Armando Sánchez Vargas

Ricardo Mansilla Sánchez

Alonso Aguilar Ibarra

Diego Ali Roman Cedillo

**Abstract**

The relationship between environmental regulation and productivity is subject of intense academic debate.

Here, we propose the use of the exponential Gumbel distribution in order to study potential nonlinear relationships. This tool is applied to examine the causal effect of environmental regulation on manufacturing productivity in Mexico, using a data set at the plant level. Our empirical results show that the link between environmental regulation and productivity is in fact nonlinear and that there exists a decreasing trade-off between productivity and environmental regulation at the manufacturing industry in Mexico. We also find that such trade-off is larger for small firms, but almost negligible for large companies. Thus, we argue that much of the debate on different effects is due to the heterogeneity of the industry. This empirical result might be useful for the design of national policies devoted to enhancing environmental performance, and for optimizing the allocation of financial resources and investments for the industry’s productivity.

**JEL classification codes: **C46; C51; D24; Q52

**Key-words: **exponential Gumbel distribution, Porter Hypothesis, nonlinear relationships, Mexico’s manufacturing industry, environmental performance

**I. Introduction**

The relationship between environmental regulation and productivity is controversial. Although several
studies have dealt with this issue since the late 1970s, the academic debate has been centered on explaining the so-called Porter Hypothesis (Porter and van der Linde 1995) since the 1990s. The weak version of the Porter Hypothesis (PH) maintains that the positive effects of environmental regulation on firms’ economic performance are linked to innovation or to another cause. The strong version states that there is a positive relationship between regulation and economic performance but it does not look for the cause of such positive effects. Thus, this relationship is positive, according to the PH.

However, this conclusion has been severely questioned by economists as it challenges the paradigm of profit maximization on which corporate rationality is based. Thus, a controversy exists among economists, who have found that environmental regulations tend to reduce firms’ productivity, and business strategists, who sustain that environmental regulations enhance productivity (Reviews on the subject have been presented by (Jaffe, et al. 1995; Wagner 2003; Ambec and Barla 2006; Brannlund and Lundgren 2009), among others).

This debate has given rise to abundant empirical studies regarding the direction and magnitude of such relationship. In fact, the review of Brannlund & Lundgren (2009) points out that empirical research concerning the PH has been carried out under three main approaches concerning environmental regulations: on R&D, on financial impacts, and on efficiency and productivity. For the first two categories, there is no conclusive evidence for supporting the PH. In contrast, the latter approach has found mixed results, maybe in part, due to the larger number of studies performed on the subject since the late 1970s. For example, on the one hand, early studies (Barbera and McConnell 1986; Crandall 1981; Denison 1979; Gray 1987; Haveman and Christainsen 1981; Norsworthy, et al. 1979), based on aggregate data, show that environmental regulations account for a slow-down in productivity growth in the US. In a more recent work, Lanoie et al. (2007), conclude that the contemporaneous direct effect of environmental policy stringency on business performance is negative and that innovation does not offset the costs of complying with regulations.

On the other hand, there is also evidence that environmental regulations might be favorable for firms productivity. For instance, Berman and Bui (2001) show that refineries in the Los Angeles area have higher productivity levels than other US refineries despite the more stringent regulation in this area. Alpay (2002) finds that more stringent regulation seems to increase the productivity of the Mexican food processing industry. Isaksson

(2005) examined the impact of regulation on costs functions of 114 combustion firms, finding that extensive emission reductions have taken place at zero cost. Darnall (2007) demonstrated that better environmental performance enhances business performance, but that a stricter environmental regulation has a negative impact. Obviously, the debate on this issue is not closed (e.g. Brannlund and Lundgren 2009; Ambec et al. 2010) and, given the important policy implications, further research is thus needed on a number of aspects, including new forms of modeling relationships between environmental regulation and businesses performance (Wagner 2003).

Hence, in this paper, we propose a new empirical approach to face this dilemma. Specifically, we apply a nonlinear regression model in order to study the nature of such relationship. Given that this type of variables are often skewed to the right (asymmetric distributions), we hypothesize that the relationship between environmental regulation (measured as the plant’s pollution abatement expenditures) and productivity is non-linear and can be well represented by the exponential Gumbel/nonlinear/ heteroskedastic regression model (Gumbel 1960). The use of a nonlinear tool to model such a relationship has an advantage over the traditional normal linear regression model, since it does not assume a constant marginal effect of the explanatory variable over the entire distribution of the dependent variable. In fact, it estimates different marginal effects (heterogeneous partial effects) of regulation (pollution abatement expenditures) at different points in the conditional productivity distribution.

Thus, the objective of this study is twofold. First, we propose the use of the exponential Gumbel distribution and its associated regression curves to assess potential nonlinear relationships in the field of economics. And second, we add an illustrating example of the use of this tool for investigating the effects of environmental regulation on productivity at the firm level in Mexico. In other words, we aim at contributing with the implementation and application of an unexplored statistical tool, in the field of economics, to help elucidating the controversial arguments of the PH.

**II. Methods**

A. Data set and model specification

We used data from the 2002 national industrial survey in Mexico (INEGI 2003). This survey contains data for nearly 6,600 manufacturing plants; however our analysis used 563 plants for which we have investment in
pollution-control data2. It also contains an ample set of economic variables (e.g. exports, gross production, value added, investment), and data on pollution abatement expenditures at the plant level.

In order to assess the effects of environmental regulation (measured by the plant’s pollution abatement expenditures) on productivity, we constructed a pollution abatement costs indicator (the sum of investments on machinery, equipment and salaries aimed at reducing pollution at the plant level) and a labor productivity indicator (output per worker). We also used other variables as controls, included in the survey, such as: size of the plant, technological capabilities, level of energy use, and others (INEGI
2003).

As a preliminary step to propose a proper conditional model for the relationship between manufacturing productivity and environmental regulation in Mexico, we first discuss the statistical features of our data. This description is useful since in selecting a suitable model we should take into account not only theoretical issues, but also all the statistical systematic information in the data by using a set of graphical techniques (Spanos
1986). Hence, figures 1 and 2 show the productivity levels and the pollution control expenditures (as a proxy for environmental regulation) for a set of 563 manufacturing plants in Mexico for 2002. A brief analysis, of both graphs, suggests that the data seem to contain a big number of “outliers”, which allows us to state that our variables are highly skewed (with asymmetric distributions) and leptokurtic.

FIGURE 1 ABOUT HERE

FIGURE 2 ABOUT HERE

In figures 3 and 4 we show kernel estimates of the univariate densities of such variables (Silverman 1998), confirming that both variables are not normally distributed. Thus, given that the univariate empirical distributions (kernel density estimates) are similar and highly skewed to the right, we concluded that normality would not be a reasonable statistical assumption in specifying a conditional model of manufacturing productivity in Mexico.

FIGURE 3 ABOUT HERE FIGURE 4 ABOUT HERE

Furthermore, the data seem to be drawn from a random sample given that the sample mean does not appear
to change systematically3. From this evidence, we are able to conclude that both variables are independent. However, the variation around the mean, of both variables, is not always constant, which reveals that a homoskedastic model would not be an appropriate choice to model the relationship between environmental regulation and manufacturing productivity.

In addition to the previous features we can get more information regarding the joint distribution of both series
by looking at the Bivariate kernel estimate of the density function, shown in graph 5 below, and the probability contour plot with a potential regression curve of the data in graph 6 .

FIGURE 5 ABOUT HERE FIGURE 6 ABOUT HERE

These two graphs allow us to confirm that the data are highly asymmetric (skewed to the right) and non normal. Such statistical properties imply that the potential regression function must be nonlinear (see figure 6). Besides, comparing figure 5 with the bivariate normal (graph 7a) and exponential Gumbel (graph 7b) densities, there is no doubt that the latter provides a better description of the multivariate distribution of the data.

FIGURE 7 ABOUT HERE

At this point, we might infer that the joint distribution of the relationship between productivity and pollution abatement expenditures exhibits asymmetry, which make such distribution of the data closer to the exponential Gumbel rather than to the normal. Finally, in order to supplement the graphical evidence, table 1 reports several descriptive statistics of our series.

TABLE 1 ABOUT HERE

These statistics reveal that the sample´s kurtosis and skewness coefficients reinforce the evidence of non- normality nature of our data. In sum, we can say that an adequate conditional model of productivity should account for the leptokurticity and asymmetry exhibited by the data. This finding gives further support to the hypothesis that a better conditional model of the relationship between environmental regulation and productivity warrants the use of the exponential Gumbel regression model, which can be derived from the bivariate or multivariate exponential Gumbel distribution (Gumbel 1960).

Even more, based on the previous empirical evidence, we might also anticipate that the statistical features of the data are not compatible with the classic normal linear regression model, which assumes a constant marginal effect of the explanatory variable over the entire distribution of the dependent variable, in equation (1) below:

(1)

Where *y*
is the productivity level, *x* is the level of pollution abatement expenditures, and *e* is a Normal, Independent, and Identically Distributed (NIID) error. E(y/x) is the conditional expectation of *y *given *x*. Equation
(2) shows the marginal effect of *x *over *y*.

On the contrary, the stylized facts reveal that a more appropriate conditional model, in its error form
(Wooldrige 2002) would be an exponential Gumbel, nonlinear4 and heteroskedastic specification, which does not assume a linear regression curve with a constant marginal effect (see figure 6 above and equation (6) below) as follows (Gumbel 1960; (Kotz, et al. 2000)):

(3)

<

With the following conditional variance and marginal effect respectively:

(5) (6)

Equation (3) shows that the stochastic variable y (productivity) can be decomposed, by definition, into two parts: a non linear conditional expectation:

and an error term, e. This probabilistic setting thus suggests that the mean or average causal effect of the explanatory variable on the dependent variable is not linear. Moreover, the conditional variance in equation (5) is heteroskedastic.

The negative marginal effect in equation (6) is heterogeneous (depends on the values of the independent variable) and decreasing. Therefore, we see that the model in equation (3), is very different to the one in equation
1, since the Gumbel regression model does not assume a constant effect of the explanatory variable over the entire
distribution of the dependent variable.

To sum up, this section allows us to conclude that the data suggest that a better description of the relationship between regulation and productivity might be provided by a Gumbel, non-linear, heteroskedastic model.

The economic meaning of a negative nonlinear relationship, in this context, is that the mean or average causal effect of regulation (as measured by plant’s pollution abatement expenditures) on productivity is negative and decreasing. That is, the average value of productivity does not change at a constant rate as pollution abatement costs change, which means that we have heterogeneous marginal effects (changing partial effects). Even more important is the fact that the suggested model implies that there might be a decreasing trade off between productivity and regulation at the manufacturing industry in Mexico.

**B. Estimation method**

From the previous analysis we can conclude that the statistical features of our data suggest the specification
and estimation of a conditional model derived from an exponential Non-normal/nonlinear/heteroskesdastic distribution. We propose this type of models because previous research work reveal the existence of a non linear relationship and our graphical analysis suggests that the data exhibits asymmetry(skewness), leptokurticity, and stationarity.

More specifically, we propose the use of the exponential Gumbel regression model with conditional heteroskedasticity as a better model to model the relationship between regulation and productivity given the nature of our data. These types of models imply to model not only the conditional mean of the stochastic processes behind the data, but also the conditional variance.

As we discuss above, the exponential Gumbel regression model takes the form:

Where e is distributed GumbelIID (0, ω2 ) and ω2 is the conditional variance.From these equations, we can infer that the conditional mean of the Gumbel model is not linear in the conditioning variables and parameters, and that the conditional variance is heteroskedastic. Under stationarity, the maximum likelihood estimate of δ can be obtained by solving the following equation ((otz, et al. 2000):

(7)

**III. Results**

In this section we are interested in responding how the average value of productivity changes as environmental
regulation becomes stricter (as the level of pollution abatement expenditures increases). In order to do so, we estimate equation (3) using a maximum likelihood method, and controlling for some other important factors,
such as the origin of the plant, the level of energy use, and the technological capabilities of the firms5. The estimated model and misspecification tests are reported in tables 2 and 3 respectively. Table 2 shows that the parameter of interest (δ) is positive (0.97), but this estimate implies a negative correlation between productivity and environmental regulation of around -0.4 (Gumbel 1960). A negative correlation, given the functional form of the model, might suggest the existence of a trade off between our variables.

TABLE 2 ABOUT HERE

Table 3 shows the results of a set of misspecification tests for the model, which confirm that or model is appropriate, since they indicate no departures from the underlying assumptions of the Gumbel model (Spanos
2006).

TABLE 3 ABOUT HERE

It is worth noting that, in this case, the relevant estimates would be the marginal effects of regulation on productivity. Given that the Gumbel regression model is not linear, we can estimate different marginal effects of environmental regulation at different points in the conditional distribution of productivity. Specifically, we can compute the different marginal effects of regulation evaluated at the 10th, 25th, 50th, 75th and 90th percentiles and at the mode of the distribution. The results are presented in table 4.

TABLE 4 ABOUT HERE

These estimates suggest that the marginal effect of environmental regulation on productivity is clearly negative but decreasing when regulation is measured by the firm’s pollution abatement expenditures. For comparison purposes, we also report the least squares estimate of the relationship, which clearly underestimates the marginal impact of regulation on productivity. The results in table 4 imply that increasing pollution abatement expenditures lead to higher productivity costs for firms located in the 10th percentile (-0.18), compared to the costs of firms located in the 25th and 50th percentiles. That is, the cost of regulation, in terms of productivity, is decreasing for those plants located in the upper percentiles. For instance, for firms located in the 90th percentile, the effect is almost zero (-0.0003). This finding implies that there exists a decreasing trade-off between environmental regulation and productivity in the Mexican manufacturing industry.

This finding might lead us to an even more interesting conclusion: given that there is empirical evidence of a positive correlation between plant size and pollution abatement expenditures in Mexico (see Dominguez
2006), we can then infer that the negative effect of regulation on productivity is larger for small establishments but almost negligible for large companies. This outcome implies that the PH effect depends on a number of factors, among which no only market structure is involved (Greaker 2006) but also is plant size. In fact, Perez Espejo et al. (2011) point out that differentiated agri-environmental policies are warrant also in Mexico’s agricultural sector, as producers perceptions vary according to farm size.

**IV. Concluding remarks**

This paper provides a new approach to modeling potential nonlinear relationships in the field of economics,
which might arise in the presence of highly skewed and non normal data. Specifically, we propose the use of a Gumbel conditional model associated to a distribution with exponential Gumbel margins and whose regression curves are not straight lines and which do not intersect at the common mean.

We put forward that this approach is useful for studying nonlinear phenomena, such as the relationship between environmental regulation and manufacturing productivity in Mexico. Thus, we use the exponential Gumbel regression model to investigate the link between regulation and productivity at the plant level in Mexico. More specifically, we examine the effect of environmental regulation, as measured by pollution abatement expenditures, on manufacturing productivity among a set of Mexican industries.

Our empirical results show that the link between environmental regulation and productivity is in fact nonlinear and the proposed model provides a robust way to describe it. Specifically, we find that there exists a decreasing trade- off between productivity and environmental regulation at the manufacturing industry in Mexico. We also find that such trade off is more important for small firms and almost negligible for large companies, given that there is empirical evidence of a positive correlation between plant size and pollution abatement expenditures in Mexico. This empirical result might be useful for the design of national programs to enhance environmental performance (e.g. mitigation actions against climate change), given that knowledge of the magnitude of such effects could help to optimize the allocation of financial resources and investments, for the very different industries. Should this decreasing trade-off is real, then national environmental policies would need to take into account the size of manufacturing plants for allocating financial resources devoted to environmental programs such as climate change mitigation.

**Appendix**

**A. The Gumbel model**

The joint cumulative distribution function of the Gumbel model was presented by (Gumbel, 1960): F (x,y) = 1 –e–x –e–y –e–(x+y+δxy) ; 0 ≤ x ; 0 ≤ y ; (0 ≤ δ ≤ 1) (A1)

where δ is the parameter describing the association between the two random variables *x *and *y*. The maximum likelihood estimator (MLE) of δ is a solution of the equation:

(A2)

A moment estimator of δ can be obtained as the solution of the equation:

*(sample correlation coefficient) *(A3)

Thus, the δ parameter has a close relationship with the classical correlation coefficient that stands as: (A4)

Where *Ei *is the well-known exponential integral, that’s why the correlation parameter is also given by: (A5)

When δ = 0, the correlation parameter ρ is equal to zero. This represents the independent case and the bivariate distribution splits into the product of the two marginal distributions, and becomes:

F (x,y) = F (x) F (y) (A6)

x y

When δ = 1, the association parameter ρ reaches its lower limitation and is equal to -0.4036. The model is
only suitable for representing joint distribution of two correlated Gumbel distributed variables whose correlation parameter is –4.0 ≤ ρ ≤ 0.

The joint probability density function (pdf) of the Gumbel mixed model can be derived by differentiating
Equation (A1) as follows:

The conditional pdf of x given y can be derived as follows:

(A8)

The conditional expectation of *x *given *y *can be given by:

Similarly, the conditional pdf f y|x (y|x) and the conditional expectation E[y|x] of y given x can be expressed by equivalent formulae.

**B. Exponential regression misspecification tests**

The misspecification tests applied to the exponential regression model are based on the following F type tests:

Additional non-linearity in the conditional mean To test for the presence of additional non-linearities in the conditional mean we can test if α = 0 in the following regression: yi = α + α ŷ + α ŷ 2 + u (A10) 0 1 i 2 i i Where ŷ are the Gumbel model fitted values. Furthermore, we can also expect that α = 1 if the pre-specified i 1 is the correct model. Trend in conditional mean if γ To test for the presence of additional non-linearities, like a linear trend in the conditional mean, we can test = 0 in the following regression: yi = γ + γ ŷ + γ t + u (A11) 0 1 i 2 i Where ŷ are the Gumbel model fitted values. Furthermore, we can also expect that γ = 1 if the pre-specified i 1 is the correct model.

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