ARIMA models

1. ARIMA: Some Examples
ARIMA ( 0,1,0) is (1-L)yt = µ+et
ARIMA ( 0,1,1) is (1-L)yt = µ+(1- L)et
ARIMA ( 0,1,2) is (1-L)yt = µ+(1- 1L- 2L2
)et
ARIMA ( 1,1,0) is (1-L)yt = µ+ L yt + et
ARIMA ( 2,1,0) is (1-L)yt = µ+( 1L+ 2L2
) yt +et
ARIMA ( 1,1,1) is (1-L)yt = µ+(1- 1L)et+ Lyt
ARIMA ( 2,1,2) is (1-L)yt = µ+(1- 1L- 2L2
)et +( 1L+ 2L2
) yt
Theoretical Pattern: In time series models, the basic assumption is weakly stationary or
Stationary. In weakly stationary time series, the mean and variance are constant and
covariance is time invariant. But in practical situation/reality, many econometric time series
are non-stationary that is they are integrated. Now if the time series is integrated of order 1or
I (1), then its first difference is stationary.
Now if a time series is I (2), then its second differences are stationary. In general if a time
series is I(d), after differencing it d times we obtained I(0) series.
 To difference a time series d times to make it stationary and then apply the ARMA (p,
q) model; we can say the original time series model is ARIMA (p, d, q).
 The important point to note is that to use the Box-JenKins Methodology, we must
have either a stationary time series or a time series that is stationary after one or more
differencing.
Box-JenKins (BJ) methodology
Popularly known as the Box-JenKins (BJ) methodology, but technically known as the
ARIMA methodology. The objective of B-J methodology is to identify and estimate a
statistical model which can be interpreted as having generated the sample data. If this
estimated model is then to be used for forecasting we must assume that the features of this
model are constant through time and particularly over future periods.
The Box-JenKins (BJ) Methodology
Looking at time series,
 How does one know whether it follows a purely AR process (P=?)
 How does he know whether it follows a purely MA process (q=?)
 How does he know whether it follows a ARMA process (P=?, q=?)
 How does he know whether it follows a ARIMA process (P=?, d=?, q=?)

2. The Box-JenKins (BJ) methodology comes in handy in answering the preceding question.
This method consists of four steps;
Step 1:
Identification.
That is find out the appropriate values of p, d and q. Correlogram → ACF graph helps to do
this task.
Step 2:
Estimation of the parameters.
Having identified the appropriate p and q values, the next stage is to estimate the parameters
of the auto regressive and moving average terms included in the model. To estimates the
parameters, we can use OLS but NLS is common to use.
Step 3:
Diagnostic checking.
Having chosen a particular ARIMA model and having estimated its parameters, then check
the chosen model fits the data reasonably well. If we see the residuals estimated from this
model are white noise or pure random error term then chosen model is to accept the particular
fit, if not, we must start from step1. Thus the BJ methodology is an iterative process.
Step 4:
Forecasting.
By using ARIMA modeling, the forecast obtained by this method are more reliable than some
other traditional method.
Flowchart
Step1: Identification of the model (choosing tentative p,d,q)
↓
Step2: Parameter estimation
↓
Step3: Diagnostic checking ( Are residuals white noise?)
↓
Step4: if Residuals are not pure random, go to step1
if Residuals are pure random, go to step5
↓
Forecasting

3. Theoretical Patterns of ACF and PACF
Type of model Patterns of ACF Patterns of PACF
MA (q) Cuts off after lags q Declines exponentially
AR (p) Decays Exponentially Cuts off after lag p
ARMA (p, q) Exponential Decay Exponential Decay
ACF and PACF’S of AR (p) and MA (q) process have opposite patterns; in the AR (p) case
the ACF declines geometrically or exponentially but the PACF cuts off after a certain number
of lags, whereas the opposite happens to an MA (q) process.
Since we do not observe the theoretical ACF’S and PACF’S and rely on their sample counter
parts, the estimated ACF and PACF’S with not match exactly with their theoretical counter
parts.
SEASONAL PROCESSES
Sometime time series data exhibit strong periodic patterns, say
Yt= St+Nt
St is the deterministic component with periodicity s and Nt is the stochastic component that
may be modeled as an ARMA process.
Since St is the deterministic and has periodicity s, so
St=St+s=St-s
St - St-s= (1-Ls
) St=0
And ARIMA(0,1,1) model with s=4can be defined as
(1-L) (1-L4
)yt= (1- L- *L4
+ *L5
) et
Now ARIMA(0,1,1) model with s=12 can be defined as
(1-L) (1-L12
)yt= (1- L- *L12
+ *L13
) et
Problem: Draw the correlogram/ACF function for AR(1) with 1= 0.5
Problem: Draw the correlogram/ACF function for MA(2) with 1= 0.5 and 2=0.3.
Problem: Draw the correlogram/ACF function for AR(2) with 1= 0.5 and 2=0.3.
By using Yale-Walker equation you can solve this problem.