ARIMA is a statistical model used to forecast future values in a time series by examining patterns in past observations. It stands for autoregressive integrated moving average. The ARIMA model incorporates autoregressive (AR) terms to model dependence on past values, differences (I) to make the data stationary, and moving average (MA) terms to model lagged forecast errors. The ARIMA(p,d,q) parameters include p for AR order, d for differencing, and q for MA order. ARIMA is useful when data exhibits trends or seasonality and can be applied after differencing to make the time series stationary.

1. ARIMA Model
ARIMA is an acronym for “autoregressive
integrated moving average.”
It’s a model used in statistics and econometrics
to measure events that happen over a period of
time.
The model is used to understand past data or
predict future data in a series.
It’s used when a metric is recorded in regular
intervals, from fractions of a second to daily,
weekly or monthly periods.

2. Stationary Time Series
• A key role in time series analysis is played by
processes whose properties, or some of them, do
not vary over time.
• Such a property is illustrated in the following
important concept, stationary.
• We then introduce the most commonly used
stationary linear time series models–the
autoregressive integrated moving average
(ARIMA) models.

3. ARIMA in Time Series Analysis
• An autoregressive integrated moving average – ARIMA
model is a generalization of a simple autoregressive
moving average – ARMA model.
• Both of these models are used to forecast or predict
future points in the time-series data.
• ARIMA is a form of regression analysis that indicates the
strength of a dependent variable relative to other
changing variables.
• The final objective of the model is to predict future time
series movement by examining the differences between
values in the series instead of through actual values.
• ARIMA models are applied in the cases where the data
shows evidence of non-stationary.
• In time series analysis, non-stationary data are always
transformed into stationary data.

4. • The common causes of non-stationary in time
series data are the trend and the seasonal
components.
• The way to transformed non-stationary data to
stationary is to apply the differencing step.
• It is possible to apply one or more times of
differencing steps to eliminate the trend
component in the data.
• Similarly, to remove the seasonal components
in data, seasonal differencing could be applied.

5. • we can split the model into smaller components as follow:
• AR: an AutoregRegressive model which represents a type of
random process. The output of the model is linearly
dependent on its own previous value i.e. some number of
lagged data points or the number of past observations [2].
• MA: a Moving Average model which output is dependent
linearly on the current and various past observations of a
stochastic term [3].
• I: integrated here means the differencing step to generate
stationary time series data, i.e. removing the seasonal and
trend components [1].
• ARIMA model is generally denoted as ARIMA(p, d, q) and
parameter p, d, q are defined as follow:
• p: the lag order or the number of time lag of autoregressive
model AR(p)
• d: degree of differencing or the number of times the data
have had subtracted with past value
• q: the order of moving average model MA(q)

6. Interpretation of sample ACF and PACF plot
• ACF: The autocorrelation coefficient function, define
how the data points in a time series are related to the
preceding data points.
• PACF: The partial autocorrelation coefficient function,
like the autocorrelation function, conveys vital
information regarding the dependence structure of a
stationary process.
• Now let’s explain how to select a model based on ACF
and PACF plots.
• Given observations of a time series, one approach to the
fitting of a model to the data is to match the sample ACF
and sample PACF of the data with the ACF and PACF of
the model, respectively.
• We can use the sample ACF plot to see if the specific
time series data is stationary or not

7. • Load readxl
• Read Dataset
• View(data)
• Class(dataset)
• class(gdp)
• gdptime<-
ts(gdp$GDP,start=min(gdp$Date),frequency = 4)
• class(gdptime)

8. • class(gdp)
• gdptime<-ts(gdp$GDP,start=min(gdp$Date),frequency = 4)
• class(gdptime)
• library(forecast)
• library(tseries)
• plot(gdptime)
• acf(gdptime)
• pacf(gdptime)
• adf.test(gdptime)
• gdpmodel=auto.arima(gdptime,ic="aic",trace = TRUE)
• gdpmodel
• acf(ts(gdpmodel$residuals))
• pacf(ts(gdpmodel$residuals))
• mygdpforecast=forecast(gdpmodel,level = c(95),h=10*4)
• mygdpforecast
• plot(mygdpforecast)
• Box.test(mygdpforecast$residuals,lag = 5,type = "Ljung-Box")
• Box.test(mygdpforecast$residuals,lag = 15,type = "Ljung-Box")
• Box.test(mygdpforecast$residuals,lag = 30,type = "Ljung-Box")