The goal of `esemifar`

is to provide an easy way to
estimate the nonparametric trend and its derivatives in
trend-stationary, equidistant time series with long-memory stationary
errors. The main functions allow for data-driven estimates via local
polynomial regression with an automatically selected optimal
bandwidth.

You can install the released version of `esemifar`

from CRAN with:

`install.packages("esemifar")`

This is a basic example which shows you how to solve a common
problem. The data `airLDN`

in the package contains daily
observations of the air quality index of London (Britain) from 2014 to
2020. The data was obtained from the European Environment Agency. To
make use of the `esemifar`

package, it has to be assumed that
the data follows an additive model consisting of a deterministic,
nonparametric trend function and a zero-mean stationary rest with
long-range dependence.

`library(esemifar) # Call the package`

```
<- airLDN # Call the 'airLDN' data frame
data <- log(data$AQI) # log-transform the data and store the actual values as a vector
Yt
# Estimate the trend function via the 'esemifar' package
<- tsmoothlm(Yt, p = 1, qmax = 1, pmax = 1, InfR = "Var")
results
# Easily access the main estimation results
<- results$b0 # The optimal bandwidth
b.opt <- results$ye # The trend estimates
trend <- results$res # The residuals
resid
b.opt#> [1] 0.2423177
coef(results$FARMA.BIC) # The model parameters
#> d ar
#> 0.1032687 0.5435804
```

An optimal bandwidth of 0.2423177 was selected by the iterative
plug-in algorithm (IPI) within `tsmoothlm()`

. A FARIMA
(1, *d*, 0) was obtained from the trend-adjusted residuals with
*d̂* = 0.1032687 and *ϕ̂* = 0.5435804. Moreover, the
estimated trend fits the data suitably and the residuals seem to be
stationary.

The trend estimation function can also be used for the implementation of semiparametric generalized autoregressive conditional heteroskedasticity (Semi-GARCH) models with long memory in Financial Econometrics (see Letmathe et al., 2021).

In `esemifar`

three functions are available.

**Original functions since version 1.0.0:**

`dsmoothlm`

: Data-driven Local Polynomial for the Trend’s Derivatives in Equidistant Time Series`critMatlm`

: FARIMA Order Selection Matrix`tsmoothlm`

: Advanced Data-driven Nonparametric Regression for the Trend in Equidistant Time Series

**Data Sets**

`airLDN`

: Daily observations of individual air pollutants from 2014 to 2020`gdpG7`

: Quarterly G7 GDP between Q1 1962 and Q4 2019