Geoscience Reference
In-Depth Information
where y
t
is the response variable, say the crop yield at time t;
Y
t−1
, Y
t−2
, …, Y
t−p
are the respective variables at different time
with lag; β
0
, β
1
, …, β
p
are the regression coefficients; and ε
t
is
the error term.
Similarly, qth-order MA model, that is, MA (q) model may
be specified as
MA(q): Y
t
=+ +
θθε
θ ε
+
+
θ ε
+
ε
(4.7)
0
1t1
−
2
t2
−
qtq
−
t
where θ
0
, θ
1,
…, θ
q
are the coefficients.
AR and MA models can be effectively combined to form a
more general and useful class of time series models known as
ARMA models. However, they can only be used successfully
when the data are stationary. This model class can be extended
to non-stationary series by allowing differencing of the data
series. These are known as ARIMA models (Box and Jenkins,
1970). There are many varieties of ARIMA models available
in the literature. The general no-seasonal model is known as
ARIMA (p,d,q), where p is the order of the AR part, d is the
degree of first differencing involved and q is the order of the
MA part. The value of p and q may be inferred by looking
at auto-correlation function (ACF) and partial auto-correlation
function (PACF) as given in Table 4.3.
Table 4.4 highlights the production (in million tonnes) of
wheat in India from 1949-1950 to 2010-2011.
From Figure 4.2, it is clear that the data are non-stationary.
So, in order to make the data stationary, one has to perform
a differencing operation. After the first differencing, the data
become stationary. Graphical representation of stationary data
has been presented in Figure 4.3. Once the data become station-
ary, one can easily fit the ARIMA model.
In Figures 4.4 and 4.5, ACF and PACF operations have been
performed. Based on this figure, one can easily say that for the
table 4.3
Primary distinguishing characters of theoretical
ACFs and PACFs for stationary process
Process
ACF
PAC F
AR
Tails off towards zero
(exponential decay or
damped sign wave)
Cuts off to zero (after lag p)
MA
Cuts off to zero (after lag q)
Tails off towards zero
(exponential decay or
damped sign wave)
