This document provides an introduction to ARIMA (AutoRegressive Integrated Moving Average) models. It discusses key assumptions of ARIMA including stationarity. ARIMA models combine autoregressive (AR) terms, differences or integrations (I), and moving averages (MA). The document outlines the Box-Jenkins approach for ARIMA modeling including identifying a model through correlograms and partial correlograms, estimating parameters, and diagnostic checking to validate the model prior to forecasting.

2. Contents
 Introduction to ARIMA
• Assumptions
 ARIMA Models
 Pros & Cons
 Procedure for ARIMA Modeling
(Box Jenkins Approach)

3. Introduction To ARIMA
 Acronym for Auto Regressive Integrated Moving
Average
 It is a prediction model used for time series
(time series is a collection of observations of
well-defined data items obtained through
repeated measurements over time)analysis &
forecasting.
Ex: measuring the level of unemployment each
month of the year would comprise a time series.

4.  A time series can also show the impact of
cyclical, seasonal and irregular events on the
data item being measured.
 Here the terms are:
Auto Regressive : lags of variables itself
Integrated :Differencing steps required to make
stationary
Moving Average :lags of previous information
shocks

5.  A non seasonal ARIMA model is classified as an
"ARIMA(p , d , q)" model, where:
p is the number of autoregressive terms,
d is the number of non seasonal differences
needed for stationarity, and
q is the number of lagged forecast errors in the
prediction equation.

6. Assumptions
 The data series used by ARIMA should be
stationary-by stationary it means that the
properties of the series doesn’t depend on the
time when it is captured. A white noise series
and series with cyclic behavior can also be
considered as stationary series.
 A non stationary series is made stationary by
differencing.

7.  Data should be univariate - ARIMA works on a single
variable. Auto-regression is all about regression with
the past values.

8. ARIMA Models
 Auto Regressive (AR) Model:
 Value of a variable in one period is related to
the values in previous period.
 AR(p) - Current values depend on its own p-
previous values
 P is the order of AR process
 Ex : AR(1,0,0) or AR(1)
 Moving Average (MA) Model:
 Accounts for possibility of a relationship b/w
a variable & residuals from previous period.

9.  MA(q) - The current deviation from mean
depends on q- previous deviations
 q is the order of MA process
 Only error terms are there
 Ex: MA(0,0,1) or MA(1)
 ARMA Model: both AR and MA are there,i.e,
ARMA(1,0,1) or ARMA(1,1)
 ARIMA Model : if differencing term is also
included ,i.e, ARIMA(1,1,1)=ARMA(1,1) with
first differencing
 ARIMAX: if some exogenous variables are also
included.

10. ARIMA+X=ARIMAX
ARIMA with environmental variable is very
important in the case when external variable
start impacting the series
Ex. Flight delay prediction depends not only
historical time series data but external variables
like weather condition (temperature , pressure,
humidity, visibility, arrival of other flights,
weighting time etc.)

11. Pros & Cons
 Pros :
1.Better understand the time series patterns
2.Forecasting based on ARIMA
 Cons : Captures only linear relationships ,
hence , a neural network model or genetic
model could be used if a non linear
associations(ex: quadratic relation) is found in
the variables.

12. Procedure for ARIMA Modeling
• Ensure Stationarity :Determine the appropriate values
of d .
• Make Correlograms (ACF & PACF): PACF indicate the AR
terms & ACF will show the MA terms.
• Fit the model :Estimate an ARIMA model using values
of p, d, & q you think are appropriate.
• Diagnostic Test : Check residuals of estimated ARIMA
model ; pick best model with well behaved residuals.
• Forecasting : use the fitted model for forecasting
purpose.

13. The Box-Jenkins Approach
1.Differencing the
series to achieve
stationary
2.Identify the model
3.Estimate the
parameters of the
model
Diagnostic checking.
Is the model
adequate?
No
Yes4. Use Model for forecasting

14. Step-1: Stationarity
 In order to model a time series with the Box-
Jenkins approach, the series has to be stationary.
 If the process is non-stationary then first
differences of the series are computed to
determine if that operation results in a stationary
series.
 The process is continued until a stationary time
series is found.
 This then determines the value of d.

15. Testing Stationarity
 Dickey-Fuller test
 P value has to be less than 0.05 or 5%
 If p value is greater than 0.05 or 5%, you
accept the null hypothesis, you conclude
that the time series has a unit root.
 In that case, you should first difference the
series before proceeding with analysis.

16.  What DF test ?
 Imagine a series where a fraction of the
current value is depending on a fraction of
previous value of the series.
 DF builds a regression line between fraction
of the current value Δyt and fraction of
previous value δyt-1
 The usual t-statistic is not valid, thus D-F
developed appropriate critical values. If P
value of DF test is <5% then the series is
stationary

17. Step-2:Making Correlograms
 AutoCorrelation Function (ACF):it is a
correlation coefficient. However, instead of
correlation between two different variables,
the correlation is between two values of the
same variable at times Xi and Xi+k.
 Correlation with lag-1, lag2, lag3 etc.,
 The ACF represents the degree of persistence
over respective lags of a variable.

18. ACF Graph
-0.50
0.000.501.00
Autocorrelationsofpresap
0 10 20 30 40
Lag
Bartlett's formula for MA(q) 95% confidence bands

19.  Partial Autocorrelation Function (PACF):
 The exclusive correlation coefficient
 the "partial" correlation between two variables is the
amount of correlation between them which is not
explained by their mutual correlations with a specified
set of other variables.
 For example, if we are regressing a variable Y on other
variables X1, X2, and X3, the partial correlation
between Y and X3 is the amount of correlation
between Y and X3 that is not explained by their
common correlations with X1 and X2.
 Partial correlation measures the degree of
association between two random variables, with the
effect of a set of controlling random variables removed.

20. PACF Graph-0.50
0.000.501.00
0 10 20 30 40
Lag
95% Confidence bands [se = 1/sqrt(n)]

21. Fit the Model
 Fit model based on AR & MA terms.
 Make use of auto.arima(x) function ,where x is
data series. It will do various combination of
AR & MA terms and find the best model based
on lowest AIC(Acyle Information Criteria ).
 For fitting model use arima(x,order=c(p,d,q))
function.Ex: fit=arima(x,order=c(4,0,2)).
 Order=c(p,d,q) is model received from
auto.arima(x) function.

22. Diagnostic Test
 First find the residuals: use residuals(model)
function.Ex: fit_resid=residuals(fit).
 Now do diagnostic on all these residuals(A
residual in forecasting is the difference
between an observed value and its forecast
based on other observations: ei=yi−y^i. For
time series forecasting, a residual is based on
one-step forecasts; that is y^t is the forecast
of yt based on observations y1,…,yt−1.).
 If residuals are IID(i.e, having no auto
correlation ) then model is fit..

23.  For diagnostic use different tests ,ex,Ljung Box
test.Make use of Box.test() function to find p.
 Ex:Box.test(fit_resid,lag=10,type=“Ljung-Box”)
 If p-value is non zero then no serial correlation is
there & model is fit & can be used for forecasting
purpose

24. Thanks