**nortsTest** is an R package for assessing normality of stationary process, it tests if a given data follows a stationary Gaussian process. The package works as an extension of the **nortest** package that performs normality tests in random samples (*independent data*). The four principal package’s functions are:

epps.test() function that implements the Epps test,

lobato.test() function that implements the Lobato and Velasco’s test,

vavra.test() function that implements the Psaradaki and Vavra’s test,

rp.test() function that implements the random projections test of Nieto-Reyes, Cuesta-Albertos and Gamboa’s test.

Additionally, inspired in the function *check.residuals()* of the forecast package, we provide the **check_residuals** methods for checking model’s assumptions using the estimated residuals. The function checks stationarity, homoscedasticity and normality, presenting a report of the used tests and conclusions.

Classic hypothesis tests for normality such as *Shapiro & Wilk, Anderson & Darling*, or *Jarque & Bera*, do not perform well on dependent data. Therefore, these tests should not be used to check whether a given time series has been drawn from a Gaussian process. As a simple example, we generate a stationary ARMA(1,1) process simulated using an t student distribution with 7 degrees of freedom, and perform the Anderson-Darling test from the *nortest package*.

```
x = arima.sim(100,model = list(ar = 0.32,ma = 0.25),rand.gen = rt,df = 7)
nortest::ad.test(x)
#>
#> Anderson-Darling normality test
#>
#> data: x
#> A = 0.50769, p-value = 0.1954
```

The null hypothesis is that the data has a normal distribution and therefore, follows a Gaussian Process. At *α* = 0.05 significance level the alternative hypothesis is rejected and wrongly concludes the data follows a Gaussian process. Applying the Lobato and Velasco’s test of our package, the null hypothesis is correctly rejected.

```
lobato.test(x)
#>
#> Lobatos and Velascos test
#>
#> data: x
#> lobato = 16.864, df = 2, p-value = 0.0002177
#> alternative hypothesis: x does not follow a Gaussian Process
```

In the next example we generate a stationary AR(2) process, using an exponential distribution with rate of 5, and perform the *epps* and *rp* with k = 5 random projections tests. With a significance level at *a lpha* = 0.05, the null hypothesis of non-normality is rejected.

```
set.seed(298)
# Simulating the AR(2) process
x = arima.sim(250,model = list(ar =c(0.2,0.3)),rand.gen = rexp,rate = 5)
# tests
epps.test(x)
#>
#> epps test
#>
#> data: x
#> epps = 38.158, df = 2, p-value = 5.178e-09
#> alternative hypothesis: x does not follow a Gaussian Process
rp.test(x,k = 5)
#>
#> k random projections test
#>
#> data: x
#> k = 5, lobato = 188.771, epps = 28.385, p-value = 0.0007823
#> alternative hypothesis: x does not follow a Gaussian Process
```

As an example, we analyze the monthly mean carbon dioxide (*in ppm*) from the **astsa package**, measured at Mauna Loa Observatory, Hawaii. from March, 1958 to November 2018. The carbon dioxide data measured as the mole fraction in dry air, on Mauna Loa constitute the longest record of direct measurements of CO2 in the atmosphere. They were started by C. David Keeling of the Scripps Institution of Oceanography in March of 1958 at a facility of the National Oceanic and Atmospheric Administration.

```
library(astsa)
data("cardox")
autoplot(cardox,xlab = "years",ylab = " CO2 (ppm)",color = "darkred",
size = 1,main = "Carbon Dioxide Levels at Mauna Loa")
```

The time series clearly has trend and seasonal components, for analyzing the *cardox* data we proposed a Gaussian linear state space model. We use the model’s implementation from the forecast package as follows:

```
library(forecast)
#>
#> Attaching package: 'forecast'
#> The following object is masked from 'package:astsa':
#>
#> gas
model = ets(cardox)
summary(model)
#> ETS(M,A,A)
#>
#> Call:
#> ets(y = cardox)
#>
#> Smoothing parameters:
#> alpha = 0.5591
#> beta = 0.0072
#> gamma = 0.1061
#>
#> Initial states:
#> l = 314.6899
#> b = 0.0696
#> s = 0.6611 0.0168 -0.8536 -1.9095 -3.0088 -2.7503
#> -1.2155 0.6944 2.1365 2.7225 2.3051 1.2012
#>
#> sigma: 9e-04
#>
#> AIC AICc BIC
#> 3136.280 3137.140 3214.338
#>
#> Training set error measures:
#> ME RMSE MAE MPE MAPE MASE
#> Training set 0.0232403 0.312003 0.2430829 0.006308831 0.06883992 0.1559102
#> ACF1
#> Training set 0.07275949
```

The best fitted model is a *multiplicative level, additive trend and seasonality* state space model. If the model’s assumptions are satisfied, then the model’s errors behave like a Gaussian stationary process. These assumptions can be checked using our *check_residuals* functions.

In this case, we use an Augmented Dickey-Fuller test for stationary assumption, and a random projections test for normality.

```
check_residuals(model,unit_root = "adf",normality = "rp",plot = TRUE)
#>
#> ***************************************************
#>
#> Unit root test for stationarity:
#>
#> Augmented Dickey-Fuller Test
#>
#> data: y
#> Dickey-Fuller = -9.7249, Lag order = 8, p-value = 0.01
#> alternative hypothesis: stationary
#>
#>
#> Conclusion: y is stationary
#> ***************************************************
#>
#> Goodness of fit test for Gaussian Distribution:
#>
#> k random projections test
#>
#> data: y
#> k = 2, lobato = 3.8260, epps = 1.3156, p-value = 0.3328
#> alternative hypothesis: y does not follow a Gaussian Process
#>
#>
#> Conclusion: y follows a Gaussian Process
#>
#> ***************************************************
```

Now that all the model’s assumptions are checked, the model is accepted and can be used to forecast.

```
autoplot(forecast(model,h = 12),include = 100,xlab = "years",ylab = " CO2 (ppm)",
main = "Forecast: Carbon Dioxide Levels at Mauna Loa")
```

The current development version can be downloaded from GitHub via

```
if (!requireNamespace("remotes")) install.packages("remotes")
remotes::install_github("asael697/nortsTest",dependencies = TRUE)
```

The **nortsTest** package offers additional functions for descriptive analysis in univariate time series.

*uroot.test*: performs unit root test for checking stationary in linear time series. The Ljung-Box, Augmented Dickey-Fuller, Phillips-Perron and Kpps tests can be selected with the*unit_root*option parameter.*seasonal.test*: performs seasonal unit root test for stationary in seasonal time series. The hegy, ch and ocsb tests are available with the seasonal option parameter.*arch.test*: for checking the ARCH effect in time series. The Ljung-Box and Lagrange Multiplier tests can be selected from the*arch*option parameter.*normal.test*: for normal distribution check in time series and random samples. The tests presented above can be chosen for stationary time series. For random samples (*independent data*), the Anderson & Darling, Shapiro & Wilks, and Jarque-Bera tests are available with the normality option parameter.

For visual diagnostic, we offer ggplot2 methods for numeric and time-series data. Most of the functions were adapted from Rob Hyndman’s forecast package.

*autoplot*: For plotting time series objects (*ts class*).*gghist*: histograms for numeric and univariate time series.*ggnorm*: quantile-quantile plot for numeric and univariate time series.*ggacf & ggpacf*: partial and auto correlation functions plots for numeric and univariate time series.*check_plot*: summary diagnostic plot for univariate starionary time series.

Currently our check_residuals() and check_plot() methods are valid for the current models and classes:

**ts**: for uni variate time series**numeric**: for numeric vectors**arima0**: from the stats package**Arima**: from the forecast package**fGARCH**: from the fGarch package**lm**: from the stats package**glm**: from the stats package**Holt and Winters**: from the stats and forecast package**ets**: from the forecast package**forecast methods**: from the forecast package.

For overloading more functions, methods or packages, please make a pull request or send a mail to: asael_am@hotmail.com

Epps, T.W. (1987). Testing that a stationary time series is Gaussian.

*The Annals of Statistic*. 15(4), 1683-1698.Nieto-Reyes, A., Cuesta-Albertos, J. & Gamboa, F. (2014). A random-projection based test of Gaussianity for stationary processes.

*Computational Statistics & Data Analysis, Elsevier*. 75(C), 124-141.Lobato, I., & Velasco, C. (2004). A simple test of normality for time series.

*Journal Econometric Theory*. 20(4), 671-689.Psaradakis, Z. & Vavra, M. (2017). A distance test of normality for a wide class of stationary process.

*Journal of Econometrics and Statistics*. 2, 50-60.Hyndman, R. & Khandakar, Y. (2008). Automatic time series forecasting: the forecast package for R.

*Journal of Statistical Software*. 26(3), 1-22.