Open Journal of Statistics, 2012, 2, 460-468
http://dx.doi.org/10.4236/ojs.2012.24058 Published Online October 2012 (http://www.SciRP.org/journal/ojs)
An Application of Non-Linear Cobb-Douglas Production
Function to Selected Manufacturing Industries in
Bangladesh
Md. Moyazzem Hossain1, Ajit Kumar Majumder2, Tapati Basak2
1Department of Statistics, Islamic University, Kushtia, Bangladesh
2Department of Statistics, Jahangirnagar University, Dhaka, Bangladesh
Received May 14, 2012; revised June 20, 2012; accepted July 2, 2012
ABSTRACT
Recently, businessmen as well as industrialists are very much concerned about the theory of firm in order to make cor-
rect decisions regarding what items, how much and how to produce them. All these decisions are directly related with
the cost considerations and market situ ations where the firm is to be operated. In this rega rd, this paper should be help-
ful in suggesting the most suitable functional form of production process for the major manufacturing industries in
Bangladesh. This paper considers Cobb-Douglas (C-D) production func tion with additive error and multiplicative erro r
term. The main purpose of this paper is to select the appropriate Cobb-Douglas production model for measuring the
production process of some selected manufacturing industries in Bangladesh. We use different model selection criteria
to compare the Cobb-Douglas production function with additive error term to Cobb-Douglas production function with
multiplicative error term. Finally, we estimate the parameters of the production function by using optimization
subroutine.
Keywords: Cobb-Douglas Production Function; Newton-Raphson Method; Manufacturing Industry; Bangladesh
1. Introduction
A developing country like Bangladesh which is facing
enormous problems so far as industrialization policy is
concerned does not follow the po licy of Marxian Economy,
neither thus it strictly follow the policy of a Capitalist
country. The economy of Bangladesh actually turned out
to be a mixed economy since a long time. The industry
sector was severely damaged during the war of liberation
in 1971. After a series of adjustments and temporary
changes in state policy, the government finally adopted a
new industrial policy in 1982. Every industrialist tries to
produce goods with maximum profit but with minimum
cost. In order to do this, it has to be decided what to
produce, how much to produce and how to produce. The
industries need various inputs such as labor, raw material,
machines etc. to produces goods. An i ndust ry ’s production
cost depends on the quan tities of inputs it buy and on the
prices of each input. Thus an industry needs to select the
optimal combination of inputs, that is, the combination
that enables it to produce th e desired level of output with
minimum cost and hen ce with maximum profitab ility. So
the main object ive of this pap er is to selec t the a p p r o p r i a t e
Cobb-Douglas production function. We use differ en t mod el
selection criteria to compare the Cobb-Douglas producti on
function with additive error term to Cobb-Douglas pro-
duction function with multiplicative error term. Finally,
we estimate the parameters of the production function by
using optimization subroutine .
2. Cobb-Douglas Production Function
The Cobb-Douglas production function is the widely
used function in Econometrics. A famous case is the wel l-
known Cobb-Douglas production function introduced by
Charles W. Cobb and Paul H. Douglas, although antici-
pated by Knut Wicksell and, some have argued, J. H.
Von Thünen [1]. They have estimated it after studying
different industries in the world, for this it is used as a
fairly universal law of production.
The Cobb-Douglas production function with multipli-
cative error term can be represented as,
3 2
1tttt
pLKu
 
 
t
pL
(2.1)
where, is the output at time t; t is the Labor
input; t
K
is the Capital input; 1

is a constant; t is
the random error term. u
2

and 3

are positive pa-
rameters.
C
opyright © 2012 SciRes. OJS

MD. M. HOSSAIN ET AL. 461
The Cobb-Douglas production function with additive
error term can be represented as,
3 2
tttt
pLKu
 

t
p L
1
 (2.2)
where, is the output at time t; t is the Labor
input; t
K
is the Capital input; 1

is a constant; t is
the random error term. u
2

and 3

are positive pa-
rameters.
3. Literature Review
Production functions for the industrial sector as a whole
as well as for seven important industries in India are
worked out based on cross-section data relating to indi-
vidual firms for the two years 1951 and 1952 (V. N.
Murti and V. K. Sastry) [2]. De-Min Wu [3], derives the
exact distribution of the indirect least squares estimator
of the coefficients of the Cobb-Douglas production func-
tion within the context of a stochastic production model
of Marschak-Andrews type. The stochastic term in Cobb-
Douglas type models is either specified to be additive or
multiplicative (Stephen M. Goldfeld and Richard E.
Quandt) [4]. They developed a model in which a Cobb-
Douglas type function is coupled with simultaneous
multiplicative and additive errors. This specification is a
natural generalization of the “pure” models in which either
additive or multiplicative stochastic terms are intro-
duced. A Cobb-Douglas type function with both multi-
plicative and additive errors has been proposed by Gold-
feld and Quandt [5]. They suggested a maximum likely-
hood approach to the estimation of a Cobb-Douglas type
model when the model includes both multiplicative and
additive disturbance terms. As expected, an analytical
expression for the solution to the maximization problem
did not exist. Indeed, because of the complexity of the
likelihood function, their maximization algorithm had to
be used in conjunction with a numerical integration tech-
nique.
Kelejian [6] generalize and simplify the work by
Goldfeld and Quandt [5]. Specifically, an estimation
technique is suggested which does not require the speci-
fication of the disturbance terms beyond their means and
variances, which does not require the compounding of a
maximization algorithm with a numerical integration
technique, but yet leads to asymptotically efficient esti-
mates of the parameters of the regression function. In
addition, the procedure readily lends itself to interpreta-
tion. For instance, it will become evident that if the dis-
tribution of the multiplicative disturbance term is not
known, the scale parameter of the model (unlike the
other parameters) will not be identified. The origins of
the Cobb-Douglas form date back to the seminal work of
Cobb and Douglas [1], who used data for the US manu-
facturing sector for 1899-1922 (although, as Brown [7]),
Sandelin [8], and Samuelson [9], indicate, Wicksell
should have taken the credit for its “discovery”, for he
had been working with this form in the 19th century).
Cheema [10], Kemal [11], and Wizarat [12], showed
the performance of large-scale manufacturing sector of
Pakistan. Moreover tax holidays also encourage invest-
ment in the industrial sector. For example, Azhar and
Sharif [13], and Bond [14], empirically proved the posi-
tive correlation between tax holidays and industrial out-
put. But it is also worth to note that tax rate and output of
manufacturing sector are inversely related. The authors
used the data of West Virginia State of United States.
However, Radhu [15], showed the positive correlation
between indirect tax rates an d prices of the commodities.
Vijay K. Bhasin and Vijay K. Seth [16], estimate produc-
tion functions for Indian manufacturing industries and to
find whether plausible and meaningful estimates can be
obtained for returns to scale, substitution, distribution,
and efficiency parameters. Some studies are based on
data collected through surveys specially designed for
estimate the levels of technical efficiency (TE) (e.g., Lit-
tle et al., [17], Page [18]). Many of the studies are con-
cerned with estimating and explaining variations in TE
only in Small-Scale Industrial Units by fitting either a
deterministic or a stochastic production frontier (e.g.,
Bhavani [19], Goldar [20], Neogi and Ghosh [21], Ni-
kaido [22]). A review of other studies in this area may be
found in Goldar [23].
All these studies, however, use data relating to years
prior to the economic reforms. For instance, Bhavani [19]
uses data collected under the first Census of small scale
industrial units in 197 3 to estimate the TE of firms at the
4-digit level indu stries of metal product groups by fitting
a deterministic translog production frontier with three
inputs-capital, labor and materials- and observes a very
level of average efficiency across the four groups. Simi-
larly, on the basis of the data made available by the Sec-
ond All India Census of Small Scale Industrial Units in
1987-1988, Nikaido [22] fits a single stochastic produc-
tion frontier, considering firms under all the 2-digit in-
dustry groups and using intercept dummies to distinguish
different industry groups. He finds little variation in TEs
across industry groups and a high level of average TE in
each industry groups. Neogi and Ghosh [21] examine the
inetrtemporal movement of TE using panel industry-level
summary data for the year s 1974-1975 to 1987-1988 and
observe TEs to be falling over time.
In recent years, there has been an increasing interest in
the examination of productivity from different parts of
the economy such as industry, agriculture, and services.
Numerous studies have attempted to explain produ ctivity
in the economic sector, for example, productivity growth
in Swedish manufacturing (Carlsson [24]), the impact of
regional investment incentives on employment and pro-

MD. M. HOSSAIN ET AL. 462
ductivity in Canada (M. Daly, Gorman, Lenjosek,
MacNevin, & Phiriyapreunt [25]), productivity and im-
perfect competition in Italian firms (Contini, Revelli, &
Cuneo [26]), explaining total factor productivity differ-
entials in urban manufacturing of US (Mullen & Wil-
liams, [27]). Total factor productivity growth in manu-
facturing has been examined by applied parametric and
non-parametric approaches. In most of the studies have
used non-parametric approach, wherein total factor pro-
ductivity growth has decomposed into efficiency change
and technological change. Efficiency change measures
“catching-up” to the isoquant while technological change
measures shifts in the isoquant. For example, see Weber
and Domazlicky [28]; Nemoto and Go to [29]; (Maniada-
kis and Thanassoulis [30] and Radam [31].
The studies by Golder et al. [32], Lall and Rodrigo
[33], and Mukherjee and Ray [34], however, relate to
the post-reform era. Using panel data for 63 firms in the
engineering industry from 1990-1991 to 1999-2000
drawn from the Prowess database (version 2001) of the
Centre for Monitoring Indian Economy, Goldar et al.
[32] fit a translog stochastic production frontier to esti-
mate firm-level TE scores in each year. They find the
mean TE of foreign firms to be higher than that of do-
mestically owned firms but do not find any statistically
significant variation in mean TE across public and pri-
vate sector firms among the latter group. They can at-
tempt to explain variation in TEs in terms of economic
variables, including export and import intensity and the
degree of vertical nitration. Lall and Rodrigo [33] ex-
amine TE variation across four industrial sectors in In-
dia during 1994 and consider TE in relation to scale,
location extent of infrastructure investment and other
determinants.
Md. Zakir Hossain, M. Ishaq Bhatti, Md. Zulficar Ali,
[35], reviews some models recently used in the literature
and selects the most suitable one for measuring the pro-
duction process of 21 major manufacturing industries in
Bangladesh. In particular, they estimates and tests the
coefficients of the production inputs for each of the se-
lected manufacturing indu stries using Banglad esh Bureau
of Statistics annual data over the period 1982-1983
through 1991-1992. Cheng-Ping Lin [36] analyzes the
cost function of construction firms with due considera-
tion of their available resources by using Cobb-Douglas
Production and Cost Func tions. Moosup Jung, et al. [37]
made a study on Total Factor Productivity of Korean
Firms and Catching up with the Japanese Firms. They
measured and compared the TFP of both Korean and
Japanese listed firms of 1984 to 2004. They found that
the average TFP of Korean firms grew about 44.1% be-
tween 1984 and 2005, with 2.1% annual growth rates.
Industry was observed to b e outstanding.
Danish A. Hashim [38] made research on “Cost and
Productivity in Indian Textiles” for Indian Council for
Research on International Economic Relations. His ob-
servations and findings are: there is an inverse relation-
ship between the unit cost and productivity: Industry and
States, which witnessed higher productivity (growth)
experienced lower unit cost (growth) and vice-versa.
Better capacity utilization, reductio ns in Nominal Rate of
Protection and increased availability of electricity are
found to be favorably affectin g the productivity in all the
three industries. M. Z. Hossain and K. S. Al-Amri, [39]
find that for most of the selected industries the C-D func-
tion fits the data very well in terms of labor and capital
elasticity, return to scale measurements, standard errors,
economy of the industries, high value of R2 and rea-
sonably good Durbin-Watson statistics. The estimated
results suggest that the manufacturing industries of Oman
generally seem to indicate the case of increasing return to
scale. Of the nine industries, seven exhibit increasing
return to scale and only the rest two show decreasing
return to scale. They also find that no industry with con-
stant return to scale.
4. Estimation Procedure
Equation (2.1) is nearly always treated as a linear rela-
tionship by making a logarithmic transformation, which
yields:
12 3
logloglogloglog
tttt
pLKu



 
logu
1
log
(3.1)
where, is treated as an additive random error with
a zero mean. In this form the function is a single equation
which is linear in the unknown parameters:

, 2

and 3.

In the case of Equation (2.2), the minimization of,
2
u

is no longer a simple linear estimation problem.
To estimate the production function we need to know
different types of non-linear estimation. In non-linear
model it is not possible to give a closed form expression
for the estimates as a function of the sample values, i.e.,
the likelihood function or sum of squares cannot be
transformed so that the normal equations are linear. The
idea of using estimates that minimize the sum squared
errors is a data-analytic idea, not a statistical idea; it
does not depend on the statistical properties of the
observations. Newton-Raphson method is one of the
method which are used to estimate the parameters in
non-linear system.
4.1. Newton-Raphson Method
Newton-Raphson is one of the popular Gradient methods
of estimation. In Newton-Raphson method we find the
values of
j

that maximize a twice differentiable con-
cave function, the objective function

g

. In this me-

MD. M. HOSSAIN ET AL. 463

g


t
thod we approximate

at

by Taylor series
expansion up to the qu adratic terms



 

1
2
tt
gg

t
tt t



 

G
H
where,

t
i
g
t

 G






is the gradient vector and

2
t
ik
g
t


H







g
is the Hessian matrix. This Hes-
sian matrix is positive definite, the maximum of the ap-
proximation

occurs when its derivative is zero.
 
tt
tt

 
GH
 
 
1
0
t
t

 




HG
1t
(3.11)
This gives us a way to compute


 
1
t t
, the next value
in iterations is,
1tt







HG
The iteration procedures continue until convergence is
achieved. Near the maximum the rate of convergence is
quadratic as define by
2
ˆˆ
tt
ii
c
1
ii

  
0

for some when ct
i

is near i
ˆ

for all . Thus
we get estimates i
ˆi

by Newton-Raphson methods.
For the model (2.2), to estimate the parameters we
minimize the following error sum squares

1
n
t
Sp

 
3 2
2
1ttt
LK
 
 
In case of nonlinear estimation we use the score vector
and Hessian matrix. The elements of score vector are
given below:


1
1
1
2* n
tt
t
SpLK 


 


33 22
*
t tt
LK
 




 
3 2
2
1
1
2** ln*
n
ttt t
t
S
pLK L
 







 

 
3 2
1
tt
LK
 








3 2
3
1
1
2** ln*
n
ttt t
t
S
pLK K
 











3 2
1
tt
LK
 



Also the elements of Hessian matrix are given below:

3 2
2 2
21
1
2* n
tt
t
SLK
 







 





 
33 22
33 22
2
1
1
12
1
2*ln **
*ln*
n
ttttt
t
tttt tt
SLLKLK
pLK LLK
 
 








 




 


 


33 22
33 22
2
1
1
13
1
2* ln**
*ln *
n
ttttt
t
tttt tt
SKLKLK
pLK KLK
 
 








 



 



 


3 2
33 22
22
1
21
2
2
11
2*ln *
*ln *
n
ttt
t
tttt tt
SLLK
pLKL LK
 








 







 

 



   
 
33 22
33 22
2
23
11
1
11
2*ln** ln*
*ln*ln *
n
tttttt
t
ttttt tt
S
KLK LLK
pLKKLLK
 
 










 



 



 


3 2
33 22
22
1
21
3
2
11
2* ln*
*ln *
n
ttt
t
tttt tt
SKLK
pLKK LK
 
 













 
Hence the Score vector is
 
123
,,
SSS












G


and Hessian matrix is
 

 
 
222
212 13
1
222
2
12 23 2
222
2
13 233
SSS
SSS
SSS




 


 









 




 

  








 


H
model. Some of these criteria are discuss below.
4.2. Model Selection Criteria
To find the appropriate production function we use
model selection criterion. The model that minimizes the
criterion is the best model. The recent years several criteria
for choosing among models have proposed. These entire
take the form of residual sum of squares (ESS) multiplied
by a penalty factor that depend on the complexity of the

MD. M. HOSSAIN ET AL. 464
4.2.1. Finite Prediction Error (FPE)
Akaike (1970) developed Finite Prediction Error proce-
istic of this proce- dure, which is known as FPE. The stat
dure can be represente d as,
FPE ESS TK
TT



K



whe of observations and re, T is the number
K
is the
number of estimated parameter (Ranathan [40]).
Akaike (1974) also developed another procedure which
The form of
am
4.2.2. Akaike Inf orma ti on Cri teria (AI C )
is known as Akaike Information Criteria.
this statistic is given below,
2
AIC e
K
T
T


n me variables are
droppe d ( R amanathan [40]).
Q) Criterion
Hunnan and Quinn (1979) developed a procedure which
is procedure
ESS 



The value of AIC deceases wheso
4.2.3. Hunnan and Quinn (H
is known as HQ criteria. The statistic of th
can be represented as


2
HQ kT
ESS

lnT
T


The value of HQ will decrease provided there are at
observations (Ramanathan [40]).
Craven and Wahba (1978) developed a procedure which

C criteria. The form of
least 16
4.2.4. SCHWARZ Criterion
is known as

SCHWARZ BI
this procedure is represented as
SCHWARZ
K
T
E S



ST
T
also decrease provided
there are at least 8 observations (Ramanathan [40]).
Craven and Wahba (1981) developed a procedure which
teria. The form of this pro-
The value of SCHWARZ will
4.2.5. SH IBATA Criterion
is known as SHIBATA cri
cedure is represented as
2
SHIBATA ESS TK

TT

 .
When some vari ables dropped SHIBATA will increase
(Ramanathan [40]
ross Validatio n (GC V )
Generalized Cross Validation (GCV) is another proce-
(1979).
).
4.2.6. Generali zed C
dure which is developed by Craven and Wahba
The form of the statistic is given below
2
GCV 1
ESS K
TT




 


If one or more variables are dropped ten GCV will
decrease (Ramanathan [40]).
The model selection criteria Rice developed by Craven
e form of this criterion can be rep-
h
4.2.7. Rice Criterion
and Wahba (1984). Th
resented as
1
2ESSK  
RICE 1
TT


 


.
(Ramanathan [40]).
iterion
The form of this criterion can be represented as
4.2.8. SGMASQ Cr
1
SGMASQ 1
ESSK





.
TT
 

If SGMASQ decreases (that is 2 R incses) when
one or more variable dropped, then GCV and RICE
will also decreas).
stries of
Bangladesh for This Study
ang-
ladladesh Census of
Leather Products;
r;
s (Wooden);
;
cts;
Industries;
s;
6.
In case of Cobb-Douglas production function with multi-
rea
es (Ramanathan [40]
5. Selected Manufacturing Indu
In recent publications of “Statistical Yearbook of B
esh [41]” and “Report on Bang
Manufacturing Industries (CMI) [42]” published by BBS,
we get the p ublished secondary d ata for the majo r manu-
facturing industries of Bangladesh over the period 1978-
2002. We have chosen the following manufacturing in-
dustries for the ongoing analysis.
1) Manufacturing of Textile;
2) Manufacturing of Leather &
3) Manufacturing of Leather Footwea
4) Manufacturing of Wood & Cork Products;
5) Manufacturing of Furniture & Fixture
6) Manufacturing of Paper & Paper Products;
7) Manufacturing of Printing & Publications;
8) Manufacturing of Drugs & Pharmaceuticals
9) Manufacturing of Industrial Chemical;
10) Manufacturing of Plastic Products;
11) Manufacturing of Glass & Glass Produ
12) Manufacturing of Iron & Steel Basic
13) Manufacturing of Fabricated Metal Product
14) Manufacturing of Tra ns p ort Eq uipment;
15) Manufacturing of Beverage;
16) Manufacturing of To bacc o.
Results and Discussion

MD. M. HOSSAIN ET AL. 465
Con
HQ WA
plicative error terms i.e., for intrinsically linear model
and additive errors i.e., for intrinsically nonlinear model,
we get the following estimates by using different model
selection criteria discussed in Section 4.2.
From Table 1, we observe that, the Cobb-Douglas
production function with additive error (2.2) performs
bertter fo the selected manufacturing industries based on
the data under study period. Thus the strictly nonlinear
models (which are nonlinear with additive error terms)
seem to be better than intrinsically linear model (which
are nonlinear with multiplicative error terms).
Now we estimate the parameters of the Cobb-Douglas
production function with additive errors by using op-
timization subroutine. The estimates are given in Table
2. There are economies of scale in the manufacturing of
Drugs & pharmaceuticals, Furniture & fixtures (wood-
en), Iron & steel basic, Leather footwear, Fabricated
metal products, Plastic products, Printing & publica-
tions, Tobacco since 1

 for these industries. There
are diseconomies of scale in the Beverage, Chemical,
Glass & glass products, Leather & leather products,
Paper & paper products, Textile, Wood & crock prod-
ucts industries, Transport equipment since 1

 for
these indus tries.
Table 1. Values of different model selection criter two
models under studia of y.
FPE AIC
Name of
the industry Model (2.1) Model (2.2) Model (2.1) Model (2.2)
Beverage 1360488 13 1358701 32 21372134
Chemical
1 1
Leathear 9 1 1
Leather
Pr
8 6 6
1 1
T
1614804 957644 1612683 956386
Drugs 58414018 3029017 58337287 3025038
Furniture 4256943 193762 4251351 193507
Glass 16250 9608 16228 9595
Iron 20604099 892909420577034 8904229
er footw9730649672619959964964676
products1589512 1371610 1587424 1369808
Fabricated metal 1192242 885584 1190676 884420
Paper 1980514 1613608 1977913 1611488
Plastic 121323 75752 121164 75653
inting 1065074 409231 1063675 408694
Textile 7418640991446487303810 9822627
Tobacco 24044986 114384024013401 1129201
ransport 22949982 19106587 22919836 19081489
Wood 69177 30634 69086 30593
tinued
SCHRZ
Name Model
(2.1) Model (2.2) Model (2.1) Model (2.2)
of
the industry
Beverage 141283236 1579422194260 2472
Chemical 11
1 2
r footwear 12 12
er products
g
9 718
1 1
t
V
6376939 9448987 68531180811
Drugs 606614693145557 675925793504964
Furniture 4420727201217 4925834 224208
Glass 16875 9977 18803 11117
Iron 213968329657382238416091903411
Leathe0356773 0429501540126276375
Leath16506681424382 183 9271 1587130
Fabricated metal 1238113919656 1379578 1024735
Paper 20567141675691 2291712 1867153
Plastic 12599178667 140386 87655
Printin1106052424976 1232428 473534
Textile 07820302604389 011546810900084
Tobacco 249701051572593278231602894865
Transpor2383297219841704 2655610022108794
Wood 71838 31812 80047 35447
SHIBATA GC
Name of
the industry Model
(2.1) Model (2.2) Model (2.1) Model (2.2)
Beverage 132277 1382083 0569620772171
Chemical 11
1 1
r footwear 91 11
er products
g
8 687
1 1
t
5869949 3104366 4043972844
Drugs 567914062944877 593412253077096
Furniture 4138695188380 4324514 196838
Glass 15798 9341 16508 9760
Iron 200317638403286209311489229556
Leathe696034 912614 0131367998487
Leath15453591333510 161 4743 1393382
Fabricated metal 1159124860984 1211166 899641
Paper 19255001568786 2011951 1639221
Plastic 11795373648 123249 76955
Printin1035488397864 1081980 415727
Textile 49903457972396 88062381024218
Tobacco 233770690834288244266521320726
Transpor2231248318575849 2331426819409866
Wood 67255 29783 70275 31120

MD. M. HOSSAIN ET AL. 466
E ASQ
Continued
RICSGM
Name o
7. Hypothesis Testing
To investigate the model that is the model is well fitted
or not, we have consider the following the null hypothe-
sis, 0
f Model
(2.1) Model (2.2) Model (2.1) Model (2.2)
the industry :0H

Beverage 1410876 29 1209967 2216322 189
Chemical 1
6057 2692
1682
ar 1 2 8 1
er
1236399 918383 1059771 787186
521 424388 946732 363761
9065 6214
2493 9905
t
674611993112 1435381851239
Drugs 7500 3141203 51923572 459
Furniture 4414
Glass
608
16851 9964 1444
200938 3783949
4
172233
8540
Iron
Leather
21367214 19630171 18314755 5861
footwe 0342437 040122864946748676
Leath
products
Fabricated
1648383 1422411 1412900 1219209
metal
Paper 2053867 1673371 1760457 1434318
Plastic
Printing 1104
125817 78558 107843 67335
Textile 6368 72503889 77705458 6190
Tobacco 5541 11556574 21373321 635
Transpor23799982 19814239 20399984 16983633
Wood 71739 31768 61491 27230

, i.e., the model is not fitted well, against
the alternative hypothesis, 0:0H

, i.e., the model is
fitted well, where

is the vector of parameters, i.e.,

123




for the model (2.2).
Under the null hypothesis, the test statistic is,


2
2
1
1
Rk
FRnk




kn
0
where, is the number of parameter and is the
number of obse rvations.

We reject
H
, if



2
0.05,1 ,
2
1
1knk
Rk
FF
Rnk





2
R
0:0H
,
which implies that model is fitted well.
The analytical results of the hypothesis testing are
presented in Table 3.
From Table 3, we observe that is highly signifi-
cant for all the manufacturing industries, we can say that
the intrinsically nonlinear model (2.2) is fitted well ac-
cording to the null hypothesis


he ef gtwith additive error term (intrinsically nonlinear) of the indus-
der study
.
Table 2. Tstimates oCobb-Doulas producion function
tries un.
Industry name Intercept p-valueCapital elasticity (2

) p-value Labor elasticity (3

) p-value Return to scale (23

 )
23
1





Beverge 5.51 0.06500.683362 0.00010.230199 0.04580.913561 1.094618 a8489
Chemical 6.552999 0.07200.567255 0.00010.239483 0.01130.806738 1.23956
r footwear
e
g 47
t
Drugs 1.418816 0.01070.578490 0.03500.583740 0.03211.16223 0.860415
Furniture 0.136145 0.37511.583382 0.00010.323816 0.00011.907198 0.524329
Glass 10.8587850.00480.446118 0.00010.267905 0.07920.7140231.400515
Iron 5.432328 0.57020.317029 0.06410.825566 0.02651.142595 0.875201
Leathe9.975966 0.00010.168618 0.12960.851867 0.00011.020485 0.979926
Leathr pr o d uc ts 149.5248 0.00080.273520 0.02930.396121 0.01850.669641 1.4933 37
Fabricated metal 1.560328 0.13570.282128 0.07530.979802 0.00011.26193 0.792437
Paper 36.90303 0.46610.154744 0.50830.593256 0.00020.748 1.336898
Plastic 10.04537 0.00010.081875 0.55270.962046 0.00011.0439210.957927
Printin0.761334 0.00901.062223 0.00010.215724 0.00071.27790.782505
Textile 33.44288 0.29790.503446 0.00010.237309 0.08740.740755 1.349974
Tobacco 5.828218 0.13610.867991 0.00010.257396 0.00011.125387 0.888583
Transpor35.21922 0.46050.037898 0.79960.873132 0.01240.91103 1.097659
Wood 45.73787 0.07890.054236 0.67310.566334 0.00010.62057 1.611422

MD. M. HOSSAIN ET AL. 467
Table 3. The values of test statistic of intrinsically nonlinear
model for selected manufacturing industries.
ndustry F Name of I R2
Beverage 350.3247 0.9709
Chemical 0.9733 382.7584
Drugs 0.9956 2375.864
13 .9973
footwear
ted metal
o
rt
Furniture 0.98550
Glass 0.9545 220.2692
Iron 0.752 31.83871
Leather0.9885 902.5435
Leather products0.9642 2 82.7961
Fabrica0.954 217.7609
Paper0.795640.86986
Plastic 0.9155 113.7604
Printing 0.991 1156.167
Textile 0.958 239.5
Tobacc0.9711 352.8218
Transpo0.7434 30.41972
Wood 0.931 141.6739
In order to forecast the production of manufacturing
industries, we use the product ion f unctio n (Table 4).
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