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:

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.

```
data <- airLDN # Call the 'airLDN' data frame
Yt <- log(data$AQI) # log-transform the data and store the actual values as a vector
# Estimate the trend function via the 'esemifar' package
results <- tsmoothlm(Yt, p = 1, qmax = 1, pmax = 1, InfR = "Var")
# Easily access the main estimation results
b.opt <- results$b0 # The optimal bandwidth
trend <- results$ye # The trend estimates
resid <- results$res # The residuals
b.opt
#> [1] 0.2422682
coef(results$FARMA.BIC) # The model parameters
#> d ar
#> 0.1032687 0.5435804
```

An optimal bandwidth of \(0.2422682\) was selected by the iterative plug-in algorithm (IPI) within `tsmoothlm()`

. A FARIMA (\(1, d, 0\)) was obtained from the trend-adjusted residuals with \(\hat{d} = 0.1032687\) and \(\hat{\phi} = 0.5435804\). Moreover, the estimated trend fits the data suitably and the residuals seem to be stationary.

The trend estimation in combination with a fitted FARIMA model for the trend-adjusted values as in Example 1 is an ESEMIFAR model. With version 2.0.0 of the package, forecasting for based on such fitted ESEMIFAR models can be conducted using a designated predict-method. When applying that method, the user has the option between forecasting intervals based on the normality assumption for the innovations or based on a bootstrap.

```
h <- 200
# Via a bootstrap
set.seed(123)
fc <- predict(results, n.ahead = h, method = "boot")
head(fc$mean, 2)
#> [1] 4.135976 3.995659
head(fc$lower, 2)
#> 95% 99%
#> [1,] 3.725928 3.534129
#> [2,] 3.510523 3.330463
head(fc$upper, 2)
#> 95% 99%
#> [1,] 4.589879 4.749304
#> [2,] 4.531115 4.723409
# Under normality
fc <- predict(results, n.ahead = h, method = "norm")
head(fc$mean, 2)
#> [1] 4.135976 3.995659
head(fc$lower, 2)
#> 95% 99%
#> [1,] 3.717548 3.586068
#> [2,] 3.497323 3.340734
head(fc$upper, 2)
#> 95% 99%
#> [1,] 4.554404 4.685883
#> [2,] 4.493994 4.650583
```

Moreover, a quick plot of the results can be generated using a designated plot-method.

```
t_step <- diff(tail(t, 2))
t2 <- c(t, seq(from = tail(t, 1) + t_step, by = t_step, length.out = h))
plot(fc, t = t2, xlab = "Year", ylab = "AQI",
main = "Point forecasts with 95%- and 99%-forecasting intervals")
```

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., 2023).

In `esemifar`

three functions are available.

**Original functions since version 1.0.0:**

`dsmoothlm`

: Data-driven Local Polynomial for the Trendâ€™s Derivatives in Equidistant Time Series`critMatlm`

: FARIMA Order Selection Matrix`tsmoothlm`

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

**Original functions since version 2.0.0:**

`predict.esemifar`

: ESEMIFAR Prediction Method`arma_to_ma`

: MA Representation of an ARMA Model`arma_to_ar`

: AR Representation of an ARMA Model`d_to_coef`

: Filter Coefficients of the Fractional Differencing Operator`farima_to_ma`

: MA Representation of a FARIMA Model`farima_to_ar`

: AR Representation of a FARIMA Model

`airLDN`

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

: Quarterly G7 GDP between Q1 1962 and Q4 2019