Discussion Paper No. 543

TESTING MULTIPLE NON-NESTEDFACTOR DEMAND SYSTEMS

Matteo Manera
andMichael McAleer

June 2001

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Testing Multiple Non-nested Factor Demand Systems

Matteo Manera
Department of Economics
Bocconi University, Milan, Italy

Michael McAleer
Department of Economics
University of Western Australia
Revised: June 2001

Keywords. Systems of equations; Factor demands; Flexible functional forms; Paired non-nested
tests; Joint non-nested tests.
JEL classifications. C52; C30.
Corresponding author. Matteo Manera – Istituto di Economia Politica – Università Commerciale“L. Bocconi” – Via R. Sarfatti, 25 – 20136 Milano – Italia – office (direct): +39-02-58365413 –fax: +39-02-58365439 – e-mail: matteo.manera@uni-bocconi.it
Acknowledgements. Seminar participants at the Agricultural and Resource Economics (ARE)Group and the Department of Economics at the University of Western Australia (UWA) offeredhelpful comments. The first author would like to thank the ARE Group and the Department of
Economics of UWA for financial support. The second author wishes to acknowledge the financial
support of the Australian Research Council and the Institute of Social and Economic Research at
Osaka University.

Abstract. Empirical factor demand analysis typically involves making a choice from among several
competing non-nested functional forms. Each of the commonly used factor demand systems, such as
Translog, Generalized Leontief, Quadratic, and Generalized McFadden, can provide a valid anduseful empirical description of the underlying production structure of the firm. As there is no
theoretical guidance on selecting the model which is best able to capture the relevant features of thedata, formal testing procedures can provide additional information. Paired and joint univariate nonnested tests of a null model against both single and multiple alternatives have been discussed at
length in the literature, whereas virtually no attention has been paid to either paired or joint
multivariate non-nested tests. This paper shows how some multivariate non-nested tests can bederived from their univariate counterparts, and examines how to use these tests empirically tocompare alternative factor demand systems. The empirical application involves the classical BerndtKhaled annual data set for the U.S. manufacturing sector over the period 1947-1971. A statisticallyadequate empirical specification is determined for each competing factor demand system. The
empirical results are interpreted for each system, and the models are compared on the basis ofmultivariate paired and joint non-nested procedures.

1. Introduction
Empirical factor demand analysis typically involves making a choice from among several competing
non-nested functional forms. Each of the commonly used factor demand systems, such as Translog,
Generalized Leontief, Quadratic and Symmetric Generalized McFadden, can provide a valid anduseful empirical description of the underlying production structure of the multi-input neoclassicalfirm. A common feature of flexible functional forms is that they are non-nested (or separate). Thus,
given two or more systems of factor demands, it is not possible to obtain one system by imposing
suitable parametric restrictions on the other(s). Moreover, as there is no a priori theory suggestingthat the specification of one system should be preferred over another, it is necessary to choose fromamong the competing models using empirical considerations.
The important task of model determination can be accomplished using a formal non-nested
testing procedure. Paired and joint univariate non-nested tests of a null model against both single
and multiple alternatives have been discussed at length in the literature. However, virtually no
attention has been paid to either paired or joint multivariate non-nested tests. This paper shows how
some multivariate non-nested tests can be derived from their univariate counterparts, and examines
how to use these tests empirically to compare alternative factor demand systems.
As the outcome of a non-nested test is influenced by the type of misspecification affecting thecompeting models, it is essential to investigate the performance of each factor demand systemagainst real data. The empirical application presented is very popular in the applied production
literature, and contains annual data on aggregate output of U.S. manufacturing industries, and prices
and quantities for a capital-labour-energy-materials (KLEM) technology over the period 1947-1971(see Berndt and Khaled, 1979). A statistically adequate empirical specification is determined foreach competing factor demand system. Estimation results and some diagnostic statistics arepresented for each factor demand system, and each is used to calculate some classical indicators of
the production structure of an economic sector, such as price and output elasticities. The systems arethen compared on the basis of multivariate paired and joint non-nested testing procedures. Finally,
the empirical results are interpreted for each system, and some practical issues regarding model
selection and testing of systems of equations in applied research are discussed.

1This paper is organized as follows. In Section 2, alternative factor demand systems are presented
using some of the most popular flexible functional forms, namely Translog, Quadratic, Generalized
Leontief and Symmetric Generalized McFadden (for further details, see Diewert and Wales, 1987).
In Section 3, multivariate extensions of some well-known and pedagogically appealing univariate
paired non-nested tests (namely, the J, P0 and P1 tests of Davidson and MacKinnon, 1981; the PE
statistic of MacKinnon, White and Davidson, 1983; and the Bera and McAleer (1989) test, hereafter
the BEM test), and the multivariate joint non-nested test of Barten and McAleer (1997), hereafter
the BAM test, are described. Section 4 reports the main empirical results from estimation anddiagnostic testing of the competing factor demand systems. Estimated price and output elasticities
from each model are also presented. In Section 5, multivariate paired and joint non-nested tests areapplied to compare the alternative factor demand systems. The empirical results are interpreted foreach system. Section 6 provides some concluding comments.

2. Alternative factor demand systems
In this section attention is focused on the four most widely used flexible functional forms in thecontext of cost function estimation: Translog, Quadratic, Symmetric Generalized McFadden andGeneralized Leontief. As is customary in applied factor demand analysis, the cross-equation
symmetry restrictions are maintained for each model. Several techniques are available for imposing
the appropriate curvature conditions on the cost function (see Morey, 1986). In this paper, we haveleft unconstrained the matrix of second-order partial derivatives of the cost function with respect tofactor prices in each model, and have checked ex post if the negative semi-definiteness of the
Hessian of the cost function is satisfied over the sample period.
Consider the following specification for the logarithm of the firm's cost function C(·):

nln C (⋅) = α 0 + ∑ α i ln Pi + α y ln Y + α t t + α yt t ln Y
i =1

+

n
1 n n
α
ln
P
ln
P
+
α iy ln Pi ln Y
∑ ∑ ij i j ∑2 i =1 j =1
i =1

n
1
1
2
+ ∑ α it t ln Pi + α yy (ln Y ) + α tt t 2
2
2
i =1

2

(2.1)

where Pi indicates the price of the i-th input (i,j=1,...,n), Y is output, t=1,...,T is a time trend, and thesymmetry condition αij=αji for all i,j, is imposed. Necessary and sufficient conditions for C(·) to belinearly homogeneous in input prices are given by:

n

n

n

n

i =1

i =1

i =1

i =1

∑ α i = 1; ∑ α ij = 0; ∑ α iy = 0; ∑ α it = 0 (i=1,...,n).

(2.2)

By using the Translog specification (2.1) and Shephard’s lemma, the following expressions forthe input shares are obtained (see Berndt and Christensen, 1973, p. 85):nShi = α i + ∑ α ij ln Pj + α iy ln Y + α it t (i=1,...,n)

(2.3)

i =1

with αij=αji for all i,j=1,…,n.
The Quadratic second-order approximation to the firm's true cost function can be defined asfollows:n

C (⋅) = α 0 + ∑ α i Pi + α y Y + α t t + α yt Yt

(2.4)

i =1

+

n
1 n n
α
P
P
+
α iy Pi Y∑ ∑ ij i j ∑2 i =1 j =1
i =1

n
1
1
+ ∑ α it Pi t + α yy Y 2 + α tt t 2
2
2
i =1

with αij=αji for all i,j=1,...,n. A direct application of Shephard’s lemma yields the system of factor
demands:n

X i = α i + ∑ α ij Pj + α iy Y + α it t (i=1,...,n).

(2.5)

i =1

The major drawback of the Quadratic functional form is that linear homogeneity restrictions inprices cannot be imposed parametrically, which means that the cost function under linear
homogeneity in factor prices has to be respecified and a new separate demand system obtained.
3

When linear homogeneity in prices is imposed, the cost function and the related factor demands
have the following forms:
n
P
C (⋅)
= α 0 + ∑ α i i + α y Y + α t t + α yt Yt
P1
P1
i =2

+

(2.6)

n
Pi Pj
P
1 n n
α
+
α iy i Y∑∑∑
ij
2 i =2 i =2
P1 P1 i = 2
P1

Pi
1
1
t + α yy Y 2 + α tt t 2
P1
2
2

n

+ ∑ α it
i =2

and
n
Xi
P
= α i + ∑ α ij i + α iy Y + α it t (i=2,...,n).
P1
P1
j =2

(2.7)

Consider the following functional form for a cost function, with the usual symmetry conditions
imposed:C (⋅) = Y

1 n
∑ σ ij Pi Pj
2 i =1
n

∑θ P
i

n

+ ∑ α i Pi

(2.8)

i =1

i

i =1

n

n

i =1

i =1

+ ∑ α ii Pi Y + ∑ α it Pi t

The cost function defined by expression (2.8) is a generalization of the functional form due to
McFadden (1978, p. 279), as suggested by Diewert and Wales (1987, p. 53). It should be noted thatthe σij parameters have to sum to zero in order to identify all the coefficients in expression (2.8).
The related system of Symmetric Generalized McFadden factor demands can be derived:n∑ σ ij Pj

Xi
j =1
= n
Y

∑θ P
i

i =1

i

−θi

1 n n
∑∑ σ ij Pi Pj
2 i =1 j =1 n ∑ θ i Pi 
 i =1


(2.9)

2

4

+

αi
α t
+ bii + it .
YYThe traditional Generalized Leontief cost function is a functional form in the square roots of input
prices. In this paper, we consider the following version:

n
 n n
 n
C (⋅) = Y  ∑ ∑ α ij Pi Pj  + ∑ α i Pi + ∑ α it Pi ti =1
 i =1 j =1
 i =1(2.10)

where αij=αji for all i,j is a maintained hypothesis. The system of factor demands is derived from
(2.10), as follows:n
Pj α i α it tXi= ∑ bij
+
+
Y
Pi
Y
Y
i =1

(i=1,...,n).

(2.11)

3. Alternative non-nested testing proceduresThis section presents some alternative non-nested testing procedures for competing systems ofequations. In particular, modifications of the paired J, P0, P1, PE and BEM tests are presented as
extensions of the corresponding univariate statistics.
3.1. Multivariate paired non-nested tests
Some variants of both the J and the P tests (see Davidson and MacKinnon, 1981; MacKinnon,
White and Davidson, 1983) may be applied to the case of multivariate regression models. For thispurpose, the null and the alternative hypotheses are given as:
H0: yi = fi(Xi,ai) + u0i

(3.1)

and
H1: yi = gi(Zi,bi) + u1i

(3.2)
5

where i denotes equation i=1,...,n, in a system of equations, u0 and u1 are distributed as N(0,Ω0) andN(0,Ω1), respectively, and Ωj is the contemporaneous covariance matrix of the error term
corresponding to hypothesis Hj, j=0,1. Equations (3.1) and (3.2) are multivariate, non-simultaneousmodels, such as factor demand systems or systems of cost share equations.
A straightforward extension of the J test to a multivariate context is based on the following
auxiliary equation:
H c : y i = f i (⋅) + λ i gˆ i (⋅) + u i (i=1,...,n)(3.3)

which is obtained by combining linearly equations (3.19) and (3.20) and replacing gi(·) with thefitted values gˆ i (⋅) . Alternatively, Hc in (3.3), which is a composite model, is equivalent to H0 in(3.1) with the addition of gˆ i (⋅) . The null hypothesis H0 is not rejected if the parameters λi are zerofor all i. A Wald-type statistic enables testing of the null hypothesis H0 against the alternative H1that the λi are all jointly zero. If the roles of H0 and H1 are reversed, it is possible to test the validityof H1 against H0.
Davidson and MacKinnon (1982, p. 555) discuss two multivariate versions of the P test. The firsttest is based on the following composite model:[]

H c : y i − fˆi (⋅) = λ i gˆ i (⋅) − fˆi (⋅) + Fˆi (⋅)d i + u i(3.4)

where Fˆi (⋅) is the row vector of derivatives of fi(·) with respect to ai evaluated at â i andd i = a i − aˆ i . Under H0 the vector ui is distributed as N(0,Ω0), so that model (3.4) must be estimatedby a systems generalized least squares procedure using a covariance matrix which is proportional toΩ̂0 . The Wald test of λ1=…=λn=0 is the multivariate extension of Davidson and MacKinnon’s P0
test. They also propose the following auxiliary regression equation:

(

)[

]

ˆ Ω
ˆ −1 gˆ (⋅) − fˆ (⋅) + u
H c : y i − fˆi (⋅) = Fˆi (⋅)d i + λ i Ω
0
1
i
i
i

6

(3.5)

where the variables are defined using the same notation as above. The Wald test of λ1=…=λn=0 is
the multivariate extension of Davidson and MacKinnon’s P1 test.
The multivariate analogue of the PE test for models with dependent variables subject to different
transformations is given by the following auxiliary regression equation:[]

H c : y i − fˆi (⋅) = λ i gˆ i (⋅) − hi ( fˆi ) + Fˆi d i + u i(3.6)

where the variables are defined as in the extension of the P test.
The BEM approach (see Bera and McAleer, 1989) can be readily adapted to the multivariate case.
Let the null and the alternative hypotheses be given by expressions (3.1) andH1: hi(yi) = gi(Zi,bi) + u1i(3.2')

where hi(·) is a known transformation of yi, and u0i and u1i are NID(0,σ20IT) and NID(0,σ21IT),
respectively. Combining the disturbances u0i and u1i linearly, with weights (1-λi) and λi, yields:(1-λi)[yi-fi(·)] + λi[hi(yi)-gi(·)] = ui

(3.7)

where ui is NID under both H0: λi=0 and H1: λi=1. Dividing expression (3.7) by (1-λi) gives:
yi = fi(·) + θ0iu1i + ui/(1-λi)(3.8)

where θ0i = -λi/(1-λi), while dividing (3.7) by λi gives:
hi(yi) = gi(·) + θ1iu0i + ui/λi(3.9)

where θ1i = -(1-λi)/λi. A test of θ0i=0 (i=1,…,n) in (3.8) corresponds to a test of λi=0 in (3.7), so thatif θ0i=0 is not rejected, λi=0 is not rejected. In a similar manner, a test of θ1i=0 in (3.9) is equivalent
to a test of λi=1 in (3.7), so that if θ1i=0 is not rejected, λi=1 is not rejected. A serious problem isthat H0 and H1 are not testable because u0i and u1i in (3.8) and (3.9) are not observable. It is possible
to replace the disturbances from H0 and H1 in (3.8) and (3.9), respectively, with some estimated7

residuals as follows. First, systems (3.1) and (3.2') are estimated to obtain the fitted values ŷ i and[]hˆi ( y i ) , respectively. Second, ŷ i and hˆi ( y i ) are transformed as hi ( yˆ i ) and hi−1 hˆi ( y i ) , respectively.
Third, the models given by:hi ( yˆ i ) = g i ( Z i , bi ) + η1i(3.10)

and[]hi−1 hˆi ( y i ) = f i ( X i , a i ) + η 0i

(3.11)

are estimated to obtain the residuals η̂1i and η̂ 0i , respectively. Finally, the models given by:

y i = f i ( X i , a i ) + θ 0 iηˆ i1 + ε i(3.12)

andhi ( y i ) = g i ( Z i , bi ) + θ 1iηˆ 0i + ε i(3.13)

are estimated. The BEM test is a Wald-type test of θ 01 = ... = θ 0 n = 0 for H0 in (3.12) andθ 11 = ... = θ 1n = 0 for H1 in (3.13), respectively.
3.2. Multivariate joint non-nested tests
The test procedures considered so far belong to the class of multivariate paired non-nested tests,
whose common feature is that the null hypothesis is tested against a specific non-nested alternative,
after which the roles of null and alternative are reversed. Recently, some univariate tests have been
proposed in the literature in which the null hypothesis is tested against several alternatives
simultaneously, leading to joint tests. This section concentrates on the multivariate version of one ofthese procedures, namely the BAM test of Barten and McAleer (1997).
Consider m non-nested non-linear systems of equations with different non-linear datatransformations on the dependent variable yi, i=1,…n, as follows:8H1: h1i(yi) = g1i(X1i,b1i) + u1i(3.14)

H2: h2i(yi) = g2i(X2i,b2i) + u2i•••
Hm: hmi(yi) = gmi(Xmi,bmi) + umi
where uji is NID(0,σ2ji) for j=1,…m. Suppose that H1 is, in the first instance, chosen as the null
hypothesis. It is then possible to combine the disturbances in (3.14) as follows:m
1 − ∑ λ ji u1i + λ 2 i u 2 i + ... + λ mi u mi = u i .


j =2



(3.15)

Under the null hypothesis H1: λ 21 = ... = λ 2 n = λ 31 = ... = λ 3n = ... = λ m1 = ... = λ mn = 0 , u1i=ui somthat the first model given by H1 is valid. Setting µ ji = − λ ji 1 − ∑ λ ji  , (3.15) can be rewritten
j =2



as:

m

[

]

h1i ( y i ) = g i1 ( X i1 , bi1 ) + ∑ µ ji h ji (⋅) − g ji (⋅) + v i
j=2(3.16)mwhere vi = u i 1 − ∑ λ ji  . Testing the null hypothesis (3.16) involves verifying if the µji arej=2


jointly zero. Unfortunately, a problem with this procedure is that the parameters µji are not
identified. It is possible to resolve the problem by extending McAleer’s (1983) univariate joint
testing procedure in the following manner. First, since h1i(yi) = g1i(X1i,b1i) + u1i under the null
hypothesis H1, replace yi in hji(yi), j=2,...,m, with:[ (yˆ 1i = h1−i 1 g 1i X 1i , bˆ1i)]

(3.17)

9where bˆ1i is the generalized least squares estimate of b1i under the null hypothesis H1. It is worth
noting that yˆ 1i under H1 is asymptotically uncorrelated with u1i, and hence with vi. Second, estimate
the auxiliary system:h ji ( yˆ i1 ) = g ji ( X ji , b ji ) + η1 ji (j=2,...m, i=1,…,n)(3.18)

to obtain the residuals(ηˆ1 ji = h ji ( yˆ i1 ) − g ji X ji , bˆ1 ji)(3.19)

where bˆ1 ji are the generalized least squares estimates of bji from (3.18) and gji(·) includes anintercept term. Third, use the residuals in (3.19) to compute the following modification of (3.16):m

h1i ( y i ) = g 1i ( X 1i , b1i ) + ∑ µ jiηˆ1 ji + v i1 .

(3.20)

j=2

Finally, upon estimating the parameters in (3.20), test the extent to which the residuals η̂1 ji in(3.20) contribute to the empirical performance of H1 through a standard Wald-type test.

4. Empirical evidence on factor demand systems
In order to illustrate the usefulness of the non-nested tests presented above, the systems of factor
demands introduced in Section 2 are estimated using Berndt and Khaled's (1979) classical annual
data set for the U.S. manufacturing sector over the period 1947-1971. It is assumed that U.S.
manufacturing can be described by a regular aggregate production function relating the flows ofgross output Y to the services of four inputs, namely capital (K), labour (L), energy (E) andmaterials (M). Corresponding to such a production function, there exists a dual cost function
summarizing all the characteristics of the representative firm's technology.
When output quantity and input prices are exogenous, the dual cost function can be written as:

10C = C (Y , Pk , Pl , Pe , Pm , t )(4.1)

where C(·) represents total input costs, Pi, i=K,L,E,M, are the factor prices, and t is an index oftechnical progress.
For purposes of empirical implementation, the existence of random errors in the cost minimizing
behaviour of the firm is such that each equation in each demand system has an additive disturbanceterm which reflects the firm’s errors in deciding the optimal level of inputs. First-order serial
correlation for each system is accommodated using a Cochrane-Orcutt transformation for eachequation (see, e.g., Berndt, 1991, pp. 476-9). The estimated single-equation serial correlation
coefficients have been used to estimate the system.
The Translog system (TLG) comprises the cost equation (4.1) specified by the functional form
(2.1), and the three share equations (2.3) for labour, energy and materials, in addition to the linear
homogeneity restrictions (2.2) and symmetry conditions. It is well known that only n-1 shareequations are estimated because the four cost shares (2.3) sum to unity, so that the sum of the
disturbances across the four equations is zero for each observation. Consequently, the covariancematrix is singular and non-diagonal (Berndt and Wood, 1975, p. 261). The disturbance from the Shk
equation is omitted and the vector u comprising the disturbances of the remaining share equations
and the cost function is specified as a multivariate normal distribution with E(u)=0 and E(uu')=Ω,
where Ω is constant over time (Diewert and Wales, 1987, p. 58).
The Quadratic demand system is given by the cost equation (4.1), specified by equation (2.6) andthe three demand equations (2.7) for labour, energy and materials. As already noted, the reason for
excluding the capital equation is that, since linear homogeneity in prices cannot be imposedparametrically, a normalization with respect to an arbitrarily chosen factor price is required. QDR
denotes the Quadratic demand system with linear homogeneity in prices imposed, NHQDR1
denotes the same system with non-homogeneity, that is, when linear homogeneity in prices is not
imposed, and NHQDR denotes the Quadratic demand system formed from the four demand
equations for capital, labour, energy and materials without imposing linear homogeneity in prices.
Finally, the Symmetric Generalized McFadden demand system (SGM) is formed from the four
equations (2.9), and the Generalized Leontief model (GLT) is given by equation (2.11). In both the11

SGM and GLT specifications, the dependent variables are input levels divided by output, as thismakes the assumption of homoskedasticity of the disturbances more plausible. The cost function is
not estimated since it does not contain any additional information.
The main characteristics of the factor demand systems used in the empirical application aresummarized in Table 1. All systems are estimated with the multivariate least squares routine Lsq
implemented in Tsp 4.4 (for details, see Hall, Cummins and Schnake, 1997). Linear disembodied
technical change is accommodated by the presence of linear and quadratic trends in the estimatedequations. The estimated parameters are reported in Appendix 1, and Appendix 2 shows the resultsof some diagnostic statistics for the estimated models.
[Table 1, Appendix 1 and Appendix 2]
In particular, Appendix 2 indicates the absence of both first-order serial correlation andheteroskedasticity, and that the curvature properties of the firm’s cost function are satisfied for eachfunctional form.
Factor demand systems are typically used to calculate indicators which can be useful for
describing the production structure of an economic sector. For each estimated model, Appendix 3
reports the mean values of input demand elasticities with respect to input prices and output over theperiod 1948-1971.
[Appendix 3]
The magnitudes and signs of these elasticities depend crucially on the selected model. This isparticularly true for the price elasticities of capital, which are roughly comparable for GLT, SGM,NHQDR and TLG, but appear quite different for NHQDR1 and QDR, that is, for the Quadratic
functional form where the demand for capital is not directly estimated. In general, direct priceelasticities are negative and output elasticities are positive, as suggested by theory, and the crossprice elasticities are all below one in absolute value. From their signs, it is possible to obtain
information about factor substitution and complementarity, which is also not independent of thefunctional form. For example, capital and energy are complements, according to GLT, NHQDR1,
SGM and TLG, but substitutes on the basis of NHQDR and QDR. Capital and labour are substitutes
12for all models, except for NHQDR and NHQDR1. Labour and materials are complements according
to only GLT, SGM and NHQDR1. Energy and materials are substitutes in all systems, with the
exception of NHQDR1. Finally, materials and labour are complements for GLT, NHQDR1, SGM
and TLG, whereas they are substitutes for NHQDR and QDR. In summary, all estimated systems
seem to offer plausible interpretations of the production structure of the US manufacturing sector
over the period 1948-1971, but the interpretations depend on the chosen specification. Systems
QDR and NHQDR suggest the existence of substitution and complementarity relationships among
the factors, which do not agree, in general, with the indications of the other models.

5. Empirical evidence on multivariate non-nested tests
Alternative factor demand systems based on competing non-nested flexible functional forms were
estimated in Section 4. Each model was shown to be statistically adequate and to capture therelevant features of the data. In addition, the conclusions drawn in terms of price and output
elasticities were not unique, and depended crucially on the chosen model. Thus, economic theory isof little assistance in discriminating among the competing models, and the empirical evidence
suggests that the in-sample performance of each model is acceptable. Moreover, the choice of model
has important implications for economic analysis. In this case, non-nested testing procedures canprovide useful additional information. The factor demand systems of Sections 2 and 4 are comparedin this section on the basis of the multivariate paired and joint non-nested tests illustrated in Section
3.
In Table 2 the results of preliminary systems RESET tests for each competing model are reported.
These tests are calculated by adding the corresponding squared fitted values to each equation of thesystem and by testing their joint significance using a Wald statistic. Two versions of the systems
RESET test are presented. The first test is calculated under the condition that the coefficients of thesquared fitted values in each equation are different, leading to a Wald statistic with a χ2(n)distribution, where n is the number of equations in the system (in our case, n=4). The second
version of the test is based on the condition that the coefficients of the squared fitted values in eachequation are identical, and is a Wald test with a χ2(1) distribution. Rejection of the null hypothesisof correct model specification by the χ2(n)-RESET test is interpreted as misspecification of at least13

one equation of the system, but not necessarily of the whole system, whereas rejection of the null
hypothesis by the χ2(1)-RESET test indicates that the system itself is misspecified.
[Table 2]
The results show that NHQDR and NHQDR1 are rejected at the 1% significance level by boththe χ2(4)- and χ2(1)-RESET tests, and QDR is rejected by both versions of the test, but at different
levels of significance. Specification TLG is rejected at the 1% significance level by the χ2(4)RESET test, but is not rejected by the χ2(1)-RESET test. This is a contradiction, since the test
indicates that functional form misspecification affects a sub-set of the system, but that the systemitself does not suffer from misspecification. Finally, GLT and SGM are rejected only at the 5%
significance level and only by the χ2(4)-RESET test. Thus, the systems RESET tests are incapableof determining a single model which performs best, although NHQDR and NHQDR1 appear to be
more problematic than the others. This last evidence is in line with the empirical results of Section 4.
Table 3a shows the results obtained by comparing non-nested systems of equations with the samedependent variables, namely SGM and GLT on the one hand, and QDR and NHQDR1 on the other.
The three different paired non-nested tests used in the empirical application are the J, P0 and P1
tests, as discussed in Section 3, which are implemented according to equations (3.3), (3.4) and (3.5),
respectively. Each test is presented in two versions, with the χ2(4)-version being based on thecondition that the coefficients λi (i=1,…,4) are different from each other, whereas the χ2(1)-version
imposes the condition that λ1=…=λ4=λ. The interpretation of the non-nested tests is analogous tothose of the systems RESET tests. Rejection of the null hypothesis of correct specification by theχ2(4)-test is interpreted as misspecification of at least one equation of the system, but not necessarilyof the whole system, against the chosen non-nested alternative. Conversely, a rejection of the null
hypothesis by the χ2(1)- test indicates that the system itself is rejected against its non-nested
counterpart.
[Table 3a]
The SGM system is rejected against GLT at the 1% significance level by the χ2(4)-version ofeach of the J, P0 and P1 tests. Each non-nested test suggests that there is a problem in at least one
equation of the SGM system when compared with its GLT counterpart. The empirical evidence ismixed when the χ2(1)-version of the test is considered as, in this case, SGM is rejected at the 1%
14

level only by the P0 test, whereas the J test rejects SGM at the 5% level, and the P1 test suggests
non-rejection of SGM. In particular, the P1 test denotes an inconsistency as the χ2(1)-version
implies that SGM as a whole is correctly specified as compared with its GLT counterpart. When theroles of the null and alternative are reversed, GLT is rejected at the 5% level by the χ2(4)-version of
the J and P0 tests, whereas the P1 test rejects GLT at the 1% level. The behaviour of the J and P1
tests is contradictory, as the χ2(1)-version of these tests suggests that there are no problems with thespecification of the GLT system as a whole, whereas the χ2(1)-version of the P0 test rejects the nullat the 5% level. In summary, the multivariate paired non-nested tests suggest that both the SGM andGLT specifications suffer from problems of misspecification.
In Table 3a, the second pair of competing models is given by QDR and NHQDR1. When QDR isthe null hypothesis, it is strongly rejected by all three non-nested tests, under both the χ2(4)- andχ2(1) versions. However, when NHQDR1 is the null, only the P1 test strongly rejects NHQDR1,
regardless of which version of the test is used. The J test marginally rejects NHQDR1 with theχ2(4)-version, whereas it does not reject the null with the χ2(1)-version. The P0 test does not reject
NHQDR1 at all. In this case, the results from these multivariate paired non-nested tests areinterpreted as indirect evidence against the hypothesis of linear homogeneity in input prices.
Table 3b reports the results from the comparison of pairs of systems of equations with different
dependent variables, where the competing pairs of systems are TLG and QDR, SGM and NHQDR,
GLT and NHQDR, and TLG and NHQDR1. The multivariate non-nested tests used are the PE test(see equation (3.6)) and the BEM test (see equations (3.12) and (3.13)).
[Table 3b]Both the χ2(4)- and χ2(1)-versions of the tests are presented. In general, all competing models are
rejected by all tests, although there are a few interesting cases. When TLG and QDR are comparedusing the χ2(1)-version of the PE test neither is rejected. Moreover, the χ2(1)-version of both the PE
and BEM tests is unable to reject TLG against the alternative system NHQDR1. One possible
interpretation is that the χ2(1)-version of the test is too restrictive to detect specification problems
which are likely to affect only a sub-set of each system. A similar comment applies in testing TLG
against NHQDR1 using the χ2(1)-version of both the PE and BEM tests.

15

Tables 4a-4b present the empirical results obtained by using the multivariate joint BAM test (seeequation (3.20)). The three competing systems are NHQDR, SGM and GLT, and each null model istested jointly against two alternatives. Four versions of the BAM test are available, according to therestrictions placed on the λji (j=1,2; i=1,…,4) in equation (3.20). If the λij are unrestricted, thisyields the χ2(8)-version of the BAM test, with four parameters across each of two alternative
systems. The χ2(4)-version is obtained when the following restrictions hold: λ11=λ21, λ12=λ22,
λ13=λ23, λ14=λ24, with the same coefficient for the two non-nested alternatives across each of thefour equations. The χ2(2)-version is given by the following restrictions: λ11=λ12=λ13=λ14,
λ21=λ22=λ23=λ24, with the same coefficient for each equation across each of two non-nested
alternatives. Finally, the χ2(1)-version of the BAM test is obtained by imposing the following
restrictions: λ11=λ21=λ12=λ22=λ13=λ23=λ14=λ24=λ, with the same coefficient across four equations
and two alternatives. Notice that the χ2(8)- and χ2(2)-versions do not impose restrictions across thealternative models, whereas the χ2(4)- and χ2(1)-versions do impose cross-alternative restrictions.
[Tables 4a and 4b]
In general, all three models are strongly rejected, regardless of which version of the test is used. As
before, there are a few cases worth highlighting. When the χ2(2)-version of the BAM test is used,
TLG is not rejected against NHQDR1 and QDR jointly. Thus, the two alternative systems, eachconsidered as a whole, do not add useful information to the TLG null model as a whole. When theχ2(1)-version of the test is used, neither QDR nor TLG is rejected against the other two models
jointly. In this case, the added information given by a linear combination of both equations andalternatives is empirically irrelevant. Finally, if the χ2(4)-version of the test is considered, the TLG
system is rejected only at the 5% significance level, whereas all other models are rejected at the 1%
level. In summary, these results suggest that the information contained in a linear combination of
the corresponding equations across the two alternatives is statistically important in rejecting eachmodel against two non-nested alternatives jointly, at least at the 5% level.

6. Conclusion
The key points of this paper can be summarized as follows. Alternative factor demand systems havebeen presented using some of the most popular flexible functional forms, namely Translog,
16

Quadratic, Generalized Leontief and Symmetric Generalized McFadden. Each system has been
estimated using Berndt and Khaled's (1979) classical annual data set for the U.S. manufacturingsector over the period 1947-1971. The important task of model determination has beenaccomplished using a formal non-nested testing procedure. Multivariate extensions of some wellknown and pedagogically appealing univariate paired non-nested tests (namely, the J, P0 and P1
tests of Davidson and MacKinnon, 1981; the PE statistic of MacKinnon, White and Davidson,
1983; and the Bera and McAleer (1989)), and the multivariate joint non-nested test of Barten and
McAleer (1997), have been applied to compare alternative factor demand systems. Preliminary
systems RESET tests for each competing model have also been reported. Systems RESET tests andmultivariate paired non-nested tests were each presented in two versions, namely χ2(4) and χ2(1).
Four versions of the multivariate joint non-nested test were developed, namely χ2(8), χ2(4), χ2(2)and χ2(1).
The main results are as follows. Each model has been shown to be statistically adequate and to
capture the relevant features of the data. In addition, the conclusions drawn in terms of price andoutput elasticities were not unique, and depended crucially on the chosen model. Systems RESET
tests were incapable of determining a single model which performs best, although NHQDR and
NHQDR1 appeared to be more problematic than the others. The multivariate paired extensions of
the J, P0 and P1 tests suggested that both the SGM and GLT specifications suffered from problemsof misspecification. When the same tests were used to compare QDR and NHQDR1, the results
were interpreted as indirect evidence against the hypothesis of linear homogeneity in input prices.
Systems involving different dependent variables were compared using the multivariate non-nested
PE and BEM tests. In general, all competing models were rejected by all tests, although there were a
few interesting cases. When TLG and QDR were compared using the χ2(1)-version of the PE test,
neither was rejected. Moreover, the χ2(1)-version of both the PE and BEM tests was unable to reject
TLG against the alternative system NHQDR1. Finally, the multivariate joint BAM test was used tocompare systems NHQDR, SGM and GLT. In general, the results suggested that the information
contained in a linear combination of the corresponding equations across the two alternatives was
statistically important in rejecting each model against two non-nested alternatives jointly.

17

References
Barten, A.P. and M. McAleer (1997), "Comparing the empirical performance of alternative demandsystems", L’Actualité Économique, 73, 27-45, reprinted in Économétrie Appliquée, ed. by C.
Gouriéroux and C. Montmarquette, Paris, Éditions Economica, 27-45.
Bera, A.K. and M. McAleer (1989), "Nested and non-nested procedures for testing linear and loglinear regression models", Sankhya B, 51, 212-224.
Berndt, E.R. (1991), The Practice of Econometrics: Classic and Contemporary, Reading, AddisonWesley, 449-506.
Berndt, E.R. and L.R. Christensen (1973), "The translog function and the substitution of equipment,
structures and labor in U.S. manufacturing 1929-68", Journal of Econometrics, 1, 81-114.
Berndt, E.R. and M.S. Khaled (1979), "Parametric productivity measurement and choice among
flexible functional forms", Journal of Political Economy, 87, 1220-1245.
Berndt, E.R. and D.O. Wood (1975), "Technology, prices and the derived demand for energy",
Review of Economics and Statistics, 57, 259-268.
Davidson, R. and J.G. MacKinnon (1981), "Several tests for model specification in the presence ofalternative hypotheses", Econometrica, 49, 781-793.
Davidson, R. and J.G. MacKinnon (1982), "Some non-nested hypothesis tests and the relations
among them", Review of Economic Studies, 49, 551-565.
Diewert, W.E. and T.J. Wales (1987), "Flexible functional forms and global curvature conditions",
Econometrica, 55, 43-68.
Hall, B.H., Cummins C. and R. Schnake (1997), Time Series Processor. Version 4.4. User's Guide,
Stanford, TSP International.
McAleer, M. (1983), "Exact tests of a model against non-nested alternatives", Biometrika, 70, 285288.
McFadden, D. (1978), "The general linear profit function", in Production Economics: a DualApproach to Theory and Applications, ed. by M.A. Fuss and D.L. McFadden, Amsterdam,
North-Holland, 409-453.
MacKinnon, J.G., White, H. and R. Davidson (1983), "Tests for model specification in the presenceof alternative hypotheses: Some further results", Journal of Econometrics, 21, 53-70.
Morey, E.R. (1986), "An introduction to checking, testing and imposing curvature properties: Thetrue function and the estimated function", Canadian Journal of Economics, 19, 207-235.

18

Table 1. Factor demand systems
SystemEquations

Transformations on the dependent variables

GLT

4 factor demands (K,L,E,M)

Ratios of levels (K/Y,L/Y,E/Y,M/Y)

NHQDR

4 factor demands (K,L,E,M)

Levels (K,L,E,M)

NHQDR1

1 cost function (C)3 factor demands (L,E,M)

Levels (C)Levels (L,E,M)

QDR

1 cost function (C)3 factor demands (L,E,M)

Levels (C)Levels (L,E,M)

SGM

4 factor demands (K,L,E,M)

Ratios of levels (K/Y,L/Y,E/Y,M/Y)

TLG

1 cost function (C)3 factor demands (L,E,M)

Logarithms (C)Shares (PlL/C,PeE/C,PmM/C)

19

Table 2. Systems RESET tests
SystemGLT
NHQDR
NHQDR1
QDR
SGM
TLG

χ2(4)
11.427*(0.022)85.996**(0.000)160.788**(0.000)20.202**(0.000)11.647*(0.020)15.676**(0.003)

χ2(1)
3.395
(0.065)32.553**(0.000)116.171**(0.000)5.915*(0.015)3.425(0.064)0.915(0.339)Notes: The systems RESET test is calculated by adding the squared fitted values to each equation of the system and by
testing their joint significance under the assumptions that each coefficient attached to the fitted values is different (χ2(4))
or identical (χ2(1)) across all equations; * denotes rejection of the null hypothesis of correct functional form at the 5%
significance level; ** denotes rejection of the null hypothesis of correct functional form at the 1% significance level; Pvalues are given in parentheses.

20

Table 3a. Multivariate paired non-nested tests: J, P0 and P1
Test
J

P0

P1

H0
SGM

H1
GLT

GLT

SGM

QDR

NHQDR1NHQDR1

QDR

SGM

GLT

GLT

SGM

QDR

NHQDR1NHQDR1

QDR

SGM

GLT

GLT

SGM

QDR

NHQDR1NHQDR1

QDR

χ2(4)16.103**(0.003)12.818*(0.012)254.161**(0.000)9.802*(0.044)25.140**(0.000)11.843*(0.019)201.006**(0.000)8.223(0.084)34.884**(0.000)23.564**(0.000)2191.842**(0.000)207.978**(0.000)χ2(1)6.157*(0.013)3.0005(0.083)250.495**(0.000)1.452(0.228)7.089**(0.008)3.904*(0.048)196.511**(0.000)1.513(0.219)3.167(0.075)1.457(0.227)2172.844**(0.000)203.181**(0.000)Notes: * denotes rejection of the null hypothesis H0 against the paired alternative H1 at the 5% significance level; **denotes rejection of the null hypothesis H0 against the paired alternative H1 at the 1% significance level; P-values are
given in parentheses.

21

Table 3b. Multivariate paired non-nested tests: PE and BEM
Test
PE

BEM

H0
TLG

H1
QDR

QDR

TLG

SGM

NHQDR

NHQDR

SGM

NHQDR

GLT

GLT

NHQDR

TLG

NHQDR1NHQDR1

TLG

TLG

QDR

QDR

TLG

SGM

NHQDR

NHQDR

SGM

NHQDR

GLT

GLT

NHQDR

TLG

NHQDR1NHQDR1

TLG

χ2(4)43.533**(0.000)22.394**(0.000)114.973**(0.000)21.583**(0.000)40.532**(0.000)130.980**(0.000)23.299**(0.000)34.810**(0.000)10.419*(0.034)44.825**(0.000)74.299**(0.000)30.161**(0.000)45.783**(0.000)61.417**(0.000)51.168**(0.000)307.541**(0.000)

χ2(1)
0.134
(0.714)1.486(0.223)103.424**(0.000)9.397*(0.002)
13.489**(0.000)112.255**(0.000)0.488(0.485)8.908**(0.003)2.880(0.090)43.371**(0.000)71.625**(0.000)14.297**(0.000)17.621**(0.000)52.231**(0.000)0.144(0.705)50.291**(0.000)Notes: * denotes rejection of the null hypothesis H0 against the paired alternative H1 at the 5% significance level; **denotes rejection of the null hypothesis H0 against the paired alternative H1 at the 1% significance level; P-values are
given in parentheses.

22

Table 4a. Multivariate joint non-nested test: BAM
H0
NHQDR

H1
SGM

H2
GLT

SGM

NHQDR

GLT

GLT

NHQDR

SGM

QDR

NHQDR1TLG

TLG

NHQDR1QDRNHQDR1

TLG

QDR

χ2(8)
136.063**(0.000)102.234**(0.000)74.306**(0.000)52.809**(0.000)72.120**(0.000)377.083**(0.000)χ2(2)22.387**(0.000)71.896**(0.000)52.565**(0.000)42.687**(0.000)3.712(0.156)51.890**(0.000)Notes: * denotes rejection of the null hypothesis H0 against the multiple alternatives H1 and H2 at the 5% significancelevel; ** denotes rejection of the null hypothesis H0 against the multiple alternatives H1 and H2 at the 1% significancelevel; P-values are given in parentheses.

23

Table 4b. Multivariate joint non-nested test: BAM
H0
NHQDR

H1
SGM

H2
GLT

SGM

NHQDR

GLT

GLT

NHQDR

SGM

QDR

NHQDR1TLG

TLG

NHQDR1QDRNHQDR1

TLG

QDR

χ2(4)
37.245**(0.000)74.236**(0.000)61.389**(0.000)26.982**(0.000)11.393*(0.022)23.186**(0.000)χ2(1)15.928**(0.000)71.601**(0.000)52.239**(0.000)0.365(0.546)1.121(0.290)9.582**(0.002)

Notes: * denotes rejection of the null hypothesis H0 against the multiple alternatives H1 and H2 at the 5% significancelevel; ** denotes rejection of the null hypothesis H0 against the multiple alternatives H1 and H2 at the 1% significancelevel; P-values are given in parentheses.

24Appendix 1. Factor demand systems: Estimation
Parameters
α0

GLT
-

NHQDR
-

αk

-0.288(0.353)αy0.619**(0.110)0.196*(0.077)0.761**(0.093)0.012(0.043)
-

αt

-

-

αkk

αky

0.003**(0.0007)0.002**(0.0003)-0.0007(0.0004)-0.0007**(0.0002)
-

αkt

-

αll

-0.00002(0.0007)0.003**(0.0008)-0.0006(0.0004)--0.037(0.075)0.211**(0.047)0.088(0.050)-0.075*(0.034)0.001**(0.0003)-0.038*(0.017)-0.391**(0.058)0.662**(0.063)0.026(0.050)0.003**(0.0003)
-

αl
αe
αm

αkl
αke
αkm

αle
αlm
αly
αlt
αee
αem
αey

-0.003(0.004)-0.003**(0.001)0.002**(0.0006)--0.033(0.140)-0.058(0.104)--0.195(0.120)0.236**(0.083)0.0009**(0.0003)

NHQDR145.882(29.815)241.786**(62.258)
0.983**(0.157)1.893**(0.218)1.150**(0.130)0.045(0.168)-QDR-18.090(59.415)--566.152**(86.485)-0.063(0.073)-0.382**(0.106)-0.251**(0.071)0.777**(0.183)8.055**(1.379)-0.635**(0.091)0.195*(0.079)-0.162**(0.054)0.003**(0.0004)
0.030**(0.008)-0.770**(0.114)-0.110*(0.055)0.0008(0.0005)-25

0.575**(0.125)1.014**(0.129)-0.008(0.058)0.960**(0.147)-SGM
0.652**(0.111)0.209**(0.078)0.790**(0.092)-0.007(0.04)-

TLG
0.748**(0.088)0.411**(0.078)0.198**(0.016)0.127(0.098)0.840**(0.015)--0.003**(0.0004)
-

-

-

-

-

-

-

-

-

-

-

-

-

-

-0.739**(0.073)0.303**(0.064)0.242**(0.052)0.003**(0.0004)
0.012*(0.005)-0.625**(0.118)0.104(0.089)0.001*(0.0005)

0.004**(0.0004)-

0.116*(0.046)0.007(0.010)-0.126*(0.050)-0.030*(0.015)0.0002(0.001)0.011*(0.005)-0.013(0.008)-0.029**(0.003)-0.005(0.004)0.001*(0.0005)--

Appendix 1. Factor demand systems: Estimation (continued)
Parameters
αet
αmm
αmy

GLT
0.023**(0.006)0.004**(0.0006)
-

NHQDR
-

QDR
0.026**(0.006)-0.235*(0.099)0.004**(0.0001)-0.004(0.002)-

SGM
0.021**(0.006)0.005**(0.0002)--

NHQDR1
0.054**(0.008)-0.533**(0.083)0.005**(0.0004)
0.025**(0.005)-0.003(0.002)-

TLG
0.0007**(0.0002)0.180**(0.057)
0.098**(0.019)-0.002(0.001)--0.046(0.106)0.004**(0.0002)
-

αyy

0.002
(0.002)
-

αyt

-

-

-

-

-

-

αtt

-

-

-

-

-

-

σll

-

-

-

-

-

σle

-

-

-

-

σlm

-

-

-

-

σee

-

-

-

-

σem

-

-

-

-

σmm

-

-

-

-

ρc

-

-

0.471

0.855

-0.009**(0.002)0.008**(0.002)-0.004**(0.001)-0.015**(0.002)0.008**(0.002)-0.001(0.002)
-

0.537

ρk
ρl
ρe
ρm

0.468
0.405
0.464
-0.1340.728
0.092-0.140-0.149

0.492
0.227
0.260

0.717
0.405
-0.162

0.486
0.458
0.456
-0.112-0.097
0.203
0.014

αmt

-

Notes: * denotes rejection of the null hypothesis of a zero coefficient at the 5% significance level; ** denotes rejectionof the null hypothesis of a zero coefficient at the 1% significance level; ρi (i=c,k,l,e,m) are the single-equation serial
correlation coefficients; standard errors are given in parentheses.

26Appendix 2. Factor demand systems: Diagnostic statistics
Statistic
R2 c
R2 k
R2 l
R2 e
R2 m
HETc

GLT
0.676
0.963
0.820
0.806
-

NHQDR
0.994
0.977
0.992
0.998
-

HETk

1.029
[0.310]0.350[0.554]0.004[0.952]0.166[0.684]-0.029[0.864]1.175[0.278]3.034[0.082]1.055[0.304]-0.354[0.095]0.084[0.697]-0.006[0.977]0.062[0.771]24
00.326[0.111]-0.122[0.571]-0.204[0.337]-0.160[0.449]
24
0

HETl
HETe
HETm
ARc
ARk
ARl
ARe
ARm
NOB
Concavity
violations

NHQDR1
0.993
0.972
0.985
0.991
0.012
[0.914]
-

QDR
0.955
0.956
0.984
0.996
5.989
[0.014]-0.424[0.515]0.012[0.914]0.127[0.721]0.182[0.435]-0.632[0.427]2.420[0.120]2.034[0.154]0.419[0.091]-

0.182[0.388]0.092[0.660]0.164[0.505]24
00.281[0.210]0.280[0.190]0.371[0.089]
24
0

SGM
0.691
0.961
0.830
0.793
1.078
[0.299]0.070[0.791]0.007[0.934]0.023[0.880]0.354[0.096]0.068[0.753]-0.019[0.928]0.022[0.918]
24
0

TLG
0.999
0.832
0.947
0.815
0.746
[0.388]0.283[0.594]0.613[0.433]0.142[0.706]0.182[0.410]0.068[0.749]0.078[0.719]0.220[0.314]240Notes: R2i (i=c,k,l,e,m) values are computed as the squared correlation coefficients of actual and fitted values of thedependent variables for each equation; HETi are LM-type heteroskedasticity tests calculated by regressing the squared
residuals on a constant term and the squared fitted values of the dependent variables for each equation; ARi are the serial
correlation coefficients for each equation, namely the estimated coefficients of the regression of the residuals from eachequation on their one-period lagged counterparts; NOB indicates the total number of observations; Concavity violations
refer to the number of principal minors which do not satisfy the conditions for the Hessian matrix of the cost function to
be negative semi-definite; P-values are given in parentheses.

27Appendix 3. Factor demand systems: Price and output elasticities (mean values)
Elasticities
εkk
εkl
εke
εkm
εlk
εll
εle
εlm
εek
εel
εee
εem
εmk
εml
εme
εmm
εky
εly
εey
εmy

GLT
-0.071(0.018)0.196(0.022)-0.064(0.003)-0.062(0.003)0.184(0.015)-0.441(0.019)0.317(0.009)-0.060(0.003)-0.064(0.006)0.337(0.040)-0.495(0.047)0.222(0.013)-0.068(0.005)-0.070(0.007)0.246(0.003)-0.107(0.011)0.656(0.065)0.877(0.038)0.314(0.075)0.977(0.005)NHQDR-0.025(0.003)0.207(0.005)0.068(0.013)-0.055(0.008)0.195(0.015)-0.541(0.060)0.713(0.079)0.027(0.002)0.067(0.009)0.748(0.014)-0.175(0.036)0.201(0.031)-0.061(0.008)0.032(0.001)0.225(0.045)-0.042(0.006)0.220(0.015)0.796(0.096)0.181(0.009)0.887(0.024)NHQDR1-2.222(0.434)-0.0004(0.00004)-0.002(0.0003)-0.001(0.0002)-0.059(0.005)-0.879(0.101)0.210(0.025)-0.167(0.013)-0.293(0.046)0.221(0.011)-0.692(0.152)-0.094(0.016)-0.205(0.031)-0.196(0.012)-0.105(0.022)-0.485(0.077)0.787(0.088)0.816(0.099)0.168(0.010)0.967(0.031)

QDR-0.002(0.0006)
0.002(0.0007)0.0008(0.0008)-0.0008(0.0002)
0.374(0.091)-0.882(0.135)0.289(0.085)0.219(0.054)0.125(0.115)0.299(0.064)-0.504(0.202)0.079(0.028)-0.151(0.020)0.250(0.045)0.088(0.032)-0.188(0.059)0.994(0.075)0.783(0.101)
0.236(0.012)0.886(0.028)SGM-0.062(0.017)0.219(0.043)-0.068(0.025)-0.090(0.005)0.195(0.036)-0.389(0.055)0.315(0.018)-0.122(0.012)-0.054(0.025)0.361(0.045)-0.523(0.024)0.215(0.018)-0.078(0.003)-0.110(0.011)
0.246(0.027)-0.058(0.016)0.638(0.067)0.884(0.046)0.306(0.081)0.980(0.011)

TLG-0.120(0.067)0.331(0.010)-0.062(0.009)-0.148(0.060)0.064(0.004)-0.301(0.006)0.069(0.004)0.167(0.009)-0.074(0.009)0.427(0.019)-0.696(0.014)0.343(0.025)-0.012(0.004)0.074(0.008)0.025(0.003)-0.086(0.006)0.100(0.061)
0.724(0.003)0.186(0.048)0.991(0.004)

Notes: εsw denotes the demand elasticity of input s to the price of input w; εsy indicates the demand elasticity of input swith respect to output y; mean values are calculated over the period 1948-1971; standard deviations are reported inparentheses.

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