| normCast {smoots} | R Documentation |
Forecasting Function for ARMA Models under Normally Distributed Innovations
Description
Point forecasts and the respective forecasting intervals for autoregressive- moving-average (ARMA) models can be calculated, the latter under the assumption of normally distributed innovations, by means of this function.
Usage
normCast(
X,
p = NULL,
q = NULL,
include.mean = FALSE,
h = 1,
alpha = 0.95,
plot = FALSE,
...
)
Arguments
X |
a numeric vector that contains the time series that is assumed to follow an ARMA model ordered from past to present. |
p |
an integer value |
q |
an integer value |
include.mean |
a logical value; if set to |
h |
an integer that represents the forecasting horizon; if |
alpha |
a numeric vector of length 1 with |
plot |
a logical value that controls the graphical output; for
|
... |
additional arguments for the standard plot function, e.g.,
|
Details
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 under the assumption
of normally distributed innovations. Let q(c) be the 100c-percent
quantile of the standard normal distribution. The 100a-percent
forecasting interval at a point n + k, where k = 1, 2, ..., h,
is given by
[\hat{X}_{n+k} - q(a_r)s_k, \hat{X}_{n+k} + q(a_r)s_k]
with s_k being the standard deviation of the forecasting error
at the time point n + k and with
a_r = 1 - (1 - a)/2. For ARMA models with normal
innovations, the variance of the forecasting error can be derived from the
MA(\infty) representation of the model. It is
\sigma_{\epsilon}^{2} \sum_{i=0}^{k - 1} d_{i}^{2},
where d_i are the coefficients of the MA(\infty)
representation and \sigma_{\epsilon}^{2} is the
innovation variance.
The function normCast 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.
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. 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
forecasting intervals will be obtained.
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.
Value
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.
Author(s)
Dominik Schulz (Research Assistant) (Department of Economics, Paderborn University),
Package Creator and Maintainer
References
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. (2020). Forecasting economic growth processes for developing economies. Unpublished.
Examples
### Example 1: Simulated ARMA process ###
# Simulation of the underlying process
n <- 2000
n.start = 1000
set.seed(21)
X <- arima.sim(model = list(ar = c(1.2, -0.7), ma = 0.63), n = n,
rand.gen = rnorm, n.start = n.start) + 7.7
# Application of normCast()
result <- normCast(X = X, p = 2, q = 1, include.mean = TRUE, h = 5,
plot = TRUE, xlim = c(1971, 2005), col = "deepskyblue4",
type = "b", lty = 3, pch = 16, main = "Exemplary title")
result