Modeling Barley Production in Punjab

Barley has been an important commodity of the country. In this paper Forecast model for production of Barley in Punjab has been fitted. Methodologies for fitting of the model has been used these include ARIMA model. Diagnostic test has been carried out to see the adequacy of fitted models. Forecasted production has been obtained for coming five years.


Introduction
Pakistan has a rich and vast natural resource base covering various ecological and climatic zones; hence the country has great potential for producing all types of commodities.Agriculture is the hub of economic activity in Pakistan.It lays down foundation for economic development and growth of the economy.It directly contributes 25% to Gross Domestic Product (GDP) and provides employment of 44% of the total labour force of the country.
During the last five years (2000-01 to 2004-05), against the normal surface water availability at canal heads of 103.5 million acre feet (MAF), the overall (both for Kharif and Rabi) water availability has been less in the range of 5.9 percent (2003-04) to 29.4 percent (2001-02).(Source: Ministry of Food, Agriculture and Livestock; Bureau of Statistics).
Barley (Hordeum vulgare) is a major food and animal feed crop, a member of the grass family.Barley is the fifth largest cultivated cereal crop in the world (530,000 km² or 132 million acres).Cultivated barley is descended from wild barley, which still can be found in the Middle East.Both forms are diploid (2n=14 chromosomes).All variants of barley have fertile bastards and are thus considered to belong to one and the same species today.The major difference between wild and domesticated barley is the brittle rachis of the former, which is conductive to self-propagation.The earliest finds of barley come from Epi-Paleolithic sites the Levant beginning in the Natufien.The first domesticated barley has been found in the aceramic neolithic layers (PPN B) of Tell Abu Hureyra in Syria.The domestication seems to be contemporaneous to that of wheat.
The most proper seed season for spring barley is any time in March or April, though good crops produced have seen, the seed of which was sown at a much later period.Barley is widely adaptable and is currently a major crop of the temperate and tropical areas.

Stationarity Test for Barley
The stationarity of the data has been checked by using the Unit Root test the results are given below:

P-value
From the above table it is concluded that production of Barley in Punjab become stationary at the level, so ARIMA model with d at "0" will be used.

ACF and PACF for Production of Barley in Punjab
For the data of production of Barley for Punjab the data is stationary at level so the "ACF" and "PACF" of original data are plotted and from the plot it is observed that for this the value of "p" and "q" are respectively "1", "1".The ARMA (p, q) model applied on the 'd' differences of t Y is called Auto-Regressive Integrated Moving Average (ARIMA) Model.It is denoted by ARIMA (p, d, and q).Where "p" is the order of AR process, "q" is the order of MA process and "d" is the order of differencing.
The ARMA models are generalization of the simple AR model that uses three tools for modeling series correlation in the disturbance.
The model can also be checked for adequacy by doing a chi-square test, known as the Box-Pierce Q statistic, on the autocorrelations of the residuals.The test statistic is: Which is approximately distributed as a chi-square variate with "k-p-q" degree of freedom.In this equation N = length of the time series.K = First k autocorrelation being checked.M = Maximum no. of lags checked.If the calculated value of Q is larger than the chi-square for k-p-q degree of freedom, the model should have been considered inadequate.It is possible that two or more models have been judge to be approximate, yet none of the models may be an exact fit for the data.In this case, the principle of parsimony should prevail, and simpler model should have chosen.

ARMA Models
These are the mixture of AR and MA process or models.The time series t x for t = 0, 1, 2, .............. ± ± ± is said to be ARMA (p, q) if t x is stationary and The parameter p and q are called the auto regressive and moving average orders.If t x has non-zero mean μ then ARMA (p, q) can be written as given bellow The ARMA models become AR if q=0 and if p=0 these become MA models.The ARMA models can be written as bellow: ( ) ( )

ARIMA Models
ARIMA stands for Auto-Regressive Integrated Moving Average.These models deal with non-stationary time series, while ARMA (p, q), AR (p) and MA (q) models are used to deals with second order stationary time series.By using different operation on non-stationary population the population becomes stationary.The ARIMA (p, d, q) models assume that the th d difference ( ) is a stationary ARMA (p, q) process.

Analysis
The analysis of the data has been carried out in order to obtain suitable model for forecasting production of Barley in Punjab.The data is used for this purpose is from 1981 to 2004.The data is taken from "fifty years of Pakistan in statistics" and "statistical year book".The production is taken to be dependent variable, and Area, Temperature and Rain are taken to be independent variables.
The Wald chi-square test is 31.40 with a p-value of 0.000, indicating that the overall model for Punjab is significant and therefore it can be used to forecast the production of Barley in the province.
From the (Table-3.2)Model it can be easily seen that the coefficients of Area is significant and Temperature and Rain are insignificant.
The graphs of fitted values from the models along with the actual values are given below:

Figure 2 :
Figure 2: ACF and PACF For Production of Barley in Punjab

kr=
Sample autocorrelation function of the kth residual term.d = Degree of differencing to obtain a stationary series.

Figure 3 . 1 :
Figure 3.1: Fitted and Actual values for Production of Barley in Punjab

Figure 3 . 2 :
Figure 3.2: Figures of ACF and PACF of Error of Barley in Punjab