ESTIMATING THE PRODUCTIVITY OF PUBLIC INFRASTRUCTURE USING MAXIMUM ENTROPY ECONOMETRICS
Preliminary version: October 25, 2005
Jorge RODRIGUEZ-VALEZ *
Department of Economics-University of Leon (Spain) The Lawrence R. Klein Centre at Autonoma de Madrid University
(jorge.rodriguez@unileon.es) Antonio ALVAREZ
Department of Economics-University of Oviedo (Spain) (alvarez@uniovi.es)
Carlos ARIAS
Department of Economics-University of Leon (Spain) (deecas@unileon.es)
Esteban FERNANDEZ
Department of Applied Economics-University of Oviedo (Spain) (evazquez@uniovi.es)
ABSTRACT
The empirical analysis of public infrastructure productivity has been limited by several econometric problems. These problems are probably the cause of the great variability in results reported in the literature. One of these econometric problems is the existence of multicollinearity when infrastructure is included as an input in a production function. In this paper, we propose to deal with multicollinearity by using a priori information on parameter values. For this purpose, we explore the use of maximum entropy econometrics to estimate the parameters of a regional production function in Spain. Our results show that infrastructure plays an important role on production. This result is not affected by quite large changes in priors on parameter values.
Keywords: Infrastructures, productivity, maximum entropy, multicollinearity Jel Codes: C13, C51, H54, R11, O47.
(*) Facultad de Economicas, Campus de Vegazana, 24071-Leon (Spain); Phone 34 987 291 719.
1. INTRODUCTION
Empirical analysis in economics is very often hampered by the characteristics of data.
For example, when there is a high correlation among explanatory variables in a model.
In fact, multicollinearity produces high sensitivity of the estimates to the specification of the model, low levels of significance, and unreasonable estimates in sign or magnitude [Greene (2000), p. 256]. Under these circumstances, it is reasonable to claim that results are of little if any value.
The high correlation between private and public capital causes multicollinearity in the estimation of regional production functions which include both types of capital as inputs. Ai and Cassou (1997) and Vijverberg, Vijvergerg and Gamble (1997) point to multicollinearity (together with other econometric problems that sometimes have not been fully addressed in empirical work) as the cause of the disparity of estimates obtained in the literature. In fact, a non negligible number of papers in the literature fail to find evidence in favour of a positive contribution of infrastructure to growth [Holtz- Eakin (1994), Baltagi and Pinnoi (1995), Garcia-Milà, McGuire and Porter (1996)]
while others find clear evidence of such contribution [Aschauer (1989), Munnell (1990) and Garcia-Milá and McGuire (1992)]1.
Goldberger (1991, p. 246) shows the similarities between multicollinearity in multivariate analysis and a problem of small sample in a univariate population. In other words, he sees multicollinearity as a problem of lack of data to estimate the parameters of a model. In this setting, it is reasonable to tackle the problem using out of sample information in the form of “priors” on parameter values. The use of this type of information is the basis of the estimation by maximum entropy (ME).
In this paper, we estimate an aggregate production function by ME to obtain evidence on the contribution of public capital to output growth in the Spanish regions. Since the estimation by maximum entropy uses priors on parameter values, it is important to check the influence of these priors in the estimates. Our results show that the estimates change only moderately under quite extreme changes in the priors. In fact, in all cases, the output elasticity of public capital is always larger than 0.16.
1 For a survey of this large literature seee Gramlich (1994) and de la Fuente (2002).
In the present paper, we use an innovative estimation method (ME) and compare the results obtained with the estimates provided by more conventional econometric methods. Additionally, an original contribution is the exploration of the opportunities that the particular characteristics of the method (use of priors and sensitivity of estimates to changes in priors) provide for the empirical analysis of the productivity of public capital. The results of this exercise point out to ME as a method that provides a venue for empirical analysis when multicollinearity is an issue.
The remainder of this paper is organized as follows. In section 2, we discuss the main features of maximum entropy econometrics. In section 3, we specify a production function estimable by ME. In section 4, we describe the data and present the results of the estimation by ME. The paper ends with some preliminary conclusions.
2. MAXIMUM ENTROPY ECONOMETRICS: AN OVERVIEW
We want to estimate the parameters of a linear model in which a variable y depends on H explanatory variablesxh:
e X
y= + [1]
where y is a
(
7 ×)
vector of observations, X is the(
T×H)
matrix of explanatory variables for each observation, is the(
H×1)
vector of parameters to be estimated, and e is a(
7 ×)
vector of random disturbances. We estimate this model using the maximum entropy estimator (Golan, Judge and Miller, 1996) 2.The starting point is the specification of each parameter as a discrete random variable h that can take (0 ≥) different values which are grouped in the so called support vector. The possible outcomes of the discrete variable are chosen using priors on the value of the parameter. For example, in some instances, the priors determine a range of possible values for the parameters. In this case, the support vector contains M values chosen from this interval. Therefore, for each parameter we choose a support vector h
2 There are several applied papers that used this estimator. See, for example, Paris and Howitt (1998), Fraser (2000), Golan, Perloff and Shen (2001) and Gardebroek and Oude Lansink (2004).
containing a finite number (M) of possible values for each parameter
(
h h hM)
h = b1,b 2,,b
b . Each support vector has a probability vector
(
h h hM)
h = p 1,p 2,...,p
′
p such that p h
M
m
hm = ∀
∑
=, 1
1
. Therefore, each parameter in the model can be written as a linear combination of the values of the support vector (chosen) weighted by its probability (unknown):
∑
==
= M
m
hm hm h
h
h b p
1
b p [2]
The vector of H unknown parameters in the model can be written as:
=
=
=
H H 2
1
H 2 1
p ...
p p
b . 0 0
. . . .
0 . b 0
0 . 0 b ... Bp
2
1
[3]
A similar procedure can be used to specify the vector of random disturbance of the model (e). For each element etof the vector, we assume the existence of a support vector v=
(
v1,v2,...,vJ)
3 with probabilities w′t =(
wt1,wt2,...,wtJ)
where J ≥2. The vector of random disturbances e can be written as:
=
=
=
T 2 1
w ...
w w
v . 0 0
. . . .
0 . v 0
0 . 0 v ... Vw
e eT
e e
2 1
[4]
The value of the random disturbance, for an observation t is:
∑
==
= J
j tj j
t v w
e
1
vwt [5]
Finally, the model in equation [1] can be written as:
Vw XBp
y= + [6]
The common practice is to choose a set of values centered around zero such that:vi = −vJ− −(i 1).
In this specification, the problem of estimating the unknown vector of parameters is transformed into the estimation of M×H + J×T probabilities of the values of the support vectors (ph and wt). Using the estimated probabilities, an estimate of the parameters can be recovered as:
∑
== M
m
hm hm
h p b
1
ˆ ˆ ; ∀h=1,...,H [7]
ME consists of estimating the probabilities in [6] by maximizing an entropy function.
The entropy function is a measure of uncertainty or ignorance about the outcomes of an event represented by a random variable. The entropy function defined by Shannon (1948) can be written as:
∑
=−
= M
m
m
m p
p EF
1
ln )
(p [8]
where EF is the value of the entropy function and p=
[
p1,p2,...,pM]
are the probabilities of M possible outcomes x1, x2,...,xM of a discrete random variable x, suchthat 1
1
∑
== M
m
pm .
The function EF(p) goes to zero when the probability of one of the possible outcomes of the random variable goes to one. In other words, the function goes to zero as uncertainty vanishes. This is the minimum value of the entropy function since it can not take negative values. In the other hand, the entropy function achieves its unrestricted
maximum for the uniform distribution:
= ∀m= M
pm M1 , 1,...,
. This is a quite intuitive result since the uncertainty about the outcome of an event is maximum when all outcomes have the same probability.
Intermediate cases (between minimum and maximum uncertainty) are determined by the available information. The existence of data consisting of several observations of the outcomes of the random variable reduces the uncertainty and, therefore, the entropy. In fact, it is possible to reject probability distributions that could not have generated these data. For example, two different realizations of a random variable mean that the sample
could not come from a probability distribution that attributes a probability of one to a realization and zero to the others. Therefore, it is necessary to choose among probability distributions compatible with the available sample information. Among these, ME chooses as the random generating process the probability distribution in which entropy is maximum. Operationally, this probability distribution is obtained by maximizing the entropy function subject to a set of constraints defined by the dataset.
The maximization of entropy could seem, at first, counterintuitive because its value goes to zero as uncertainty decreases. However, the maximization of entropy corresponds to the conventional criteria in empirical analysis. In fact, it is a conservative approach aiming to choose a probability distribution that does not imply more knowledge (less entropy) that the strictly supported by the available data.
The estimation of the model in [1], requires the estimation of the probabilities of the elements of the support vector. The probabilities can be calculated using the following optimization program:
∑ ∑ ∑∑
= = = =
−
−
= H
h
T
t J
j
tj tj M
m
hm
hm p w w
p EF
Max
1 1 1 1
, ( wp, ) ln( ) ln( )
w
p [9a]
Subject to:
T t
y w v p
b
x t
J
j tj j H
h
hm hm M
m
ht , 1,...,
1
1 1
=
∀
= +
∑
∑∑
= = =[9b]
1,...,
, 1
1
H h
p
M
m
hm = ∀ =
∑
=[9c]
T t
w
J
j
tj 1, 1,...,
1
=
∀
∑
==
[9d]
The equation in [9a] is the entropy function of Shannon adapted to the estimation of M×H + J×T probabilities. The equation [9b] contains sample information in terms of the model in [6]. These constraints assure that the estimated probabilities are compatible with the T available observations. The equations in [9c] and [9d] assure that probabilities add-up to one. The Lagrangian associated to the constrained maximization can be written as:
∑ ∑
∑ ∑
∑ ∑ ∑ ∑
∑ ∑
∑ ∑
= =
= =
= = = =
= =
= =
−
+
− +
− −
+
−
−
=
T
t
J
j tj t
H
h
M
m hm h
T
t
J
j
j tj H
h
m hm M
m ht t
t T
t J
j
tj tj H
h M
m
hm hm
w p
v w b
p x y
w w p
p L
1 1
1 1
1 1 1 1
1 1
1 1
1 1
ln ln
γ µ
λ
[10]
The solution of the optimization program in [9] gives the estimates of the probabilities of the elements of the support vectors. The parametersof the model can be recovered using expression [7].4
3. EMPIRICAL MODEL
The role of infrastructure in economic growth has usually been studied by including private inputs and public capital as explanatory variables in a production function. The basic model can be written as:
) , , ,
(A K L G f
Y = [11]
where Y is the production, K is the stock of private capital, L is Labor, G denotes the stock of public infrastructure and A is a measure of total factor productivity.
In log form, a technology represented by a Cobb-Douglas production function can be written as:
it it G it L it K i
it t t K L G e
Y = + + + ln + ln + ln +
ln τ ττ 2 [12]
where subscript i denotes regions, subscript t denotes time5, and e is a random disturbance. Exogenous technical change is introduced in the model as a quadratic function of time. The productivity of infrastructure depends on the value of the parameter . A value of the parameter greater than zero can be interpreted as evidence G of a positive contribution of infrastructure to private production.
The model in [12] is estimated using data on 17 Spanish regions observed for 19 years (1980-1998). There are 22 parameters to estimate: 17 regional effects, 3 output
7KHSURSHUWLHVRIWKH0(HVWLPDWRUVDUHDQDO\]HGLQGolan, Judge and Miller (1996; pp. 96-123 and 131- 133).
5 In the previous section, t was used to denote observations. Here, we use the notation in the panel data literature.
elasticities and two parameters that model technical change. The support vectors are chosen using priors on parameter values obtained from published research. Table 1 shows results on output elasticities obtained in empirical research that uses a model similar to ours: primal approach, data of Spanish regions, Cobb-Douglas technology with regional effects and a trend.
TABLE 1:OUTPUT ELASTICITIES IN SPANISH REGIONS
Paper CRS* K
L
G
Mas, Maudos, Pérez and Uriel (1994) K, L 0.63 0.37 0.24 Mas, Maudos, Pérez and Uriel (1994) K, L, G 0.44 0.37 0.19
García-Fontes and Serra (1994) -- 0.49 0.46 0.06
Mas, Maudos, Pérez and Uriel (1996) K, L 0.42 0.58 0.07 Mas, Maudos, Pérez and Uriel (1996) K, L, G 0.40 0.53 0.07
Dabán and Murgüi (1997) -- 0.06** 0.44 0.05**
Moreno et al. (1997) K, L, G, Gs 0.51 0.35 0.05
Delgado and Álvarez (2000) -- 0.29 0.40 0.20
Freire and Alonso (2002) K, L, G 0.31 0.57 0.13
Álvarez, Orea and Fernández (2003) -- 0.07** 0.5 0.00**
Cantos, Gumbau-Albert and Maudos (2005) -- 0.34 0.32 0.04
* Constant Returns to Scale in the inputs shown in the colum. Gs denotes Social Public Capital.
** Estimates not significantly different from zero.
There is quite a lot of variability on the results (due to different definitions of variables, different periods of analysis or different data bases). However, the results can be used to choose an interval for the support vector of output elasticities for the three inputs. The support vector of the regional effects and the trend are chosen based in the estimates of model [12] obtained using the within estimator for panel data. Finally, we choose the support vector for the random disturbances using the current practice in the literature.
Golan, Judge and Miller (1996: p.88) suggest to set the lower and upper values of the support vector of the error as ± 3 times the standard deviation of the dependent variable6. Following this rule, we choose a vector with three values centered in zero (J=3). The support vectors chosen are:
6 This interval covers 99% of the range of the dependent variable. Therefore, the range of the error includes likely includes even the limiting case in which the independent variables do not have any explanatory power. For a detailed discussion of this issue see Pukelsheim (1994).
(-12.5; 0.0; 12.5) (0.0; 0.05; 0.1)
( 0.05; 0.0; 0.05) (0.3; 0.4; 0.5) (0.4; 0.5; 0.6) (0.0; 0.15; 0.3) (-2.6; 0.0; 2.6)
i
K L G
i 1,2,...17
= ∀ =
=
= −
=
=
=
=
τ ττ
b b b b b b v
[13]
The model in [2] written as in [6] requires to estimate 66 probabilities for the values of the support vector of the parameters (3 probabilities for each of the 22 parameters).
Additionally, the sample contains 323 observations (17 regions observed from 1980 to 1998), which implies that it is necessary to estimate 969 probabilities for the values of the support vector of the random disturbance (3 probabilities for each error). Therefore, the estimation consists of the maximization of the following entropy function:
∑∑∑
∑
∑
∑
∑
∑
∑∑
= = =
=
=
=
=
=
= =
⋅
−
⋅
−
⋅
−
⋅
−
−
⋅
−
⋅
−
⋅
−
=
17 1
19 1
3 1 3
1 3
1 3
1
3 1 3
1 17
1 3
1
) ln(
) ln(
) ln(
) ln(
) ln(
) ln(
) ln(
) , (
i t m
itm itm
m
Gm Gm
m
Lm Lm
m
Km Km
m
m m
m
m m
i m
im im
w w
p p
p p
p p
p p
p p
p p
EF p w τ τ ττ ττ
[14a]
Subject to:
- 323 constraints stemming from the sample using expression [12]:
∑
∑
∑
∑
∑
∑
∑
=
=
=
=
=
=
=
⋅ +
⋅
⋅ +
⋅
⋅ +
+
⋅
⋅ +
⋅
⋅ +
⋅
⋅ +
⋅
=
3 1 3
1 3
1
3 1 3
1 3 2
1 3
1
ln ln
ln ln
m
itm m m
Gm Gm it m
Lm Lm it
m
Km Km it m
m m m
m m m
im im it
w v b
p G b
p L
b p K b
p t b
p t b
p
Y τ τ ττ ττ
[14b]
- 22 adding-up constraints for the probabilities of the support vectors of the parameters:
22 , , 1 1
3 1
=
∑
==
h p
m
hm [14c]
- 323 adding-up constraints for the probabilities of the support vector of the random disturbances.
19 , , 1
; 17 , , 1 1
3 1
=
=
∑
==
t i
w
m
itm [14d]
- An adding-up constraint to the errors.
∑∑∑
= = ==
17 ⋅
1 19
1 3
1
0
i t m
itm
m w
v [14e]
4. DATA AND ESTIMATION
We have used the following variables for the estimation of equation [12]. The private production (Y), measured by the Gross Value Added, is taken from the official Regional Accounting of Spain [INE (2003)], linked with data on previous periods published in Cordero and Gayoso (1997). We subtract the part of the production corresponding to services not for sale to obtain a measure of private production. The public7 and private capital (K) have been collected by Fundación BBVA (1998). Labor is the number of employees in the private sector as reported by Mas et al. (2000).
The estimates of the parameters of equation [12] using the support vector defined in [13] are shown in the third column of Table 2.8 Public capital appears as a relevant variable to explain private production since the output elasticity of public capital is close to 0.19. The ME estimates can be compared with the estimates obtained using the within estimator in the fourth column of Table 2. In this case, the output elasticities of public and private capital are not significantly different from zero. These results illustrate the difficulties of using econometric methods that do not use priors on the value of parameters in samples with multicollinearity.
7 Social Capital (education and health care infrastructure) is not included as part of Public Capital. Several papers have shown the negligible effect of these expenditures on production. Baltagi y Pinnoi (1995) claim that social public capital stock may not be the best index of education and health services.
8 The estimates were obtained using the CONOPT2 algorithm for nonlinear optimization in GAMS.
TABLE 2:ESTIMATES OF THE PARAMETERS OF THE PRODUCTION FUNCTION
Variable Parameter ME* Within ME**
Private Capital
K 0.419 0.014
(0.35) 0.342
Labor
L 0.507 0.483
(13.24) 0.667
Public Capital
G 0.186 -0.046
(-1.88) 0.308
t
τ 0.044 0.029
(12.97) 0.042
t2
ττ -0.002 -0.0003
(-3.39) -0.002 t-ratios in parenthesis
*: Estimates obtained using the support vector in [13]
**: Estimates using the support vector in [15]
The dependence of ME estimates on priors on the parameter values is an interesting feature of the method but also a potential weakness. Therefore, it is reasonable to be concerned about the relative contribution of support vectors and sample data to the estimates by ME. In this paper, we explore this issue by estimating the model in [12]
using several ranges in the support vectors. For this purpose, a unique support vector is defined for the three output elasticities (between zero and two). This support vector does not seem very informative. In fact, this choice of support values disregards previous research results showing that the elasticity of private capital is lower than the elasticity of labor and greater than the elasticity of public capital. For the remaining parameters in the model, we keep the support vector used in the previous estimation. Therefore, the support vectors used are:
(-12.5; 0.0; 12.5) (0.0; 0.05; 0.1)
( 0.05; 0.0; 0.05) (0.0; 1.0; 2.0) (0.0; 1.0; 2.0) (0.0; 1.0; 2.0) (-2.6; 0.0; 2.6)
i
K L G
i 1,2,...17
τ ττ
= ∀ =
=
= −
=
=
=
= b b b b b b v
[15]
The results of the estimation are shown in the last column of Table 2. In spite of the wide support vector, the changes in the estimates are small given the large swings in the values of the support vector. The width of the support vector of output elasticity of private capital and labor changes ten fold (from 0.2 to 2) while the changes in the estimates are -0.08 and 0.16 respectively. The change in the estimate of the output
elasticity of public capital is 0.12 while the width of the support vector goes from 0.3 to 2. The estimate of the output elasticity of public capital changes more than the estimates of labor and private capital, but the estimate remains in the (upper) interval of estimates obtained previously in the literature [de la Fuente (2002)]. Therefore, the information included through the support vectors affects the estimation by ME but the empirical evidence obtained is not completely determined by these vectors.
The previous exercise shows how the point estimates are affected by the choice of the support vector. This result is usually seen as a problem of the methodology. However, in this paper, we propose to use the change in the estimates for changes in the support vector to check the strength of the empirical evidence obtained by ME. In particular, we analyze the likelihood of finding evidence of a negative productivity of public capital by using different support vectors.
In fact, the results in Table 2 support the idea of a positive productivity of public capital under two very different sets of priors on parameter values. In the first estimation, we construct the support vector using seemingly relevant information on parameter values while in the second case the support vectors are so wide that they do not seem very relevant in guiding the estimation. However, both estimates suggest a positive productivity of public capital.
A null or negative productivity of public capital is a counterintuitive result which has been found sometimes in empirical research. Using ME, we are able to show that the data do no support this hypothesis. We do that by including such possibilities in the support vector of the output elasticity of public capital. For this purpose, we estimate the model in [12] using the support vectors a, b and c in Table 3. These vectors are constructed fixing the higher value in 0.3 and choosing negative values for the lower value. For the remaining parameters, we use the support vectors in [13]. The results of this exercise are shown in Table 3. The output elasticity of public capital has remained above 0.16 in all instances. In other words, ME does not produce a null or negative estimate of the output elasticity of public capital when the support vector clearly contemplates that outcome. This result can be interpreted as additional evidence favorable to the existence of a positive and relevant effect of public capital in production.
TABLE 3:ESTIMATION BY ME WITH DIFFERENT SUPPORT VECTORS Variable Parameter Support vector
a Support vector
b Support vector c Private Capital
K 0.421 0.422 0.422
Labor
L 0.508 0.508 0.508
Public Capital
G 0.172 0.172 0.169
t
τ 0.045 0.045 0.045
t2
ττ -0.002 -0.002 -0.002
Support vector a : bG = (-0.1; 0.1; 0.3) Support vector b: bG = (-0.2; 0.05; 0.3) Support vector c: bG = (-0.7; -0.2; 0.3)
The ME and Within estimates have been used above to highlight the differences between conventional econometrics and ME. The basic difference is that ME uses out- of-sample information to deal with samples with multicollinearity. However, the comparison can be strengthened by assuming the existence of constant returns to scale (CRS) in the production function. In this case, the econometric estimation is done under the assumption that the output elasticities of private inputs add up to one (K +L =1).
In ME, the same parametric restriction can be used to estimate the parameters without a support vector for these two parameters. The parameters can be estimated directly as probabilities since they add-up to one. Therefore, under CRS, both methods use similar a priori information on the parameters. In a model with CRS, the entropy function becomes:
∑∑∑
∑
∑
∑
∑∑
= = =
=
=
=
= =
⋅
−
⋅
−
⋅
−
⋅
−
−
⋅
−
⋅
−
⋅
−
=
17 1
19 1
3 1 3
1
3 1 3
1 17
1 3
, 1
) ln(
)
ln(
ln ln
) ln(
) ln(
) ln(
) , (
i t m
itm itm
m
Gm Gm
L L K K
m
m m
m
m m
i m
im w im
p
w w
p p
p p
p p
p p
EF
Max p w τ τ ττ ττ
[16]
where, K +L =1 is now one of the constraints of the optimization program.
The results of estimating equation [12] under CRS by both methods are shown in Table 4. The support vector for the regional effects have been constructed using the Within estimator under CRS (-5.5; 0.0; 5.5). As before, we use a wide support vector for the output elasticity of public capital that, apparently, does not contain relevant information
for the estimation by ME. The support vectors of the trend and the random disturbances are the ones shown in [13] and [15].
TABLE 4: PRODUCTION FUNCTION UNDER CRE
Variable Parameter ME Within
Private Capital
K 0.343 0.347
(7.54) Labor
L 0.657 0.653
(7.54) Public Capital
G 0.290 -0.035
(-1.07) t
τ 0.043 0.024
(8.06) t2
ττ -0.002 -0.0005
(-5.10) t-ratios in parenthesis
The results in Table 4 show that the ME estimates under CRS are very similar to the estimates with wide support vectors (last column of Table 2). Additionally, under CRS the estimates of the output elasticities of private capital and labor are very similar in ME and Within. However, both methods give different estimates of the output elasticity of public infrastructure. The within estimator gives a negative (although not significantly different from zero) while by ME gives a positive and substantive value for the output elasticity of public capital.
As a summary, the results in this section illustrate the opportunities of empirical analysis raised by ME in models with multicollinearity and provide fresh empirical evidence about the productivity of public capital in the Spanish regions.
5. CONCLUSIONS
In this paper, we have explored the use of maximum entropy estimation for empirical analysis when multicollinearity is an issue. This problem emerges in the estimation of a Cobb-Douglas production function for the Spanish regions in which public capital is included as an input. The key element of the estimation by maximum entropy is the introduction of out of sample information through priors on parameter values. As a result, it is important to analyze the changes in estimates with changes in priors. For that purpose, we first estimate the production function with two quite different sets of priors on parameter values. In both cases, the output elasticity of public capital is positive and
quite substantive. Using a third set of priors on parameter values, we have shown that the data do not seem to support the hypothesis of a null or negative output elasticity of public capital.
Although, the result obtained with maximum entropy estimation need of a detailed analysis of sensibility, the methodology appears as a useful procedure to gather empirical evidence when the use of more common econometric methods is impeded or precluded by multicollinearity. This conclusion stems mainly from the thorough comparison of maximum entropy estimation with the within estimator. Due to multicollinearity, the within estimator fails to provide evidence of a positive productivity of public capital while that evidence is quite strong estimating the model by maximum entropy.
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