# 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. ```r 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 ```r 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

hA_calc function