will be shown
in the console, if method = "boot".
an integer value >0 that states the number of (logical) cores to
use in the bootstrap (or NULL); the default is the maximum number of
available cores
(via future::availableCores); for
cores = NULL, parallel computation is disabled.
a character object; defines the forecasting method used
for the nonparametric trend; for np.fcast = "lin" the trend is
is extrapolated linearly based on the last two trend estimates; for
np.fcast = "const", the last trend estimate is used as a constant
estimate for future values; is set to "lin" by default.
a single logical value; if the argument is set to
TRUE and if also method = "boot", a list is returned instead
of a matrix (FALSE); the first element of the list is the usual
forecasting matrix whereas the second element is a matrix with h
columns, where each column represents the calculated forecasting errors for
the respective future time point \(n + 1, n + 2, ..., n + h\); is set to
FALSE by default.
a logical value that controls the graphical output; for
plot = TRUE, the original series with the obtained point forecasts
as well as the forecasting intervals will be plotted; for the default
plot = FALSE, no plot will be created.
additional arguments for the standard plot function, e.g.,
xlim, type, ... ; arguments with respect to plotted graphs,
e.g., the argument col, only affect the original series X;
please note that in accordance with the argument x (lower case) of the
standard plot function, an additional numeric vector with time points can be
implemented via the argument x (lower case). x should be
valid for the sample observations only, i.e.
length(x) == length(obj$orig) should be TRUE, as future time
points will be calculated automatically.
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
This function is part of the smoots package and was implemented under version 1.1.0. The point forecasts and forecasting intervals are obtained based on the additive nonparametric regression model $$y_t = m(x_t) + \epsilon_t,$$ where \(y_t\) is the observed time series with equidistant design, \(x_t\) 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 (see also Beran and Feng, 2002). Thus, we assume \(y_t\) to be a trend-stationary time series. Furthermore, we assume that the rest term \(\epsilon_t\) follows an ARMA(\(p,q\)) model $$\epsilon_t = \zeta_t + \beta_1 \epsilon_{t-1} + ... + \beta_p \epsilon_{t-p} + \alpha_1 \zeta_{t-1} + ... + \alpha_q \zeta_{t-q},$$ where \(\alpha_j\), \(j = 1, 2, ..., q\), and \(\beta_i\), \(i = 1, 2, ..., p\), are real numbers and the random variables \(\zeta_t\) are i.i.d. (identically and independently distributed) with zero mean and constant variance.
The point forecasts and forecasting intervals for the future periods \(n + 1, n + 2, ..., n + h\) will be obtained. With respect to the point forecasts of \(\epsilon_t\), i.e., \(\hat{\epsilon}_{n+k}\), where \(k = 1, 2, ..., h\), $$\hat{\epsilon}_{n+k} = \sum_{i=1}^{p} \hat{\beta}_i \epsilon_{n+k-i} + \sum_{j=1}^{q} \hat{\alpha}_j \hat{\zeta}_{n+k-j}$$ with \(\epsilon_{n+k-i} = \hat{\epsilon}_{n+k-i}\) for \(n+k-i > n\) and \(\hat{\zeta}_{n+k-j} = E(\zeta_t) = 0\) for \(n+k-j > n\) will be applied. In practice, this procedure will not be applied directly to \(\epsilon_t\) but to \(y_t - \hat{m}(x_t)\).
The point forecasts of the nonparametric trend are simply obtained following the proposal by Fritz et al. (forthcoming) by $$\hat{m}(x_{n+k}) = \hat{m}(x_n) + Dk(\hat{m}(x_n) - \hat{m}(x_{n-1})),$$ where \(D\) is a dummy variable that is either equal to the constant value \(1\) or \(0\). Consequently, if \(D = 0\), \(\hat{m}(x_{n})\), i.e., the last trend estimate, is used as a constant estimate for the future. However, if \(D = 1\), the trend is extrapolated linearly. The point forecast for the whole component model is then given by $$\hat{y}_{n+k} = \hat{m}(x_{n+k}) + \hat{\epsilon}_{n+k},$$ i.e., it is equal to the sum of the point forecasts of the individual components.
Equivalently to the point forecasts, the forecasting intervals are the sum of the forecasting intervals of the individual components. To simplify the process, the forecasting error in \(\hat{m}(x_{n+k})\), which is of order \(O(-2/5)\), is not considered (see Fritz et al. (forthcoming)), i.e., only the forecasting intervals with respect to the rest term \(\epsilon_t\) will be calculated.
If the distribution of the innovations is non-normal or generally not further
specified, bootstrapping the forecasting intervals is recommended. If they
are however normally distributed or if it is at least assumed that they are,
the forecasting errors are also approximately normally distributed with a
quickly obtainable variance. For further details on the bootstrapping
method, we refer the readers to bootCast, whereas more
information on the calculation under normality can be found at
normCast.
In order to apply the function, a smoots object that was generated as
the result of a trend estimation process needs to be passed to the argument
obj. The arguments p and q represent the orders of the
of the ARMA(\(p,q\)) model that the error term
\(\epsilon_t\) is assumed to follow. If both arguments are
set to NULL, which is the default setting, orders will be selected
according to the Bayesian Information Criterion (BIC) for all possible
combinations of \(p,q = 0, 1, ..., 5\). Furthermore, the forecasting
horizon can be adjusted by means of the argument h, so that point
forecasts and forecasting intervals will be obtained for all time points
\(n + 1, n + 2, ..., n + h\).
The function also allows for two calculation approaches for the forecasting
intervals. Via the argument method, intervals
can be obtained under the assumption that the ARMA innovations are normally
distributed (method = "norm"). Alternatively, bootstrapped intervals
can be obtained for unknown innovation distributions that are clearly
non-Gaussian (method = "boot").
Another argument is alpha. By passing a value
to this argument, the (\(100\)alpha)-percent confidence level for
the forecasting intervals can be defined. If method = "boot" is
selected, the additional arguments it and n.start can be
adjusted. More specifically, it regulates the number of iterations of
the bootstrap, whereas n.start sets the number of 'burn-in'
observations in the simulated ARMA processes within the bootstrap that are
omitted.
Since this bootstrap approach for method = "boot" generally needs a
lot of computation time, especially for
series with high numbers of observations and when fitting models with many
parameters, parallel computation of the bootstrap iterations is enabled.
With cores, the number of cores can be defined with an integer.
Nonetheless, for cores = NULL, no cluster is created and therefore
the parallel computation is disabled. Note that the bootstrapped results are
fully reproducible for all cluster sizes. The progress of the bootstrap can
be observed in the R console, where a progress bar and the estimated
remaining time are displayed for pb = TRUE.
Moreover, the argument np.fcast allows to set the forecasting method
for the nonparametric trend function. As previously discussed, the two
options are a linear extrapolation of the trend (np.fcast = "lin") and
a constant continuation of the last estimated value of the trend
(np.fcast = "const").
The function also implements the option to automatically create a plot of
the forecasting results for plot = TRUE. This includes the feature
to pass additional arguments of the standard plot function to
modelCast (see also the section 'Examples').
NOTE:
Within this function, the arima function of the
stats package with its method "CSS-ML" is used throughout
for the estimation of ARMA models. Furthermore, to increase the performance,
C++ code via the Rcpp and
RcppArmadillo packages was
implemented. Also, the future and
future.apply packages are
considered for parallel computation of bootstrap iterations. The progress
of the bootstrap is shown via the
progressr package.
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.
Fritz, M., Forstinger, S., Feng, Y., and Gries, T. (forthcoming). Forecasting economic growth processes for developing economies. Unpublished.
# \donttest{
X <- log(smoots::gdpUS$GDP)
NPest <- smoots::msmooth(X)
modelCast(NPest, h = 5, plot = TRUE, xlim = c(261, 295), type = "b",
col = "deepskyblue4", lty = 3, pch = 20, main = "Exemplary title")
# }
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