The Malmquist Index Regression Equations
as a Dynamic System
JINGHAI ZHENG1
Department of International Economics
Norwegian Institute of International Affairs, Norway
Paper prepared for the BITE project
Preliminary, January 2009

Abstract
This article demonstrates that the popular DEA-based Malmquist productivity
indexes when used in regression analysis the set of linear equations involved
can be treated as a system. With reference to the special structure of theknowledge production function, the regression equations can be further
specified as a dynamic system. The properties of the Malmquist Index
regression equations provide rich microstructures for the relationship between
productivity growth, productivity growth components, and the determinants ofproductivity growth. Knowledge of these structures will be very helpful inapplied work. A feasible scheme for linearizing the system is proposed to help
interpret empirical results and facilitate efficient econometric estimations.

JEL classification: L10; L96; O30

1Jinghai Zheng is also Associate Professor at the Department of Economics, Gothenburg University,
Sweden, and Guest Research Fellow at the Center for China Studies, Tsinghua University, Beijing,
China.

1

2

1. Introduction
Simar and Wilson (2007) note that internet search engines in 2004 revealed
more than 800 hits mostly associated with working papers using the two-stage
procedure for analysis on determinants of DEA scores, in addition to the morethan 40 published papers included in their survey of the literature. The two-stepprocedure in fact has been well received in the literature on both theoretical andstatistical grounds. Ray (1991) gave two reasons ― one theoretical and one
practical ― for using the procedure: First, researchers can treat the nonproduction factors as fixed and “indivisible non-discretionary” inputs and do notneed the free disposability assumption with respect to these inputs; Second, it
gives researchers substantial flexibility to choose non-production inputs inregression analyses. Recently, Chilingerian and Sherman (2004), Ray (2004),
and Ruggiero (2004), and Da Silva e Souza (2007) all indorsed the two-stepprocedure in the analysis of the technical efficiency and productivity
determinants.
The two-step procedure also possesses some favorable statistical properties.
Banker (1993) discusses the statistical basis for the DEA efficient scores anddemonstrates that DEA estimators of best practice are maximum likelihood
estimators under rather general assumptions and exhibit the desirable asymptotic
property of consistency. Banker and Natarajan (2008) further show that the twostep procedure consisting of DEA followed by ordinary least squares (OLS)
regression analysis yields consistent estimators of the impact of contextual
variables. In other words, the two-step procedure can be seen as a stochastic
production frontier with both one-sided inefficiency deviations as well as twosided random noise. Their study thus provides a formal statistical basis forregressing DEA efficiency scores and Malmquist productivity indexes on
explanatory variables.
For analysis using small samples, advances in the bootstrapping DEA scores(Simar and Wilson 1998, 2000a, and 2000b) and Malmquist indexes (Simar andWilson 1999) have further made the non-parametric estimation of technicalefficiency and productivity growth more attractive among researchers. As to thetwo-step procedure, Simar and Wilson (2007) have resolved a key problem withthe second step regression analysis. It has been known in the literature that DEA
efficiency scores are dependent on each other in the statistical sense (Xue andHarker 1999). The reason for the dependency problem is that the DEA
efficiency score is a relative efficiency index instead of an absolute efficiency
measure. Similar concern can be raised about the dependency of the efficiencychange and technical progress indexes with the Malmquist productivity index
3decomposition since these indexes are calculated on the basis of the DEA
efficiency scores. Simar and Wilson (2007) took care of the problem by using adouble bootstrap technique.
While studies using the two step procedure for analyzing DEA efficiency
scores have been plenty such as those found in Destefanis and Sena (2007),2
Madden and Savage (2001), and Zheng, Liu, and Bigsten (1998), there havebeen much fewer applications in which the methodology was applied to theMalmquist productivity index and its components. Examples using Malmquist
indexes as explanatory variables in regression analysis include, Castellacci andZheng (2008), Zheng, Liu, and Bigsten (2003), and Dismuke and Sena (1999).3
Since the early 1980s economists have started to decompose total factor
productivity growth into technical progress and technical efficiency change
components (Nishimizu and Page 1982). In recent years the DEA-based
Malmquist index model of productivity decomposition (Färe et al 1994) appears
to have become standard in the investigation on productivity performance offirms and regions. A major advantage of this approach is that it allows
decomposing the change in total factor productivity (TFP) into technicalprogress and technical efficiency change; loosely speaking, a shift in the bestpractice production frontier is an indication of technical change, and technicalefficiency change is associated with learning by doing, improved managerial
practices, and change in the efficiency when using an existing technology(Felipe, 1999). Once TFP growth is decomposed into efficiency change andtechnical progress, we can identify the explanatory variables affecting a specificcomponent. By separating the best practice firms from inefficient ones, we canevaluate the performance of the firms with different characteristics.
Combining the Malmquist index estimates with regression analyses forms a
multi-equation system for examination of the links between TFP indexes (inboth level and growth rate) and their determinants in a two-step procedure (forexample in Zheng, Liu, and Bigsten 2003). The advantages of such regressions
are obvious since multiple equations would allow researchers to identify thechannels through which a specific factor affects TFP growth if different factors
are behind different productivity growth components. Supplemented withtechnical efficiency (level) index and a binary variable representing best practice2

Destefanis and Sena (2007) used DEA technical efficiency scores and a binary variable to investigate
their determinants with the OLS regression and estimation of a Logit model.
3
Dismuke and Sena (1999) regressed Malmquist productivity indexes on a set of explanatory
variables using SURE estimation methods without explicit knowledge about the structure of such asystem.

4

firms, the relationships between the regression equations can become even more
complicated. Therefore, the decomposition of TFP and the possibility ofinteractions between TFP level and growth rate, and between technical progress
and technical efficiency change components present both challenges andopportunities for empirical analyses.
Although the DEA-based Malmquist indexes have become increasingly
popular in empirical works, so far it is not clear how regression analysis should
be carried out if all the indexes are used simultaneously regressing on a set of
explanatory variables. In this study we demonstrate that given the structure of
the Malmquist index model, the popular DEA-based Malmquist indexes whenused in regression analysis the set of linear equations involved can be treated asa system. With reference to the particular structure of the knowledge production
function, the regression equations can be further specified as a dynamic system.
The properties of the Malmquist Index regression equations provide richmicrostructures for the relationship between productivity growth, productivity
growth components, and the determinants of productivity growth. Knowledge ofthese structures will be very helpful in applied work. A feasible scheme forlinearizing the system is proposed to help interpret empirical results andfacilitate efficient econometric estimations.
The order of our presentation is as follows. Section 2 introduces the basic
assumptions concerning the structure of the knowledge production function thatis consistent with the Malmquist indexes as measures of knowledge flows.
Conditions required of the Malmquist indexes to be reated as measures of theknowledge flows are presented in Section 3. Taking account of the special
structure for the knowledge production function, in Section 4 we demonstratethat the Malmquist index regression equations can be regarded a dynamic
system with important theoretical implications and useful properties for
empirical analysis. Section 5 provides an empirical example using Norwegianfirm data and we summarize in Section 6.
2. Basic Assumptions
Investigation on the determinants of productivity is to endogenize total factor
productivity (TFP) either in level or growth rate or both with respect to factorsthat may impact productivity performance. In endogenous growth literature thisrelationship is often referred to as the knowledge production function as in
Romer (1990). However, specifications involved so far in the literature are often
simplistic and restrictive as they often take the form of Cobb-Douglas. Several

5

examples can be found in Jones (1999). Since any specification for the
relationship between TFP and its determinants can be seen as a knowledge
production function, we would like to see what major assumptions are involved
at the basic level.
Assuming that the production unit is the firm, its production function with yas output and x as inputs takes the standard Hicks-neutral form as follows:y = AF (x)(2.1)
and more specifically with reference to the growth literature,y = AF(K, LY)(2.1B)
where Y stands for the ordinary output, K and LY refer to capital and labor, and A
is the level of technology or useful knowledge stock that can be employed in the
production process. Although the specification was used in the macro context, it
is equally valid for firm level analysis since what is interested here is thefunctional form or the microstructure of the knowledge production function.
Following Romer (1990), the knowledge production is measured as the changeiin knowledge stock A , and the basic knowledge production function can be
expressed as follows.4
i

A = f ( x)(2.2)

or more specifically as in Phelps (1966),i

A = f ( K A , LA , L, A)(2.2B)
where KA and LA stand for technology capital or research capital stock andresearch labor input, and L = LA + LY is the labor force of the firm. Generally weassume the knowledge production function with properties as follows.
A1.
A2.

The knowledge production function is monotone in inputs
The input set of the knowledge production is convex

A3.
A4.
A5.

Researcher is the essential input, i.e., A = f ( K A , 0, L, A) = 0
The input set is nonempty and closed
The knowledge production function is finite, nonnegative, realvalued, and single valued for all nonnegative and finite inputs

iand note that the knowledge production function f(•) is related to the“technology function” of Phelps (1966) through the relationship as follows:tA(t ) = ∫ f ( E ( LA (v), L(v), K A (v)), A(v − ω ))dv
(2.3)
−∞
4

Phelps (1966) uses the term “technology function” in referring to the relationship between research
inputs and research outcomes; Gumolka (1970) mentioned “the production function of innovations”;
and Jones (1999, 2002, and 2006) prefers to phrase the relationship as “the idea production function.”

6Moreover, Phelps (1966) suggested the dynamic properties required of a valid“technology function”, i.e., the knowledge production function in stock form,
A(t) as follows:
B1.
B2.
B3.

B4.

Diminishing returns
Diminishing marginal rate of substitution
The marginal effectiveness of current research is an increasing
function of the level of technology recently attained (technicalprogress in research)
Exponential growth of researchers will produce an exponential
increase of the level of technology

Due to the dynamic nature of the knowledge production function not all
specifications are feasible for theoretical analyses and empirical estimations.
Therefore, one possible specification is a Hicks-neutral form as follows:i

A = f ( K A , LA , A) = Aϕ ( K A , LA )

(2.4)
which makes the knowledge level multiplicatively separable from the rest of thefunction. Interestingly, ϕ ( K A , LA ) is simply the “effective research function” that
we have seen in Phelps (1966). This form allows separate analysis of ϕ ( K A , LA )instead of dealing with the f ( K A , LA , A) as a whole. This property is very
important since the analysis of the former is much simpler. Note that theformulation (2.4) is not usually homogeneous of degree one in its arguments
unless ϕ ( K A , LA ) is homogeneous of degree zero (which is unlikely). This
property of (2.4) will have consequences in the dynamic setting.
Like in Nordhaus (1969), to secure linear homogeneity of the entire
knowledge production function it is necessary to consider a more flexible form
as in Jones (1999) as follows:i

A = Aφϕ ( K A , LA )(2.5)
where φ < 1 is imposed. This specification presents “increasing returns in the
production of new ideas” when φ > 0 . It also allows for diminishing returns if
5
φ < 0 . Its Solow residual form (Solow 1957) is given by

5

Concerning the “the future of the new economy”, Jones (2001) notes “Economists cannot say what it
takes to generate knowledge at a permanently faster rate and thereby raise the productivity growth rate
permanently” and “It is certainly possible that the economy becomes increasingly better at producing
new ideas, … However, it is also possible that it becomes increasingly difficult to discover new ideas,
as the most obvious ideas are discovered first.”

7

i

A
= Aφ −1ϕ ( K A , LA )
A

(2.6)

Log linearization of (3.17) yields:iAln = (φ − 1) A + ln ϕ ( K A , LA )
A

(2.7)

This result is interesting in that knowledge stock enters the relationship linearlyand log ϕ ( K A , LA ) can be specified with a log linear function such as CobbDouglas and translog, which makes the empirical formulation easy on one hand.
For example in the Cobb-Douglas case we will have:iA
ln = δ + (φ − 1) ln A + α ln K A + β ln LA
A

(2.8)
i

A
On the other hand the appearance of Solow residual in log, ln presents someAdifficulties in empirical work because in practice Solow residual can be
negative.
3. Malmquist Productivity Indexes as Measures of Knowledge Flow
Economists have long known that TFP growth can be interpreted as technicalprogress under certain assumptions (Chambers, 1988). This interpretation was
typically adopted in Solow (1957) for the well-known Solow residual, while theconcept of technical efficiency was developed independently in Farrell (1957).
Nishimizu and Page (1982) demonstrated that TFP growth or the Solow residual
could be partly interpreted as technical progress and partly as change intechnical efficiency since the definitions of the two concepts are mutually
exclusive. Technical progress is defined as the shift of the production frontier
over time while technical efficiency change refers to the movement towards the
production frontier. However, more meaningful interpretations are required in
the situation when TFP growth is regarded as endogenous.
For example, R&D expenditure alone will not be able to shift production
frontiers or improve technical efficiency. It has to be used to add new expertiseto the knowledge stock of the firm, and the new knowledge has to be usedefficiently during which process additional knowledge (experience) are also
accumulated. In the endogenous growth literature it is well accepted that theknowledge stock in turn can also be used to generate new knowledge, which isinterpreted as the “standing on the shoulder of the giant” effect (Romer 1990).
We next show that the popular Malmquist TFP growth index can not only be
decomposed into technical progress and technical efficiency change
components, but it can also be used as measure of knowledge flow that is8

consistent with the interpretation of TFP in the endogenous growth literature.
Another important and useful result we would like to show is that although the
Malmquist productivity index components are derived as multiplicatively
separable, it is also approximately additively separable if a simple linearization
scheme is applied.
Following Färe et al. (1994) the output-based Malmquist index of
productivity change is defined by assuming that, for each time period t=1, …, T,
the production possibility set St represents the transformation of inputs, xt∈R +N ,into outputs, yt∈R +M , as follows:
St = {(xt, yt): xt can produce yt}.
(3.1)
The output distance function at t is defined as:
Dot (xt, yt) =inf{λ : (xt, yt/λ )∈ St}.
= (sup {λ : (xt, λ yt)∈ St})-1 .
(3.2)
t t
t
Two important properties can be derived. One is that Do (x , y )≤ 1 if and only if(xt, yt)∈ St, and the other is that Dot (xt, yt) =1 if and only if (xt, yt) is on theboundary or frontier of the technology. Production is considered technicallyefficient when Dot (xt, yt) =1 (Farrell 1957). In empirical work, an efficient
observation is referred to as the best practice in the sample and can be identified
using DEA techniques.
Under constant returns to scale, it can be shown that the output distancefunction is homogenous of degree 1 in output and –1 in input (Färe and
Grosskopf, 1996). The distance function can involve two different time periods
as follows:
Dot (xt+1, yt+1) =inf{λ : (xt+1, yt+1/λ)∈ St}.
(3.3)
This function measures the maximal proportional change in outputs required tomake (xt+1,yt+1) feasible in relation to the reference or benchmark technology att. Similarly, a distance function that measures the maximal proportional change
in output required to make (xt, yt) feasible in relation to the technology at t+1,
denoted Dot+1 (xt, yt) may be defined. To avoid arbitrary benchmarking between t
and t+1, a common practice is to specify the output-based Malmquist
productivity change index as the geometric mean of two Malmquist productivity
indexes, one with technology at t and the other at t+1 as benchmarks, as follows:
1/ 2

 D t (xt +1 , y t +1 )  D t +1 (x t +1 , y t +1 ) M o (x , y , x , y ) =  o t t t  o t +1 t t  .
 Do (x , y )  Do (x , y ) 
t +1

t +1

t

t

(3.4)

In all definitions concerning Malmquist indexes, constant returns to scale isassumed for the technology as suggested in Färe and Grosskopf (1996). The

9

Malmquist productivity index in (2.4) can be further decomposed into two
components:
EFFCH =

Dot +1 (xt +1 , y t +1 ),Dot (xt , y t )

(3.5)

and
1/ 2

 Dot (xt +1 , y t +1 ) Dot (xt , y t ) 
TECH =  t +1 t +1 t +1 t +1 t t  Do (x , y ) Do (x , y ) 

,

(3.6)

where the expression in (3.5) measures the change in efficiency between periods
t and t+1, which we denote efficiency change. Expression (3.6) captures shifts
in the frontier technology, which we denote to be the technical change
component; values less than one in both cases signify deterioration in
productivity.
Another important property with the Malmquist index defined in (3.4) is that
it is exactly the straight measure of productivity change when there are only one
input and one output (Färe and Grosskopf, 1996). Define TFP change in the oneinput and one-output case as follows:
TFPCH =

y t +1 / xt +1
y t / xt

Note that output distance function is homogeneous of degree 1 in output and –1
in input under constant returns to scale,Dot ( x t , y t ) = ( y t / x t ) Dot (1,1)

This leads to
TFPCH =

y t +1 / x t +1 y t +1 / xt +1 Dot (1, 1) Dot ( xt +1 , x t +1 )= t t=
yt / xty / x Dot (1, 1)
Dot ( x t , y t )(3.7)

which explains the first term in (3.4) as a productivity index. Subtracting 1 fromthis productivity index, it coincides with the TFP growth rate or Solow residual,•

A/ A in (2.6), which was interpreted in Romer (1990) as growth rate in the stockof knowledge.
A more formal proof on the equivalence of the Malmquist index and theusual Solow residual can be found in Fried, Knox, and Schmidt (2008) as
follows. In the one output case, production function can be defined as
F (x) = max( y : (x, y ) ∈ S)

Again assuming constant returns to scale, this expression can be rewritten interms of the output distance function:
F (x) = y : / Do (x, y ) = y / [ yDo (x,1)] = 1/ Do (x,1)
10orDo (x, y ) = y / F (x)On the other hand, assuming technical change is Hicks-neutral as in Solow(1957), a discrete version can be expressed as follows:y t = F t (x t ) = A(t )ϕ (xt ),t = 0,1D (x , y ) = y A(t )ϕ (x ) ,
t
o

t

t

t

t = 0,1t

Substitute the terms in the Malmquist productivity index (3.4) with these
expressions:
1/ 2

 y1 / A(0)ϕ (x1 )   y1 / A(1)ϕ (x1 )  
M o ( x , y , x , y ) =  0
0  00  y / A(0)ϕ (x )   y / A(1)ϕ (x )  
1

1

0

0

= A(1) / A(0)Thus the Malmquist productivity index also qualifies as a unit free measure ofknowledge flow upon which a knowledge production function can be defined,
assuming of course constant returns to scale, Hicks-neutrality, and technicalefficiency. In general:i
TFPCH ≈ ( A/ A) + 1 iEFFCH ≈ ( AE / AE ) + 1i
TECH ≈ ( AT / AT ) + 1 


(3.8)

and note that TFPCH=EFFCH ⋅ TECH , therefore define,iii[( A/ A) + 1]=[ ( AE / AE ) + 1] ⋅ [( AT / AT ) + 1]

(3.9)

The following results will be very useful in both theoretical analysis andempirical work:

11

i

i

i

A + A ( AT + AT ) ( AE + AE )
=
⋅
A
AT
AE
i

i

( A + AT )( AE + AE )
= T
AT AE
i

(3.10)ii

i( A A + AE AT + AT AE + AT AE )
= T E
AT AE

Define A = AT AE , then we have
i

i

i

i

i

A = AE AT + AT AE + AT AE(3.11)

Divide both sides of the equation by A
i

i

i

i

i

A AT AE AT AE=++⋅
A AT AE AT AE(3.12)

Ignore the last second-order term,the Mamlquist index based TFP growth rate is given byiii

A AT AE≈+A AT AE(3.13)

given that A = AT AE . The importance of the result lies in that the multiplicatively
separable Malmquist productivity indexes is approximately additively separable
in terms of productivity growth rates. Take one more step,
i

i

i

A
A
A
+ 1 ≈ [ T + 1] + [ E + 1] − 1
AAT
AE(3.14)

i.e.,
TFPCH ≈ TECH + EFFCH − 1(3.15)

In fact the multiplicatively separable Malmquist productivity index
decomposition relationship can be linearized directly using Talor series
expansion around (1,1) as follows:
TFPCH=TECH iEFFCH≈ 1+(TECH-1)+(EFFCH-1)(3.16)=TECH + EFFCH − 1

12One gets the same result as in (3.15).
Strictly speaking, the knowledge production measured, say, with number ofpatents is only the supply side while the demand side decides whether theknowledge are useful or not. This is an issue we have touched upon in Zheng(2008) with reference to Gordon’s studies. Productivity outcomes are the resultsof balancing supply of with the demand for useful knowledge.
4. The Malmquist Index Regression Equations as a Dynamic System
Many studies have estimated the Malmquist index and its components, but it
was not clear how analysis of determinants of these indexes should be carriedout. In some cases, only level variables such as technical efficiency were used inregression analysis due to the lack of structural guidance with respect to theregressions using the Malmquist productivity growth index and its components.
For example, one issue encountered in Zheng, Liu, and Bigsten (2003) was
whether the first-differences instead of the levels of the explanatory variables
should be involved in the regression since Malmquist productivity growthindexes are intrinsically first-differenced indicators. Fortunately, the knowledge
production function defined in Section 2 provides convenient theoreticalguidance in this regard. In fact the framework of the knowledge production
function can be very useful in many other important aspects. We now propose astraightforward approach for conducting analysis in this context, which is to sortout the properties of the Malmquist indexes that have been largely ignored in theexisting literature. Incorporation of the dynamic aspects of the knowledge
production function in the Malmquist index regression equations generates
desirable model properties and brings new opportunities for detailed analysis on
the determinants of the productivity performance. We will examine these usefulproperties by means of propositions. Proves are also provided and discussed
whenever necessary.
In Zheng (2009) some attempts have been made to infer the microstructureon the link between innovation and productivity, drawing insights from theendogenous growth literature. However, the endogenous growth literature
basically says nothing about productivity decomposition, except in Nelson andPhelps (1966) in a somewhat different context. Zheng (2009) notes that thelinear nature of the Nelson and Phelps specification can be treated as a system if
one is to exploit the cross-equation relations in econometric estimations. This
was to take advantage of the cross-equation restrictions in the Malmquist index
regressions. It is now clear that the multiplicatively separable Malmquist index
decomposition can be conveniently turned into additively separable indexes as13

we have already shown in (3.15) and a formal treatment of the issue will be
provided in this section later.
Given the dynamic nature of the knowledge production function,
endogenizing simultaneously multiple productivity indicators such as the
Malmquist index estimates requires working with differential equation systems.
In what follows, we will demonstrate that specifications of the system reveal
relationships that were not so clearly seen when the Malmuqist index regression
equations are not treated as a system. For example, since knowledge stock is
considered as a productive input in the process of acquiring new knowledge,
does there exist a division of labor between different knowledge? In the context
of productivity growth decomposition, the question is whether knowledge
accumulated through technical efficiency improvement will be useful when theobjective is to produce a new product? On the other hand, we ask if one needs
knowledge on past technologies when working to improve technical efficiency
with newly installed machineries. If one is to decide the specification of the
dynamic system, it is inevitable that these considerations need to be taken into
account. A fully specified dynamic system based on the Malmquist index
regression equations will provide new opportunities for in depth studies of therelationship between total factor productivity and its determinants.
To begin with, we present a “full” list of the Malmquist index regression
equations as follows:(1) Technical efficiencyt

A = Dot (x, y ) ⋅ ∫ f (x, AT (v), AE (v))dv0(2) Best practice (binary)tAfront = ∫ f (x, AT (v), AE (v))dv
0(3) Total factor productivity growthi[( A/ A) + 1]=1 + f ( x, AT , AE )(4) Technical efficiency change

14

i[( AE / AE ) + 1] = 1 + f E (x, AT , AE )(5) Technical progressi[ AT / AT + 1] = 1 + fT (x, AT , AE )where AT stands for the level of technology and AE the level of technicalefficiency. When treating the equations (1) through (5) as a dynamic system, thefollowing properties hold.
Proposition 1 The system of linear regression equations on the basis of (3.4) (3.6) is not independent.
This is obvious since multiplication of (3.5) and (3.6) is equal to (3.4).
Estimation of any two equation enables researcher recover the third equation.
Assume that( AT / AT ) + 1 = ∑ βTi xi
i( AE / AE ) + 1 = ∑ β Ei xi
iwhere x’s are explanatory variables. The matrix form is given by, i ( AT / AT ) + 1 =  βT 1 i  β E1
+
(
A
/
A
)
1
 E E


βT 2
βE2

 x1 
... βTn   x2 ... β En   ...   xn 

and note thatiii[( A/ A) + 1]=[ ( AE / AE ) + 1] ⋅ [( AT / AT ) + 1]
= ( ∑ βTi xi ) ⋅ ( ∑ β Ei xi )= ∑ βTi β Ei xi x jCollecting the equations as a system, we have[( A/ A) + 1]= ∑ βTi β Ei xi x j
i[( AT / AT ) + 1] = ∑ βTi xi
i(P1.1)[( AE / AE ) + 1] = ∑ β Ei xi
i15which is a system of equations with cross-equation restrictions. This implies thatparameters in (P1.1) can be derived from parameter estimates of (P1.2) and(P1.3), which means that only two equations need to be estimated.
Proposition 2 If the firm is already on the production frontier then the only way
for it to raise productivity is through technical progress.
The proof is rather straightforward. If a firm is already on the production frontier
at time t, then the best it can do at time t+1 is to be on the production frontier inorder to avoid negative contributions to productivity growth in terms oftechnical efficiency change defined in (3.5). Second, if technology does not
improve at t+1, the second term in (3.6) is one, and the first term will also be
one since frontier does not shift. Combining the two cases, equation (3.4) will beequal to one, indicating zero productivity growth (a contradiction).
This property is rather useful in the context of multiple regressions involving
several productivity indexes defined here. Factors that always manage to push
the firms to the frontier will not be effective in the regression for technicalefficiency change. The efficiency change regression equation becomesredundant.
Proposition 3 If the stock of a firm’s knowledge is itself a factor of production,then the Malmquist Index regression equations can be treated as a dynamic
system.
Proof:
Denoting the Mamlquist total factor productivity (TFP) index Mo with thenotation in Jones (1999) asii

A+ A A
= +1
M0 ≈
A
A

(P3.1)

Since Mo can be further decomposed into MT and ME, which are mutually
exclusive, we have
i
i
 i
 i

AT + AT AE + AE  AT
A
M 0 = M T iM E ≈i=+ 1i E + 1
 AT
A

A`T
AE

 E


(P3.2)The specification for the determinants of both MT and ME are complicated. We
have in depth discussions with reference to the structure of the knowledge
production function in Zheng (2009). One possible choice is as follows:

16

i

AT
( M T − 1) ≈= AT(φ −1) fT (x) 
AT


i

A
( M E − 1) ≈ E = AE(θ −1) f E (x) AE

(P3.3)

which is a Hicks-neutral case for simplicity and it says that the rate of technicalprogress depends on the stock of technology itself, but not on the stock ofknowledge associated with technical efficiency improvement and vise versa.
Alternatively,iAT(φ −1)( M T − 1) ≈=A
fT ( x ) 
AT


i

AE
( M E − 1) ≈= A(θ −1) f E (x) AE

(P3.4)

which specifies the entire knowledge stock of the firms can be used for new
knowledge acquisition, i.e., the knowledge on technology itself and knowledge
about how to improve efficiency. Zheng (2008) adopted a specification like(P3.4). Since this specification could be a result of casual inclusion of
explanatory variables when conducting regression analysis on the estimates ofthe Malmquist indexes we will sometimes refer to it as the “naïve” regression
below. Notice that A = AT ⋅ AE , hence a more flexible specification might be given
by
i

AT
( M T − 1) ≈= AT(φT −1) AEθT fT (x) 
AT


i

A
( M E − 1) ≈ E = ATφE AE(θE −1) f E (x) AE

(P3.5)and φ ≠ θ in general. When φ = θ the system becomes (P3.4), and it is reduced to(P3.3) with φE = θT = 0 . We take the commonly encountered specification of(P3.4) in empirical applications as an example:

17

i

i

i

A
A
A
M 0 ≈ [ + 1] = [ T + 1] ⋅ [ E + 1] = ( A(φT −1) fT (x) + 1)( A(φE −1) f E (x) + 1)
A
AT
AE

(P3.6)(φT −1)= 1+ A(φE −1)fT ( x) + A(φT +φE − 2)

f E ( x) + A

fT ( x ) f E ( x)which presents a modestly complicated dynamic structure. This proves the
proposition.
Proposition 4 The “naïve” regression system (P3.4) is not stable but can besaddle-path stable.
Proof:
Given thatiAT
( M T − 1) ≈= A(φ −1) fT (x) 
AT


i

A
( M E − 1) ≈ E = A(θ −1) f E (x) AE

(P3.4)

Define A = AT ⋅ AE , theniAT = ATφ AE(φ −1) fT (x) i(θ −1) θ
AE = AT AE f E (x) (P4.1)

To simplify the derivation we assume f (1) = 1 . Taylor linearization at the point 1
leads toiAT ≈ δ T 0 + φ AT + (φ − 1) AE + ∑ βTi xi iAE ≈ δ E 0 + (θ − 1) AT + θ AE + ∑ β Ei xi (P4.2)(P4.2) forms a system of ordinary differential equations. For stability of thesystem, it requires that the two eigenvalues of the parameter matrix
(φ − r ) (φ − 1)  (θ − 1) (θ − r ) 

(P4.3)

18

to be real and negative. The determinant of the characteristic matrix is given by(φ − r )(φ − 1)(θ − 1) (θ − r )=0(P4.4)

Rewrite slightly(1 − r ) + (φ − 1)

(φ − 1)(θ − 1)(1 − r ) + (θ − 1)

=0

(P4.5)Let R = (1 − r ) , φ ' = φ − 1 , and θ ' = θ − 1 , thenR +φ '

θ'φ'=0
R +θ '(P4.6)( R + φ ')( R + θ ') − φ 'θ '=  R 2 + (φ '+ θ ') R + φ 'θ ' − φ 'θ '= R 2 + (φ '+ θ ') R = 0
R1 = 0, r1 = 1(P4.7)

R2 = −(φ '+ θ '),r2 = φ '+ θ '+ 1 = φ − 1 + θ − 1 + 1'
= φ + θ −1

Therefore, for saddle path stable solution, it requires (φ + θ ) < 1 or φ < (1 − θ ) . This
proposition allows Efficiency change equation and technical progress equation
to have different structures with respect to the level of technology. But it does
not rule out the possibility of a similar structure for the two equations. For
example, if we apply the structure of the knowledge production function (2.5) totechnical progress index (3.6), with φ < 0 the higher the level of technology theslower the speed of technical progress. The structure implied in the formulationof Benhabib and Spiegel (1994) is somewhat similar and pointing to a similar
direction. The lower the level of technical efficiency, the stronger the incentive
for catching up with the best practice, i.e., θ < 0 . An empirical example of thistype of results can be found in Zheng (2008). We next take a look at the moregeneral case of (P3.5).

19

Proposition 5 The system (P3.5) and (P3.3) can be stable.
Proof:
Since we havei
AT = ATφT AEθT fT (x) i
AE = ATφE AEθE f E (x) (P3.5)

Taylor linearization leads toiAT ≈ δ T 0 + φT AT + θT AE + ∑ βTi xi iAE ≈ δ E 0 + φE AT + θ E AE + ∑ β Ei xi (P5.6)For stability of the system, it requires that the two eigenvalues of the parameter
matrixφT θT 
φ θ 
 E
E

(P5.7)

to be real and negative. The determinant of the characteristic matrix is given by
φT − rθT=0φE
θE − r

(P5.8)

where r is the characteristic root. This means:(φT − r )(θ E − r ) − θT φE = 0(P5.9)

r 2 − (φT + θ E )r + (φT θ E − θT φE ) = 0r=− [ −(φT + θ E ) ] ± (φT + θ E ) 2 − 4(φT θ E − θT φE )2

After Simplification

20(P5.10)r=(φT + θ E ) ± (φT − θ E ) 2 + 4θT φE
2(P5.11)For stability it is required that(φT − θ E ) 2 + 4θT φE ≥ 0(P5.12)(φT + θ E ) ± (φT − θ E ) 2 + 4θT φE
r=<0
2(P5.13)andwhich ends our proof for Proposition 5.
If we apply Proposition 5 to the simplest scenario described in (P3.3), with
φE = θT = 0 (P5.11) reduces tor=(φT + θ E ) ± (φT − θ E )2so
r1 = φT = φ and r2 = θT = θ

For system stability it requires that both eigenvalues to be negative and sadlepath stability is also possible in this simple case.
To further simplify the system, one alternative might be to make thedecomposition of Malmquist index additive instead of multiplicative by taking
logarithm of both sides of (3.2). However, since log cannot be used for negativevalues of Malmquist TFP index, we pursue a linearization of (3.2) using thefirst-order Taylor expansion at the point zero as follows.6
Proposition 6 (P3.2) can be conveniently linearized.
Proof:

6

According to the basic assumption A5 TFP growth cannot be negative, so log linearization inprinciple should be allowed. But in practice, it’s better to be able to accommodate negative values of
TFP change or growth at this stage. Taking positive values restricts data to those observations that can
only be generated with a well-defined knowledge production function.

21

We have already seen this in (3.16) from a slightly different perspective. Expand(P3.2) around the point of (0,0) with respect to growth rates in AT and AE yields,
i
i
 i
 i

AT + AT AE + AE  AT
A
M 0 = M T iM E ≈i=+ 1i E + 1
 AT
A

A`T
AE

 E

i

(P6.1)

i

AT AE
+
AT AE

≈ 1+

For example, if we refer to specification in (P3.4), we haveiA
M 0 − 1 = ≈ A(φ −1) fT (x) + A(θ −1) f E (x)A(P6.2)andi

A ≈ Aφ fT (x) + Aθ f E (x)(P6.3)

which is rather convenient and consistent with the Nelson and Phelps (1966)
theoretical and Benhabib and Spiegel (1994) empirical formulations.
In Zheng (2009), we proposed to treat the additive version of the productivity
decomposition in Benhabib and Spiegel (1992), which was an empiricalreformulation of Nelson and Phelps (1966). However, since the Malmquist
index decomposition is multiplicative, we here suggest to linearize (P3.5)
directly as follows:iA`T≈ δ T 0 + δ T (φT − 1) AT + δ TθT AE + δ T ∑ βTi xi 
AT


i

AE
≈ δ E 0 + δ EφE AT + δ E (θ E − 1) AE + δ E ∑ β Ei xi AE
(P6.4)For simplicity we set δ = 1 . According to (P6.1)iii

A AT AE≈+A AT AE

Then we haveiA
≈ (δ T 0 + δ E 0 ) + [δT φT + δ E (φE − 1)] AT + [ (δ TθT + δ E (θ E − 1) ] AEA+ ∑ (δ T βTi + δ E β Ei ) xi

Taking account (P3.5), we have22

(P6.5)iA≈ (δ T 0 + δ E 0 ) + [δT φT + δ E (φE − 1)] AT + [ (δ TθT + δ E (θ E − 1)] AE A+ ∑ (δ T βTi + δ E β Ei ) xi

i


A`T
≈ δ T 0 + δ (φT − 1) AT + δθT AE + δ ∑ βTi xi

AT

i

AE

≈ δ E 0 + δφE AT + δ (θ E − 1) AE + δ ∑ β Ei xiAE(P6.6)

which is a linear dynamic system with linear cross-equation constraints. The
advantage of cross-equation linear constraints is that one can impose these
constraints in linear regressions to achieve better efficient parameter estimations.
For example, in Zheng, Liu and Bigsten (2003), TFP growth and its components
were used in regressions on the same set of the explanatory variables. Althoughno cross-equation restrictions were applied there, one can see in their Table 5that the sum of the parameters in the two TFP growth component equations wasvery close to the parameter estimate in the TFP growth equation. Not allparameter estimates were statistically significant without imposing crossequation restrictions. The situation may change if one does so.
Proposition 7 The long run solution.
This is straightforward.
5. An Empirical Example of Norwegian Firms
We now update the results in Zheng (2008) in which a Norwegian firm panel
and CIS3 and CIS4 data are used jointly. The complete set of the six regression
equations is listed as follows:(1) Technical efficiency(2) Best practice (binary)(3) Technical efficiency change(4) Positive technical efficiency change (binary)(5) Technical progress(6) Positive technical progress (binary)
We thus use technical efficiency, technical efficiency change, technical progress
as dependent variables; and level TFP, innovation intensity, and measure of
R&D personnel are used as independent variables together with other auxiliary

23variables of the CIS datasets. We will focus on the two Malmquist index
component equations as follows:
EFFCH = α 2 + δ 2 (θ − 1)TFP + δ 2 λ2 LA + ρ 2

α 2 = (1 + δ 2 ) − δ 2 (θ − 1) − δ 2 λ2
TECH = α 3 + δ 3 (φ − 1)TFP + δ 3λ3 LA + ρ3

α 3 = (1 + δ 3 ) − δ 3 (φ − 1) − δ 3λ3

∆R
Y

∆R
Y

(5.1)

(5.2)

As in Zheng (2008), (5.1) and (5.2), with all parameters identified, are the basic
microstructure we apply for the regression analyses. Note that since LA = sL both
the ratio of researchers to the total employed and the total labor can be included
in a regression. This, on one had, can make the magnitude between explanatory
variables more comparable, and on the other hand can be used to test if theparameter associated with the ratio is statistically different from that associatedfrom the total labor L. In empirical estimations, in the case of labor input
sometimes it is better to use a unit of measure that is comparable with the
productivity index on the right hand side of the equation. For example labor can
be standardized using a “mean transformation” or log transformation. In Zheng(2008) log transformation was used, but in the updated results in Table 1 wegave used the “mean transformation.”24

Table 1 Regression results for the Malmquist dynamic regression(Norwegian Firms 1995-2004)
Best practice Efficiency Change Technical progress
Intercept-3.39***

0.419***

1.829***(-5.84)(2.73)(21.26)

0.631***-0.547***(6.11)(-8.50)0.072***-0.008-0.009*(3.79)(-0.90)(-1.81)2.02***

0.045

0.026

(2.83)(0.22)(0.20)-0.7490.045-0.196(-0.84)(0.21)(-1.35)

R&D Purchase intensity 0.376

0.200*

0.007(0.89)(1.68)(0.09)

0.222-0.0060.010(1.16)(-0.10)(0.28)

0.338*-0.0290.025(1.69)(-0.43)(0.58)-0.213-0.073**-0.026(-1.47)(-2.10)(-1.28)0.178*

0.018-0.004(1.90)(0.68)(-0.28)-0.011-0.011-0.018(-0.14)(-0.58)(-1.64)0.091-0.0260.002(1.06)(-0.97)(0.14)

96.06(Sigma) 0.376(Sigma) 0.376<.0001<.0001

TFP level (A)Employment
R&D Personnel
Innovation intensity

PatentExport intensity
Continuous R&D
Market orientation
Product life
Market location
Likelihood Ratio
P-value
N

939

923

25

923

6. Summary

The DEA-based Malmquist productivity index model has become increasingly
popular in empirical estimations since its introduction to the literature about two
decades ago. However, regression analyses based on the Malmquist productivity
index and its components are far less in numbers than those based on straightestimates of DEA efficiency scores. As the progresses with the understanding of
the statistical properties of the DEA frontier estimates and advances inbootstrapping DEA scores are making the two-step procedure of DEA followedby OLS regression more attractive, the need for a systematic treatment of the
Malmquist index regression equations has become an urgent task. In this study
we have approached the issue through the knowledge production function
framework suggested in Zheng (2009) and constructed the multiple regression
equations into an interactive, interrelated, and fully specified dynamic system.
The immediate new knowledge acquired through this process is that the system
consisting of the “naïve” Malmquist regression equations can be saddle-path
stable.
On the other hand, the dynamic system constructed in this study in general
has three attractive properties. First, the Malmquist index is consistent with the
Solow residual under rather standard assumptions, which paves the way fordeveloping the set of Malmquist index regression equations into a dynamic
system under the framework of the knowledge production function. Second,cross-equation restrictions not only exist but also can be imposed in empiricalestimations using a linearised version of the Malmquist index. Third, thedynamic nature of the equation system widens the scope for the knowledge
production function to be specified. As we have mentioned in the study, theendogenous growth literature basically says nothing about the productivity
decomposition with respect to technical progress on the one hand and technicalefficiency improvement on the other. But our specifications of the knowledge
production function reveal that several different systems can be considered; andknowledge stocks can be further classified into those that may be effective in
improving technical efficiency and those that might help push forward the
production frontier, i.e., the application of new technologies. This property ofthe system enriches the structures that can be informative of the knowledge
acquisition process, and should have important implications for micro studies on
the relationships between innovation and productivity and for macro modelingof endogenous economic growth.
Although the two-step procedure under which our dynamic system wasdeveloped possesses desirable statistical properties for large samples, the26

method might be subject to small sample bias. To improve the implementations
in both the first and second step procedures the application of bootstrapping
techniques recommended in Simar and Wilson (2007) would be sometimes
worth the effort. The technique may enable researchers to achieve better
precision for parameter estimates and more reliable statistical inference.
Combining a fully specified dynamic system of Malmquist productivity index
regression equations with the bootstrap methodology for DEA efficiency scoresand Malmquist productivity indexes may prove to be useful for systematic
empirical analysis of productivity performance.

27

References
Benhabib, J. and Spiegel, M. (1994), The role of human capital in economic
Development: Evidence from aggregate cross-country data. Journal of Monetary
Economics, 34, 143-173.
Banker, Rjiv D. and Natarajan, Ram, Evaluating Contextual Variables Affecting
Productivity Using Data Envelopment Analysis. Operations Research, Vol. 56
Issue 1, p48-58, Jan/Feb 2008.
Banker, Rajiv D., Maximum Likelihood, Consistency and Data Envelopment
Analysis: A Statistical Foundation. Management Science, Vol. 39 Issue 10,
p1265-1273, October 1993.
Bjurek, Hans, Urban Kjulin, and Bjorn Gustafsson, (1992), "Efficiency,
productivity and determinants of inefficiency at public day care centers in
Sweden - Comment", Scandinavian Journal of Economics 94.
Castellacci and Zheng (2008),
Da Silva e Souza, Geraldo and Staub, Roberta Blass, Two-stage inference using
data envelopment analysis efficiency measurements in univariate production
models. International Transactions in Operational Research, Vol. 14 Issue 3,
p245-258, May 2007.
Dismuke, Clara Elizabeth and Sena, Vania, Has DRG Payment Influenced theTechnical Efficiency and Productivity of Diagnostic Technologies in Portuguese
Public Hospitals? An Empirical Analysis Using Parametric and Non-Parametric
Methods. Health Care Management Science, v. 2, iss. 2, pp. 107-16, May 1999.
Fried, Harold O., C. A. Knox Lovell, and Shelton S. Schmidt, The Measurementof Productive Efficiency and Productivity Change. Oxford University Press, Jan2008.
Nelson, R. and Phelps, E., 1966. Investment in humans, technological diffusion,and economic growth. American Economic Review: Papers and Proceedings 51(2), 69-75.

28

Simar, Leopold and Wilson, Paul W., Estimation and Inference in Two-Stage,
Semi-Parametric Models of Production Processes. Journal of Econometrics, v.
136, iss. 1, pp. 31-64, January 2007.
Simar L, Wilson P. 2000a. A general methodology for bootstrapping in
nonparametric frontier models. Journal of Applied Statistics 27: 779–802.
Simar L, Wilson P. 2000b. Statistical inference in nonparametric frontiermodels: the state of the art. Journal of Productivity Analysis 13: 49–78.
Simar L, Wilson P. 1999. Estimating and bootstrapping Malmquist indices.
European Journal of Operational Research 115: 459–471.
Simar L, Wilson P. 1998. Sensitivity of efficiency scores, how to bootstrap in
non-parametric frontier models. Management Science 44: 49–61.
Xue, M., Harker, P.T., 1999. Overcoming the inherent dependency of DEA
efficiency scores: a bootstrap approach. Unpublished Working Paper, Wharton
Financial Institutions Center, University of Pennsylvania.
Zheng, Jinghai (2009a), The Structure of the Knowledge Production Function,
working paper for the BITE project. Forthcoming, Feb. 2009.
Zheng, Jinghai, Innovation, Technology and Productivity Performance inNorwegian Firms (1995-2004), International Workshop on Chinese
Productivity, September 20-21, 2008, Hangzhou, China.
Zheng, J., Liu, X. and Bigsten, A. (2003): ‘Efficiency, technical progress, andbest practice in Chinese state enterprises (1980–1994)’, Journal of Comparative
Economics, 31 (1), 134-152.
Zheng, J., Liu, X. and Bigsten, A. (1998), Ownership Structure andDeterminants of Technical Efficiency: An Application of Data Envelopment
Analysis to Chinese Enterprises (1986-1990). Journal of Comparative
Economics 26 (3), 465-484.

29

References on Determinants of productivity
Akinci, Fevzi, (1999), "Determinants of technical efficiency in federally funded
community health centers", Ph.D. dissertation, Saint Louis University.
Arnold, Victor L., Indranil R. Bardhan, William Wager Cooper and Subal C.
Kumbhakar, (1996), "New uses of DEA and statistical regression for efficiency
evaluation and estimation - with an illustrative application to public secondary
schools in Texas", Annals of Operations Research 66, pp 255-277. 9/25 to 9/27.
Bardhan, Indranil R., (1995), Data envelopment analysis and frontier regression
approaches for evaluating the efficiency of public sector activities : Applications
to public school education in Texas. Ph.D. dissertation, University of Texas,
Austin.
Bardhan, Indranil R., William Wager Cooper and Subal C. Kumbhakar, (1998),
A simulation study of joint uses of data envelopment analysis and statisticalregressions for production function estimation and efficiency evaluation. The
Journal of Productivity Analysis 9(3), pp 249-278.
Bjurek, Hans, Urban Kjulin, and Bjorn Gustafsson, (1992), Efficiency,
productivity and determinants of inefficiency at public day care centers in
Sweden – Comment. Scandinavian Journal of Economics 94.
Bowlin, William Frank, Abraham Charnes, William Wager Cooper and H.
David Sherman, (1985), Data envelopment analysis and regression approaches
to efficiency estimation and evaluation. Annals of Operations Research 2, pp
113-138.
Chang, Hsi-Hui, (1998), "Determinants of hospital efficiency: The case ofcentral government-owned hospitals in Taiwan", Omega, International Journalof Management Science 26(2), pp 307-317.
Destefanis, Sergio and Sena, Vania, Patterns of Corporate Governance andTechnical Efficiency in Italian Manufacturing. Managerial and Decision
Economics, v. 28, iss. 1, pp. 27-40, January 2007.
Elam and Charles Terrance Clark, (1988), Efficiency frontier determination by
constrained facet analysis. Operations Research 36(5), pp 785-796, Set-Oct.
Lovell, C. A. Knox, Lawrence Clayton Walters and Lisa L. Wood, (1994),"Stratified models of education production using modified DEA and regression
analysis", in (Charnes, Abraham, William Wager Cooper, Arie Y. Lewin and30

Lawrence M. Seiford, ed.), Data envelopment analysis: Theory, methodology
and applications, Kluwer Academic Publishers, Boston, pp 329-351.
Schinnar, Arie P., E. Kamis-Gould, N. Delucia and A. B. Rothbard, (1990),"Organizational determinants of efficiency and effectiveness in mental healthpartial care programs", Health Services Research 25(2), pp 387-420.
Tian, Weiming and Guang Hua Wan, (2000), "Technical efficiency and its determinants in
China's grain production", The Journal of Productivity Analysis 13(2), pp 159-174.

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