hA_calc function

Calculation of Theoretically Optimal Bandwidth and Its Components

Allows to calculate the theoretically optimal bandwidth for estimating the trend and the seasonality in an equidistant time series with short-range dependence using locally weighted regression, if the trend function and the exact ARMA dependence structure of the errors are known.

hA_calc(
  m,
  arma = list(ar = NULL, ma = NULL, sd_e = 1),
  p = c(1, 3),
  mu = c(0, 1, 2, 3),
  frequ = c(4, 12),
  n = 300,
  cb = 0.1
)

Arguments

• m: an expression that defines the trend function in terms of x, where x is the rescaled time on the interval $[0,1]$.
• arma: a list with the elements ar, ma and sd_e; ar is a numeric vector with the AR-coefficients, ma is a numeric vector with the MA-coefficients and sd_e is the innovation standard deviation.
• p: the order of polynomial to use locally for the trend estimation.
• mu: the smoothness parameter of the second-order kernel function used in the weighting process.
• frequ: the frequency of the theoretical time series (4 for quarterly and 12 for monthly time series).
• n: the number of observations.
• cb: the part of observations to drop at each boundary.

Returns

This function returns a list with various elements. See the documentation of deseats to understand, what each quantity signifies.

• b: This is the theoretical quantity $\beta_{(k)}$.
• hA: The theoretically asymptotically optimal global bandwidth for locally weighted regression applied to the theoretical time series under consideration.
• Imk: This is the theoretical quantity $I[m^{(k)}]$.
• RK: This is the theoretical quantity $R(K)$.
• RW: This is the theoretical quantity $R(W)$.
• sum_autocov: This is the theoretical quantity $2\pi c_f$.

Details

For simulation studies of the function deseats one may be interested in obtaining the theoretically optimal bandwidth for local regression first for a given theoretical process (from which realizations will be drawn in the simulation). This function assists in obtaining this theoretical bandwidth.

Examples

arma <- list(ar = 0.8, sd_e = 0.01)
m_f <- expression(13.1 + 3.1 * x + (dnorm(x / 0.15 - 0.5 / 0.15) / 0.15) / 4)
n <- 500
p <- 1
mu <- 1
frequ <- 4
cb <- 0.05

hA_calc(
 m = m_f,
 arma = arma, 
 p = p,
 mu = mu,
 frequ = frequ,
 n = n,
 cb = cb
)

t <- 1:n
xt <- t / n
mxt <- 13.1 + 3.1 * xt + dnorm(xt, mean = 0.5, sd = 0.15) / 4

S2 <- rep(c(0, 1, 0, 0), length.out = n)
S3 <- rep(c(0, 0, 1, 0), length.out = n)
S4 <- rep(c(0, 0, 0, 1), length.out = n)
sxt <- -0.5 + 0.25 * S2 + 0.5 * S3 + 1.25 * S4

set.seed(123)
et <- arima.sim(model = list(ar = 0.8), sd = 0.01, n = n)
yt <- ts(mxt + sxt + et, frequency = frequ)
plot(yt)

est <- deseats(yt)
est@bwidth
est@sum_autocov

Author(s)

• Dominik Schulz (Research Assistant) (Department of Economics, Paderborn University),

  Author and Package Creator