








































American Journal of Agricultural Science, Engineering and Technology 

Fitting and Forecasting of Trend Models for the HYV Boro Yields of 

Dinajpur District 

Provash Kumar Karmokar*, Rangan Kumar and Ranjan Kumar Kundu 

Department of Statistics 

University of Rajshahi, Rajshahi-6205, Bangladesh 

*Corresponding Authors Email: sprovash@yahoo.com

†Research Student 

Abstract 

Bangladesh is a densely populated country and the main food of the country is rice. Although the 

High Yielding Variety (HYV) Bro rice is being cultivated in almost all areas of Bangladesh it is 

enormously cultivated in Dinajpur district of Bangladesh. Trend is very important to know the HYV 

Boro rice yields of the country. Hence the objective of this research is to fit and forecast of trend models 

of HYV Bro rice yields of Dinajpur district for the year from 1971 to 2007. The regression diagnostics 

revealed in this study indicate that the data is autocorrelated.  Hence the Cochrane-Orcutt method was 

employed to fit such data set. Finally, we forecasted the yields of the three trend models for HYV Boro 

rice yields of Dinajpur up to 2021. On basis of the regression diagnostics, the quadratic trend model is an 

appropriate model for this data set and in this model would be useful for the decision makers for their 

agriculture and food policy formulation. 

Keywords: HYV Boro rice, Trend Models, Ordinary Least Square, Cochrane and Orcutt method, 

Forecasting. 

61AJASET, ISSN: 2158-8104 (Online), 2164-0920 (Print), Vol. 2, Issue. 1

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American Journal of Agricultural Science, Engineering and Technology 

1. Introduction

Bangladesh is an agro-based developing country and the economic development of the country is 

mainly based on agriculture which contributes about 20.24% to the Gross Domestic Product. About 

43.53% of the labor force is employed in agriculture (BER, 2010).  Therefore, agriculture is a most 

important sector of Bangladesh and rice is the staple food of the country. Rice is the principal sources of 

food, calorie, and protein intake for most of the people.  Rice contributes positively in the economy of 

the country and about 84% people of Bangladesh are directly or indirectly engaged in a wide range of 

agricultural activities (Rahman 2004). In the agricultural sector rice is a major source of livelihood in 

terms of providing food, income and employment in Bangladesh. Rice occupies 10.58 million hectares 

of land which is about 77 percent of the cultivated area (BBS, 2008). In Bangladesh, a country currently 

experiencing rapid population growth and serious food shortages, an effort to increase crop yields through 

the introduction of High Yielding Varieties (HYV) of rice was initiated in 1966. Among the all other rice 

varieties, Boro is the major rice crop of Bangladesh which provides about 55% of the total rice production. 

Boro is a variety of rice which is cultivated in almost every area of Bangladesh. This type of rice is 

cultivated in nearly 35% of the 10.80 million ha of rice harvested area, and contributed 50% of the 

38.7 million tons of rice produced in 2001/2002 (Singh et al 2003b). Normally rice is grown in 

Bangladesh in three distinct seasons like, Aus (from April to August), and Amon (from August to 

December) and Boro (from January to June). Irrigated rice or Boro rice is a potential area for 

increasing rice yield, which currently accounts for about 57% of total rice production (BBS estimate 

2008). 

In the literature, a number of trend analyses have been carried out to measure the secular trend of some 

agricultural productions (e.g., Sahu (2003); Gupta et al (1999); Karmokar and Imon (2008)). These 

studies mainly based on the traditional OLS method of estimation. However, in this study we found that 

the rice production data of the selected area are affected with autocorrelation problem. Hence the 

objective of the study is to analyse and validate results of the High Yielding Variety (HYV) Boro rice 

production data of Dinajpur district of Bangladesh in presence of autocorrelation 

This paper is organised as follows. The description of the data and methods are presented in Section 2. 

The results and discussions are reported in Section 3 and finally Section 4 concludes the paper. 

2. Data and Methods

The rice production data for Dinajpur district of Bangladesh comprising of 37 years HYV Boro rice yields 

(y) from 1971 to 2008 have been used in this study. The data have been collected from Bangladesh Bureau

of Statistics (BBS), government of Bangladesh.

A varieties of nonparametric tests like Kruskal–Wallis’s test, Wald–Wolfowitz’s test, Whitney-Pettitt’s

test (Pettitt, 1979), Mann–Kendall’s test, Spearman’s test are seen in the literature to detect the trend of

a data. To investigate the trend in a data set, Kendall and Stuart (1961) suggested Mann–Kendall’s test

and Spearman’s test as a powerful test when the most likely alternative to randomness is linear or non

linear trend.  In this study we measure the existence of trends using the non-parametric tests Mann-

Kendall rank tests and Spearman’s rank-correlation.  The Mann-Kendall rank test and Spearman’s rank-

correlation (see Kendall and Stuart, 1961; Mann, 1945) are briefly discussed bellow.

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American Journal of Agricultural Science, Engineering and Technology 

The Mann-Kendall rank statistic tm  is computed, by first replacing the observations xi’ s by their ranks ki 

s  such that each term is assigned a number ranging from 1 to n which reflects its magnitude relative to 

the magnitudes of all other terms. For each element ki the number Ni is calculated as the number of kj 

terms preceding it such that kj > ki .  Then tm is given by, 

)1(

4
1

1









nn

N

t

n

i

i

m

  (1) 

tm is distributed very nearly as a normal distribution for large n and can be used as the basis of a 

significance test, 

)1(9

104






nn

n
rr gm

     (2) 

where rg is the desired probability point of the normal distribution appropriate to a two-tailed test. If tm 

lies inside the range ± rm then the time series does not contain a trend.  

Spearman’s rank-correlation method is simple and distribution-free which is also used to test the trends 

among the data sets. The test statistics of Spearman rank-correlation test is defined as: 

)1(

6

1
2

11

2






nn

d
n

i

     (3) 

where, iii kykxd  when two or more observations, id , have the same value, the average rank iky  is 

calculated. 

A test-statistic tt is used to test the null hypothesis  0:0 H  (there is no trend)

against the alternative hypothesis 0:1 H (there is a trend). The test statistic is defined as 

21

2






n
t t   (4) 

For large n, the value of rs can be tested for significance by calculating the quantity ts given by 

the equation 

21

2

s

ss
r

n
rt




   (5) 

The computed ts indicate the boundary value for the test. 

Method of Ordinary Least Square (OLS) is an estimation technique in statistics (Harper, 1974) that was 

first described by German mathematician Carl Friedrich Gauss around 1794 (Bretsche, 1995). It is more 

convenient and widely used method among various method of estimation of unknown parameters on 

regression analysis technique. In the OLS the estimation of parameters of a linear regression model is 

based on the assumption that the errors are normally independently distributed with zero means and 

constant variance. Sometime data of a regression model involve regressors and response variables that 

have a natural sequential order over time.  Further, the errors of the regression model may have a special 

form of relationship which indicator of autocorrelation. In such cases the traditional OLS estimators may 

have some faulty solution of the parameters and consequently the result may mislead to explain the whole 

scenario. Because the OLS estimators are still unbiased and consistent this is because both unbiasedness 

and consistency do not depend on assumption σ2 which is in this case violated. The OLS estimators will 

be inefficient and the best linear unbiased estimator (BLUE) property will be not maintained. The 

estimated variances of the regression coefficients will be biased and inconsistent, and therefore  

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American Journal of Agricultural Science, Engineering and Technology 

hypothesis testing may not be valid. Since  2

iu is affected in an autocorrelated situation the R2 will also

be affected.  Thus the formal autocorrelation can be tested by DW statistics which may be falls in the 

inconclusive region or may falls in the positive or negative autocorrelation region. The use of the 

Cochrane and Orcutt method (see Cochrane and Orcutt, 1949) may be a solution of such problem may  

gives a meaningful estimate and interpretation. Consequently we can get the best fit of the model.  In 

these viewpoints in this study we employ the Cochrane and Orcutt method to forecast the HYV Boro 

yields of Dinajpur district of Bangladesh. 

To investigate the trend of yield of HYV Boro rice for Dinajpur district of Bangladesh we consider Linear, 

Compound and Quadratic trend models. The mathematical model of Linear, Compound and Quadratic 

models are as follows. 

i. linear trend model

iij tbay  ,    (6) 

where iy is the yield of HYV Boro, t is the time and  i is the random error.

ii. quadratic model

ii tctbay  2
    (7) 

iii. compound trend model

i
t

i aby     (8) 

Taking logarithmic on both sides of Eq.(8) we get, 

****

ii tbay     (9) 

where ii yy ln
*
 , aa ln*  , bb ln*  , ii  ln

*
. 

3. Results and Discussions

A time series consists of a trend component can be ensured using any time series (TS) plot.  The TS plot 

of the yield of HYV Boro rice of the selected district for the year from 1971 to 2007 is shown in Fig.1. 

Fig 1. TS Plot of HYV Boro production for the district Dinajpur 

The descriptive statistics of yield of HYV Boro rice of Dinajpur District is shown in Table 1. 

Table 1.  Descriptive Statistics for HYV Boro yields of Dinajpur 

Min Max Mean SE CV (%) Skewness Kurtosis 

2.03 3.81 2.715 0.420 15.45 0.917 3.859 

y 

0.00 
0.50 
1.00 
1.50 
2.00 
2.50 
3.00 
3.50 
4.00 
4.50 

1971 1975 1979 1983 1987 1991 1995 1999 2003 2007 

y 

 

Year of Yields 

Y
ie

ld
 o

f 
H

Y
V

 B
o

ro
 r

ic
e 

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From the Table 1, the skewness and Kurtosis are 0.917 and 3.859 respectively which indicate that the 

data are positively skewed and the kurtosis value is greater than 3 imply that the distribution of the 

frequency curve is leptokurtic.  Since we are dealing with the yield data it is important to determine either 

the values of the yields increase or decrease over time or not. Basically it is a matter of investigation that 

whether the probability distribution of such variable is changes over time or not.  It is interesting to note 

whether or not there is a trend in the time series data and for this purpose, we used the Mann-Kendal and 

Spearman rank correlation tests.  Specially we wish to test the hypothesis that,  

H0 : There is no trend  in the data set 

H1 : There exits trend  in the data set. 

The results of these tests appear in Table 2.   

Table 2. Mann-Kendal and Spearman test of trend 

Test Range 

(Years) 

Test statistic Value at 95% 

level 

± rm 

Trend Exist 

Mann-Kendal 

tm 

37 −0.935 0.338 Yes 

Spearman rank 

tS 

37 −2.146 0.834 Yes 

The 95% level of ± rm values will indicate that either there exist a trend or not. If the computed test 

statistics falls outside the band (– rm, + rm) would indicate that the trend exist in the data set.  From Table 

2 the value of Mann-Kendal statistic, tm is −0.935 and its rm value at 95% is 0.338. Therefore the test 

statistic value falls outside of 95% interval of rm indicating that the H0 is rejected. Therefore, there exists 

a trend in the data set of HYV Boro rice yields of Dinajpur. The Spearman rank statistic, tS and rm values 

are −2.146 and 0.834 respectively. Since the tS value also falls outside of the 95% level value indicating 

the existence of trend by Spearman rank test. 

The OLS results of the coefficients of the three fitted models together with their corresponding p values 

are presented in Table 3 to Table 5. 

Table 3.  OLS estimates of parameters for Linear Trend model 

Variable Co-efficient Standard  Error T Ratio P-Value 

t 0.019 0.004 4.458 0.000 

Constant 2.292 0.867 26.440 0.000 

Table 4.  OLS estimates of parameters for Compound Trend model 

Variable Co-efficient Standard Error T Ratio P-Value 

t 0.007 0.002 4.457 0.000 

Constant 0.831 0.033 25.140 0.000 

Table 5.  OLS estimates of parameters for Quadratic Trend model 

Variable Co-efficient Standard  Error T Ratio P-Value 

t −0.020 0.017 −1.221 0.231 

t
2 0.001 0.000 2.460 0.020 

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Constant 2.528 0.126 20.150 0.000 

The regression diagnostics R2, adjusted R2, Durbin-Watson statistic and Jarque-Bera (JB) statistic ( Jarque 

and Bera in 1981) with p values are reported in the Table 6 for the selected data set. 

Table 6.   R2, adjusted R2, JB and DW statistic for the trend models by OLS 

Model R
2

Adjusted R
2 JB DW 

Statistic P value 

Linear 0.383 0.364 3.882 0.144 1.373 

Quadratic 0.484  0.451 1.049 0.592 1.624    

Compound 0.383 0.364 2.074 0.355 1.403 

We observe from the results presented in Table 6 that the quadratic trend model possesses the highest 
2R

and adjusted 
2R . The Jarque-Bera clearly shows that the three trend models obey the normality 

assumption of errors. 

The presence of autocorrelation in a data set is measured by the DW statistics of the residuals is compared 

to respective critical values (see, the tables in Draper and Smith (1981)).  If the DW value is less than a 

lower limit then the null hypothesis, H0 of no serial correlation in the residuals can be rejected, but if DW 

exceeds an upper limit, the null hypothesis, H0 can’t be rejected.  But if the DW statistic is found to be 

in between these boundary levels it will be indeterminate.  If the DW statistic is found to be near to 2 

mean that the data are not affected by autocorrelation problem.   

With regards to the measurement of the OLS parameters of selected models the DW test revealed that in 

the three trend model, the H0 of non-existence of serial correlation in the respective residuals were 

accepted. So the data set has autocorrelation problems and autocorrelation may complicate the application 

of statistical tests by reducing the number of independent observations.  It can also affect the predictions 

because future values depend on current and past values.  Durbin-Watson (Durbin and Watson, 1950) test 

to detect the presence of autocorrelation which affects the efficiency of the estimators can be used in 

autocorrelated data set. To overcome the autocorrelation we employed the Cochrane and Orcutt method 

for three trend models according to our objectives of the study.   

The results of Cochrane and Orcutt method for Linear, Quadratic and Compound trend models are 

displayed in Table 7- Table 9. Finally the regression diagnostics and forecasted yield of HYV Boro rice 

for Dinajpur is presented in Table 10 and Table 11 respectively. 

Table 7.  Cochrane and Orcutt estimates by Linear Trend model 

Variable Co-efficient Standard  Error T Ratio P-Value 

t 0.020 0.006 3.526 0.001 

Constant 2.286 0.114 20.000 0.000 

Table 8.  Cochrane and Orcutt estimates by Compound Trend model 

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Variable Co-efficient Standard  Error T Ratio P-Value 

t 0.008 0.021 3.531 0.001 

Constant 0.830 0.043 19.270 0.000 

Table 9.  Cochrane and Orcutt estimates by Quadratic Trend model 

Variable Co-efficient Standard  Error T Ratio P-Value 

t -0.021 0.019 -1.080 0.289 

t
2 0.001 0.001 2.171 0.038 

Constant 2.528 0.145 17.390 0.000 

In Table 7, the estimated constant is 2.286 with standard error 0.114. The estimated coefficient for t is 

0.020 with standard error 0.006. The t-ratio of time t and constant (estimates are 3.526 and 20.000 

respectively) are significant for the Linear trend model. From Table 8, the estimated constant is 0.830 

with standard error 0.043. The estimated coefficient for time is 0.008 with standard error 0.021. The t-

ratio of time t and constant (estimates are 3.531 and 19.270 respectively) are significant for the Compound 

trend model.  In the Table 9, the estimated constant is 2.528 with standard error 0.145. The estimated 

coefficient for t and t2 are –0.021 and 0.001 respectively with standard errors 0.019 and 0.001. The t-

ratio of t, t2 and constant are 17.390, 2.171 and -1.080. These coefficients are significant. 

Table 10.  Regression diagnostics by Cochrane-Orcutt method 

Model R
2

Adjusted R
2 JB DW 

Statistic P value 

Linear 0.431 0.413 2.073 0.355 1.867    

Quadratic 0.499     0.467 1.776 0.411 1.935    

Compound 0.429    0.411 1.401 0.496 1.877    

Table 11 Method wise forecasted HYV Boro rice production of Dinajpur 

Year of 

Production 

Actual 

Production 

(yhdin) 

Forecasted by 

Linear Trend 

Model 

Forecasted by 

Compound  

Trend Model 

Forecasted by 

Quadratic 

Trend Model 

2004 3.39 2.965 2.963 3.156 

2005 3.52 2.985 2.985 3.215 

2006 3.75 3.005 3.008 3.276 

2007 3.81 3.025 3.031 3.339 

2008 - 3.044 3.054 3.405 

2009 - 3.064 3.077 3.473 

2010 - 3.084 3.100 3.543 

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American Journal of Agricultural Science, Engineering and Technology 

2011 - 3.104 3.123 3.616 

2012 - 3.124 3.147 3.691 

2013 - 3.144 3.171 3.768 

2014 - 3.164 3.195 3.848 

2015 - 3.184 3.219 3.930 

2016 - 3.204 3.243 4.014 

2017 - 3.224 3.268 4.100 

2018 - 3.244 3.293 4.189 

2019 - 3.264 3.318 4.280 

2020 - 3.284 3.343 4.373 

2021 - 3.304 3.368 4.469 

MSE  0.056  0.053  0.051 

AME  0.184  0.183  0.184 

It is observed from Table 10 that the 
2R and adjusted 

2R values for Linear trend model are 0.431 and 

0.413 respectively. The 
2R and adjusted 

2R values for Quadratic trend model are 0.499 and 0.467. For

Compound trend model the
2R and adjusted 

2R values are 0.429 and 0.411.  Therefore the quadratic 

trend model possesses the highest 
2R  and adjusted 

2R  values.  The Jarque-Bera clearly shows that the

three trend models obey the normality assumption of errors.  The DW statistics of the three models are 

close to 2 indicating that the data set is not affected by autocorrelation problem.  

4. Conclusion

In this research, the objective was to fit and forecast of HYV Bro rice yields for Dinajpur. It is evident 

from the statistical tests that the data is autocorrelated.  Hence the Cochrane-Orcutt method has been 

employed to fit such data set. Finally, we forecasted the yields of the three trend models for HYV Boro 

rice yields of Dinajpur up to 2021. On basis of the regression diagnostics, the quadratic trend model is an 

appropriate model for this data set and in this model would be useful for the decision makers for their 

agriculture and food policy formulation. 

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