Point forecasts and the respective forecasting intervals for autoregressive-moving-average (ARMA) models can be calculated, the latter via bootstrap, by means of this function.
bootCast(
X,
p = NULL,
q = NULL,
include.mean = FALSE,
n.start = 1000,
h = 1,
it = 10000,
pb = TRUE,
cores = future::availableCores(),
alpha = 0.95,
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 \(h\) matrix forecasting matrix with point forecasts and bounds of the forecasting intervals.
a 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.
a numeric vector that contains the time series that is assumed to follow an ARMA model ordered from past to present.
an integer value \(\geq 0\) that defines the AR order
\(p\) of the underlying ARMA(\(p,q\)) model within X; 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.
a logical value; if set to TRUE, the mean of the
series is also also estimated; if set to FALSE, \(E(X_t) = 0\) is
assumed; is set to FALSE by default.
an integer that defines the 'burn-in' number
of observations for the simulated ARMA series via bootstrap; is set to
1000 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.
an integer that represents the total number of iterations, i.e.,
the number of simulated series; is set to 10000 by default; decimal
numbers will be rounded off to integers.
a logical value; for pb = TRUE, a progress bar will be shown
in the console.
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 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.
a single logical value; if the argument is set to
TRUE, 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(X) should be TRUE, as future time
points will be calculated automatically.
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. For a given time series \(X_t\), \(t = 1, 2, ..., n\),
the point forecasts and the respective forecasting intervals will be
calculated. It is assumed that the series follows an ARMA(\(p,q\)) model
$$X_t - \mu = \epsilon_t + \beta_1 (X_{t-1} - \mu) + ... + \beta_p
(X_{t-p} - \mu) + \alpha_1 \epsilon_{t-1} + ... + \alpha_{q}
\epsilon_{t-q},$$
where \(\alpha_j\) and \(\beta_i\) are real
numbers (for \(i = 1, 2, .., p\) and \(j = 1, 2, ..., q\)) and
\(\epsilon_t\) are i.i.d. (identically and independently
distributed) random variables with zero mean and constant variance.
\(\mu\) is equal to \(E(X_t)\).
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 \(\hat{X}_{n + k}\), where \(k = 1, 2, ..., h\), $$\hat{X}_{n + k} = \hat{\mu} + \sum_{i = 1}^{p} \hat{\beta}_{i} (X_{n + k - i} - \hat{\mu}) + \sum_{j = 1}^{q} \hat{\alpha}_{j} \hat{\epsilon}_{n + k - j}$$ with \(X_{n+k-i} = \hat{X}_{n+k-i}\) for \(n+k-i > n\) and \(\hat{\epsilon}_{n+k-j} = E(\epsilon_t) = 0\) for \(n+k-j > n\) will be applied.
The forecasting intervals on the other hand are obtained by a forward
bootstrap method that was introduced by Pan and Politis (2016) for
autoregressive models and extended by Lu and Wang (2020) for applications to
autoregressive-moving-average models.
For this purpose, let \(l\) be the number of the current bootstrap
iteration. Based on the demeaned residuals of the initial ARMA estimation,
different innovation series \(\epsilon_{l,t}^{s}\) will
be sampled. The initial coefficient estimates and the sampled innovation
series are then used to simulate a variety of series
\(X_{l,t}^{s}\), from which again coefficient estimates will
be obtained. With these newly obtained estimates, proxy residual series
\(\hat{\epsilon}_{l,t}^{s}\) are calculated for
the original series \(X_t\). Subsequently, point forecasts for the
time points \(n + 1\) to \(n + h\) are obtained for each iteration
\(l\) based on the original series \(X_t\), the newly obtained
coefficient forecasts and the proxy residual series
\(\epsilon_{l,t}^{s}\).
Simultaneously, "true" forecasts, i.e., true future observations, are
simulated. Within each iteration, the difference between the simulated true
forecast and the bootstrapped point forecast is calculated and saved for each
future time point \(n + 1\) to \(n + h\). The result for these time
points are simulated empirical values of the forecasting error. Denote by
\(q_k(.)\) the quantile of the empirical distribution for the
future time point \(n + k\). Given a predefined confidence level
alpha, define \(\alpha_s = (1 -\) alpha\()/2\). The
bootstrapped forecasting interval is then
$$[\hat{X}_{n + k} + q_k(\alpha_s), \hat{X}_{n + k} + q_k(1 -
\alpha_s)],$$
i.e., the forecasting intervals are given by the sum of the respective point
forecasts and quantiles of the respective bootstrapped forecasting error
distributions.
The function bootCast allows for different adjustments to
the forecasting progress. At first, a vector with the values of the observed
time series ordered from past to present has to be passed to the argument
X. Orders \(p\) and \(q\) of the underlying ARMA process can be
defined via the arguments p and q. If only one of these orders
is inserted by the user, the other order is automatically set to 0. If
none of these arguments are defined, the function will choose orders based on
the Bayesian Information Criterion (BIC) for
\(0 \leq p,q \leq 5\). Via the logical argument
include.mean the user can decide, whether to consider the mean of the
series within the estimation process. By means of n.start, the number
of "burn-in" observations for the simulated ARMA processes can be regulated.
These observations are usually used for the processes to build up and then
omitted. Furthermore, the argument h allows for the definition of the
maximum future time point \(n + h\). Point forecasts and forecasting
intervals will be returned for the time points \(n + 1\) to \(n + h\).
it corresponds to the number of bootstrap iterations. We recommend a
sufficiently high number of repetitions for maximum accuracy of the results.
Another argument is alpha, which is the equivalent of the confidence
level considered within the calculation of the forecasting intervals, i.e.,
the quantiles \((1 - \) alpha\()/2\) and \(1 - (1 - \)
alpha\()/2\) of the bootstrapped forecasting error distribution
will be obtained.
Since this bootstrap approach 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.
If the argument export.error is set to TRUE, the output of
the function is a list instead of a matrix with additional information on
the simulated forecasting errors. For more information see the section
Value.
For simplicity, the function also incorporates the possibility to directly
create a plot of the output, if the argument plot is set to
TRUE. By the additional and optional arguments ..., further
arguments of the standard plot function can be implemented to shape the
returned plot.
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.
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.
Lu, X., and Wang, L. (2020). Bootstrap prediction interval for ARMA models with unknown orders. REVSTAT–Statistical Journal, 18:3, 375-396.
Pan, L. and Politis, D. N. (2016). Bootstrap prediction intervals for linear, nonlinear and nonparametric autoregressions. In: Journal of Statistical Planning and Inference 177, pp. 1-27.
### Example 1: Simulated ARMA process ###
# Function for drawing from a demeaned chi-squared distribution
rchisq0 <- function(n, df, npc = 0) {
rchisq(n, df, npc) - df
}
# Simulation of the underlying process
n <- 2000
n.start = 1000
set.seed(23)
X <- arima.sim(model = list(ar = c(1.2, -0.7), ma = 0.63), n = n,
rand.gen = rchisq0, n.start = n.start, df = 3) + 13.1
# Quick application with low number of iterations
# (not recommended in practice)
result <- bootCast(X = X, p = 2, q = 1, include.mean = TRUE,
n.start = n.start, h = 5, it = 10, cores = 2, plot = TRUE,
lty = 3, col = "forestgreen", xlim = c(1950, 2005), type = "b",
main = "Exemplary title", pch = "*")
result
### Example 2: Application with more iterations ###
if (FALSE) {
result2 <- bootCast(X = X, p = 2, q = 1, include.mean = TRUE,
n.start = n.start, h = 5, it = 10000, cores = 2, plot = TRUE,
lty = 3, col = "forestgreen", xlim = c(1950, 2005),
main = "Exemplary title")
result2
}
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