Modelling and forecasting of tur production in India usingARIMA model

SUMMARY : The paper describes an empirical study of modeling and forecasting time series data of
tur production in India.Yearly tur production data for the period of 1950-1951 to 2014-2015 of India were
analyzed by time-series methods.Autocorrelation and partial autocorrelation functions were calculated
for the data. The Box Jenkins ARIMA methodology has been used for forecasting. The diagnostic
checking has shown that ARIMA(1, 1, 1) is appropriate. The forecasts from 2015-2016 to 2024-2025
were calculated based on the selected model. The forecasting power of autoregressive integrated
moving average model was used to forecast tur production for ten leading years. These forecasts
would be helpful for the policy makers to foresee ahead of time the future requirements of tur seed,
import and/or export and adopt appropriate measures in this regard.
How to cite this article : Borkar, Prema andBodade, V.M. (2017). Modelling and forecasting of tur production
in India using ARIMA model. Agric. Update, 12(2): 252-257; DOI : 10.15740/HAS/AU/12.2/252-257.
BACKGROUND AND OBJECTIVES
Tur is an important pulse crop in India. It
is also known as pigeonpea, red gram and
Arhar. It is mainly cultivated and consumed
in developing countries of the world. This crop
is widely grown in India. India is the largest
producer and consumer of tur in the world.
Tur is the second largest pulse crop in India
accounting for about 20 per cent of total pulse
production. The production of tur during2014-
15 as per fourth advance estimates in the
country was about 2.78 million tonnes. The
area under tur in current year 2014-15 is
estimated about 39.05 lakh ha which is about
6.4 per cent less than the previous year.
Forecasts have traditionally been made
ModellingandforecastingofturproductioninIndia
usingARIMA model
PREMA BORKAR AND V.M. BODADE
HIND AGRICULTURAL RESEARCH AND TRAINING INSTITUTE
ARTICLE CHRONICLE :
Received :
07.03.2017;
Revised :
22.03.2017;
Accepted :
04.04.2017
RESEARCH ARTICLE :
KEY WORDS :
ACF - autocorrelation
function, ARIMA -
autoregressive
integrated
moving average,
PACF - partial
autocorrelation
function, Tur
using structural econometric models.
Concentration have been given on the
univariate time series models known as auto
regressing integrated moving average
(ARIMA) models, which are primarily due to
world of Box and Jenkins (1970). These
models have been extensively used in practice
for forecasting economic time series,
inventory and sales modeling (Brown, 1959
and Holt et al., 1960) and are generalization
of the exponentially weighted moving average
process. Several methods for identifying
special cases of ARIMA models have been
suggested by Box- Jenkins and others.
Makridakis et al. (1982) and Meese and
Geweke (1982) have discussed the methods
of identifyingunivariate models.Amongothers
Author for correspondence :
PREMA BORKAR
Gokhale Institute of
Politics and Economics,
PUNE (M.S.) INDIA
See end of the article for
authors’ affiliations
Agriculture Update
Volume 12 | Issue 2 | May, 2017 | 252-257
e ISSN-0976-6847
Visit us : www.researchjournal.co.in
DOI: 10.15740/HAS/AU/12.2/252-257
AU

253
Hind Agricultural Research and Training Institute
Agric. Update, 12(2) May, 2017 :
Jenkins and Watts (1968);Yule (1926 and 1927); Bartlett
(1964); Quenouille (1949), Ljunge and Box (1978) and
Pindycke and Rubinfeld (1981) have also emphasized
the use of ARIMA models. In this study, these models
were applied to forecast the production of tur crop in
India. This would enable to predict expected tur
production for the years from 2013 onward. Such an
exercise would enable the policy makers to foresee ahead
of time the future requirements for tur storage, import
and/or export of tur thereby enabling them to take
appropriate measures in this regard. The forecasts would
thus help save much of the precious resources of our
country which otherwise would have been wasted.
RESOURCES AND METHODS
Respective time series data for this study were
collected from various Government Publications of India.
Box and Jenkins (1976) linear time series model was
applied. Auto Regressive Integrated Moving Average
(ARIMA) is the most general class of models for
forecasting a time series. Different series appearing in
the forecasting equations are called “auto-regressive”
process. Appearance of lags of the forecast errors in
the model is called “moving average” process. The
ARIMA model is denoted by ARIMA (p,d,q),
where,
“p” stands for the order of the auto regressive
process,
“d” is the order of the data stationary and
“q” is the order of the moving average process.
The general form of the ARIMA (p,d,q) can be
written as described by Judge et al. (1988).
d
yt
=+1
d
yt-1
+2
d
yt-2
+----- + p
yt-p
+et-l
 et-1
-2
et-2
q
e t-2
(1)
where,
d
denotes differencing of order d, i.e., yt
=yt
-yt-1
,
2
yt
=yt
-t-1
and so forth,
Y t-1
-------- yt-p
are past observations(lags),
,1
------- p
are parameters (constant and co-
efficient) to be estimated similar to regression co-
efficients of the auto regressive process (AR) of order
“p” denoted by AR (p) and is written as
Y = +1
yt-1
+2
yt-2
+ -------- + p
yt-p
+et
(2)
where,
et
is forecast error, assumed to be independently
distributed across time with mean  and variance 2
e,
et-1
, et-2
——— et-q
are past forecast errors,
1
, ————— q
are moving average (MA) co-
efficient that needs to be estimated.
While MA model of order q (i.e.) MA (q) can be
written as
Yt
= et
-1
t-1
- 2
et-2
-------------------- q
et-q
(3)
The major problem inARIMAmodeling technique
is to choose the most appropriate values for the p, d, and
q. This problem can be partially resolved by looking at
the auto correlation function (ACF) and partial auto
correlation functions (PACF) for the series (Prindycke
and Rubinfeld, 1981). The degree of the homogeneity,
(d) i.e. the number of time series to be differenced to
yield a stationary series was determined on the basis
where the ACF approached zero.
After determining “d” a stationary series d yt
its
auto correlation function and partial autocorrelation were
examined to determined values of p and q, next step was
to “estimate” the model. The model was estimated using
computer package “SPSS”.
Diagnostic checks were applied to the so obtained
results. The first diagnostic check was to draw a time
series plot of residuals. When the plot made a rectangular
scatter around a zero horizontal level with no trend, the
applied model was declared as proper. Identification of
normality served as the second diagnostic check. For
this purpose, normal scores were plotted against residuals
and it was declared in case of a straight line. Secondly, a
histogram of the residuals was plotted. Finding out the
fitness of good served as the third check. Residuals were
plotted against corresponding fitted values: Model was
declared a good fit when the plot showed no pattern.
Using the results ofARIMA(p, q, d), forecasts from
2013 upto 2025 were made. These projections were based
on the following assumptions.
– Absence of random shocks in the economy,
internal or external.
– Agricultural price structure and polices will
remain unchanged.
– Consumer preferences will remain the same.
OBSERVATIONS AND ANALYSIS
The results obtained from the present study as well
as discussions have been summarized under following
heads:
Building ARIMA model for tur production data in
India :
To fit anARIMAmodel requires a sufficiently large
252-257

254
data set. In this study, we used the data for tur production
for the period 1950-1951 to 2014-15.As we have earlier
stated that development of ARIMA model for any
variable involves four steps: identification, estimation,
diagnostic checking and forecasting. Each of these four
steps is now explained for tur production. The time plot
of the tur production data is presented in Fig. 1.
The newly constructed variable Xt
can now be
examined for stationarity. The graph of Xt
was stationary
in mean. The next step is to identify the values of p and
q. For this, the autocorrelation and partial autocorrelation
co-efficients of various orders of Xt are computed (Table
1). The ACF and PACF (Fig. 2 and 3) shows that the
order of p and q can at most be 1.We entertained three
tentative ARIMA models and chose that model which
has minimum AIC (Akaike Information Criterion) and
BIC (Bayesian Information Criterion). The models and
corresponding AIC and BIC values (Table A).
Table A : AIC and BIC value
ARIMA (p, d, q) AIC BIC
1 0 0 47.03 51.28
1 1 0 38.77 42.99
1 1 1 25.02 31.35
Table 1 : Autocorrelations and partial autocorrelations
Lag Autocorrelation Std. error Lag Partial autocorrelation Std. error
1 0.619 0.124 1 0.619 0.127
2 0.566 0.123 2 0.297 0.127
3 0.608 0.122 3 0.316 0.127
4 0.566 0.121 4 0.144 0.127
5 0.551 0.120 5 0.123 0.127
6 0.489 0.119 6 -0.023 0.127
7 0.441 0.118 7 -0.052 0.127
8 0.390 0.117 8 -0.098 0.127
9 0.399 0.116 9 0.032 0.127
10 0.324 0.114 10 -0.083 0.127
11 0.375 0.113 11 0.159 0.127
12 0.347 0.112 12 0.047 0.127
13 0.230 0.111 13 -0.127 0.127
14 0.237 0.110 14 -0.053 0.127
15 0.240 0.109 15 -0.003 0.127
16 0.166 0.108 16 -0.113 0.127
Case Number
61575349454137332925211713951
ValueProduction
3.5
3.0
2.5
2.0
1.5
1.0
3.5
3.0
2.5
2.0
1.5
1.0
Case number
Valueproduction
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61
Fig. 1 : Time plot of tur production data
The above time plot indicated that the given series
is nonstationary. Non-stationarity in mean is corrected
through appropriate differencing of the data. In this case
difference of order 1 was sufficient to achieve stationarity
in mean.
So the most suitable model is ARIMA(1, 1, 1) this
model has the lowest AIC and BIC values.
Model parameters were estimated using SPSS
package. Results of estimation are reported in Table 2.
The model verification is concerned with checking the
residuals of the model to see if they contain any
systematic pattern which still can be removed to improve
on the chosen ARIMA. This is done through examining
the autocorrelations and partial autocorrelations of the
residuals of various orders. For this purpose, the various
correlations upto 16 lags were computed and the same
MODELLING & FORECASTING OF TUR PRODUCTION IN INDIA USING ARIMA MODEL
252-257

255
Table 2 : Estimates of the fitted ARIMA model
Estimates Std Error t Approx sig
Non- Seasonal lag AR1 -0.139022 0.170830 -0.8138 0.41908686
MA1 0.715422 0.130183 5.49549 0.00000091
Constant 0.018101 0.009625 1.88055 0.06505709
Number of residuals 61
Number of parameters 2
Residual df 58
Adjusted residual sum of squares 4.8710246
Residual sum of squares
Residual variance
5.1143186
0.08271759
Model Std. error 0.28760667
Log-Likelihood -9.4831015
Akaike’s information criteria (AIC) 24.96
Schwarz’s bayesian criterion (BIC) 31.29
Fig. 2 : ACF of differenced data
1.0
.5
0.0
-.5
-1.0
Lag number
ACF
1 3 5 7 9 11 13 15
2 4 6 8 10 12 14 16
Co-efficient
Confidence limits
Fig. 3 : PACF of differenced data
1.0
.5
0.0
-.5
-1.0
Lag number
PACF
1 3 5 7 9 11 13 15
2 4 6 8 10 12 14 16
Co-efficient
Confidence limits
alongwith their significance which istested byBox-Ljung
test are provided in Table 3.As the results indicate, none
of these correlations is significantly different from zero
at a reasonable level. This proves that the selected
ARIMA model is an appropriate model. The ACF and
PACF of the residuals (Fig. 4 and 5) also indicate ‘good
fit’ of the model.
The last stage in the modeling process is forecasting.
ARIMA models are developed basically to forecast the
corresponding variable. There are two kinds of forecasts:
sample period forecasts and post-sample period
forecasts. The former are used to develop confidence in
the model and the latter to generate genuine forecasts
foruse in planningand other purposes.TheARIMAmodel
can be used to yield both these kinds of forecasts. The
residuals calculated during the estimation process, are
considered as the one step ahead forecast errors. The
forecasts are obtained for the subsequent agriculture year
from 2014-15 to 2024-2025.
Conclusion :
In our study, the developed model for tur production
was found to be ARIMA (1,1,1). The forecasts of tur
production, lower control limits (LCL) and upper control
limits (UCL) are presented in Table 4. The validity of
the forecasted values can be checked when the data for
the lead periods become available. The model can be
used by researchers for forecasting of tur production in
India. However, it should be updated from time to time
with incorporation of current data.
This paper discloses the production of tur from1950-
1951 to 2014-2015 and also shows the future movement.
252-257

256
Fig. 4 : ACF of residuals of fitted ARIMA model
1.0
.5
0.0
-.5
-1.0
Lag number
ACF
1 3 5 7 9 11 13 15
2 4 6 8 10 12 14 16
Co-efficient
Confidence limits
Fig. 5 : PACF of residuals of fitted ARIMA model
1.0
.5
0.0
-.5
-1.0
Lag number
PartialACF
1 3 5 7 9 11 13 15
2 4 6 8 10 12 14 16
Co-efficient
Confidence limits
Table 3 : Autocorrelations and partial autocorrelations of residuals
Lag Autocorrelation Std. error Box- Ljung df Sig. Lag Partial autocorrelation Std. error
1 -0.032 0.125 0.067 1.000 0.796 1 -0.032 0.128
2 -0.120 0.124 1.000 2.000 0.607 2 -0.121 0.128
3 0.108 0.123 1.766 3.000 0.622 3 0.101 0.128
4 0.038 0.122 1.861 4.000 0.761 4 0.030 0.128
5 0.168 0.121 3.794 5.000 0.579 5 0.199 0.128
6 0.026 0.120 3.841 6.000 0.698 6 0.036 0.128
7 -0.036 0.119 3.933 7.000 0.787 7 0.004 0.128
8 -0.086 0.117 4.469 8.000 0.813 8 -0.131 0.128
9 0.006 0.116 4.472 9.000 0.878 9 -0.031 0.128
10 -0.155 0.115 6.287 10.000 0.791 10 -0.236 0.128
11 0.021 0.114 6.319 11.000 0.851 11 0.022 0.128
12 0.023 0.113 6.362 12.000 0.897 12 -0.011 0.128
13 -0.178 0.112 8.895 13.000 0.781 13 -0.081 0.128
14 0.021 0.111 8.932 14.000 0.835 14 0.043 0.128
15 0.015 0.109 8.950 15.000 0.880 15 0.060 0.128
16 -0.088 0.108 9.607 16.000 0.886 16 -0.063 0.128
Table 4 : Forecasts for tur production (2015-16 to 2024-2025) (Million Tonnes)
Years Forecasted production Lower limit Upper limit
2015-2016 2.89461 2.26349 3.52574
2016-2017 2.91268 2.26125 3.56411
2017-2018 2.93079 2.25916 3.60242
2018-2019 2.94889 2.25711 3.64067
2019-2020 2.96699 2.25511 3.67887
2020-2021 2.98509 2.25316 3.71703
2021-2022 3.00319 2.25124 3.75514
2022-2023 3.02130 2.24937 3.79322
2023-2024 3.03940 2.24753 3.83127
2024-2025 3.05750 2.24572 3.86928
MODELLING & FORECASTING OF TUR PRODUCTION IN INDIA USING ARIMA MODEL
ACF of residuals PACF of residuals
252-257

257
To formulate future development plan for tur production,
it is essential to know the previous condition and also
see the future trend. In this study, forecasting is done by
using some sophisticated statistical tools so that the
government and policy makers can easily realize about
the future development of tur production and could take
initiatives to improve the production.
Authors’ affiliations :
V.M. BODADE, Department of Agricultural Economics and Statistics,
Dr. Panjabrao Deshmukh Krishi Vidyapeeth, AKOLA (M.S.) INDIA
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