metadata: authors: 0: author: Andree,Bo Pieter Johannes
1: author: Spencer,Phoebe Girouard
2: author: Chamorro,Andres
3: author: Dogo,Harun
count: World
docna: 0: docna: Environment and Development : Penalized
            Non-Parametric Inference of Global Trends in Deforestation,
            Pollution and Carbon
docty: Policy Research Working Paper
subtopic: Global Environment,Inequality,Environmental Disasters & Degradation,Common Carriers Industry,Food & Beverage Industry,Plastics & Rubber Industry,Business Cycles and Stabilization Policies,Textiles, Apparel & Leather Industry,Pulp & Paper Industry,Construction Industry,General Manufacturing,Nutrition
teratopic: Poverty Reduction,Environment,Health, Nutrition and Population,Macroeconomics and Economic Growth,Industry
abstracts: cdata!: This paper revisits the issue of
            environment and development raised in the 1992 World
            Development Report, with new analysis tools and data. The
            paper discusses inference and interpretation in a machine
            learning framework. The results suggest that production
            gradually favors conserving the earth's resources as
            gross domestic product increases, but increased efficiency
            alone is not sufficient to offset the effects of growth in
            scale. Instead, structural change in the economy shapes
            environmental outcomes across GDP. The analysis finds that
            average development is associated with an inverted $U$-shape
            in deforestation, pollution, and carbon intensities. Per
            capita emissions follow a $J$-curve. Specifically, poverty
            reduction occurs alongside degrading local environments and
            higher income growth poses a global burden through carbon.
            Local economic structure further determines the shape,
            amplitude, and location of tipping points of the
            Environmental Kuznets Curve. The models are used to
            extrapolate environmental output to 2030. The daunting
            implications of continued development are a reminder that
            immediate and sustained global efforts are required to
            mitigate forest loss, improve air quality, and shift the
            global economy to a 2°pathway.
repnme: repnme: Environment and Development : Penalized
            Non-Parametric Inference of Global Trends in Deforestation,
            Pollution and Carbon
display_title: Environment and Development :
            Penalized Non-Parametric Inference of Global Trends in
            Deforestation, Pollution and Carbon
listing_relative_url: /research/2019/02/30858024/environment-development-penalized-non-parametric-inference-global-trends-deforestation-pollution-carbon
topicv3: Manufacturing Sectors m1326521 1281,Poverty m1326687 695,Environment, Natural Resources and Blue
            Economy m1327391 566,Deforestation m1503714 3386,Malnutrition m1627088 4737
publishtoextweb_dt: 2019-02-25T00:00:00Z
historic_topic: Poverty Reduction,Environment,Health, Nutrition and Population,Macroeconomics and Economic Growth,Industry
no_of_pages: 51
author_keys: 0: author_key: 1539948
1: author_key: 1539158
2: author_key: 1278233
model_extractions: Policy Research Working Paper 8756 Environment and Development Penalized Non-Parametric Inference of Global Trends in Deforestation, Pollution and Carbon _Bo Pieter Johannes Andrée Phoebe Spencer Andres Chamorro Harun Dogo_ Environment and Natural Resources Global Practice February 2019
Policy Research Working Paper 8756

# **Abstract**

This paper revisits the issue of environment and development raised in the 1992 World Development Report, with new analysis tools and data. The paper discusses inference and interpretation in a machine learning framework. The results suggest that production gradually favors conserving the earth’s resources as gross domestic product increases, but increased efficiency alone is not sufficient to offset the effects of growth in scale. Instead, structural change in the economy shapes environmental outcomes across GDP. The analysis finds that average development is associated with an inverted $U$-shape in deforestation, pollution, and carbon intensities. Per capita emissions follow a $J$-curve. Specifically, poverty reduction occurs alongside degrading local environments and higher income growth poses a global burden through carbon. Local economic structure further determines the shape, amplitude, and location of tipping points of the Environmental Kuznets Curve. The models are used to extrapolate environmental output to 2030. The daunting implications of continued development are a reminder that immediate and sustained global efforts are required to mitigate forest loss, improve air quality, and shift the global economy to a 2°pathway.

This paper is a product of the Environment and Natural Resources Global Practice . It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/research. The authors may be contacted at bandree@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
Environment and Development: Penalized Non-parametric Inference of Global Trends in Deforestation, Pollution and Carbon Bo Pieter Johannes Andr ́ee<sup>1,2,3,*</sup> , Andres Chamorro<sup>1</sup> , Phoebe Spencer<sup>1</sup> , and Harun Dogo<sup>1</sup> 1 _World Bank Group_

> 2 _Department of Spatial Economics/SPINlab, VU Amsterdam, Netherlands_

> 3 _Tinbergen Institute_

> * _Corresponding Author: email bandree@worldbank.org or b.p.j.andree@vu.nl_

**Keywords:** Environment, Development, Penalized Inference, Non-parametric models, Kernel Regularized Least Squares.
# **1 Introduction**

Will continuation of economic development increase pressure on the earth’s finite resources, or does the increase in income provide the basis for environmental improvement? This question asked by (Grossman and Krueger, 1995) and highlighted earlier in (World Bank, 1992) is critical to the design of sustainable development strategies but continues to remain at the center of debate. Evidence from past development suggests that increases in wealth and income occur simultaneously with a structural transformation process in which the composition of inputs and methods of production used in the economy gradually shift in favor of less destructive production. We revisit the empirical issue using panel data on environmental output and economic development for a large set of countries using a flexible kernel model that allows dependencies to vary smoothly throughout the data.

Interactions between the economic and natural systems are dominated by complex pathways, and the processes that produce the levels of environmental output are high dimensional. This is because environmental output is determined not only by the efficiency of production, which may improve nonlinearly across GDP, but also by the total production size, which varies across panels of countries (Stern et al., 1996; Stern, 2004). If the scale of the economy is large, minute changes in the efficiency of production can result in large differences in output levels. Therefore, if a panel is constructed that includes economies of widely different scales, the variance in environmental output levels can be expected to vary with the GDP levels of the countries. To cope with this, one should acknowledge in the model design that environmental output is de facto a result of both a _scale_ component, and a _technology_ component.

Many empirical approaches have tried to model degradation levels directly, not distinguishing between the role of _scale_ and a _technology_ separately, and therefore assume an unrealistic degree of homogeneity of the samples. Vollebergh et al. (2005) pay particular focus to homogeneity assumptions in their environmental output regressions, and conclude that correctly modeling heterogeneity is essential to prevent spurious correlation in reduced-form panel estimations. Other issues relate simply to inappropriately dealing with the serial dependence that results from misspecification and may bias inference. This has partly been addressed by adding control variables as in studies reviewed by (Stern, 1998), or by deploying fixed-effect approaches Stern (2004). Time series approaches are also widespread, for example (Perman and Stern, 2003; Stern, 2004) that claim that the error correction approach provides appropriate diagnostic statistics and specification tests 1
for the environment-economic relationship. However, even in time, linearity and constant variance assumptions break for moderate time dimensions because nonlinearities result from the income dependence of the derivatives of the function mapping changes in income to changes in degradation. From that perspective, the non-parametric error correction approach (Shahbaz et al., 2017) improves on previous work. More discussion on nonlinear cointegration can be found in (Wagner, 2015), the general conclusion is that the diagnostics available in the standard framework are not appropriate in the nonlinear case because powers of integrated processes are themselves not integrated. In the non-parametric context on the other hand, causality and other correct-specification arguments are tightly related to the penalization technique, or bandwidth setting, that may take the limit criterion away from the _true_ parameter.

In this paper, we focus on a cross-comparable _technology_ component represented by degradation intensities to cope with the heteroskedasticity related to economic scales. To allow for a wide variety of potential nonlinearities with minimal parametric assumptions, we deploy a machine learning method that learns from similarities in the data using kernels. The method is known as Kernel Regularized Least Squares (Hainmueller and Hazlett, 2014). The key reason behind this choice is that apart from flexibility and taking full advantage of the kernel learning framework, it is still straightforward to back out marginal effects.

Machine learning methods are often developed with different applications in mind than the classical regression models that have been developed primarily for economic inference. Because the limit results in non-parametric models depend on externally set tuning parameters that are not part of the vector of estimated parameters, it is not immediately clear whether the estimates can be interpreted similarly as those obtained using parametric methods. Penalization in the non-parametric context differs from penalization in the GLM case, such as in the popular LASSO. While penalized GLM’s require the penalty to vanish asymptotically for generality claims, positive penalization in the limit may be a necessity to ensure identifiable uniqueness for non-parametric models. Regularization or penalization, while primarily known for dealing with over-fitting, is in fact a way to flexibly establish simple subspaces in which consistency theorems hold. As a result, the consistency and normality limits are uniquely defined for every level of penalization, which makes it less straightforward to interpret the estimator, and it’s derivatives, with usual confidence. However, penalties may still be found that yield estimates that conform to standard interpretation. Specifically, penalties that result from minimizing an out-of-sample criterion 2
pull the consistency limit toward the result that induces the optimal conditional distribution implied by the weighted kernel across all penalties and weights, as judged by the out-of-sample criterion.<sup>1</sup> We use the framework together with out-of-sample selection of fixed effects, to model remotely sensed and reported environmental data for 95 countries that include 85% of the world’s population, 83% of global carbon output and 72% of all forest cover. We find that the production gradually favors conserving the earth’s finite resources as GDP increases, but our results are not supportive of a single Environmental Kuznets Curve. Rather, countries follows individual curves with heterogeneity in the location, shape, and height of tipping points, depending on the composition of the economy. Focusing the policy dialog on a single average curve may thus result in harmful economic outcomes, as the traditional EKC may provide a poor estimate of local conditions. Across different possible paths of development, we do observe that average development is associated with an inverted _U_ -shape in deforestation, pollution and carbon intensities of production units. Per capita carbon emissions, follow a _J_ -curve. We use the models to extrapolate environmental output forward to 2030 under alternative growth scenarios, allowing each country to progress further on it’s specific development path. These results highlight the daunting implications of continued development under the current standards of practice and remind us once more that immediate and sustained global efforts are required on multiple fronts to mitigate forest loss, improve air quality, and shift the global economy to a 2<sup>_◦_</sup> pathway.

The remainder of this paper is as follows. We discuss estimation methods in Section 2. Additional discussion on the role of the penalty in establishing identifiable uniqueness, and the interpretation of the estimation results are provided in our appendix. Section 3 details the data used for our empirical analysis in section 4. Finally in section 5, we use our empirical descriptions to explore the implications of continuation of growth on environmental output. Section 6 concludes.

# **2 Methods**

We use the Kernel Regularized Least Squares estimator developed in Hainmueller and Hazlett (2014). We aim to summarize the key features of non-parametric kernel estimation.

> 1That result is the same as the standard convergence toward the minimizer of the expected KullbackLeibler divergence Kullback and Leibler (1951) between the true conditional density of the data and the conditional density implied by the estimated model White (1994). Under that result, the estimator converges to the result that delivers the best approximation to the true distribution of the data.

3
We do not introduce new theoretical results, instead we aim to provide an overview that highlights differences with respect to parametric estimation. Particularly, we detail the role that regularization plays in correct inference. Readers that are familiar with penalized kernel models may proceed directly to section 2.4, in which we explain how we treat fixed in the estimation.

In many applications, the postulated model is assumed to consist of a finite set of parameters. This imposes strong assumptions about the behavior of process being modeled. Specifically, linear models assume that the relationship between two variables _Y_ and _X_ described by a parameter _β_ is constant across levels of _Y_ and _X_ . Such strong assumptions about the DGP are rarely if ever justified by economic theory and can lead to seriously erroneous conclusions. The single parameter elasticities in linear models can at most approximate the average of the nonlinear elasticities locally on a function. Linear approaches can yield useful evidence within a relatively narrow range of the overall state space, particularly if large parts of the population are likely to pass through that state. However, in general, local approximations fall short when building global arguments. The errors induced by fixing the relationship at the average increase with the divergence between the average observation and the values of the observations of interest. This is undoubtedly the case in the analysis of economic development and environmental output, in which the structural behavior of the outliers, such as the poorest or most polluted, are often of foremost concern to policymakers.

Finite dimensional nonlinear parametric models may address several of these issues, but require strong predictions from underlying economic theory on the implied form of the structural relationship for the parameters to be economically meaningful. Such functions may be difficult to parameterize. Finite series approximators may also provide flexibility, but the resulting conclusions are often different from models in which the order of approximation is allowed to vary along the sample size, see (Horowitz, 2011) for discussion.

Non-parametric models make fewer assumptions about the DGP, and can produce approximations with varying flexibility. The key question in this case is how flexible the empirical function should be given the data that have been observed. A regularized nonparametric model exposed to growing data is in a sense an approximator that adjusts its belief of what an appropriate description of the DGP is according to the number of observations that it has seen. Key is that this belief changes appropriately as the data grow. A non-parametric model in which the size of the model is appropriately regulated 4
results in a small size when samples are few, but may increase in dimensionality as the data grows. As a result, the approximation error declines with growing data. Regulation of the order of approximation in non-parametric models occurs through tuning parameters. Correct inference is therefore strongly dependent on values that are not estimated by the criterion, but instead set by the researcher. While the relationship between standard loss minimization and correct parameter inference, as in the linear Least Squares literature, is a basic concept well-known to many researchers, inference based on the estimators of a non-parametric model has to consider the effect of the external parameters for which result do not follow under the same consistency and normality theorems. For example, while Hainmueller and Hazlett (2014) provide consistency and normality results for their model, they state explicitly that these results are different for every level of penalization. This does not make the relationship between the penalty and the DGP clear.

# **2.1 Assumptions about the DGP**

Suppose, we observe an _nx_ -variate _T_ -period sequence **x** _T_ := _{_ **x** _}_<sup>_T_</sup> _t_ =1<sup>that describes the state</sup> of one economy throughout time. At each point in time, we observe _N_ trajectories of this _nx_ -variate sequence, i.e., we focus on repeated cross-sectional vectors of length _N_ describing the evolution of _N_ economies. Each vector contains observations of for example income and the composition of the economy, all indexed over a set of locations _i, ..., N_ . The matrix **X** _t_ consisting of _nx_ columns describing different variables and _N_ rows describing the different locations, is indexed by time. We consider a second, repeated cross-sectional sequence **y** , of degradation levels generated by: We can observe **y** _T_ , a subset of the results of this process **y** _T_ := _{_ **y** _}_<sup>_T_</sup> _t_ =1<sup>.Thefunction</sup> _h_ 0 : _X →Y ⊆_ R produces environmental output for every coordinate **x** _t ∈X_ .<sup>2</sup> We assume that _h_ 0 is a unique measurable function that for each coordinate **x** _t ∈X_ assigns a _true_ value _yt ∈Y_ for all _t ∈_ Z. In a sense, by assuming this particular form, we assume that the environment does not endogenously degrades itself, i.e. that _y_ does not spontaneously generate itself. Instead, this assumes that the evolution of degradation levels for each economy _y_ is symptomatic to external, local economic development variables **x** . This does

> 2Particularly a B( _X_ ) _/_ B( _Y_ )-measurable mapping as kernels with a universal approximating property require at least that the target function is measurable, see for example (Micchelli et al., 2006).

5
not exclude the possibility that _y_ may in part affect elements of **x** , it requires however that feedback effects are invertible, in turn implied by some form of stability, and follow levels in **x** such that _h_ 0 describes the net relationship between **x** and _y_ .<sup>3</sup> We also assume that _h_ 0 is smooth, particularly that it maps similar coordinates in _X_ to similar values in _Y_ . This implies that for each state of the economy at each point in time **x** _t_ , we observe a level of deforestation, pollution or carbon _yt_ that is induced by the state of the economy through the function _h_ 0, and that for small changes in the state of an economy we expect to see small changes in environmental output. Furthermore, it assumes that for two economies that are similar in terms of composition and scale, we expect similar environmental output.

# **2.2 Kernel Regularized Least Squares for panels**

In certain situations, economic theory suggests a particular form of _h_ 0. For example in the studies of demand curves, Phillips curves, and Engel curves, there may be a strong prior belief that _h_ 0 is nonlinear of a specific form. In more general situations, logic only results in stylized facts about local or global behavior of _h_ 0 and we may want to impose little structure on _h_ 0. In the current case, the Environmental Kuznets theory suggests an inverted _U_ -shape between degradation and economic development. The relationships may of course be of a completely other form or differ across environmental variables, while ideally we keep the analysis of both relationships within a similar regression framework. We therefore postulate a very flexible regression of the form Our modeled function _h_ is defined as a mapping _h_ : _X ×_ Θ _→Y_ , where Θ is the parameter space. In parametric regressions, Θ is assumed to be compact and finite dimensional. This immediately imposes structure on _h_ , thus translating into assumptions about _h_ 0 if we maintain a belief that **_θ_** 0 _∈_ Θ. By reducing the size of Θ we simplify the possible structure of _h_ , i.e., chances that **_θ_** 0 _∈_ Θ become increasingly slim. While we minimize assumptions about _h_ 0 by working with Θ as an infinite dimensional space, some assumptions about _h_ 0 are unavoidable as Θ has to be parameterized eventually. In our example, as we shall see, one still has to specify radial basis functions.

> 3If _y_ = _f_ 0( **x** ) + _g_ 0( _y_ ) with _g_ 0 describing simultaneous feedback and _f_ 0 describing the contemporaneous exogenous effects, then one can also write _y_ = _h_ 0( **x** ) if _g_ 0 is invertible, with _h_ 0( **x** ) = ( _I − g_ 0)<sup>_−_1</sup> ( _f_ 0( **x** )), hence _h_ 0 arises from the composite ( _I − g_ 0)<sup>_−_1</sup><sup>_◦_</sup> _f_ 0 and describes the combined effect of exogenous impulses and feedback.

6
In parametric regressions, Θ plays a key role as the Euclidean space containing all the possible coordinates of potential parameter vectors **_θ_** . In the non-parametric case, there is a subtle difference. Suppose that for every **_θ_** _∈_ Θ, there is a function _h_ ( _·_ ; **_θ_** ) : _X →Y_ that is B( _X_ ) _/_ B( _Y_ )-measurable. We can define _H_ Θ( _X_ ) as the Hilbert space containing an infinite collection of functions _{h_ ( _·_ ; **_θ_** ) : **_θ_** _∈_ Θ _}_ generated by Θ. We shall use a simplified notation to reduce cluttering and instead write that **_θ_** indexes the functions _h_ **_θ_** _∈H_ Θ. The common notation **_θ_** 0 _∈_ Θ is thus equivalent to saying _h_ 0 _∈H_ Θ, i.e., **_θ_** 0 _∈_ Θ : _h_ ( **x** _t_ ; **_θ_** 0) = _h_ 0( **x** _t_ ) _∀_ **x** _t ∈X_ . This clarifies that, while a in a parametric regression problem where we are fore-mostly concerned with searching a compact parameter space Θ for the parameter vector **_θ_** _→_ **_θ_** 0, in the current framework we are explicitly interested in searching across a space of functions produced under some process of generating flexible functions from simple parameter vectors given the sample space, _hX_ , for infinite **_θ_** _∈_ Θ, for the function that best resembles the target function _h_ **_θ_** _→ h_ 0.<sup>4</sup> Hence, we can write the estimator also as<sup>5</sup> There are many ways to generate _H_ Θ. In the current framework, we focus on using a kernel _k_ together with a local parameter _θi_ that weights the surface to produce any flexible functional form.

The functions _h_ **_θ_** _∈H_ Θ, are allowed to follow any kernel that has the universal approxima- _−∥_ **x** _i −_ **x** _<u>j∥</u>_ 2 tion property, in this paper we adopt a Gaussian kernel _k_ ( **x** _i,_ **x** _j_ ; _nx_ ) = exp � _nx_ � with _∥_ **x** _i −_ **x** _j∥_ being the Euclidean distance, and _nx_ being a fixed bandwidth equal to the dimension of **x** _T_ . We count the constant as being part of **x** _T_ .

The kernel _k_ can be understood as a measure of similarity, which is seen by applying a Cauchy-Schwarz inequality revealing that if _xi_ and _xj_ are similar, then _k_ ( _xi, xj_ ) will be close to 1, and close to 0

> 4Specifically, each **_θ_** indexes a member in _H_ Θ( _X_ ) according to the map _hX_ : Θ _→H_ Θ( _X_ ) with _hX_ ( **_θ_** ) := _h_ ( _·_ ; **_θ_** ) _∈H_ Θ( _X_ ) _∀_ **_θ_** _∈_ Θ.

> 5The criterion function _QT_ can also be written as _QT_ ( **X** _T , h_ 0( **X** _T_ ) _, h_ ( **X** _T_ ; **_θ_** )), as we started under the notion that **y** _T_ = _{h_ 0( **Xt** ) _}t_<sup>_T_</sup> =1<sup>=</sup><sup>_h_</sup> 0<sup>(</sup><sup>**X**</sup> _T_<sup>).Insteadofhighlightingthatwesearchforfunctions,this</sup> representation makes the direct connection between the criterion function and target function _h_ 0, that generated the data, explicit.

7
when _xi_ and _xj_ are dissimilar. For a given observed collection ( _y, x ∈_ **x** ), _h_ **_θ_** is thus a function resulting from placing kernels over _xi_ and scaling the similarity surface using local coefficients _θi_ such that the summated surface approximates the true density of the data. This produces flexible functions that can describe local relationships between _y_ and **x** by assigning similar observations a similar scaling factor that maps onto similar output.

Different parameterizations of the local coefficients _θi_ may produce equally well, e.g. perfect, fits, such that the problem of estimating the vector **_θ_** = ( _θ_ 1 _, ..., θN_ )<sup>_′_</sup> is generally ill-posed without adding further structure to the problem. The specific estimation strategy to learn about the trends in the data is therefore of the form where _π_ ( _h_ **_θ_** ) _>_ 0 _∀ h_ **_θ_** _∈H_ Θ is a strictly positive function that monotonically increases by a measure of complexity defined on _h_ **_θ_** . The penalty is critical to ensuring identifiability and consistency of the estimator within simple subset spaces of _H_ Θ. At the same time it allows to fit nonlinearities of varying smoothness while working with a fixed kernel bandwidth that produces a relatively smooth similarity surface, as _θi_ is able to scale the nonlinearities locally albeit at a cost _π_ ( _h_ **_θ_** ). Hence, it favors less complex solutions to the criterion function by penalizing the high frequency domain in _H_ Θ. Specifically, let _K_ be an _N × N_ symmetric kernel matrix with entries _k_ ( **x** _j,_ **x** _i_ ) measuring pair-wise similarities. This yields a model that is a linear combination of basis functions, each measuring similarity of one observation to another observation in the data set, and mapping it to a local output.

The need for a regularization technique is obvious, the parameters ( _θ_ 1 _t, θ_ 2 _t, ..., θNt_ ) can always rescale the similarity surface to match **y** _t_ perfectly. Instead, the penalized estimator, takes into account the complexity of the rescaling by introducing a factor _λ∥h_ **_θ_** _∥_<sup>2</sup> _K_<sup>and</sup> chooses the best fitting function by minimizing: 8
_T_ in which<sup>�</sup><sup>_N_</sup> _i_ � _t_<sup>(</sup><sup>_yit −h_(</sup><sup>**x**</sup><sup>_it_;</sup><sup>_h_</sup><sup>**_θ_**))2arethestandardsumofsquaredresiduals.</sup><sup>_λ∥h_</sup><sup>**_θ_**</sup><sup>_∥_2</sup> _K_<sup>=</sup> _⟨h_ **_θ_** _, h_ **_θ_** _⟩H_ Θ is a penalty that increases monotonically as a function of the complexity of _h_ under **_θ_** . We focus on the _L_<sup>2</sup> norm. Finally, _λ ∈_ R _>_ 0 is te parameter that determines the strength of the penalty. Using this kernel we can work with an _NT × NT_ kernel matrix by defining the dependent variable _Y_ as the _NT_ length vector resulting from stacking the time observations, _X_ as the _NT × nx_ matrix resulting from stacking the columns similarly and **_θ_** as an _NT_ length parameter vector.<sup>6</sup> Using the Gaussian kernel, eq. (7) becomes In a panel application, the functions _h_ **_θ_** that result from weighted kernels can produce interesting time-varying dynamics across levels of **X** _t_ . This is for example appropriate when a time-varying stationary processes is of interest in which the nonlinearities change throughout the data but are not depending on time itself. Alternatively, one can work with time itself as a covariate, in which case processes that are only locally stationary can be modeled. Intuitively, the kernel approach then results in similar coefficients for similar time. In the case of non-stationary data, the kernel can approximate local conditional means in the data that may vary throughout the sample space.

# **2.3 The role of the penalty**

The basic idea of penalizing the criterion function has been explored in many statistical applications, and is for example at the heart of the widely adopted LASSO estimator (Tibshirani, 1996; Zou, 2006). The added structure to the criterion function is a frequentist’s analogue to the role that the prior plays within the Bayesian framework. We note that the penalty in the current setting is not primarily a way to improve small sample perfor-

> 6Specifically:

9
mance, but that it is in fact the central feature of the learning model that determines what functional forms can be fitted. This differs from kernel approaches in which the bandwidth is the key tuning parameter. In the current approach the bandwidth is fixed to produce smooth functions, but nonlinearities are subsequently locally adjusted using the vector of weights **_θ_** to increase flexibility. The penalization approach is able to shrink the hypothesis space and flexibly establish a subspace in which consistency holds. By balancing between fit and complexity of the locally weighted kernel, the size of the subspace can be regulated by the penalty. In the case of penalized GLM’s considered in Blasques and Duplinskiy (2015), nonzero penalties take one away from **_θ_** 0 in the limit if the penalty effect does not vanish asymptotically.<sup>7</sup> In that sense, a penalized criterion delivers a pseudo- _true_ parameter with a divergence from **_θ_** 0 that is controlled by the penalty function. Setting an appropriate penalty therefore determines what one can infer from **_θ_** 0. In the current context, positive penalties are a necessity to ensure uniqueness. This might lead to the thought that penalized non-parametric estimators that require positive penalization, are biased by definition. The estimate of the weights **_θ_**<sup>ˆ</sup> obtained through eq. (9) is different for every value of _λ_ . The tuning parameter _λ_ thus represents the researcher’s predefined level of tolerance for accepting nonlinear functions. High values of _λ_ force the model to linearize it’s dependencies, whereas extreme values for _λ_ will set all coefficients to zero and describe the data using only an average expectation. Hence for every penalty, we find a different functional form _h_<sup>ˆ</sup> _λ_ induced by the estimate **_θ_**<sup>ˆ</sup> _λ_ through eq. (4) given a specified kernel. Since _λ_ itself is not an estimated parameter, it is generally difficult, if not impossible, to tell whether eq. (9) yields an estimate of _h_<sup>ˆ</sup> close to _h_ 0. Without knowing the magnitude of _∥h_<sup>ˆ</sup> _λ − h_ 0 _∥_ , the method may seem rather useless for economic inference.

Sch ̈olkopf and Smola (2001), suggests to set the penalty through an out-of-sample prediction minimization problem to remove the dependence of the results on the external influence of the researcher that determines the level of penalization a priori. Hainmueller and Hazlett (2014) suggest one such strategy for Kernel Regularized Least Squares estimates by minimizing out-of-sample prediction errors over a vector _λ ∈_ Λ based on leave-one-out predictions, noting that it performs well in practice. While practical performance and the removal of external influence on the results provide intuition to set penalties in this way, it does not focus on the question whether _∥h_<sup>ˆ</sup> _λ − h_ 0 _∥_ is in fact minimized. In our appendix, we

> 7Furthermore, **_θ_** 0 in the standard context is the _true_ parameter. In the non-parametric context, that _true_ parameterization arguably does not exist, however one can think of **_θ_** 0 as the parameterization that produces _h_ 0 through the kernel, or alternatively, selects _h_ 0 the _true_ (non)linear functional form _H_ Θ that produces the _true_ density of the data.

10
provide additional discussion on the role of the penalty in ensuring identifiable uniqueness and establishing the consistency and normality results. We discuss that the strategy to set penalties by minimizing an out-of-sample criterion naturally pulls the estimator toward the weight vector that induces the _true_ function in the limit, such that inference can be applied as usual. This is because for a given penalty _λ_ , the estimated function conditional on that penalty _h_<sup>ˆ</sup> _|λ_ provides the optimal density across all functions _h ∈H_ Θ _|λ_ induced under that penalty, so choosing the estimate from a set of results found using different penalties _h_ ˆ _|λ ∈_ Λ that provides the optimal out-of-sample density, also minimizes _∥h_ ˆ _λ − h_ 0 _∥λ∈_ Λ in the limit since _h_ 0 is the function that by definition provides the best out-of-sample density. In other words, estimating eq. (9) while setting _λ_ based on out-of-sample prediction error minimization, yields an estimated function that minimizes _∥h_<sup>ˆ</sup> _− h_ 0 _∥_ in the limit across the entire family of models generated under all weight vectors and all penalties, which is similar to the standard case in which the criterion converges to the parameter that induces the best conditional density across the entire parameter space (White et al., 1980; White, 1994).

# **2.4 Fixed effects and out-of-sample shrinkage**

Linear effects can be included by using difference estimators as detailed in Hainmueller and Hazlett (2014). Nonlinear effects can be modeled by supplying group specific trend variables and group identifiers through _X_ . In this case all coefficients may depend on time, and similarly across similar groups in the data. Nonlinear fixed effects approaches combined with non-parametric parts around the economic variables may result in models with an enormous size while often the amount of observations locally in the time dimension, remains relatively small in environmental economic panels. Model size not only relates to the complexity of functions around the economic variables, but also to the number of fixed effects in the model. In-sample selection strategies to decide on the right number of effects, are complicated in the regularized non-parametric context. While standard regressions additional variables always improve fit, this is not the case in the current context. Adding fixed effects results in different complexity of the local weights vector. Therefore the effect of the complexity penalty in the criterion may increase such that the penalized estimator adjusts the weighting vector to achieve lower complexity. While this reduces the penalty value, it may possibly lower the in-sample _R_<sup>2</sup> . Comparing models with and without fixed effects is therefore a comparison between functional forms with different complexity and 11
nonlinearities. This is a comparison of non-nested models with an unknown, possibly realvalued, difference in degrees of freedom.<sup>8</sup> To decide on the right number of effects, we start by estimating a model that includes all fixed effects. We then remove the least significant dummy, and obtain new results. We repeatedly evaluate the out-of-sample prediction performance while shrinking the effects, and select the model with the optimal out-of-sample density across all fixed effect models. Intuitively, this approach starts with a model similar to a linear Fixed Effects model, as the penalty heavily discounts the thresholds introduced by the effects resulting in flattened marginal effects, and gradually allows fixed heterogeneity to be explained by nonlinearities across covariates instead. As a result, our final estimates are guaranteed to be preferred over the standard linear Fixed Effects model, as judged by the out-of-sample criterion.

# **3 Data**

We combined measures of tree loss, air pollution concentrations and carbon emissions on one side, and GDP indicators of economic structure on the other. Our data are from a variety of sources and includes 95 countries measured over 1999 _−_ 2014 containing approximately 85% of the world’s population, 83% of the world’s carbon output, and 72% of the world’s forest cover. The selection is mainly driven by availability of data as detailed in the following sub-sections. We have removed areas below 1500 square kilometers _−_ essentially all small islands _−_ from the analysis. A summary of the data as it enters our regressions is given below. A list of all countries included in our sample can be found in fig. 1.

Table 1 summarizes the data, all predictors are mapped into the [0 _,_ 1] interval to ensure the penalization effect is not driven by differences in variance of the different variables. We scale back the estimation results for easier interpretation.

## **3.1 Forest cover data**

We use data from (Hansen et al., 2013), which contains estimates of global tree cover extent (2000) and annual tree cover loss (2001-2014) at a spatial resolution of 30 meters.<sup>9</sup> Hansen et al. (2013) analyzed satellite images from Landsat 5, 7, and 8 to identify tree cover extent, defined as vegetation taller than 5 meters in height, and loss, defined as complete

> 8Degrees of freedom is a parametric concept whose translation to the non-parametric setting is complex. One can approximate the degrees of freedom empirically, which may result in numbers that are real-valued.

> 9The data can be found at https://earthenginepartners.appspot.com/science-2013-globalforest/download ~~v~~ 1.2.html.

12
removal of tree cover canopy. The authors reported the tree cover loss data to have a false positive rate of 13 percent, a false negative rate of 12 percent, and a ratio of total forest gain to loss over 2001-2012 of 0.34. The derived data differ from statistics reported by the UN-FAO’s Forest Resource Assessment, but due to the consistent methodology and definition of forests across countries, we believe these data are better suited for a global analysis. We define forests as pixels with a minimum canopy closure density of 30 percent. Finally, we convert the data to area measures and sum the data by country to calculate tree cover loss as a percentage of tree cover extent in 2000. Our intention is to examine “natural dense forests”, but note that the data also capture forest plantations. Our loss measure is thus only a proxy for deforestation, as there may be many other natural and anthropogenic processes (storm damage, fires, mechanical harvesting) that are reflected by the data. We refrain from including the tree cover gain data as there are significant differences in methodology that limit additivity or comparison with loss.

# **3.2 Pollution data**

We use concentrations of fine particulate matter (PM2 _._ 5), coarse dust particles of 2.5 micrometers in diameter, as a proxy for broader air pollution. The (0 _._ 01<sup>_◦_</sup> _×_ 0 _._ 01<sup>_◦_</sup> resolution) data were developed by van Donkelaar et al. (2016) and include global annual ground-level PM2 _._ 5 (1999-2014) derived from a combination of satellite-, simulation- and monitor-based sources. The dataset has been developed from satellite-derived Aerosol Optical Depth reflectance values calibrated to ground-based PM2 _._ 5 observations using a Geographically Weighted Regression.

Remote sensing methods aim to observe particulate matter but are prone to capturing fine dust released from barren lands that have similar reflectance properties in the high frequency spectral wavelengths. This poses a difficulty in our analysis, countries in desert regions have high country wide average pollution levels, while large countries, or those with substantial forest cover where ambient pollution is low, have lower average concentration to what the larger population is exposed to on a regular basis. We used gridded population data that are produced using a combination of light at night data and census data, to identify patches of urban areas.<sup>10</sup> We averaged gridded pollution data that falls within urban boundaries to the country-level, defining urban areas as places where population density was higher than 300 people per square kilometer. The results in fig. 1 show that

> 10The population grids are from http://sedac.ciesin.columbia.edu/data/collection/gpw-v4, we use the 2000 grids.

13
this procedure results in higher pollution levels in large countries with known pollution problems in cities (notably China, Nepal, Pakistan) or those with forests (notably Lao PDR, Indonesia, Senegal), and in lower concentrations in areas with known deserts (notably Chad, Tunisia, Morocco).

<!-- Start of picture text --> Difference between pollution level in urban areas and country averages<br>20<br>15<br>10<br>5<br>0<br>−5<br>−10<br>Chad Tunisia Morocco Cote d'Ivoire Cameroon Ghana Guatemala Thailand EthiopiaBenin Swaziland Sri Lanka Zimbabwe TogoTurkeyGabon Botswana Solomon Islands Guinea MozambiqueFijiRwanda El Salvador Zambia Bolivia MadagascarNetherlands UgandaGermanySierra Leone Malawi NicaraguaCambodia Costa Rica Denmark Australia Venezuela, RBTanzania ParaguayDominican RepublicFinland Timor−Leste Congo, Rep.BelgiumNigeriaNamibia Panama NorwayIreland Colombia KenyaEcuador PhilippinesLesotho JapanSpainHonduras Sweden Brazil BangladeshFrance MongoliaSwitzerland Vietnam Suriname Iran, Islamic Rep.Greece United KingdomGuinea−Bissau Austria UruguayAzerbaijanUnited States Central African RepublicCanada Indonesia ArgentinaItalyMexico Uzbekistan MalaysiaSenegalAngolaSouth Africa GeorgiaLao PDR Peru Chile India Armenia TajikistanKazakhstan Pakistan NepalChina<br><!-- End of picture text --> Figure 1: Difference between average pollution in urban areas and country-wide average PM2 _._ 5 data.

# **3.3 Vegetation control data**

We use the NDVI from the Moderate resolution Imaging Spectroradiometer (MODIS) from NASA’s Terra satellite to control for effects that relate to a variety of physical characteristics and natural assets of a country.<sup>11</sup> This dataset provides spatial and temporal comparisons of global vegetation conditions. The original data have a monthly frequency at a resolution of 1km. We calculated the mean NDVI value for each year in our analysis, we use 20002015 data, and summarized the data to the country-level using the mean, minimum, and maximum value to get a broad description of the vegetation in a country.

# **3.4 Other data**

For our economic variables and data on carbon dioxide emissions, we rely on the World Bank’s World Development Indicators (WDI). To ensure cross-country comparability, we

> 11Available at https://modis.gsfc.nasa.gov/data/dataprod/mod13.php.

14
use GDP per capita in constant 2011 international dollars adjusted for purchasing power parity. The CO2 emissions estimates retrieved from WDI were produced by the U.S. Department of Energy’s Carbon Dioxide Information Analysis Center (CDIAC), and include anthropogenic emissions from fossil fuel consumption and world cement manufacturing.

# **3.5 Missing values and outliers**

Forest cover loss included two outliers of respectively a 10.8% and a 5.4% loss in Namibia (2001, and 2005), for comparison the median observation across time in this country was 1.7%, we have capped these numbers at 3% which, seemed an appropriate maximum for the range of forest loss after inspecting a kernel density. We applied a 3 period simple moving average to further smoothen outliers. Our final dataset includes the losses for countries that held 72% of forest cover in 2000. The largest missing forest patch is that of the Russian Federation.

The WDI dataset contains a wealth of information, but some important observations are missing. GDP has only _._ 12% missing, manufacturing and services GDP shares have 5 _._ 57% missing. We interpolate these values by taking a weighted average of the nearest observations in the time dimension. The dataset includes approximately 83% of global carbon emissions and should be quite representative of missing countries as it is close to the 85% of world population included in our samples.

The WDI does not report income shares in each country but sometimes reports a Gini index. We used this to back out income shares.<sup>12</sup> After using both the Gini and reported shares, 61 _._ 25% of the observations remain missing, but missingness is only over the time dimension. In the countries that have more observations in the time dimension we observe that the income shares are relatively stable over time. We interpolate the remaining missing values using weighted averages of the nearest observations in the time dimension.

Poverty rates has 69 _._ 54% missing values. A large part of the missing values are due statistics that are not produced in high income countries. We impute all missing poverty and undernourishment rates above 23 _,_ 000 GDP per capita with zero. This completes undernourishment. The highest GDP per capita with a positive poverty rate in our data was Malaysia, with 24 _,_ 500 GDP ppp per capita and, a poverty rate of 1 _._ 3%, all other

> 12We make use of the fact that income shares held by a certain share of the population can be read off the Lorenz-curve and that the Gini is a measure of dispersion of the Lorenz-curve calculated from the summed surfaces under the Lorenz-curve and under the 45<sup>_◦_</sup> line. In total we are able to collect 937 observations of both Gini coefficients and income shares held by the first two quintiles. We estimate the nonlinear inverse map with high precision ( _R_<sup>2</sup> of .99) using the penalized non-parametric estimator. We then used the Gini observations to predict the income shares.

15
countries over 23 _,_ 000 like Kazakhstan, already attained 0% poverty rates in the published data. After, 49 _._ 54% of poverty remains missing because most countries are not complete in the time dimension. We first interpolate these variables by taking a weighted average over time. This works well for most variables, but may yield poor results for poverty, as we have seen a tremendous improvement in most countries in past years. We improve the time dynamics in the interpolated poverty data by using information about time dynamics contained in our other variables. We vectorize the interpolated values, and fit the kernel model using the full set of undernourishment, logarithmic GDP per capita, the share of manufacturing, services, urban population shares, and bottom 40 income shares. The model reaches an _R_<sup>2</sup> ’s of _._ 91. We use this model to smoothen the interpolated poverty values by taking an average of the interpolated values and the values predicted by this nonlinear model.

Table 1: Summary of the data used in our empirical application. Statistics are not weighted and not necessarily representative of the world averages.

|Statistic|Mean|St. Dev.|Min|Max|
|---|---|---|---|---|
|Annual % Tree loss|0.437|0.406|0.009|2.924|
|Urban PM2_._5 mcg/m3|18.924|12.609|0.311|63.498|
|CO2 kg / $|0.239|0.199|0.014|1.990|
|CO2 ton p.c.|3.402|4.162|0.015|20.208|
|GDP ppp p.c. 2011 international $|13,468.880|15,056.710|555.560|64,979.840|
|Population density, people / sqkm|101.592|138.270|1.524|1,148.514|
|Undernourishment rate|15.514|13.546|0.000|64.500|
|Poverty 1.90$ at 2011 international $|21.177|22.040|0.000|84.740|
|Manufacturing GDP share|14.371|6.721|0.237|38.733|
|Services GDP share|70.065|12.734|29.279|93.881|
|Urban population share|53.685|22.754|12.082|97.818|
|Bottom 40% income share|15.970|4.130|7.510|28.024|
|NDVI annual mean|0.502|0.163|0.111|0.762|
|NDVI annual min|0.327|0.164|_−_0.027|0.657|
|NDVI annual max|0.655|0.161|0.170|0.862|
|Forest cover 2000 extent million ha|3.628|2.815|0.0005|9.883|
|Country area sqkm|978,152|1,999,320|15,007|9,904,700|

# **3.6 Transformation to degradation intensities**

To address homogeneity concerns related to the scales of sovereign economies, we model standardized units of deforestation, pollution, and emissions, subsequently standardized per unit of GDP per capita in 2011 international dollars adjusted for purchasing-power parity. These environmental intensities follow a nonlinear trend with variance declining 16
<!-- Start of picture text --> EL<br>FL<br>FL<br><!-- End of picture text -->
appendix contains conditional expectations together with confidence bands for each of the economic variables, holding the effects of all other variables constant at their mean values which provide a guidance throughout our discussion of the results. Variables for which the marginal effects within the inner 50% of the percentiles range have an identical and significant sign are highlighted in the tables. This reveals that many of the variables contribute both positively as well as negatively to the output intensities depending on the data levels at which effects are evaluated. This shows that nonlinearities are important. We find that income has an unambiguous effect, all three environmental intensities improve with income but not sufficiently to offset scale growth.<sup>15</sup> While increases in GDP provide a basis for the improvement of production efficiency, it appears not to lower the net environmental output. However, as GDP increases, a structural change occurs in which poverty goes down and the shares of manufacturing services, and urban population gradually increase. We will highlight several of these structural effects that are best visualized in fig. 8 and fig. 9. Figure 9 shows how poverty, the production composition, urban population shares, and the income distribution trend across GDP.

# **4.1 Individual model results**

The results for deforestation show that early increases in population density correlate with a decrease in deforestation intensity while high population densities correlate with an increase. The trend across urban populations is a weak inverted- _U_ . The effects of manufacturing and services are less ambiguous, the move out of an agricultural society and specifically an increasing share in services that occurs with increasing GDP, is a strong correlate of declining deforestation rates. There is some evidence that economies with an unequal income distribution retain a higher deforestation intensity of production. The effect of the poverty variables is however mixed. Reducing the undernourishment rate initially seems to increase deforestation, while the transition out of extreme poverty correlates with a decrease in deforestation intensity.

In contrast with the deforestation results, we see that an increase in population density unambiguously drives pollution up. The pollution intensity trend across the urbanization rate is initially flat, but after 50% of the population has urbanized, the trend becomes

> 15The log-log specification allows for a simple interpretation. To offset the scaling effects, the marginal effect of log GDP per capita needs to be smaller than -1, which we do not observe within the 25%-75% range of effects.

18
Table 2: Deforestation intensity results using the penalized kernel regression.

||(means)|(25%)|(50%)|(75%)|
|---|---|---|---|---|
||_Dependent: _|_Log deforestat_|_ion intensity of _|_1000 GDP p.c._|
|Log 1000 GDP per capita<sup>_∗∗_</sup>|-0.453***<br>(-16.962)|-0.610***<br>(-22.856)|-0.476***<br>(-17.83)|-0.289***<br>(-10.822)|
|Population density|-0.001***|-0.003***|-0.002***|0.001***|
||(-5.351)|(-18.303)|(-9.578)|(8.449)|
|Undernourishment rate|0.001|-0.008***|0.002|0.011***|
||(0.371)|(-4.48)|(0.919)|(6.171)|
|Poverty 1.90$ rate|-0.005***|-0.013***|-0.004***|0.005***|
||(-4.118)|(-10.353)|(-2.878)|(4.297)|
|Manufacturing GDP share|-0.015***|-0.055***|-0.016***|0.023***|
||(-5.459)|(-19.402)|(-5.551)|(8.04)|
|Services GDP share<sup>_∗∗_</sup>|-0.032***|-0.051***|-0.036***|-0.016***|
||(-16.57)|(-26.109)|(-18.436)|(-7.961)|
|Urban population share|0.006***|-0.008***|0.009***|0.019***|
||(5.597)|(-6.929)|(7.767)|(17.111)|
|Bottom 40% income share<sup>_∗_</sup>|-0.027***|-0.06***|-0.033***|0.003|
||(-5.416)|(-12.304)|(-6.671)|(0.55)|
|N = 1520|R<sup>2</sup>= 0.922|_λ_=0.691.|_∗_p_<_.1; _∗∗_p|_<_.05; _∗∗∗_p_<_.01|
|_Constant omitted, t-statistics _<br>_Model controls for mean, min _<br>_∗Inner 50% of signifcant mar_<br>_∗∗Inner 50% of marginal signi_|_in parenthesi_<br> _and max ND_<br>_ginal efects _<br>_fcantly excl_|_s. Optimal m_<br>_VI, forest co_<br> _same sign, bu_<br>_udes zero._|_odel contained n_<br>_ver, and country _<br>_t range includes _|_o fxed efects._<br> _size._<br> _zero._|

Table 3: Pollution intensity results using the penalized kernel regression.

||(means)|(25%)|(50%)|(75%)|
|---|---|---|---|---|
||_Dependent: _|_Log pollution _|_intensity of _|_1000 GDP p.c._|
|Log 1000 GDP per capita<sup>_∗∗_</sup>|-0.691***|-0.842***|-0.692***|-0.567***|
||(-47.899)|(-58.388)|(-48.026)|(-39.307)|
|Population density<sup>_∗∗_</sup>|0.002***|0.001***|0.002***|0.003***|
||(18.523)|(6.658)|(17.063)|(30.846)|
|Undernourishment rate|-0.002**|-0.008***|-0.003***|0.003***|
||(-2.172)|(-8.675)|(-3.256)|(3.267)|
|Poverty 1.90$ rate<sup>_∗_</sup>|0.003***|0.000|0.003|0.008***|
||(4.493)|(0.371)|(4.842)|(11.603)|
|Manufacturing GDP share|-0.009***|-0.023***|-0.01***|0.004***|
||(-6.436)|(-15.78)|(-6.756)|(2.492)|
|Services GDP share<sup>_∗_</sup>|-0.007***|-0.012***|-0.007***|-0.001|
||(-7.00)|(-11.68)|(-6.753)|(-1.286)|
|Urban population share|-0.006***|-0.015***|-0.008***|0.001**|
||(-10.746)|(-24.929)|(-14.101)|(2.547)|
|Bottom 40% income share|-0.019***|-0.045***|-0.015***|0.012***|
||(-7.776)|(-18.221)|(-6.086)|(4.682)|
|N = 1520|R<sup>2</sup>= 0.978|_λ_=0.691.|_∗_p_<_.1; _∗_|_∗_p_<_.05; _∗∗∗_p_<_.01|

_Constant omitted, t-statistics in parenthesis. Optimal model contained no fixed effects. Model controls for mean, min and max NDVI, forest cover, and country size. ∗Inner 50% of significant marginal effects same sign, but range includes zero. ∗∗Inner 50% of marginal significantly excludes zero._ 19
Table 4: Carbon intensity results using the penalized kernel regression.

|(means)|(25%)|(50%)<br>(75%)|
|---|---|---|
|_Dependent:_|_Log carbon _|_intensity of 1000 GDP p.c._|
|Log 1000 GDP per capita<sup>_∗∗_</sup><br>-0.630***<br>(-42.341)|-0.755***<br>(-50.71)|-0.635***<br>-0.519***<br>(-42.699)<br>(-34.903)|
|Population density<br>-0.000***|-0.001***|-0.001***<br>0.000***|
|(-4.812)|(-11.919)|(-5.656)<br>(3.356)|
|Undernourishment rate<br>0.006***|-0.001|0.008***<br>0.014***|
|(5.927)|(-0.523)|(8.248)<br>(14.741)|
|Poverty 1.90$ rate<br>0.002**|-0.002***|0.001**<br>0.008***|
|(2.475)|(-3.295)|(2.092)<br>(11.496)|
|Manufacturing GDP share<sup>_∗_</sup><br>0.017***|0.001|0.019***<br>0.036***|
|(10.815)|(0.366)|(12.344)<br>(23.015)|
|Services GDP share<br>0.001|-0.007***|0.002<br>0.01***|
|(1.198)|(-6.708)|(1.667)<br>(9.01)|
|Urban population share<br>0.002***|-0.005***|0.002***<br>0.008***|
|(2.826)|(-7.41)|(2.995)<br>(12.768)|
|Bottom 40% income share<br>0.006**<br>(2.269)|-0.018***<br>(-6.492)|0.002<br>0.026***<br>(0.825)<br>(9.297)|
|N = 1520<br>R<sup>2</sup>= 0.956|_λ_=0.635.|_∗_p_<_.1; _∗∗_p_<_.05; _∗∗∗_p_<_.01|
|_Constant omitted, t-statistics in parenthesis. _<br>_Model controls for mean, min and max NDV_<br>_∗Inner 50% of signifcant marginal efects sa_<br>_∗∗Inner 50% of marginal signifcantly exclud_|_Optimal mo_<br>_I, forest cov_<br>_me sign, but _<br>_es zero._|_del contained no fxed efects._<br>_er, and country size._<br> _range includes zero._|

negative. This indicates that early urbanization is polluting, but that after reaching a tipping point, the city environment becomes cleaner. The trends across manufacturing and services are also primarily downwards. Agricultural societies have a higher pollution intensity of income, while a shift into manufacturing and services reduces the environmental output per unit of production. It remains difficult to say whether the effects reduce pollution on a net basis as this structural transformation occurs jointly with an increase in total productivity. However, for an identical amount of total GDP produced, the data seems to suggest that an agricultural economy produces the highest amount of pollution. An economy with a high manufacturing share produces less pollution, while an entirely service orientated economy outputs the lowest amount of pollution. This may also relate to a differential in value produced by these sectors which may imply different quality of production processes and differential in the total amount of economic activity for a fixed level of GDP. Across poverty and undernourishment we see hyperbolic effects that suggest that the eradication of extreme hunger occurs jointly with an increase in pollution intensity while later poverty eradication eventually occurs jointly with a reduction in pollution intensity. Poverty rates are unambiguously correlated with higher pollution intensities. Again, 20
similar to the deforestation results, it seems that societies with high income inequality are also more polluting.

Carbon intensities trend also with urbanization. We find that the carbon intensities initially increase together with the urbanization process, however after the 50% urban population tipping point, the environment becomes more efficient in carbon consumption. The shift in production composition trends oppositely with those of deforestation and manufacturing. High manufacturing and high services share in the production composition both correlate with higher carbon emission intensities. The initial decline in undernourishment rates occur together with improvements in the carbon emission intensities, poverty reduction however trends with an increase. Finally, we see that equality _−_ a stronger bottom 40% - increases carbon output when everything else is held constant, which is again an opposite trend of what we observed for deforestation and pollution.

# **4.2 Further results on heterogeneity**

Combined, the results show that income and poverty reduction provide a basis for improvements in the efficiency of economies in their use of finite resources. The economic composition is not unambiguous in its effects. To understand how structural transformation, together with urbanization, poverty reduction and increases in total production, interplay to produce a commonality in environmental output trends, we track the model predictions keeping the control variables at their means. We also keep the income distribution fixed at a mean value as it does not trend clearly with GDP as seen in fig. 9, and keep population densities fixed at means. Figure 3 shows the prediction surfaces using poverty and income as Cartesian coordinates. The model predictions fit the output levels well after scaling the log intensities, see fig. 10. While this shows that all countries gradually grow out of poverty and improve their efficiencies following a common pattern, it also reveals that there is significant heterogeneity in the environmental output intensities that relates to differences in poverty and hunger rates, urban population shares and GDP composition. This highlights that the shape of the EKC strongly depends on the development path of a country across all its dimensions. Furthermore, while the progression in output intensities follows a similar path, slight deviations from the local average may result in large differences in total environmental output. This reveals that while different development paths may relate to relatively small differences in the environmental output intensities, it may produce rather large differences in actual forest loss, air quality and carbon emissions 21
depending on the scale of the economy.

<!-- Start of picture text --> Log annual tree %loss per $1000 GDP per capita Annual tree cover %loss<br>model fit with fixed control variables model fit with fixed control variables<br>G G<br>GG GGGG G G G G G G G GGGGGGG G GGGG GGGG G G G G G GGG GG GGGGG GGG GG GG GG G G G G G GG G G G G GG GG G GGG G G G G G GG G G G GG GG GGG G G G G GGG G G GGGG G GG GG G G G GGG G GG G G GG GG G G G G G G GG G G G G GGG G GGG G GGG GG G GG GG G G G G G G GG GG GG GG G G GG G G G GGGG GGG GGG G G GG G G GG G GG GGG G G GG G G G G GGGG G G G G GGG GGG GG GG GG G G GG GGGGGGG G G G G G G G G G G G G G G G GG G GG GGGGG G GG G GGGGG G G G G G GG G GGG G G G GG G G G GGGG GGGG GG GGG G G G G G G G G GGG G G G G GGGG G G G G G G GG GGG G G G GGGGGG GG G G G G G GGGGG G G G G G GG G GGG GG GG GGGGGG G G GGGG G G GGGGG G GGG G GGG GG GGGG G G G G G GGGG G GG G GGG G G G G GG G G GG GG G G GGG GG GGG G G G G G G G G GGGGGG G G G G G G G GGGG G GGGG G G GG G G GGG GG G GG GGGG G GG G G G G G G G GGG G G GG G GGGGGGGGGGG G G G GGGG G GG G GG GGG G GG GG G GG G G GGGG G G G G G GGGGGGGGGGGGG G GGGG G GGGG G G G GGGGGG G GG GG G GG G GG G GG GG GGGG G G G G G G G GG GGG GG GGGGGG G G G G GG G G GG GG G G G GG GG G G G G G G G GGGGGGGGGGG G GGG GGGG GGGGGGG G GG G GGGGG G G GG G GGGGGGG G G G GGGGG GGG GGG G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG G GG G G G GGG G G G G GG G G GG G G G G G GGGG GGG GGG G G GGGGGGGGGGGGGGGGGGGGGGGGGG G G G G G G GGGGGGGGGGG G G GGGGG G GGGGGG G G G GGGGGGGGGGG G G G GGGGGGG G G GGGGGGG G GGGGGGGGGG GG GGGGGGGGGGGGGGGG G GGGGGGGGGGGGGGGGGGGGG G G GGGGG G GGGG G G GGG G GGGG G GGGGGGG G GGGGGG G G G GG G GGGGG G GGGGGGGGGGGGGGGGGGG G G GGGG G GG G GGG G G GGGGGGGGGGGG G G GG G GG G GG G GGGGGG G GGG GG G G GGGG GG G G GGG G G G GG G GGGGG G G G GG GG G GGGGGG G G GGG GG GGGG G GGGGGGG GG GGGGG GG G G GG GGG G GGG G G G G GGGGGGGG G GGGG [−5.09 − −4.63] [−4.6[ [[[[ [[−1.35[−0.89 GG − − −−2.7 −− GGGGGG 4.1 3.63.22.2 1.82 GGGGGGGGG 2 − −4.17]5 9 − −3.2 59 GGGGGGGGGGGGG − − − − − − − − GGGG − − − − − GGGG − GGGGGG 3.6 2.2.1.1.36]0.89]0.42] GGG GG GGGG 23]76]3083 GGG GGG 9] GG ]] GGG GG G GG GGGG G G G G G G G GGGGGGG G GGG GGG G G G G G GG G G GG G GGGGG G G GG G GG GG G G G G G GG G G G G GG GG G GGG G G G G G GG G G G GG GG GGG G G G G GGG G G GGGG G GG GG G G G GGG G GG G G GGG GG G G G G G G G G G G G GGG G GGG G GGG GG G GG GG GG G G G G G GG GG GG GG G G GG G G GGGG GGG GGG G G GG G G GG G GG GGG G G GG GGG G G G GGGG G G G GGG GGG G GG GG G G GG GGGGGGG G G G G G G G G G G G G G G G GG G G GGGG G G G GG G GGGGG G G G G G GG G GGG G G G GG G G G GGGG GGG GG G GGG GG G G G G G G G GGGG G G GGGG G G G G G G G GGG G G G G GGGGGG GG G G G G G GGG G G G G G GG G GGG GG GG GGGGGG G G GGGG G G GGGGG G GGG G GGGGG GG GGGG G G G G G GGGG G G G GGG G G G G GG G G GG G G G G GGG GG G G GG G G G G G G G GG GGGGG G G G G G GG GGGG G GGG G GG GGG GGG GG G G GGGGG G GG G G G GG G G G GGG G G GGG G GGGGGG G G GGG GGGGGG GG G GG GG G GG GGG G G G GGGGG G G G G G GGGGGGGGGGGGGG G GGGG G GGGGG G G G GGGGG G GG GG G GG G GG G GG GG GGG G G G G G G G GG GGG GG GGGGGGG G G G G G G G G GG GG G G G G GG G G G G G G G G G GGGGGGGGGG G GGG GGGG GGGGGGGG G GG G GGGGGG G G GG G GGGGGG G G G GGGG GG GGG G G G G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG G GG G G G GGG G G G GG G G GGGG G G G G GGGGGGG G G GGGGGG GG GGGGGGGGGGGGGGGG G G G G G GGGGGGGGGGGGGGGGGGGG G GGGGG G GGGGG G GGGGGG G G G GGGGGG G G G GGGGGGG G G GGGGGGGGGGGGG G GGGGG GG GGGGGGGGGGGGGGGGGG G GGGGGGGGGGGGGGGG G G GGGGG G GGG G G GGG G GGGG G G G GGGGGG G G G GG G G GG GGGG G GGGGGGGGGGGGGGGGGGG G GGG GGGG G GG G GG GG G G G GGGGGGGGGGGG G G GG G GG G GGGG G GGGGG G GGG GG G G GGG GG G G GG G GG G GG G GGGGG G G G GG GG G GGGG G G GGGGGG GG GGG G GGGGG GG GGGGG GG GG G GGG GGG G GGG G G G G GGGGGGGG G GGGG [0.07 − 0.14] [0.15[0.23 [0.30[0.38[[ [0.63 [0.71 −[0.79 − GG 0.470.55 GGGGGG GGGGGGGG − 0.22] − 0.38] − − G − 0.3 −− 0.7 GGGGGGGGGG 0.4 0.5 0.6 0. 0.8 GGGGGGGG G GGGG 7 GGGGGGGG 0] 6]42]0]8]6] GG GGGG ] GGG GGG GG GG G G G G<br>500 1250 2250 3500 6000 7500 12500 17500 40000 66000 500 1250 2250 3500 6000 7500 12500 17500 40000 66000<br>GDP Per Capita log scale GDP Per Capita log scale<br>Log PM25 mcg/m3 per $1000 GDP per capita PM25 mcg/m3<br>model fit with fixed control variables model fit with fixed control variables<br>G G<br>GG GGGG G G G G G G G GGGGGGG G GGGG GGGG G G G G G G G GG G GGGGG GGG GG GG GG G G G G G GG G G G G GG GG G GGG G G G G G GG G G G GG GG GGG G G G G GGG G G GGGG G GG GG G G G GGG G GG G G GGG GG G G G G G GG G G G G GGG G GGG G GGG GG G GG GG GG G G G G G GG GG GG GG G G GG G G GGGG GGG GGG G G GG G G GG G GG GGG G G GG GGG G G G GGGG G G G GGG GGG G GG GG G G GG GGGGGGG G G G G G G G G GG G G G G G G G G GG GGGGG G GG G GGGGG G G G G G GG G GGG G G G GG G G G GGGG GGGG GG GGG GG G G G G G G G GGG G G G GGGG G G G G G G GG GGG G G G GGGGGG GG G G G G G GGGGG G G G G G GG GGG GGG GG GG GGG G G GGGG G G GGGGG G GGG G GGG GGG GGGGG G G G G G GGGG G GG G GG G G G G GG G G GG GG G G GGG G GGG G G G G G G G G GG GGGGGG G G G G G G GGGG G GGGG GGG G GG G G GGG G G GGG GGGG G G G GG G G G GGG G G GGG G GGGGGGGGGGG G G GGG GGGG GGG G GG GGG G GG GG G G G GGG G G G G G GGGGGGGGGGGGGG G GGGG G GGGG G GGGGGG G GG GG G GG G GG G GG GG GGGG G G G G G G G GG GGG GG GGGGGG G G G G G GGG G G GG GG G G G GG GG G G G G G G G GGGGGGGGG G GGG GGGG GGGGGGGGGGGGG G GGG G GGG G G GG G GGGGGGG G G G GGGGG GG GGG G G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG G GG G G G GGG G G G GG G G GGGGGG G G G G GGGGGG G G GGGGGGGGGGGGGGGGGGGGGGGGG G G G G G GGGGGGGGG G G GGGGG G GGGGGGGGGGGGGGGG G G G GGGGGGGGGGGG G G G GGGGGGGGG G G GGGGGGG G G GG GGGGGGGGGGGGGGGG G GGGGGGGGGGGGGGGGGGGGG G G GGGGG G GGG G G GGG G GGGG G GGGGGGG G GGGGGG G G G GG G GGGGG G GGGGGGGGGGGGGG G GG G GGGGGGG G GG G GGG G G GGGGGGGGGGGG G G GG G GG G GGGG G GGGGG G GG GG G G GGGG GG G G GG G G G GG G GGGGGG G G G GG GG G GGGGGGG G G GGG GG GGG G GGGGGGG GG GGGGG GG G G GG GGG G GGG G G G G GGGGGGGG G GGG [−1.32 − −0.85] [−0.8[ [0.10[0.56[[ [1.98 [2.45 −[2.94 − GG − 1.041.51 GGGGGGG 0.3 GGGGGGGG − 0.56] − − 4 − −0.38]7 G −− GGGGGGGGGGGGG − 0. GGGG 1.0 1.5 1.9 2. 3.4 GGG 2.4 GG 9 GGGGGGGG 2]17]5]1]0] 08 GG GGGG ] GGG GGG ] GG GGG GG G GG GGGG G G G G G G G GGGGGGG G GGG GGG G G G G G GG G G GG G GGGGG GGG GG GG GG G G G G G GG G G G G GG GG G GGG G G G G G GG G G G GG GG GGG G G G G GGG G G GGGG G GG GG G G G GG G GG G G GG GG G G G G G G G GG G G G G GG G GG G GGG GG G GG GG GG G G G G G GGG GG GG GG G G GG G G GGG GG GGG G G GG GG G GGGG G GG GGG G G GG GGG G G G GG G G G GGG GGG G GG G GG G G GG GGGGGGG G G GG G G G G G GG G G G G G G G G G GGGG G G G GG G GGGGG G G G G G GG G GGG G G G GG G G G GGGG GGGG GG GGG GG G G G G G G G GGG G G G GGGG G G G G G G G GGG G G G G GGGGGG GG G G G G G GGGGG G G G G G GG GGGG GG GG GG G G G G GGG G G GGGGG G GGG G GG GG G GG GGGGG G G G G G GGG G GG GG GG G G G G GG G G GG GG G G GGG G G GGG G G G G G G G G GG GGGGGG G G G G G G GGG G GGGG GGG GG GGG GGGG GG G GGG GGGG G G G G G G G GGG G G GGG G GGGGGGGGGG G G GGG GGGGG GGG G GG GGG G GG GG G G G GGGG G G G G G GGGGGGGGGGGGG G GGGG G GGGG G GGGGGGG G GG GG G GG G GG G GG GG GGGG G G G G G G G GG GGG GG GGGGGG G G G G GGGG G G GG GG G G G GG G G G G G G G G G GGGGGGGGG G GGG GGGG GGGGGGGG G GG G GGG G G GG G G G GGGGGG G G G GGGG GGG GGG G G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG G GG G G G GGG G G G GG G G GGGGGG G G G G GGGGGG G G GGGGGGGGGGGGGGGGGGGGG G G G G G G GGGGGGGGGGGGGGG G GGGGG G GGGGG G GGGGGG G G G GGGGGGGGGGGG G G G GGGGGGGGG G G GGGGGGG G G GG GGGGGGGGGGGGGGG G GGGGGGGGGGGGGGGGGGGGG G G GGGGG G GGGGGGG G G GGG G GGGG G GGGGGGG G GGGGGG G G G GG G GGGGG G GGGGGGGGGGGGGGGG G G GGGGGGG G GG G GGG G G GGGGGGGGGGGG G G GG G GG G GGGG G GGGGG G GGG GG G G GGGG GG G G GG G G G GG G GGGGG G G G GG GG G GGGGGGG G G GGG GG GGG G GGGGGGG GG GGGGG GG G G GG GGG G GGG G G G G GGGGGGGG G GGGG [6.27 − 10.5] [10.5[14.8 [19.1[23.4[[ [36.3 [41.0 −[44.9 − GG 27.632.0 GGGGGG GGGGGGGGG − 14.8] − 23.3] − − G − 19. −− 40. GGGGGGGGGGGGGGGG 27. 31. 36. 44 49. GGGG G GGGG . GGGGG 0] 6]92]1]6]0] GG GGGG ] GGG GGG GG GGG GG G<br>500 1250 2250 3500 6000 7500 12500 17500 40000 66000 500 1250 2250 3500 6000 7500 12500 17500 40000 66000<br>GDP Per Capita log scale GDP Per Capita log scale<br>Log CO2 kg per ppp $ of GDP per $1000 GDP per capita CO2 kg per ppp $ of GDP<br>model fit with fixed control variables model fit with fixed control variables<br>G G<br>GG GGGG G G G G G G G GGGGGG G GGGGG GGGGG G G G G G GGG G GGGGG GGG GG GG GG G G G G G GG G G G G GG GG G GGG GG G G GG GG G G GG GG GGG G G G G GGG G G GGGGG G GG GG G G GGG G GGGG G GG GG G G G G G GG G GGG G G G GG G GGG GG G GG GG GG G G G G G GGG GG GG GG G G GG G G GGGG GGGG GGG G G GG G G GGGGG GGG G G GG GGG G G G GGGGGG G GGG GGG G GG G G G GG GGGGGGG G GGGG G G G G G G G G G G G GG G GG GGGGG G GG G GGGGG G GGG G GG G GGG G GGGGG G G GGGG GGGG GG GGG GG G G G G G G G GGG G GG GGGG GG G G G G GG GGG G G G GGGGGG GG G G G G G GGGGG G G G G G GG GGGG GGG GG GG G G G GGG G GGGGGG G GGG G GG GG G GG GGGGG G G G G GGGGG G GG GG GG G G G G GG G G GG GG G G GGG GG GGG G G G G G G G GG GGGGGG G G G G G G GGGG G GGGG GGG G GG G G GGG GG G GGG GGGG G G G G G G G GGG G G GG G GGGGGG G G GGG GGGGG G GGGG GGG G GG GGGG GG GG G G G GGGGGG G G G GGGGGGGGGGGGG G GGGG G GGGG G GGGGGGG G GG GG G GG G GG G GG GG GGGG G G GG G G G G GG GG GGGGG G G G G G G G GGG G G GG GG G G G GGGGGG G G G G G G G GGGGGGGGGGGG GGGG GGGGGGGGGGG G GGG G G GGG GGGGGGG G G GGGGGGGGGGGGGGGGG G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG G G GGG G G G GGG G GGGGGG GGGGG G GGGGGGGGGGG G GGGGGGGGGGGGGGGGG G G G G G G GGGGGGGGGGG G G GGGGG G GGGGGGGGGGGG G GGGGGGGG G GGGG G GG G G GGGGGGGGGGGGGGGGGGGGGGG GG GGGGGGGGGGGGGGGGGG G GGGGGGGGGGGGGGGGG G G GGGGGGGGGG G GGGG G GGGGGGGGGGG GGGGGGGG GGGGG GG GG G GGGGGGGGGGGGGGGGGGGGG G GGGGGGGGG G GGG G GGGGGGGGGGGGGG G GGG G GG G GGGGGGGGGGGG G GGG GG GGGGGGGGGGG G GGGGGGGGGGGG G GGGGGGGGGGGG G GGGGGGGGGGGGGGGG GG GGGGGGGGG G GGG GGG G GGG G G G G GGGGGGG GG [−5.52 − −5.12] [−5.1[ [[[[ [[−2.36[−1.96 GGGGGGG − − −−3.5 −− GGG 4.7 4.33.93.1 2.75 GGGGGGGG 2 − −4.73]3 3 − −3.2 45 GGGGGGGGGGGGG − − − − − − − − GGGG − − − − − GGGG − GGGGGG 4.3 3.3.2.2.36]1.97]1.57] GGGGGGGGGGGGGGG 94]54]1575 3] GGGGGGG ]] G GG GGGG G G G G G G G GGGGGGG G GGGG GGGGG G G G G G GGG G GGGGG GGG GG GG GG G G G G G GG G G G G GG GG G GGG GG G G GG GG G G GG GG GGG G G G G GGG G G GGGGG G GG GG G G GGG G GGGG G GG GG G G G G G GG G GGGG G G G GGG G GGG GG G GG GG GG G G G G G GG GG GG GG G G GG G G GGGG GGG GGG G G GG G G GGGGG GGG G G GG GGG G G G GGGGGG G GGG GGG G GG GG G G GG GGGGGGG G GGG G G G G G G G G G G G GG G GG GGGGG G GG G GGGGG G GGG G GG G GGG G GGGGG G G GGGG GGGG GG GGG GG G G G G G G G GGGG G G GGGG GG G G G G G GG G G G G GGGGGG GGG G G G G G GGGGG G G G G G GG GGGG GGG GG GG GGG G G GGG G GGGGGG G GGG G GGG GG GGGGG G G G G GGGGG G GG GG GG G G G G GG G G GG G G G GGG GG G G GG G G G G G G G GG GGGGG G G G G G GG GGGG G GGGG GGG GG GGG GGG GG G GGG GGG G G G G G G G G GGG G G GGG G GGGGG G G GGG GGGGGG GGG GG GG GGGG GG G G G G GGGGGGGG G G G GGGGGGGGGGGGGGG G GGGG G GGGG G GGGGGGG G GG GG G GG G GG G GG GG GGGG G G G G G GG GG GGG GG GGGGG G G G G G GG G G GG GG G G G GGG G GGGG G G G G G GGGGGGGGGGGGGGG GGGG GGGGGGGGGGGGGG G GGG G G GGG GGGGGGG G G GGGGGGGGGG G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG G G GGG G G G GGG G GG G G GGGG G GGG GGG GGGGGGGGGGGGGG G GGGGGGGGGGGGGGGGG G G G G G GGGGGGGGGG G G GGGGG G GGGGGGGGGGGG G GGGGG G GGGG G GGG G G GGGGGGGGGGGGGGGGGGG GG GGGGGGGGGGGGGGG G GGGGGGGGGGGGGGGGGGG G G GGGGGGGGG G GGGG GGGGG GGGGGGGGGGGG G GGGGGGG GG GGG G GGGGGGGGGGGGGGGGGGGGGGGGG GGGG GGGGGGG G GGG G GGGGGGGGGGGGGG G GGG G G G GGGGGGGGGGG G GGGG GG GGGGGGGGGGG G GGGGGGGGGGGG G GGGGGGGGGGG G GGGGGGGGGGGGGGGG GG GGGGGGGGGGG G GGG GGG G GGG G G G G GGGGGGG GGG [0.07 − 0.10] [0.10[0.13 [0.16[0.19[[ [0.29 [0.32 −[0.35 − GGGGGGG 0.230.26 G G GGGGGGGG − 0.13] − 0.19] − − G − 0.1 −− 0.3 GGGGGGGGGGG 0.2 0.2 0.2 0. 0.3 GGGGGGGGGGG 3 GGGGGGG 6] 2]69]2]5]8] GGGGGGGGGG ] GGGGGGGGGGG<br>500 1250 2250 3500 6000 7500 12500 17500 40000 66000 500 1250 2250 3500 6000 7500 12500 17500 40000 66000<br>GDP Per Capita log scale GDP Per Capita log scale<br>85 85<br>76 76<br>66 66<br>57 57<br>47 47<br>38 38<br>Poverty Rate Poverty Rate<br>28 28<br>19 19<br>9 9<br>0 0<br>85 85<br>76 76<br>66 66<br>57 57<br>47 47<br>38 38<br>Poverty Rate Poverty Rate<br>28 28<br>19 19<br>9 9<br>0 0<br>85 85<br>76 76<br>66 66<br>57 57<br>47 47<br>38 38<br>Poverty Rate Poverty Rate<br>28 28<br>19 19<br>9 9<br>0 0<br><!-- End of picture text --> Figure 3: Model fits of degradation intensities of log GDP (left) and the rescaled environmental output levels (right) across poverty and income. Population densities and income equality as well as the control variables are held constant at the mean.

22
An important takeaway is that heterogeneity in the actual output levels (right), is primarily large around the income levels where output is also highest (around $4,000 for deforestation, $6,000 for pollution, and $8,000 for the carbon weight of a single dollar production value). This indicates that the theorized Environmental Kuznets tipping points are also the points at which an averaged result, such as obtained from a linear regression, provides the poorest indication of relationships at the individual country level. While a few general rules could be extracted from the marginal effects, such as the inequality, income and population density effects, the larger part of the environmental data seems to relate heterogeneously to economic variables.

# **4.3 Further results on average curvature**

<!-- Start of picture text --> Estimated standardized environmental degradation levels across income<br>100<br>Forest loss/ha/GDP p.c.<br>80 Pollution mcg/m3/GDP p.c.<br>Carbon kg/$/GDP p.c.<br>60<br>40<br>20<br>0<br>1100 1800 3000 4900 8100 13400 22000<br>Logarithmic GDP per Capita<br>Estimated standardized environmental degradation levels across income<br>100<br>Forest loss/ha<br>80 Pollution mcg/m3<br>Carbon kg/$<br>Carbon kg p.c.<br>60<br>40<br>20<br>0<br>1100 1800 3000 4900 8100 13400 22000<br>Logarithmic GDP per Capita<br>Normalized log intensity levels<br>Normalized degradation levels<br><!-- End of picture text --> Figure 4: Normalized predicted environmental output levels across income. Predictors are held at expectations conditional on GDP. The _R_<sup>2</sup> ’s from logarithmic GDP per capita to poverty, undernourishment, manufacturing, services and urban population shares are respectively 0 _._ 801, 0 _._ 633, 0 _._ 142, 0 _._ 573 and 0 _._ 739. The conditional expectations are plotted in fig. 9. Population density, income equality, and controls are held at their means.

23
The heterogeneity in amplitude, and location of tipping points, conditional on the economic variables, implies that a single Kuznets Curve, such as it has often been treated in the literature, is a description that applies only poorly to individual country cases. However, to do some justice to the classical concept we can still construct an average development path and explore how the models fit environmental outputs to that. To do so, we derive conditional expectations for poverty, undernourishment, GDP composition, and urban population shares, using only the logarithmic GDP per capita as an explanatory variable. We then use these conditional values to build a data set that includes all variables as local averages along with GDP itself. Again, we keep the control variables and the income distribution as well as population densities fixed. We normalize the results to compare the slopes and location tipping points across income. Figure 4 shows the curvatures associated with these development paths. We have dropped the lower 2.5% of GDP observations, and the upper 20%. We focus on this range because of its particular relevance for development policy. We note that the maximum total output associated with the average tipping point is an interesting statistic, but due to heterogeneity this may be a poor approximate to predict whether a country is close to its potential tipping point after observing only environmental output. The deforestation rate associated with the average development path attains a maximum of .66% annually, while that highest pollution concentration maxes at 28.7 mcg/m3 and the carbon weight of a dollar reaches 0.271 kg.

# **4.4 Further results on heterogeneity in curvature and tipping points**

The average pathways accurately describe the transition out of poverty but it provides less insight into the effects if compositional changes. To better understand the importance of deviations in transitional variables, we plot the degradation levels associated with the average development path with additional differences in manufacturing shares, urban population shares and poverty rates. Figure 5 shows that changing these variables, while keeping everything else at the local averages, has important impacts on the location, shape, and height of tipping points. For example, increasing the share of manufacturing by 10 points, shifts the tipping point of deforestation to the left, while economies that retain high agricultural shares reach a tipping point at higher income. This implies that an earlier transition out of agriculture may prevent high deforestation rates at higher income and lower pollution levels at its peak. This is a slightly counter intuitive result as manufacturing has traditionally been portrayed as the main source of pollution. However, since our 24
data only indicate the share of manufacturing in total GDP and not the quality or quantity of goods produced, higher rates may also correspond to differences in the number of manufacturing sites and the methods of production used. The carbon emissions associated with this structural change are higher, suggesting a trade-off between pollution-heavy and carbon-intense production. In a similar fashion, poor countries that have a high urbanization rate have higher deforestation rates and reach a pollution peak faster. Poor countries that have lower urbanization, on the other hand, eventually maintain higher pollution and carbon emissions levels at higher income. This suggests that the draw-down in pollution output is not just a matter of income and productivity, it may relate to attaining critical urban population mass combined with increased income. The effects of poverty, finally, do not impact the location and shape of the environmental output levels. Countries with high poverty rates unambiguously deforest and pollute more, but emit less carbon.

<!-- Start of picture text --> Estimated tree %loss across GDP for Estimated PM25 across GDP for different Estimated CO2 across GDP for different<br>different manufacturing shares manufacturing shares manufacturing shares<br>+10 point % +10 point % +10 point %<br>local average local average local average<br>−10 point % −10 point % −10 point %<br>1100 3000 8100 22000 1100 3000 8100 22000 1100 3000 8100 22000<br>GDP per capita log scale GDP per capita log scale GDP per capita log scale<br>R2 of underlying model = 0.922 R2 of underlying model = 0.98 R2 of underlying model = 0.956<br>Estimated tree %loss across GDP for Estimated PM25 across GDP for Estimated CO2 across GDP for<br>different urbanisation rates different urbanisation rates different urbanisation rates<br>+10 point % +10 point % +10 point %<br>local average local average local average<br>−10 point % −10 point % −10 point %<br>1100 3000 8100 22000 1100 3000 8100 22000 1100 3000 8100 22000<br>GDP per capita log scale GDP per capita log scale GDP per capita log scale<br>R2 of underlying model = 0.922 R2 of underlying model = 0.98 R2 of underlying model = 0.956<br>Estimated tree %loss across GDP for Estimated PM25 across GDP for Estimated CO2 across GDP for<br>different poverty rates different poverty rates different poverty rates<br>+10 point % +10 point % +10 point %<br>local average local average local average<br>−10 point % −10 point % −10 point %<br>1100 3000 8100 22000 1100 3000 8100 22000 1100 3000 8100 22000<br>GDP per capita log scale GDP per capita log scale GDP per capita log scale<br>R2 of underlying model = 0.922 R2 of underlying model = 0.98 R2 of underlying model = 0.956<br>0.6 30 0.25<br>0.5 25 0.20<br>%loss 0.4 PM25 20<br>0.15<br>0.3 15 CO2 kg per ppp $<br>0.2 10 0.10<br>0.7 30<br>0.6 25 0.25<br>0.5 0.20<br>%loss PM25 20<br>0.4 0.15<br>15 CO2 kg per ppp $<br>0.3<br>0.10<br>0.2 10<br>0.7 30 0.30<br>0.6<br>25 0.25<br>0.5<br>%loss 0.4 PM25 20 0.20<br>0.3 15 CO2 kg per ppp $ 0.15<br>0.2 10 0.10<br><!-- End of picture text --> Figure 5: predicted environmental output levels across income. Predictors are held at expectations conditional on GDP and one variable has been incremented +/- 10 points in each plot. The local average trend is identical to those in fig. 4.

25
# **5 Projecting 2030**

To explore whether continuation of current growth can be expected to lower environmental outputs without intervention, we extrapolate GDP into the future and calculate associated model responses under three simplistic scenarios of growth. We use the estimated models to back out the conditional expectations for the economic variables based on GDP, and use these datasets of hypothetical future economies to predict levels of environmental output at each point in time. In a base scenario, each sovereigns grows at individual median 19992014 compound rates, with the highest growth rate capped at the 90% percentile (5.27% annually). In a pessimistic future, each country continues at one asymmetric deviation unit (-3.67%) below the base rate, and in an opportunistic growth scenario countries continue one asymmetric deviation unit (+0.89%) above the base rates.<sup>16</sup> In the opportunistic scenario, rates are capped at the 95% percentile rate (5.87% annually). Finally, we construct Business As Usual (BAU) as the average of the three results to balance between possible asymmetry. Table 5 summarizes the growth rates.

To limit the complexity, we keep population growth slightly below individual country median compound rates, resulting in 8.5 billion people by 2030 (in line with United Nations projections).<sup>17</sup> We use extrapolated GDP levels to derive fits for poverty and undernourishment levels, and the GDP composition, using univariate model fits of the penalized kernel model. We let the urban population depend additionally on log population densities.<sup>18</sup> At each point in extrapolated time, we compare the conditional expectations to the predictions of our base year (2014), and compute the percentage change that we then multiply with observed 2014 values. We keep all data points within observed intervals, including a cap on the sum of agriculture and services shares. This means that effectively after reaching a level of $64,980 per capita, our projection halts both the income effect on efficiency improvements and the effect of scale increase on the environmental output for a country. In the pessimistic scenario, this does not affect any individual country result, in the base

> 16Asymmetric deviation units have been calculated as the difference between the median and respectively the 25% and 75% quantiles of growth rates. In the calculations we have dropped the two largest outliers (in absolute value) for each country.

> 17We reduced the population growth rates by 0.05 times the absolute point percentages globally to reduce growth everywhere, then reduced population growth rates by an additional .15 times the percentage rates in the top 40% income countries and additional .25 times in top 20% income countries. This simple scenario is designed to represent relatively higher growth in lower income countries and a slow down in the developed countries in line with UN projections.

> 18The _R_ 2’s of the models are, 0.632 for undernourishment, 0.785 for poverty, 0.142 for manufacturing, 0.573 for services, and 0.814 for urban population shares. The uncertainty of the impact of changes in manufacturing remains high in our projections.

26
this fixes Norway’s output at current levels and caps those of the U.S. and Switzerland in respectively the last 4 and 5 years of the projection (final 9 and 10 years in the high growth scenario.).

<!-- Start of picture text --> Global population weighted GDP scenario Global population weighted poverty projection<br>Historical<br>Pessimistic<br>BAU<br>Opportunistic<br>Historical<br>Pessimistic<br>BAU<br>Opportunistic<br>2000 2005 2010 2015 2020 2025 2030 2000 2005 2010 2015 2020 2025 2030<br>Year Year<br>Forest cover weighted % tree cover loss, Population weighted PM25 concentrations,<br>projection for the world projection for the world<br>Historical<br>Pessimistic<br>BAU<br>Opportunistic<br>Historical<br>Pessimistic<br>BAU<br>Opportunistic<br>2000 2005 2010 2015 2020 2025 2030 2000 2005 2010 2015 2020 2025 2030<br>Year Year<br>Total carbon emission output,<br>projection for the world<br>Historical<br>Pessimistic<br>BAU<br>Opportunistic<br>2000 2005 2010 2015 2020 2025 2030<br>Year<br>25000<br>20<br>20000 15<br>Poverty rate<br>GDP per capita<br>15000 10<br>10000 5<br>0.65 36<br>0.60 34<br>0.55 32<br>0.50<br>30<br>PM25 concentration<br>Annual % Tree cover loss 0.45<br>28<br>0.40<br>80<br>70<br>60<br>50<br>CO2 Billion Tons 40<br>30<br>20<br><!-- End of picture text --> Figure 6: Projection for global environmental outcomes. Table 6 contains aggregate statistics of the BAU line and highlights the distributional changes across income.

Figure 6 presents our projections at the global level made by aggregating all the countrylevel results and assuming that the average in-sample trend scales appropriately with missing areas. Results are also available for income segments in table 6. In the average scenario, global extreme poverty falls below 7.4% of the global population. The poorest 20% countries in our sample have stronger successes and go from 45% poverty to just over 33%.

While poverty reductions and GDP increases may improve livelihoods through eco27
nomic gain, air pollution remains a serious threat to wellbeing as the average global citizen remains exposed to 36 mcg /m<sup>3</sup> , nearly twice the WHO prescribed guidelines. Additionally, development comes at a cost of an annual carbon output that reaches 63GT which is nearly a doubling from the 2014 levels, and a total loss of 242 million hectares of forested land. About 58% of forest loss is in countries with poverty rates above 3% or income in the bottom 60%. Many of those countries have tropical rain forests with slow regrowth rates estimated at 27% (Hansen et al., 2013). Using those statistics, loss in these countries totals 136 million ha between 2014 and 2030, netting over 3.4% of the global 2000 dense forest cover. Other insights include that success in eradicating poverty likely slows as China and India near 0% poverty and populations in poor countries grow faster than those in the developed world. Our modeled data do not signal that development alone will result in successful slowdown in natural capital depletion. At the global level, results suggest ongoing increases in global deforestation rates and carbon emissions. Global pollution exposure stabilizes disregard the growth scenario, the results instead suggest a distributional shift toward lower income countries with improving and worsening conditions balancing out at the global scale. Air pollution concentrations rise by 28% in the bottom 20% income countries. Table 6 shows that the entire bottom 30% income countries of our sample in fact continues to face increasing pollution exposure. Projections of forest cover and carbon emissions on the other hand, are heavily dependent on the economic outlook. Growing wealth in the developing world together with rapid population growth may accelerate future global carbon output.

To focus explicitly on trends in the developing world, we aggregated the results for the bottom 60 percent per capita GDP of our samples, see fig. 7. These results suggest that deforestation rates, while high, may stabilize. Similarly to pollution at a global level, our modeled data suggest a distributional shift in which deforestation increases in the poorest countries while some improvements may occur in the third GDP quintile. The global increase depicted in fig. 6 is thus partly because decoupling of deforestation and economic activity has not fully occurred even in high income countries including Canada, the U.S. and Norway. These countries, however, have better regrowth rates (0.625% regrowth rate in the temperate climatic domain) than the tropical forests that largely lie in the developing world (Hansen et al., 2013). Pollution results suggest that development in the developing world may for the first time result in possible improvements in air quality. Disregard, pollution remains dangerously high at twice the WHO guidelines. Carbon emissions increase fast, 28
the results indicate that involving the developing world (i.e. bottom 60% per capita GDP countries), that currently contributes about 15% or 4.9 GT to total emissions, in GHG emission policy dialogs will become more important as their total output more than doubles to 11.4 GT in our BAU. This will be equivalent to 35% of current global emissions.

<!-- Start of picture text --> Bottom 60 population weighted GDP scenario Bottom 60 population weighted poverty projection<br>Historical<br>Pessimistic<br>BAU<br>Opportunistic<br>Historical<br>Pessimistic<br>BAU<br>Opportunistic<br>2000 2005 2010 2015 2020 2025 2030 2000 2005 2010 2015 2020 2025 2030<br>Year Year<br>Forest cover weighted % tree cover loss, Population weighted PM25 concentrations,<br>projection for bottom 60 projection for bottom 60<br>Historical Historical<br>Pessimistic Pessimistic<br>BAU BAU<br>Opportunistic Opportunistic<br>2000 2005 2010 2015 2020 2025 2030 2000 2005 2010 2015 2020 2025 2030<br>Year Year<br>Total carbon emission output,<br>projection for bottom 60<br>Historical<br>Pessimistic<br>BAU<br>Opportunistic<br>2000 2005 2010 2015 2020 2025 2030<br>Year<br>10000 35<br>30<br>8000<br>25<br>Poverty rate<br>GDP per capita 6000 20<br>4000 15<br>40<br>0.6<br>38<br>0.5 36<br>34<br>% Tree cover loss 0.4 PM25 concentration<br>32<br>0.3 30<br>12<br>10<br>8<br>6<br>CO2 Billion Tons<br>4<br>2<br><!-- End of picture text --> Figure 7: Projection for environmental outcomes in the developing world. Table 6 contains aggregate statistics of the BAU line and highlights the distributional changes across income.

To shed final light on the challenges that await global leadership at the fronts of global warming and reducing carbon emission outputs, we have broken down total carbon emissions in per capita GDP quintiles in fig. 11. Emissions in the bottom 20% are not significantly contributing to world aggregates. Starting at the second quintile, we see considerable emission outputs all growing at alarming rates. Even the emissions of the current top 20% 29
keep increasing in our modeled data, possibly suggesting that the 2008-2014 successes may have partly been driven by a combination of slow economies and improvements due to ongoing urbanization that will likely halt as rural areas become less populated. Improved economic conditions in the top income countries may result in renewed upward momentum, albeit the increase is much slower than what is seen in the lower income countries. The slowdown resulting from modeled data indicates that stabilized levels may possibly be reached soon in high income level countries. Disregard of the decoupling discussion in high income countries, that evidently must result from policy intervention, we see that carbon output in the fourth income quintile, and China in particular, dominates the global result. In total, 57% of average global 2030 emissions, or 35.8 GT, is produced in these countries. Without improved policies, the fourth income quintile produces more than the 2014 global output.

# **6 Discussion and Conclusion**

Through this paper, we have discussed penalized non-parametric modeling of environmental output across economic development. This type of modeling works well for nonlinear processes given that they do not result in overly complex dynamics. We deployed the framework to study environmental data in a panel of 95 countries. We specifically modeled satellitederived deforestation, satellite-derived air pollution, and reported carbon emissions. To deal with heteroskedastic variance, we transformed the data to logarithmic degradation intensity of per capita GDP. Our out-of-sample shrinkage of fixed error components did not support time fixed effects.

Our results suggest that production gradually favors conserving the earth’s finite resources as GDP increases, but that this alone is not sufficient to offset the effects of scale growth. Instead, structural change in the economy shapes environmental output curves. This process shares similarities between sovereigns, but remains largely heterogeneous. These results do not support a single Environmental Kuznets rule. Instead, the results emphasize the importance of local economic conditions on environmental results. Across all data levels, some effects hold unambiguously. Poverty and income inequality correlate with higher pollution, higher deforestation, and lower carbon emissions; agricultural GDP correlates with deforestation; population densities correlate with pollution; and higher manufacturing shares correlate with increased carbon emissions. We find various tipping points in other variables, notably across urbanization rates. While local conditions may 30
be unique, average development is associated with an inverted _U_ -shape in deforestation, pollution and carbon intensities of production units. Per capita carbon missions follow a _J_ -curve as the increase in per person productivity is not sufficiently offset by efficiency improvements. Disregarding the level of per capita GDP, we observe that at least one form of natural capital degradation is high, conflicting with the belief that countries tend to ”clean up” as they develop. One could argue that the scope of the impacts of externalities to production increases with development, with the burden falling to increasingly distant households both in time and space. Although local air pollution may be more intrusive on daily life, the consequences of climate change will remain globally impactful for generations to come.

We extrapolated our descriptions forward in time to highlight the daunting implications of development under continuation of current practices without improving policies. Our results are generally in line with emission paths associated to the high radiative forcing scenario’s considered in IPCC’s 4.9<sup>_◦_</sup> world (RCP 8.5). Our projections did not indicate successes on the fronts of reducing deforestation. Air quality improves in some currently severely polluted places, but worsens in poor regions.

In our results, deforestation follows an inverted _U_ -shape across average development in the developing countries. This confirms and extends recent results from Crespo Cuaresma et al. (2017) that provide evidence for a partial EKC for forest cover at low income. However, we find that growth alone is not sufficient to halt forest loss, and we find evidence that within the bottom 60% income countries, deforestation shifts to the bottom 30%, and that countries within the top 40% income do not fully stop deforesting. Others have similarly detailed forest loss in high-income countries, for example in the United States (Sleeter et al., 2012). Future efforts should also aim to understand the regrowth dynamics across economic development, we have only used it as a control variable in our models. Generally, the temperate zones have much better regrowth rates. Taking this and projected increases in the bottom 20% into account, the African forests seem to be at increased risk as economic successes in these areas accelerate, while the amazon faces only marginal improvements in the immediate future in our modeled projections.

On the pollution side, our model projects rising PM2 _._ 5 levels in the lowest 30% income countries, with a general decrease in PM2 _._ 5 in middle income countries. PM2 _._ 5 remains far above WHO air quality guidelines in many countries, particularly in lower and middle income groups. Given population growth, these levels will expose greater and greater popu31
lations to pollution-related health risks as exposed populations continue to grow. Currently, about 90% of the global population is exposed to air quality that does not comply with the World Health Organization’s Air Quality Guidelines (World Health Organization, 2016). Tallis et al. (2017) expect that by 2050, business-as-usual development will result in over 4.8 billion people living in countries with worse air quality than in 2010. As a comparison, in our average modeled data 52% of people currently live in places where air quality has worsened by 2030. This totals to approximately 4.4 billion by 2030.

On the carbon end, our results suggest emission levels that could lead to the high radiative forcing scenario in IPCC’s 4.9<sup>_◦_</sup> world (RCP 8.5) are largely in line with business-asusual development. Worse scenarios may in fact thus be considered as relevant possibilities. Recent studies suggest we are not alone in such a conclusion. See for example Peters et al. (2012) and comments, suggesting - in line with our findings - that reported successes in carbon reduction are short-lived and largely relate to the 2008-2009 crisis and aftermath. Emissions rapidly increased in many places with the recovery. Furthermore, Peters et al. (2013) and comments thereon reveal that recent emissions continue to track the high end of suggested emission scenarios, making it increasingly unlikely that global warming will stay below 2<sup>_◦_</sup> . This is in line with our result that continuing current development, puts the world on emissions associated with a 4.9<sup>_◦_</sup> pathway. This is further substantiated by the conclusion that developments on the fronts of negative emissions are required to reach a 2<sup>_◦_</sup> future Gasser et al. (2015). Combined, the evidence suggests that a worst-case scenario over 4.9<sup>_◦_</sup> in 2100 is both not unrealistic and overlooked in both the scientific community and the political arena.

# **References**

- Bates, C. and White, H. (1985). A Unified Theory of Consistent Estimation for Parametric Models. _Econometric Theory_ , 1(02):151–178.

- Blasques, F. (2010). Identifiable Uniqueness Conditions for a Large Class of Extremum Estimators. In _ETA International Symposium on Econometric Theory and Applications_ , Singapore.

- Blasques, F. and Duplinskiy, A. (2015). Penalized Indirect Inference.

- Crespo Cuaresma, J., Danylo, O., Fritz, S., McCallum, I., Obersteiner, M., See, L., and Walsh, B. (2017). Economic Development and Forest Cover: Evidence from Satellite Data. _Scientific Reports_ , 7:40678.

32
- Dijk, D. V., Franses, P. H., Lucas, A., and Lucas, A. (1999). Testing for Smooth Transition Nonlinearity in the Presence of Outliers. _Journal of Business & Economic Statistics_ , 17(2):217.

- Domowitz, I. and White, H. (1982). Misspecified models with dependent observations. _Journal of Econometrics_ , 20(1):35–58.

- Gasser, T., Guivarch, C., Tachiiri, K., Jones, C. D., and Ciais, P. (2015). Negative emissions physically needed to keep global warming below 2<sup>_◦_</sup> C. _Nature Communications_ , 6:7958.

- Granger, C., King, M. L., and White, H. (1995). Comments on testing economic theories and the use of model selection criteria. _Journal of Econometrics_ , 67(1):173–187.

- Grossman, G. M. and Krueger, A. B. (1995). Economic Growth and the Environment. _The Quarterly Journal of Economics_ , 110(2):353–377.

- Hainmueller, J. and Hazlett, C. (2014). Kernel regularized least squares: Reducing misspecification bias with a flexible and interpretable machine learning approach. _Political Analysis_ , 22(2):143–168.

- Hansen, M. C., Potapov, P. V., Moore, R., Hancher, M., Turubanova, S. A., Tyukavina, A., Thau, D., Stehman, S. V., Goetz, S. J., Loveland, T. R., Kommareddy, A., Egorov, A., Chini, L., Justice, C. O., and Townshend, J. R. G. (2013). High-Resolution Global Maps of 21st-Century Forest Cover Change. _Science_ , 342(6160):850–853.

- Horowitz, J. L. (2011). Applied Nonparametric Instrumental Variables Estimation. _Econometrica_ , 79(2):347–394.

- Kent, J. T. and Tyler, D. E. (2001). Regularity and Uniqueness for Constrained M - Estimates and Redescending M -Estimates. _The Annals of Statistics_ , 29(1):252–265.

- Kullback, S. and Leibler, R. A. (1951). On Information and Sufficiency. _The Annals of Mathematical Statistics_ , 22(1):79–86.

- Micchelli, C. A., Xu, Y., and Zhang, H. (2006). Universal Kernels. _Journal of Machine Learning Research_ , 7:2651–2667.

- Perman, R. and Stern, D. I. (2003). Evidence from panel unit root and cointegration tests that the Environmental Kuznets Curve does not exist. _Australian Journal of Agricultural and Resource Economics_ , 47(3):325–347.

- Peters, G. P., Andrew, R. M., Boden, T., Canadell, J. G., Ciais, P., Qu ́er ́e, C. L., Marland, G., Raupach, M. R., and Wilson, C. (2013). The challenge to keep global warming below 2<sup>_◦_</sup> C. _Nature Climate Change_ , 3(January):4–6.

33
- Peters, G. P., Marland, G., Le Qu ́er ́e, C., Boden, T., Canadell, J. G., and Raupach, M. R. (2012). Rapid growth in CO2 emissions after the 2008-2009 global financial crisis. _Nature Climate Change_ , 2(1):2–4.

- P ̈otscher, B. M. and Prucha, I. R. (1991). Basic structure of the asymptotic theory in dynamic nonlineaerco nometric models, part i: consistency and approximation concepts. _Econometric Reviews_ , 10(2):125–216.

- P ̈otscher, B. M. and Prucha, I. R. (1997). _Dynamic Nonlinear Econometric Models_ . Springer Berlin Heidelberg, Berlin, Heidelberg.

- Rothenberg, T. J. (1971). Identification in Parametric Models. _Econometrica_ , 39(3):577– 591.

- Sch ̈olkopf, B. and Smola, A. J. (2001). _Learning with kernel: Support Vector Machines, Regularization, Optimization and Beyond_ .

- Shahbaz, M., Shafiullah, M., Papavassiliou, V. G., and Hammoudeh, S. (2017). The CO2–growth nexus revisited: A nonparametric analysis for the G7 economies over nearly two centuries. _Energy Economics_ , 65:183–193.

- Sin, C.-Y. and White, H. (1996). Information criteria for selecting possibly misspecified parametric models. _Journal of Econometrics_ , 71(1):207–225.

- Sleeter, B. M., Sohl, T. L., Wilson, T. S., Sleeter, R. R., Soulard, C. E., Bouchard, M. A., Sayler, K. L., Reker, R. R., and Griffith, G. E. (2012). Projected Land-Use and LandCover Change in the Western United States. _Baseline and Projected Future Carbon Storage and Greenhouse-Gas Fluxes in Ecosystems of the Western United States_ .

- Smarzewski, R. (1986). Strongly unique best approximation in Banach spaces. _Journal of Approximation Theory_ , 47(3):184–194.

- Stern, D. I. (1998). Progress on the Environmental Kuznets Curve? _Environment and Development Economics_ , 3(2):173–196.

- Stern, D. I. (2004). The Rise and Fall of the Environmental Kuznets Curve. _World Development_ , 32(8):1419–1439.

- Stern, D. I., Common, M. S., and Barbier, E. B. (1996). Economic growth and environmental degradation: The environmental Kuznets curve and sustainable development. _World Development_ , 24(7):1151–1160.

- Tallis, H., Hawthorne, P., Polasky, S., Reid, J., Beck, M., Brauman, K., Bielicki, J., Binder, S., Burgess, M., Cassidy, E., Clark, A., Costello, C., Fargione, J., Game, E., Gerber, J., Isbell, F., Kiesecker, J., McDonald, R., Metian, M., Molnar, J., Mueller, N., O’Connell, C., Ovando, D., Troell, M., Boucher, T., and McPeek, B. (2017). Meeting Economic Growth and Multiple Environmental Elements of Sustainable Development. _Forthcoming_ .

34
- Tibshirani, R. (1996). Regression selection and shrinkage via the lasso. _Journal of the Royal Statistical Society B_ , 58(1):267–288.

- van Donkelaar, A., Martin, R. V., Brauer, M., Hsu, N. C., Kahn, R. A., Levy, R. C., Lyapustin, A., Sayer, A. M., and Winker, D. M. (2016). Global Estimates of Fine Particulate Matter using a Combined Geophysical-Statistical Method with Information from Satellites, Models, and Monitors. _Environmental Science & Technology_ , 50(7):3762– 3772.

- Vollebergh, H. R. J., Dijkgraaf, E., and Melenberg, B. (2005). Environmental Kuznets curves for CO2: heterogeneity versus homogeneity. _Ssrn_ , (November):39.

- Wagner, M. (2015). The Environmental Kuznets Curve, Cointegration and Nonlinearity. _Journal of Applied Econometrics_ , 30(6):948–967.

- White, H. (1994). _Estimation, inference, and specification analysis_ . Cambridge University Press.

- White, H., White, and Halbert (1980). Using Least Squares to Approximate Unknown Regression Functions. _International Economic Review_ , 21(1):149–70.

- World Bank (1992). World Development Report 1992. Development and the Environment. Technical report.

- World Health Organization (2016). Ambient Air Pollution: A global assessment of exposure and burden of disease. _World Health Organization_ , pages 1–131.

- Zou, H. (2006). The Adaptive Lasso and Its Oracle Properties. _Journal of the American Statistical Association_ , 101(476):1418–1429.

35
# Supplementary Appendix

Penalized Non-parametric Inference of Global Trends in Deforestation, Pollution and Carbon Bo Pieter Johannes Andr ́ee, Andres Chamorro, Phoebe Spencer, Harun Dogo

## **A Additional discussion**

Non-parametric approaches are capable of producing parameterization mappings that approximate nonlinearities arbitrarily well, but do not necessarily also produce uniquely identifiable solutions to the criterion function if the hypothesis space produces universal approximations that fit the data arbitrarily well for any sample size. Estimation is therefore problematic without additional structure to the estimator, which in our case comes in the form of a penalty to the criterion, but in other settings may relate to bandwidths or other tuning parameters. It is a challenge in its own right to understand how this complexity-penalized estimator is positioned relative to the classical least squares approximation context as considered by White et al. (1980).<sup>19</sup> In the standard context, the best approximation is produced by a unique point in the entire parameter space, while in the penalty context a unique best approximation exists for every given penalty. Hence, the divergence between the _true_ functional form and the _pseudo_ -true approximation, is not driven by boundaries to the parameter space as in the parametric case, but rather it is driven by the penalty. Ultimately penalization confines the hypothesis space to simple spaces and must reflect a prior belief that the _true_ functional form would not result in a large penalty. That prior belief carries over to the limit result if the penalty does not vanish. This produces a bias, or may even render the result completely arbitrary if the penalty is set without caution. In the current setting, our penalty arises as a function of an out-of-sample criterion. As a result, the space of functions that are viewed as acceptable solutions to the criterion is generated by the data itself, and the penalized non-parametric method is able to obtain approximations of increasing complexity as the data size tends toward infinity. In the finite sample case, this estimator is appropriate given that the relationship between environmental degradation and indicators of economic development is not dominated by high-frequency components that would result in strong complexity.

In this additional discussion, let N, Z, and R denote the sets of the natural, integer, and real numbers. R _>_ 0 includes all positive, non-zero, reals. For a set _A_ , we use B( _A_ ) to denote the Borel _σ_ -algebra over _A_ . We use _t, ..., T ∈_ Z to index time, and _i, ..., N ∈_ N to

> 19White et al. (1980) discuss convergence toward the unique least squares approximator that may differ from the _true_ parameter in the presence of misspecification bias.

i
index cross-sections, _it, ..., NT ∈_ N _×_ Z labels all locations in space-time. We use boldfaced letters, e.g., **a** _∈A_ to denote vectors. Furthermore, _×_<sup>_t_</sup> _t_<sup>=</sup> =1<sup>_TA_=</sup><sup>_AT_denotestheCartesian</sup> product of _T_ copies of _A_ , and _A∞_ = _×_<sup>_t_</sup> _t_<sup>=</sup> =<sup>_∞_</sup> _−∞_<sup>_A_istheCartesianproductofinfinitecopies.</sup> For two maps _f_ and _g_ , _f ◦ g_ is their composition resulting from a point-wise application, and _⟨·, ·⟩_ denotes the inner product space.<sup>20</sup> Finally, _∥· ∥A_ denotes a norm on _A_ .

# **A.1 Identifiability in nonlinear models**

Closely related to regulating the size of non-parametric models is the ill-posedness of unregulated non-parametric models. Before discussing the relationship between penalization and identifiability of the criterion of non-parametric estimators, we provide a simplified discussion on identifiable uniqueness and its relation to inference in the context of finite dimensional nonlinear models.<sup>21</sup> Hypothesis testing in a framework of finite parameter nonlinear models is often plagued by the problem that verification of the assumptions required for identifiability, relies itself on the outcome of a hypothesis that may be difficult to test. This is problematic as identifiable uniqueness plays a key role in establishing consistency and normality of test statistics. This is illustrated by a model of the form: in which the postulated relationship between _y_ and _x_ is assumed to follow a hyperbolic curve across levels of _x_ . In this model the linearity hypothesis _H_ 0<sup>1:</sup><sup>_γ_=0relatingtothe</sup> non-existence of the curved functional form depends on a second hypothesis _H_ 0<sup>2:</sup><sup>_δ_=0</sup> being true or false. This follows from the fact that for _δ_ = 0, _γ_ can take any value without changing the predicted density implied by the model. In this case any form of completeness required for identifiable uniqueness of the estimator, holds at most for a subset of the parameter space in which the model would in fact produce an inverted _U_ -shaped form. Distributions corresponding to different values of _γ_ are only sufficiently distinct when _δ_ is sufficiently bounded away from zero. Without establishing existence and uniqueness of a consistent estimator, it is impossible to establish normality, hence the distribution of test statistics remains unknown.<sup>22</sup>

> 20As a generalization of the dot product in the Euclidean space, to higher dimensional spaces including infinite dimensional Hilbert spaces.

> 21Identifiable uniqueness is a difficult concept, more elaborate general discussion can be found here (P ̈otscher and Prucha, 1991) formal definitions and discussion at a deeper level regarding strongly unique best approximation in Banach spaces can be found here (Smarzewski, 1986); and discussion on regulated _M_ -estimation can be found here (Kent and Tyler, 2001) and an overview of concepts is written in the monograph of (Blasques, 2010).

> 22Auxiliary test statistics may still be derived, but it is sometimes difficult to ensure that Taylor expansions do not capture nonlinearities of a type not predicted by the economic theory. See for example (Dijk et al., 1999) for a discussion in the threshold framework. Researchers may also choose to rely on information criteria to compare various descriptions of the data and decide between economic theories (Granger et al., 1995). In the limit, Penalized Likelihood Criteria select the model that minimizes Kull-Back Leibler divergence with probability 1 (Sin and White, 1996), but convergence rates depend on the penalty chosen. The acceptance of an economic theory thus relies on information outside the model. In a sense, a researcher has flexibility

ii
More intuition is found in the following two definitions adapted from Definition 1 and Definition 2 in (Rothenberg, 1971).

DEFINITION. 1. _Two points α_ 1 _∈A and α_ 2 _∈A are said to be observationally equivalent with respect to a function h evaluated over x if h_ ( _x_ ; _α_ 1) _≡ h_ ( _x_ ; _α_ 2) _∀ x ∈_ R _._ DEFINITION. 2. _A point α_<sup>1</sup> _∈A is said to be identifiable by a function h evaluated over x if there is no other point α ∈A that is observationally equivalent._ Let **_θ_** := ( _δ, c, γ_ )<sup>_′_</sup> denote a vector of parameters, with **_θ_** _∈_ Θ, and **_θ_** 0 := ( _δ_ 0 _, c_ 0 _, γ_ 0)<sup>_′_</sup> be the _true_ vector of parameters. For consistency toward the _true_ parameter, one would not only require _∥_ **_θ_**<sup>ˆ</sup> _−_ **_θ_** 0 _∥→_ 0 to be the solution _a.s._ to the criterion **_θ_**<sup>ˆ</sup> := arg min **_θ_** _∈_ Θ _Q_ ( _y, x_ ; **_θ_** ) as _N →∞_ , but it needs to be the _identifiable unique_ solution. Following the definitions above, then by the definition **_θ_** 0 as the minimizer of _Q_ ( _y, x_ ; **_θ_** ), there needs to be assurance of some form that excluding as the alternative. The standard assumption is that Θ is compact. Together with _almost sure continuity_ in **_θ_** _∈_ Θ, Weierstrass’ theorem implies that **_θ_** 0 exists as a non-empty set ˆ _a.s._ Equation (10) can result directly from the parameterized model _y_ = _h_ ( _x_ ) if such that there is no point in Θ other than **_θ_** 0 that is observationally equivalent to **_θ_** 0. Specifically the observational equivalence definition may fail to hold if Θ is high dimensional. If eq. (10) is not implied by the nature of _h_ , it can also be provided by additional structure to the criterion _Q_ ( _·_ ; **_θ_** ) conditional on regions in Θ, or by limiting the search to remain within a subset **_θ_**<sup>ˆ</sup> := arg min **_θ_** _∈_ Θ _s⊂_ Θ _Q_ ( _y, x_ ; **_θ_** ), where Θ _s_ is a compact subset of the parameter space that may possibly grow in complexity along with the sample size.

DEFINITION. 1 and DEFINITION. 2 are intuitive, but provide no testable condition to decide upon the identifiability of an estimator. One insightful definition is the following adapted from (Bates and White, 1985) that ensures that the solution to the criterion is well separated.

DEFINITION. 3. _Suppose_ **_θ_** 0 _minimizes a real-valued criterion Q∞_ ( _·_ ; **_θ_** ) _on a compact metric space_ Θ _, within a circular neighborhood א_ 0( _r_ ) _⊂_ Θ _with radius r >_ 0 _that has a compact complement א_ 0( _r_ )<sup>_c_</sup> : Θ _\ א_ 0( _r_ ) _, then_ **_θ_** 0 _is uniquely identified on_ Θ if and only if _for every r >_ 0 _,_ to corroborate specific theories by designing the information criteria to support them.

iii
# **A.2 Identifiability in non-parametric models**

Non-parametric models aim to learn from the data without assuming that _h_ is up to finitely many parameters, and work under the axiom that the parameter space Θ may in fact be infinitely dimensional. By allowing for that, we minimize the risk that our parameterization assumptions precludes **_θ_** 0 _∈_ Θ, solving for misspecification bias that results from parametric assumptions. However, without imposing further structure to the criterion it is generally not possible to establish consistency of our estimate **_θ_**<sup>ˆ</sup> _→_ **_θ_** 0 uniformly over Θ as the compactness assumption on Θ does not hold in infinite dimensions.<sup>23</sup> This poses a problem in verifying DEFINITION. 3, and subsequently establishing a consistency result, such as those of (Domowitz and White, 1982).

One solution is to focus the arguments on establishing a compact subset of the parameter space such that over the complement of the compact subset the criterion function is eventually ”large”, see (P ̈otscher and Prucha, 1997). This follows by first constructing a subset Θ _s ⊂_ Θ such that **_θ_** 0 _∈_ Θ _s_ , and such that uniformly, [ _Q∞_ ( **_θ_** ) _− Q∞_ ( **_θ_** 0)] _> ε ∀_ **_θ_** _∈_ Θ _\_ Θ _s_ where _ε_ is some positive finite constant. The subset, is closed as its complement is open, and bounded as it is contained in a ball of finite radius implied by the fact that [ _Q∞_ ( **_θ_** ) _− Q∞_ ( **_θ_** 0)] _≤ ε ∀_ **_θ_** _∈_ Θ _s_ ( _d_ ), can only hold for _d < ∞_ , hence it is compact. As a consequence it is sufficient to show that consistency holds within Θ _s_ , since any _M_ -estimator must eventually fall within the compact subset. We can summarize identifiable uniqueness of **_θ_** 0 in an open space as follows.

DEFINITION. 4 (Identifiability in an open space). _Suppose_ **_θ_** 0 _minimizes a real-valued criterion Q∞_ ( _·_ ; **_θ_** ) _on an open metric space_ Θ _. Suppose furthermore that_ **_θ_** 0 _minimizes Q∞_ ( _·_ ; **_θ_** ) _within a circular neighborhood_ Θ _s_ ( _d_ ) _⊂_ Θ _, that has finite positive radius d >_ 0 _, then uniformly over_ Θ _there exists some positive constant ε for which_ [ _Q∞_ ( **_θ_** _∈_ Θ _\_ Θ _s_ ) _− Q∞_ ( **_θ_** _∈_ Θ _s_ )] _> ε. If furthermore, there is also a circular neighborhood א_ 0( _r_ ) _⊂_ Θ _s with radius r < d that has a compact complement א_ 0( _r_ )<sup>_c_</sup> : Θ _s \ א_ 0( _r_ ) _, then_ **_θ_** 0 _is uniquely identified on_ Θ if and only if _for every r,_ 0 _< r < d,_ In a sense, we want to exert some control over the structure of _Q∞_ ( _·_ ; **_θ_** ) on Θ such that any description **_θ_**<sup>_∗_</sup> observationally equivalent to **_θ_** 0 for any **_θ_** _∈_ Θ _s_ is uniquely identifiable by the criterion within Θ _s_ , disregarding how Θ is structured outside Θ _s_ . One such an approach can be found in the well-known kernel estimator. The solution offered by the kernel method depends on selecting an appropriate bandwidth that controls for the size of local neighborhoods in the sample space throughout which nonlinearities smoothly differ. For too small bandwidths, the kernel method creates a subspace Θ _k ⊃_ Θ _s_ that allows overly-flexible fits to the data. This can create an ill-posed problem, in which multiple solutions to the criterion within Θ _k_ may still deliver equally good fits as judged by the

> 23By definition a set _A ∈_ R _d_ is compact _if and only if_ it is closed and bounded.

iv
criterion evaluated over Θ _k_ . It is obvious that DEFINITION. 4 is not applicable in such a context. For too small bandwidths, the kernel method establishes Θ _k ⊂_ Θ _s_ that is small, and while DEFINITION. 4 may work for Θ _k_ , we are not sure that in fact **_θ_** 0 _∈_ Θ _k_ due to the parameterization assumptions used to construct Θ _k_ . The role of the bandwidth is therefore extremely important, identifiable uniqueness of the estimator requires the bandwidth to be sufficiently large, while reducing miss-specification bias requires the bandwidth to be sufficiently small. In an ideal framework, both factors are balanced out and Θ _k_ grows as _N →∞_ at an appropriate rate.

# **A.3 The role of the penalty in the estimator**

The fitted nonlinearities are allowed to be of any form, but _λ >_ 0 implies the penalty is never removed completely. Positive penalization is key to ensuring that there exists a finite radius neighborhood Θ _s_ ( _d_ ) _⊂_ Θ in which any M-estimator must eventually fall uniformly over Θ as [ _Q∞_ ( **_θ_** _∈_ Θ _\_ Θ _s_ ) _− Q∞_ ( **_θ_** _∈_ Θ _s_ )] _> ε_ ( **_θ_** ; _λ_ ) _>_ 0, where _ε_ ( **_θ_** ; _λ_ ) _>_ 0 is ensured for any **_θ_** by _λ >_ 0. Penalties that vanish completely at a pre-specified rate, are interesting when the researcher wishes to impose penalties only when the estimator is confronted with small sample sizes. This requires however that the criterion is uniquely identified at _λ_ = 0 eventually. Vanishing penalties may improve inference when using estimators that have poor small sample behavior by ensuring that the estimator is relatively inert to weakly nonlinear signals and less likely to overfit the data in local regions of the sample space. Penalties that take values in R _>_ 0, can improve small sample behavior, but maintain a bias towards linear solutions that persists in the limit.

Note that eq. (9) reveals that convergence of our estimator _∥h_<sup>ˆ</sup> _NT −h_<sup>ˆ</sup> _∞∥→_ 0 to a specific target function _h_<sup>ˆ</sup> _∞ ∈H_ Θ, where _h_<sup>ˆ</sup> _∞_ is possibly the _true_ function or the best approximator as judged by the penalized limit criterion, is the same as _∥_ **_θ_**<sup>ˆ</sup> _NT −_ **_θ_**<sup>ˆ</sup> _∞∥→_ 0, which is the more common notation. Hence, we shall use the latter, but really we are interested in ensuring that _h_<sup>ˆ</sup> _∞_ is a uniquely identifiable point in _H_ Θ as close to _h_ 0 as possible. Consistency and normality theorems for eq. (7) are provided in Hainmueller and Hazlett (2014). The results ensure a limit convergence toward the best approximation of the conditional expectation function given penalization, hence the limit solution is conditional on the researcher’s choice of _λ_ . The theory provided is therefore to be understood in terms of **_θ_**<sup>ˆ</sup> _NT_ converging to a _pseudo-true_ parameter as _NT →∞_ , that by construction minimizes the penalized criterion even if the penalty does not vanish. To understand the relationship between the _pseudotrue_ parameter and the _true_ parameter conditional on the penalty, it is helpful to consider precisely how the penalty influences the criterion and delivers the identifiable uniqueness property.

Let **_θ_**<sup>ˆ</sup> _π_ be the point **_θ_**<sup>ˆ</sup> _π_ := arg min **_θ_** _∈_ Θ _Q∞_ ( **_θ_** ) + _π_ ( **_θ_** ), that minimizes the penalized criterion, and **_θ_** 0 be the point **_θ_** 0 := arg min **_θ_** _∈_ Θ _Q∞_ ( **_θ_** ) that minimizes an unpenalized outsample criterion. **_θ_** _π_ is the best approximator similar to the misspecification case studied in (White et al., 1980), whereas **_θ_** 0 is the _true_ parameter, that is the weights vector that v
induces _h_ 0 through the kernel, which is the _true_ function that provides the best out-ofsample density by its definition. The function _h_<sup>ˆ</sup> _π_ := _h_ ( **x** _t_ ; **_θ_**<sup>ˆ</sup> _π_ ) is the best approximator of _h_ 0 := _h_ ( **x** _t_ ; **_θ_** 0) as judged by the penalized criterion _Q∞_ ( **_θ_** ) + _π_ ( **_θ_** ) for a given level of penalization _π_ . The penalty does not imply that _h_ ( **x** _t_ ; **_θ_**<sup>ˆ</sup> _π_ ) = _h_ ( **x** _t_ ; **_θ_** ) _∀_ **_θ_** _∈_ Θ _\_ **_θ_**<sup>ˆ</sup> _π_ and any **x** _T ∈X_ and all _NT ∈_ N _×_ Z. However, it ensures that **_θ_**<sup>ˆ</sup> _π_ is identifiable unique as the minimizer of the limit criterion even in the case of two observationally equivalent parameterizations _h_ ( **x** _t_ ; **_θ_**<sup>ˆ</sup> _π_ ) _≡ h_ ( **x** _t_ ; **_θ_**<sup>_∗_</sup> ) for some **_θ_**<sup>_∗_</sup> _∈_ Θ _\_ **_θ_**<sup>ˆ</sup> _π_ and any **x** _T ∈X_ and all _NT ∈_ N _×_ Z.

PROPOSITION. 1 (Identifiable uniqueness). _The function h_<sup>ˆ</sup> _π_ := arg min _h_ **_θ_** _∈H_ Θ _Q∞_ ( _h_ **_θ_** ) + _π_ ( _h_ **_θ_** ) _produced by hX at point_ **_θ_**<sup>ˆ</sup> _π_ := arg min **_θ_** _∈_ Θ _Q∞_ ( **_θ_** )+ _π_ ( **_θ_** ) _is uniquely identified within H_ Θ _s a simple subset in the infinite dimensional Hilbert space H_ Θ _, if π is a strictly positive penalty function continuous on_ Θ _._ The argument follows from: _Proof._ **_θ_**<sup>ˆ</sup> _π_ := arg min **_θ_** _∈_ Θ _Q∞_ ( **_θ_** ) + _π_ ( **_θ_** ), is by definition the minimizer of _Q∞_ ( _·_ ; **_θ_** ) + _π_ ( **_θ_** ) that is by construction of the least squares function and the penalty function _π_ : Θ _→_ R _>_ 0 a real-valued criterion on an open metric space Θ. Furthermore there exists some positive constant _ε_ for which because for _Q∞_ ( **_θ_** _∈_ Θ _\_ Θ _s_ ) _≡ Q∞_ ( **_θ_** _∈_ Θ _s_ ), by the monotonicity of _π_ on Θ which implies that **_θ_**<sup>ˆ</sup> _π_ minimizes _Q∞_ ( _·_ ; **_θ_** ) + _π_ ( **_θ_** ) within a neighborhood Θ _s ⊂_ Θ. Furthermore Θ _s_ ( _d_ ) has finite radius _d < ∞_ because implies _d < ∞_ , by finiteness of _ε_ in turn implied by continuity of the penalty. Finally, Θ _s_ ( _d_ ) _⊂_ Θ is compact as it closed because its radius is finite, and its complement Θ _\_ Θ _s_ is open.

We have now established that uniformly over Θ, the estimator must fall eventually inside Θ _s_ . The rest of the argument follows from standard identifiability arguments in compact parameter spaces as in (Bates and White, 1985; Domowitz and White, 1982) focused on Θ _s_ . That is, define a circular neighborhood _אk_ ( _r_ ) _⊂_ Θ _s_ with noneggative radius _r < d_ that has a compact complement _אk_ ( _r_ )<sup>_c_</sup> : Θ _s \ אk_ ( _r_ ). **_θ_** _k_ is uniquely identified on Θ as by 0 _< r < d < ∞_ , for every ( _r, d_ ), vi
In our case, this is implied by continuity of the criterion and additionally by the fact that for any observationally equivalent point _π_ ( **_θ_**<sup>_∗_</sup> ) such that _Q∞_ ( **_θ_**<sup>_∗_</sup> ) _≡ Q∞_ ( **_θ_**<sup>ˆ</sup> _π_ ), by definition of **_θ_**<sup>ˆ</sup> _π_ as the minimizer of min **_θ_** _∈_ Θ _Q∞_ ( **_θ_** ) + _π_ ( **_θ_** ) the continuity of _π_ implies Central to the result is that _π_ ( **_θ_**<sup>ˆ</sup> _π_ ) _< π_ ( **_θ_**<sup>_∗_</sup> ), providing that for two observationally equivalent functions, the _pseudo-true_ parameter is the parameter vector that induces a less complex functional form.

So far we have treated _π_ to be fixed at a prespecified level. However, for any given level of penalization, the solution to the penalized criterion is different. We can make that more explicit by writing it as the limit estimate conditional on a penalty value ( **_θ_**<sup>ˆ</sup> _∞|π_ ), and analyzing the role of _π_ in the divergence _∥_ ( **_θ_**<sup>ˆ</sup> _∞|π_ ) _−_ **_θ_** 0 _∥_ . This displays the heavy importance on determining an appropriate penalty _π_ as it is crucial to the outcome, see Blasques and Duplinskiy (2015) for some thoughts on how to choose appropriate penalty weights in a general context. Asymptotically, if the impact of _π_ vanishes, for example by using penalties of an _o_ (( _NT_ )<sup>_−_</sup> 2<sup><u>1</u></sup> ), consistency toward **_θ_** 0 can still be met in the limit, again see Blasques and Duplinskiy (2015) for detail. However, in small samples, similar to the Bayesian case, a researcher can exert influence on the outcome by setting the value of _π_ . In the current framework, _π >_ 0 prevents a generality claim as it would follow in the parametric case, however we can still focus the argument on finding an optimal penalty that minimizes _∥_ ( **_θ_**<sup>ˆ</sup> _∞|π_ ) _−_ **_θ_** 0 _∥_ , or equivalently the function divergence _∥_ ( _h_<sup>ˆ</sup> _∞|π_ ) _− h_ 0 _∥_ .

PROPOSITION. 2 (Best approximation across penalties and weights). _The divergence between the best approximation as judged by the penalized limit criterion given a level of penalization and the_ true _function is smaller than the divergence as evaluated at all other limit estimates resulting under other penalty weights_ _and results under the penalty π_ 0 _that minimizes an out-of-sample criterion_ _for some small positive δ. Hence,_ ( _h_<sup>ˆ</sup> _∞|π_ 0) _is the best approximation of h_ 0 _over H_ Θ _s×_ Π := _{H_ Θ _s|π_ 1 _× H_ Θ _s|π_ 2 _× ... × H_ Θ _s|π} ∀ π ∈_ Π _, that is across all penalties and weights._

The argument follows from _Proof._ Let **_θ_**<sup>ˆ</sup> _π_ = **_θ_**<sup>ˆ</sup> _∞|π_ := arg min **_θ_** _∈_ Θ _Q∞_ ( **_θ_** ) + _π_ ( **_θ_** ) be the minimizer of the penalized criterion for a certain level of penalization and **_θ_** 0 := arg min **_θ_** _∈_ Θ _Q∞_ ( **_θ_** ) the minimizer

vii
of an unpenalized out-of-sample criterion. When plugging the _true_ parameter in the penalized criterion, then taking _∥_ **_θ_**<sup>ˆ</sup> _π −_ **_θ_** 0 _∥→_ 0 implies that similarly _| Q∞_ ( **_θ_**<sup>ˆ</sup> _π_ ) + _π_ ( **_θ_**<sup>ˆ</sup> _π_ ) _−_ � � [ _Q∞_ ( **_θ_** 0) + _π_ ( **_θ_** 0)] _| →_ 0 This minimization is solved if the in-sample criterion evaluates _|Q∞_ ( **_θ_**<sup>ˆ</sup> _π_ ) _−Q∞_ ( **_θ_** 0) _| →_ 0 equivalently, as then immediately also _|π_ ( **_θ_**<sup>ˆ</sup> _π_ ) _−π_ ( **_θ_** 0) _| →_ 0. Hence, either _|π_ ( **_θ_**<sup>ˆ</sup> _π_ ) _− π_ ( **_θ_** 0) _| →_ 0 or _|Q∞_ ( **_θ_**<sup>ˆ</sup> _π_ ) _− Q∞_ ( **_θ_** 0) _| →_ 0, is sufficient for _∥_ **_θ_**<sup>ˆ</sup> _π −_ **_θ_** 0 _∥→_ 0.

Any such result following from taking both penalties _π_ ( **_θ_**<sup>ˆ</sup> _π_ ) and _π_ ( **_θ_** 0) to zero simultaneously as _N →∞_ is prohibited by the fact that _π_ : Θ _→_ R _>_ 0. However _|Q∞_ ( **_θ_**<sup>ˆ</sup> _π_ ) + _π_ ( **_θ_**<sup>ˆ</sup> _π_ ) _| − |Q∞_ ( **_θ_** 0) + _π_ ( **_θ_** 0) _|_ attains a minimum when setting the penalty to minimize the criterion defined on out-of-sample errors. Specifically since **_θ_** 0 is by construction the minimum of the out-of-sample criterion in the limit, setting _π_ 0 to minimize arg min _π∈_ Π _∀⊆_ R _≥_ 0 _Q∞_ ( **_θ_**<sup>ˆ</sup> _π_ ) gives

and if _|Q∞_ ( **_θ_**<sup>ˆ</sup> _π_ ) _−Q∞_ ( **_θ_** 0) _| →_ 0, it must follow that

and, If Π _⊆_ R _≥_ 0 is constructed such that _π_ 0 _∈_ Π for which the arg min’s above reach 0, then _∥_ ( _h_<sup>ˆ</sup> _∞|π_ 0) _− h_ 0 _∥_ = 0 would follow and we reach the _true_ target function. Now Π _⊆_ R _≥_ 0 can contain penalties infinitely close to 0, in practice one must work with finite sets for a grid search across Π and construct instead a set Π _⊆_ R _≥δ_ being the set of all possible parameters bounded away from zero by some arbitrarily small positive constant _δ_ . If _π_ 0 _∈/_ Π for which _∥_ ( _h_<sup>ˆ</sup> _∞|π_ 0) _− h_ 0 _∥_ = 0, then still

thus also and therefore which induces through the definition of _h_ as the weighted kernel also implying that ( _h_<sup>ˆ</sup> _∞|π_ 0) turns out to be the best approximation of _h_ 0 for all penalties in Π that each result itself as a best approximator within the subset _H_ Θ _s|π_ within the penalized criterion necessarily falls given the level of penalization.

viii
PROPOSITION. 2 implies that if the penalty is chosen by minimizing a criterion out-ofsample, a weights vector can be estimated that produces the function closest to the target function across all penalties and weights . Effectively, a researcher is able to identify an approximation that is arbitrarily close to the _true_ curve, by solving the estimator on very large data iteratively for a sufficiently wide range of penalties and selecting the result that performs optimal as an out-of-sample predictor. This is an intuitive solution as **_θ_** 0 caries a natural interpretation as the optimal out-of-sample predictor.

# **B Additional Results**

<!-- Start of picture text --> Conditional expectations of log Tree intensity Conditional expectations of log Tree intensity Conditional expectations of log PM25 intensity Conditional expectations of log PM25 intensity<br>GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG<br>7 8 9 10 11 0 200 400 600 800 1000 7 8 9 10 11 0 200 400 600 800 1000<br>log GDP per capita Population density log GDP per capita Population density<br>Conditional expectations of log Tree intensity Conditional expectations of log Tree intensity Conditional expectations of log PM25 intensity Conditional expectations of log PM25 intensity<br>G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGG<br>0 10 20 30 40 50 60 0 20 40 60 80 0 10 20 30 40 50 60 0 20 40 60 80<br>Undernourishment rate Poverty rate 1.90$ Undernourishment rate Poverty rate 1.90$<br>Conditional expectations of log Tree intensity Conditional expectations of log Tree intensity Conditional expectations of log PM25 intensity Conditional expectations of log PM25 intensity<br>GG GGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGG GGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG<br>0 10 20 30 40 30 40 50 60 70 80 90 0 10 20 30 40 30 40 50 60 70 80 90<br>Manufacturing GDP share Services GDP share Manufacturing GDP share Services GDP share<br>Conditional expectations of log Tree intensity Conditional expectations of log Tree intensity Conditional expectations of log PM25 intensity Conditional expectations of log PM25 intensity<br>GGG GGGGGGG<br>G GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG<br>20 40 60 80 100 10 15 20 25 20 40 60 80 100 0 2 4 6 8 10<br>Urban population share Bottom 40% income share Urban population share Bottom 40% income share<br>E[Y|X] −1−3 E[Y|X] −2.5 E[Y|X] 31 E[Y|X] 2.0<br>−5 −4.0 −1 1.0<br>E[Y|X] −3.0 E[Y|X] −2.4 E[Y|X] 1.2 E[Y|X] 1.2<br>−3.6 −3.2 0.4 0.9<br>E[Y|X] −4.0 E[Y|X] −2.0−4.5 E[Y|X] 1.30.9 E[Y|X] 1.41.1<br>E[Y|X] −3.0−3.4 E[Y|X] −3.0 E[Y|X] 1.20.6 E[Y|X] 1.41.10.8<br><!-- End of picture text -->

Figure 8: Conditional expectations of deforestation (left two columns) and pollution (right two columns) intensity of income for each variable fixing other variables constant at their mean.

ix
# dd ~~s~~ aladdin shia ~~al~~ d id
Table 6: BAU-2030 and base year data aggregated by 5% percentiles of income. GDP per capita as population weighted averages, population in millions, number of poor people in millions, annual tree loss in square kilometers, PM2 _._ 5 in population weighted average concentrations, carbon emissions in million tons. World totals are scaled to world totals using multipliers (1.16 for population and carbon based on the share of population in our data, and tree loss 1.42 based on the share of tree cover in the data).

|Income|GDP p.c.|GDP p.c.|Pop.|Pop.|No. Poor|No. Poor|Treeloss|Treeloss|PM25|PM25|CO2|CO2|
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|Group|2014|2030|2014|2030|2014|2030|2014|2030|2014|2030|2014|2030|
|1|1_,_094|1_,_548|68|101|41|55|3_,_382|3_,_640|15|20|10|28|
|2|1_,_495|2_,_611|171|262|74|70|3_,_750|3_,_977|17|21|20|109|
|3|1_,_899|2_,_388|46|67|19|24|1_,_415|1_,_684|23|25|22|39|
|4|2_,_305|3_,_308|96|139|36|43|1_,_821|1_,_931|29|31|27|71|
|5|2_,_917|4_,_720|237|305|57|45|1_,_295|1_,_331|47|52|96|285|
|6|4_,_249|5_,_381|265|366|43|48|5_,_782|5_,_713|46|48|190|282|
|7|5_,_265|9_,_887|111|133|6|2|4_,_716|4_,_031|23|20|170|385|
|8|5_,_426|10_,_301|1_,_519|1_,_936|341|288|4_,_440|3_,_769|54|51|2_,_286|5_,_634|
|9|6_,_875|9_,_344|176|231|21|22|3_,_297|3_,_043|12|13|212|399|
|10|8_,_219|10_,_627|15|17|1|1|4_,_002|3_,_395|9|9|14|22|
|11|10_,_078|16_,_232|292|352|22|11|17_,_799|17_,_676|15|15|540|1_,_338|
|12|11_,_993|16_,_578|114|141|11|11|3_,_784|4_,_037|15|17|646|1_,_150|
|13|13_,_135|24_,_563|1_,_691|1_,_825|49|5|33_,_540|39_,_407|44|44|11_,_212|25_,_839|
|14|16_,_347|20_,_955|206|244|4|3|2_,_032|2_,_132|22|24|1_,_128|1_,_864|
|15|18_,_386|28_,_014|162|193|3|2|5_,_160|5_,_859|17|17|750|1_,_402|
|16|22_,_607|37_,_980|53|60|0|0|1_,_822|2_,_194|20|24|438|1_,_112|
|17|33_,_000|39_,_059|204|219|0|0|6_,_647|8_,_832|21|22|1_,_160|1_,_770|
|18|38_,_515|47_,_345|244|253|0|0|26_,_962|32_,_972|12|13|2_,_324|3_,_324|
|19|43_,_584|53_,_144|128|132|0|0|4_,_694|5_,_794|15|16|1_,_276|1_,_788|
|20|51_,_694|64_,_462|354|384|0|0|17_,_649|19_,_466|10|11|5_,_534|7_,_414|
|world|14_,_088|19_,_899|7_,_137|8_,_537|730|630|218_,_664|242_,_656|36|36|32_,_544|62_,_934|

xii
<!-- Start of picture text --> Total carbon emission output, Total carbon emission output,<br>projection for bottom 20 projection for second quintile<br>Historical Historical<br>Pessimistic Pessimistic<br>BAU BAU<br>Opportunistic Opportunistic<br>2000 2005 2010 2015 2020 2025 2030 2000 2005 2010 2015 2020 2025 2030<br>Year Year<br>Total carbon emission output, Total carbon emission output,<br>projection for third quintile projection for fourth quintile<br>Historical Historical<br>Pessimistic Pessimistic<br>BAU BAU<br>Opportunistic Opportunistic<br>2000 2005 2010 2015 2020 2025 2030 2000 2005 2010 2015 2020 2025 2030<br>Year Year<br>Total carbon emission output, Total carbon emission output,<br>projection for top 20 projection for the world<br>Historical Historical<br>Pessimistic Pessimistic<br>BAU BAU<br>Opportunistic Opportunistic<br>2000 2005 2010 2015 2020 2025 2030 2000 2005 2010 2015 2020 2025 2030<br>Year Year<br>0.35 8<br>7<br>0.25 6<br>5<br>CO2 Billion Tons 0.15 CO2 Billion Tons 4<br>3<br>0.05 2<br>40<br>3.5 35<br>3.0 30<br>2.5 25<br>20<br>2.0<br>CO2 Billion Tons CO2 Billion Tons 15<br>1.5<br>10<br>1.0 5<br>17 80<br>16 70<br>15<br>60<br>14<br>50<br>13<br>CO2 Billion Tons CO2 Billion Tons 40<br>12<br>11 30<br>10 20<br><!-- End of picture text --> Figure 11: Future carbon under three scenarios broken down by income quintiles.

xiii