dsmoothlm {esemifar}R Documentation

Data-driven Local Polynomial for the Trend's Derivatives in Equidistant Time Series

Description

This function runs through an iterative process in order to find the optimal bandwidth for the nonparametric estimation of the first or second derivative of the trend in an equidistant time series (with long-memory errors) and subsequently employs the obtained bandwidth via local polynomial regression.

Usage

dsmoothlm(
  y,
  d = c(1, 2),
  pmin = c(0, 1, 2, 3, 4, 5),
  pmax = c(0, 1, 2, 3, 4, 5),
  qmin = c(0, 1, 2, 3, 4, 5),
  qmax = c(0, 1, 2, 3, 4, 5),
  mu = c(0, 1, 2, 3),
  mu.p = c(0, 1, 2, 3),
  pp = c(1, 3),
  bStart.p = 0.15,
  InfR.p = c("Opt", "Nai", "Var")
)

Arguments

y

a numeric vector that contains the time series ordered from past to present.

d

an integer 1 or 2 that defines the order of derivative; the default is d = 1.

pmin

an integer value >= 0 that defines the minimum autoregressive order to calculate the BIC-criterion for; is set to 0 by default; decimal numbers will be rounded off to integers.

pmax

an integer value >= 0 that defines the maximum autoregressive order to calculate the BIC-criterion for; is set to 0 by default; decimal numbers will be rounded off to integers.

qmin

an integer value >= 0 that defines the minimum moving-average order to calculate the BIC-criterion for; is set to 0 by default; decimal numbers will be rounded off to integers.

qmax

an integer value >= 0 that defines the maximum moving-average order to calculate the BIC-criterion for; is set to 0 by default; decimal numbers will be rounded off to integers.

mu

an integer 0, ..., 3 that represents the smoothness parameter of the kernel weighting function and thus defines the kernel function that will be used within the local polynomial regression; is set to 1 by default.

mu.p

an integer 0, ..., 3 that represents the smoothness parameter of the kernel weighting function for the iterative process to obtain initial estimates for c_f, d and b_0; is set to 1 by default.

Number Kernel
0 Uniform Kernel
1 Epanechnikov Kernel
2 Bisquare Kernel
3 Triweight Kernel
pp

an integer 1 (local linear regression) or 3 (local cubic regression) that indicates the order of polynomial upon which c_f, d and b_0 will be calculated by tsmoothlm; the default is pp = 1.

bStart.p

a numeric object that indicates the starting value of the bandwidth for the iterative process to obtain initial estimates for c_f, d and b_0; should be > 0; is set to 0.15 by default.

InfR.p

a character object that represents the inflation rate in the form h_d = h^a of the bandwidth for the iterative process to obtain initial estimates for c_f, d and b_0; is set to "Opt" by default.

Details

The trend is estimated based on the additive nonparametric regression model for an equidistant time series

y_t = m(x_t) + \epsilon_t,

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 and is assumed to follow a FARIMA(p, d, q) model (see also Beran and Feng, 2002).

The iterative-plug-in (IPI) algorithm, which numerically minimizes the Asymptotic Mean Squared Error (AMISE), is based on the proposal of Beran and Feng (2002a).

The variance factor c_f, the long memory parameter d and the starting bandwidth b0 are first obtained from a pilot-estimation of the time series' nonparametric trend (\nu = 0) with polynomial order p_p. The estimate is then plugged into the iterative procedure for estimating the first or second derivative (\nu = 1 or \nu = 2). For further details on the asymptotic theory or the algorithm, we refer the user to Letmathe, Beran and Feng (2023).

The function itself is applicable in the following way: Based on a data input y, an order of polynomial pp for the variance factor estimation procedure, a starting value for the relative bandwidth bStart.p in the variance factor estimation procedure and a kernel function defined by the smoothness parameter mu, an optimal bandwidth is numerically calculated for the trend's derivative of order d. In fact, aside from the input vector y, every argument has a default setting that can be adjusted for the individual case. However, it is recommended to initially use the default values for the estimation of the first derivative and adjust the argument d to d = 2 for the estimation of the second derivative. The initial bandwidth does not affect the resulting optimal bandwidth in theory. However in practice, local minima of the AMISE can influence the results. For more specific information on the input arguments consult the section Arguments.

After the bandwidth estimation, the nonparametric derivative of the series is calculated with respect to the obtained optimal bandwidth by means of a local polynomial regression. The output object is then a list that contains, among other components, the original time series, the estimates of the derivative and the estimated optimal bandwidth.

The default print method for this function delivers key numbers such as the iteration steps and the generated optimal bandwidth rounded to the fourth decimal. The exact numbers and results such as the estimated nonparametric trend series are saved within the output object and can be addressed via the $ sign.

Value

The function returns a list with different components:

b0

the optimal bandwidth chosen by the IPI-algorithm.

bStart.p

the starting bandwidth for the nonparametric trend estimation that leads to the initial estimates; input argument.

cf0

the estimated variance factor.

InfR.p

the inflation rate setting.

iterations

the bandwidths of the single iterations steps

mu.p

the smoothness parameter of the second order kernel; input argument.

n

the number of observations.

niterations

the total number of iterations until convergence.

orig

the original input series; input argument.

p

the order of polynomial for the local polynomial regression used within derivative estimation procedure.

pp

the order of polynomial for the local polynomial regression used in the pilot estimation; input argument.

v

the considered order of the trend's derivative; input argument d.

ws

the weighting system matrix used within the local polynomial regression; this matrix is a condensed version of a complete weighting system matrix; in each row of ws, the weights for conducting the smoothing procedure at a specific observation time point can be found; the first [nb + 0.5] rows, where n corresponds to the number of observations, b is the bandwidth considered for smoothing and [.] denotes the integer part, contain the weights at the [nb + 0.5] left-hand boundary points; the weights in row [nb + 0.5] + 1 are representative for the estimation at all interior points and the remaining rows contain the weights for the right-hand boundary points; each row has exactly 2[nb + 0.5] + 1 elements, more specifically the weights for observations of the nearest 2[nb + 0.5] + 1 time points; moreover, the weights are normalized, i.e. the weights are obtained under consideration of the time points x_t = t/n, where t = 1, 2, ..., n.

ye

the nonparametric estimates of the derivative.

Author(s)

References

Letmathe, S., Beran, J. and Feng, Y. (2023). An extended exponential SEMIFAR model with application in R. Communications in Statistics - Theory and Methods: 1-13.

Examples


# Logarithm of test data
test_data <- gdpG7
y <- log(test_data$gdp)
n <- length(y)
t <- seq(from = 1962, to = 2020, length.out = n)

# Applied dsmooth function for the trend's first derivative
result_d <- dsmoothlm(y, d = 1, pp = 1, pmax = 1, qmax = 1, InfR.p = "Opt")
estim <- result_d$ye

# Plot of the results
plot(t, estim, xlab = "Year", ylab = "First derivative", type = "l",
 main = paste0("Estimated first derivative of the trend for log-quarterly ",
 "G7-GDP, Q1 1962 - Q4 2019"), cex.axis = 0.8, cex.main = 0.8,
 cex.lab = 0.8, bty = "n")

# Print result
result_d

# The main function "dsmoothlm"------------------------------------------
[Package esemifar version 2.0.1 Index]