I. INTRODUCTION

In recent decades, the creation of cross-country measures of economic and political institutions has led to a large literature on the effect of institutions on growth. One measure of economic institutions is the Economic Freedom of the World (EFW) index by Gwartney, Lawson, and Hall (2015). The EFW index has been used as a measure of institutions in hundreds of studies in economics and related disciplines (Hall and Lawson 2014). By far the most frequent relationship of interest to economists has been the impact of institutions on growth. The very first major empirical paper using the EFW index was on this question (Easton and Walker 1997) and subsequent years have seen dozens of papers written on the economic freedom/growth relationship. Gwartney (2009) summarizes what is known about the relationship between economic freedom (EF) and growth in his presidential address to the Southern Economic Association. (1)

We investigate the mechanism through which EF can affect economic performance, measured by a country's technical efficiency, or deviation from the production frontier. If information on price is available together with an appropriate behavioral assumption on cost minimization or profit maximization, then one can also consider allocative efficiency. For example, in the input selection case, allocative efficiency entails selecting the mix of inputs that produce a given output with given input prices at minimum costs. Allocative and technical efficiency combine to provide an overall economic efficiency measure. As we utilize country-level aggregated data, the behavioral assumption is not obvious. Thus, we measure economic performance with a country's technical efficiency. Different countries operate at different distances from the production frontier, and clearly efficiency can be influenced by institutional quality (Adkins, Moomaw, and Savvides 2002). We expect that EF can generally impact both the production frontier and efficiency, thus accuracy in estimating both will be crucial to assess a country's efficiency level. From a policy perspective, accuracy is important as policymakers should be better informed about the amount of technical inefficiency that exists in an economy and exactly how changes in institutional quality at different stages influence the marginal productivity of capital and labor as well as technical efficiency.

In this paper we add to this literature by applying a smooth coefficient stochastic frontier (SF) model to estimate the production frontier and therefore assess countries' efficiency. We believe that the impact of EF on production frontier cannot be simply captured by entering it into production function linearly as a regular input or neutrally as assumed by much of the literature. The impact depends further, potentially nonlinearly, on how EF affects output elasticities (marginal products) of human capital, labor, and physical capital. (2) First, increases in EF improve the mobility of labor and capital across countries (Ashby 2010; Azman-Saini, Baharumshah, and Law 2010; Nejad and Young 2016). For example, an increase in EF for countries with low EF would make physical capital, such as foreign direct investment, more accessible (Bengoa and Sanchez-Robles 2003; Kapuria-Foreman 2007). Economic freedom thus alters opportunity costs between labor and capital and affects output elasticities. (3) Second, EF also reduces transactions costs, which improves productivity in terms of output elasticity (Klein and Luu 2003). Clearly, the impacts of EF can be highly nonlinear, both neutrally and nonneutrally. A country's EF is not a typical input like labor, physical capital, or human capital, thus we should naturally treat it as an auxiliary/environmental variable when assessing its impact. Specifically in terms of its effect on the frontier, unlike regular inputs, EF level can have a facilitating impact, thereby shifting the frontier neutrally, but it can also influence nonneutrally by affecting the productivity of regular inputs as argued above. To ignore this effect of EF on the productivity of inputs could lead to a mismeasurement of the amount of technical inefficiency in an economy. Our semiparametric model is perfect for capturing both the neutral and nonneutral effects, which are purely nonparametric.

The effects of EF on output elasticities are largely ignored in the empirical literature on EF and growth. While a handful of studies employ a stochastic production frontier approach to this question, they do not investigate the effect of output elasticities. Instead, their focus is largely on how EF affects technical efficiency. For example, Adkins, Moomaw, and Savvides (2002) investigate the effects of EF on the production frontier and find that more EF in a country leads to decreased inefficiency. They do not, however, address the effect of EF on output elasticities on the frontier. Clearly, a misspecified frontier can lead to misleading technical efficiency estimates and therefore the policy importance of EF relative to inputs such as capital and labor. Similarly, while Klein and Luu (2003) find that increased EF reduces technical inefficiency, they do not allow the elasticities of human or physical capital to vary with the level of institutional quality and henceforth fail to recognize the nonneutral effect of institutional changes on productivity.

To address this hole in the literature we adopt the semiparametric smooth coefficient model pioneered by Li et al. (2002) and recently extended by Yao et al. (Forthcoming). This approach allows us to explicitly explore both the neutral effect of EF (the shifting of the production frontier neutrally by economic freedom) and the nonneutral effect (the shifting of the production frontier through changing output elasticities). We also allow the efficiency distribution to depend on EF through a scaling function on efficiency, thus the conditional mean and variance of the efficiency depend on economic freedom. This is in contrast with the approaches so far in the literature, which arbitrarily allow only the conditional mean to depend on economic freedom, which can result in biased estimates. Thus, we believe that our efficiency estimates are robust to potential misspecification from both the frontier and efficiency distribution. We apply the multistep procedure developed by Yao et al. (Forthcoming) and estimate a smooth coefficient stochastic production frontier with a country-level panel data from 1980 to 2010 observed at 5-year increments. We do so in order to estimate the effect of EF on both the production frontiers and technical efficiencies of countries.

Contrary to Hall, Sobel, and Crowley (2010), we find that the marginal product of human capital and labor is decreasing when countries have an intermediate level of economic freedom. (4) We find, however, that increases in EF improve the marginal product of physical capital when a country has a moderate EF but negatively affects marginal product of physical capital when EF is low or high. The result shows that for many countries, although the neutral impact is positive and decreasing with economic freedom, higher EF results in an overall outward shift in the production frontier. For example, if China in 2010 had EF at the level of the United States, its production frontier would shift upward by over 20%. We compare the estimates for our semiparametric model with three parametric counterparts (Cobb-Douglas, translog, and restricted translog) and perform a model specification test. The tests suggest that the semiparametric model is more appropriate for modeling production frontiers. Failure to account for the potentially delicate nonlinear impact of EF to the frontier results in misleading efficiency levels. Empirically, the average efficiency estimates from our semiparametric model are much higher, at least 20% higher, than those from parametric models.

The remainder of this article is organized as following. Section II introduces our methodology and model specifications. Section III describes the nature of variables and data sources. Section IV discusses empirical results, illustrating the smooth varying coefficients of stochastic production frontiers and the distribution of technical efficiency. Section V provides a robustness check with an alternative measure of EF and Section VI concludes.

II. METHODOLOGY AND MODEL SPECIFICATION

Adkins, Moomaw, and Savvides (2002) investigate the effects of EF on production frontier and technical efficiency. However, using a parametric Cobb-Douglas production function with composite error, they only examine the neutral and linear effect of EF on production frontier but ignore the nonneutral effects of EF on the marginal productivity of inputs. Klein and Luu (2003) also employ a parametric Cobb-Douglas production function approach. The dependence of the efficiency term on EF is through representing the conditional mean as a linear function of determinant variables, including economic freedom. Note that the conditional variance of efficiency is modeled as a constant. This modeling strategy is considered to be arbitrary (Parmeter and Kumbhakar 2014, 55). For example, if the goal is to study how EF affects effciency, there is no particular reason why it should be assumed to exert influence through conditional mean but not variance.

We follow Yao et al. (Forthcoming) and consider a semiparametric smooth coefficient stochastic production frontier for panel data. Specifically, for i = 1, ..., n and t = 1, ..., T,

(1) [mathematical expression not reproducible]

where [Y.sub.it] is the logarithm of output, [X.sub.it] = (Ln([H.sub.it]), Ln([L.sub.it]), Ln([K.sub.it]))' represent the logarithm of traditional inputs, including human capital, labor and capital, and [delta]([EF.sub.it]) = ([alpha]([EF.sub.it]), [beta]([EF.sub.it])')', where [beta]([EF.sub.it]) = ([[beta].sub.H]([EF.sub.it]), [[beta].sub.L]([EF.sub.it]), [[beta].sub.K]([EF.sub.it]))', is a vector of unknown smooth functions of exogenous environmental variable, the economic freedom. The composite error is [[epsilon].sub.it] = [v.sub.it] - [u.sub.it], where we specify a two-sided error [v.sub.it] ~ i.i.d.N (0, [[sigma].sup.2.sub.v]) representing random noise which independent of [EF.sub.it], [X.sub.it], and [u.sub.it]; [u.sub.it] = [u.sub.i]g([EF.sub.it];[eta]) for a one-sided error u(, scaled by a nonnegative function g([EF.sub.it];[eta]) known up to parameter [eta], capturing inefficiency. We consider [u.sub.i] ~ i.i.d. | N (0, [[sigma].sup.2.sub.u]) | and independent of [EF.sub.it] and [X.sub.it]. [EF.sub.it] enters inefficiency term through the scaling function g(.) to affect the distribution of [[epsilon].sub.it]. For [EF.sub.i] = ([EF.sub.i1], ..., [EF.sub.iT])' and [[epsilon].sub.i] = ([[epsilon].sub.i1] , ..., [[epsilon].sub.iT])', we denote the conditional density of e; given [EF.sub.i] by h([[epsilon].sub.i];[EF.sub.i], [[theta].sub.0]), where [[theta].sub.0] = ([[sigma].sup.2.sub.u],[[sigma].sup.2.sub.v],[eta]) denote the true parameters. Thus with above distribution specifications, [mathematical expression not reproducible] we employ a Gaussian kernel function [mathematical expression not reproducible] and data-driven least square cross validation method to find the optimal bandwidth for our dataset.

Since the seminar work of Aigner, Lovell, and Schmidt (1977) and Meeusen and Van Den Broeck (1977), the SF approach as a tool to model and estimate efficiency has grown exponentially (see Kumbhakar, Wang, and Horncastle 2015 for extensive reviews and applications). SF models are popular among practitioners due to the fact that these models accommodate stochastic noise as an integral part of the production technology and can separate noise from inefficiency. Furthermore, one can easily perform statistical tests on many economic hypotheses of interest. However, restrictive assumptions typically are made on either the production frontier function and/or the distributional assumptions, with respect to inefficiency and the noise terms. For example, the SF model introduced by Aigner, Lovell, and Schmidt (1977) and Meeusen and Van Den Broeck (1977) uses a parametric frontier along with a composite error term in which the one-sided inefficiency term follows a particular distribution (half-normal for example), while the noise term follows a normal distribution.

With panel data, one can relax the distribution assumption on the inefficiency term (see Schmidt and Sickles 1984, Cornwell, Schmidt, and Sickles 1990 and Lee and Schmidt 1993, Horrace and Parmeter 2011 and Parmeter, Wang, and Kumbhakar 2017 for different approaches). However, flexibility in modeling the frontier is still limited to a known parametric functional form such as Cobb-Douglas or translog. Even with a correctly specified distribution for the composite errors, an incorrectly specified frontier function can still lead to misleading conclusion regarding inefficiency levels, returns to scale, technical change, etc. On the other hand, maintaining the distribution structure in Aigner, Lovell, and Schmidt (1977) and Fan, Li, and Weersink (1996), Martins-Filho and Yao (2015) investigate a nonparametric frontier model and examine properties of the estimators. Kumbhakar et al. (2007) and Park, Simar, and Zelenyuk (2015) model and estimate the frontier and all parameters of the distribution of the composite error as smooth functions of the inputs. A common feature of these methods is that the frontier is fully nonparametric, although the rate of convergence of the proposed frontier estimator is rather slow especially when the number of inputs (conditioning variables) is large. It is the well-known curse of dimensionality problem afflicting multivariate kernel-based nonparametric estimation. Beacuse it is common to have a large number of variables in frontier models, the accuracy of the asymptotic approximation can be rather poor.

In this paper, we utilize the smooth coefficient frontier model introduced in Yao et al. (Forthcoming). The frontier function takes a more flexible functional form, that is, [alpha]([EF.sub.it])+[X.sub.it]'[beta]([EF.sub.it]), instead of just a linear or a semiparametric partially linear form. The sample size required for estimation is not as demanding as a fully nonparametric frontier model, and therefore likely to be useful to the applied researchers. Clearly, the frontier can be shifted neutrally by EF via [alpha] (EF), and also nonneutrally through [beta](EF). The semiparametric frontier model proposed in this article is different from the standard smooth coefficient regression model (Cai and Li 2008; Li et al. 2002) because the conditional mean of the composite error is not zero due to the presence of the onesided inefficiency term. We assume that the inefficiency and noise term follow the half-normal and normal distributions which depends on economic freedom. To capture the dependence, we allow that EF enters inefficiency term through the scaling function g(.) to affect the distribution of the efficiency, thus, the conditional mean and variance of the inefficiency term is a function of EF, known up to certain parameters. This is in contrast to Adkins, Moomaw, and Savvides (2002), who allow EF to arbitrarily impact only the mean, but not the variance, of inefficiency. Thus, we allow the inefficiency to vary across individual and time through its dependence on EF. We note that it is important to determine the level of inefficiency and also to understand how the inefficiency is affected by EF. Ignoring the effect of EF in the composite error term, especially in the one-sided inefficiency term, can cause biased estimates of the frontier function and technical inefficiency level. This could, in turn, potentially lead to incorrect policy inferences.

We utilize a multistep estimation procedure to consistently estimate the semiparametric frontier function (Yao et al. Forthcoming). (5) As E(ejt\EFil) i= 0, the standard smooth varying coefficient estimation as in Li et al. (2002) cannot be applied directly. Instead, subtracting conditional mean of Equation (1) on both sides, we have

(2) [Y.sub.it] - E([Y.sub.it]|[EF.sub.it]) = ([X.sub.it] - E ([X.sub.it]|[EF.sub.it]))' X [beta]([EF.sub.it]) + [mu]([EF.sub.it];[[theta].sub.0]) + [[epsilon].sub.it],

then we estimate [??] ([EF.sub.it]) and [??] with the following steps.

First, let [[??].sub.it] - [[epsilon].sub.it] + [mu] ([EF.sub.it]; [[theta].sub.0]), then E([[??].sub.it]|[EF.sub.it]) = 0. From Equation (2), we construct [[??].sub.it] = [Y.sub.it] - [??] ([Y.sub.it]|[EF.sub.it]), [[??].sub.it] = [X.sub.it]-[??]([X.sub.it]|[EF.sub.it]), where [??]([Y.sub.it]|[EF.sub.it]) and [??] ([X.sub.it]|[EF.sub.it]) are local linear estimates of conditional mean of [Y.sub.it] and [X.sub.it], respectively, evaluated at [EF.sub.it]. Then, Equation (2) transforms into

(3) [mathematical expression not reproducible]

and we apply standard smooth varying coefficient estimation (Li et al. 2002) on Equation (3) to obtain consistent estimator [mathematical expression not reproducible] such that

(4) [mathematical expression not reproducible]

where [mathematical expression not reproducible] denotes the Kronecker product.

Second, recall that [mu]([EF.sub.It];[[theta].sub.0]) is known up to the parameter [[theta].sub.0], we can construct [mathematical expression not reproducible] We estimate [theta] by [??] via pseudo-likelihood estimation. Following Pitt and Lee (1981), we write the log-likelihood function as

(5) [mathematical expression not reproducible]

where [mathematical expression not reproducible] and [phi](*) and [PHI](*) refer to the probability density function and cumulative density function of a standard normal, respectively. Furthermore, we choose [mathematical expression not reproducible] in the application, [eta] can be interpreted as the semi-elasticity of expected inefficiency with respect to [EF.sub.it].

Third, after [??] is obtained in the second step, [mu] ([EF.sub.it]; [??]) can be estimated. Adding [mu] ([EF.sub.it];[??]) to both sides of Equation (1), we have

(6) [Y.sub.it] + [mu] ([EF.sub.it];[??]) = (1,[X'.sub.it]) [delta] ([EF.sub.it]) + [mu]([EF.sub.it];[??]) + [[epsilon].sub.it],

Again, since E([[??].sub.it]|[EF.sub.it]) = 0, we estimate [??] ([EF.sub.it]) = ([??] ([EF.sub.it]), [??] ([EF.sub.it])')' with the standard smooth varying coefficient estimation. Specifically, [??] (EF) = [[??].sub.0]], where [mathematical expression not reproducible] such that (7)

[mathematical expression not reproducible]

where [mathematical expression not reproducible]

Finally, following Jondrow, Knox, and Schmidt (1982), we calculate observation-specific technical inefficiencies for our semiparametric model. With proper modifications, we derive the conditional density of [u.sub.it] given [[epsilon].sub.it] and [EF.sub.it] as

[mathematical expression not reproducible]

where [g.sub.it] = g([EF.sub.it]|[eta]), and [[sigma].sup.2] ([EF.sub.it]) = [[sigma].sup.2.sub.v] + [[sigma].sup.2][g.sup.2] ([EF.sub.it]). We estimate technical efficiency as [mathematical expression not reproducible]

We compare our semiparametric model with three benchmark models. The first is a Cobb-Douglas production function with composite error which is nested in Equation (1),

(8) [mathematical expression not reproducible]

where we only allow EF to shift the production frontier through the intercept term [alpha] ([EF.sub.it]) = [[alpha].sub.0] + [[alpha].sub.1][EF.sub.it] + [[alpha].sub.2][EF.sup.2.sub.it] , a quadratic function to capture potential nonlinearity. All coefficients of [X.sub.it] are assumed to be constants.

The second model is a translog model:

(9) [mathematical expression not reproducible]

Due to the presence of square and cross products of [X.sub.it], it is not nested in Equation (1), but it imposes no a priori restrictions on substitution possibilities among input variables.

In the third benchmark, we restrict the translog production model such that the coefficients of [Ln.sup.2]([H.sub.it]), [Ln.sup.2]([L.sub.it]), [Ln.sup.2]([K.sub.it]), Ln([L.sub.it]) Ln([K.sub.it]), Ln ([L.sub.it]) Ln([H.sub.it]), and Ln([H.sub.it])Ln([K.sub.it]) are jointly equal to zero, which makes it nested in Equation (1). After removing those terms, we estimate restricted translog model as,

(10) [mathematical expression not reproducible]

III. DATA DESCRIPTIONS

We use the country-level data from the Penn World Table (PWT) and Economic Freedom of the World: 2015 Annual Report by Gwartney, Lawson, and Hall (2015) to investigate the effects of EF on a country's production frontier and technical efficiency. Our dataset consists of 440 observations for 110 countries from 1990 to 2010 observed at 5-year increments. (6) A complete list of countries and their respective EF as of 2010 are presented in Table 1. The EFW index is based on a 0-10 scale, with higher scores representing higher levels of economic freedom.

We employ the PWT because it is the most comprehensive country-level aggregated data comprised of human capital per capita, labor, and physical capital. The 8.1 version of the PWT provides output-side real gross domestic product (GDP) in million of 2005 U.S. dollars for different countries over time, which facilitates comparison of economic productivity across a large number of countries. The combined dataset enables us to estimate production frontiers and technical efficiency, and to investigate the effects of changing EF on production frontiers.

Our dependent variable Y is the logarithm of output-side real GDP in millions of 2005 U.S. dollars. Independent variables include the logarithm of human capital per capita Ln(H), the logarithm of labor force Ln(L), and the logarithm of real capital stock Ln(K), where H is an index of human capital based on years of schooling and returns to education (Barro and Lee 2013; Psacharopoulos 1994), labor force L is in millions of persons participated in employment, and real capital stock K is in millions of 2005 U.S. dollars. Additional descriptions of variables from the PWT can be obtained in Feenstra, Inklaar, and Timmer (2015).

EFW index is our primary variable of interest in this paper and we treat it as the "environmental" variable in our estimation of production functions. Based on our hypothesis, the effects of EF on output are double-sided. Increases in EF can reduce transaction costs for productive activities, make foreign capital more accessible and domestic capital more productive, and improve the return on education, all of which boost productivity. However, on the other hand, increases in EF also relax governmental control of the population, which increases the migration of immigrants to countries with higher levels of economic freedom. While this is likely to increase the productivity of destination countries, it will lower output elasticity in the origin country.

We use the chain-linked EFW index to ensure comparability across time. EFW measures the degree of economics freedom in five major areas: (1) size of government, (2) legal system and security of property rights, (3) access to sound money, (4) freedom to trade internationally, and (5) regulation. Each of these components is based on several variables. The rating of size of government, for instance, is based on four separate components, such as government consumption as a percentage of total consumption. Each component is put on a 0-10 scale and then aggregated up to an overall score for the entire country, which also varies from 0 to 10. The detailed EFW index structure can be obtained from Gwartney, Lawson, and Hall (2015). Gwartney and Lawson (2003) provide an overview of the history and creation of the index. The summary statistics for all the variables are presented in Table 2.

IV. ESTIMATION RESULTS

The estimation results for our semiparametric smooth coefficient model are summarized in Table 3 Panel A, providing the mean values and 10th, 50th (median), and 90th percentile of our smooth coefficients of [mathematical expression not reproducible] as well as the parameter estimate [mathematical expression not reproducible] Compared with benchmark models presented in Table 4 Panel A, our semiparametric estimates give a lower [[??].sup.2.sub.u] but a higher [[??].sup.2.sub.v], 2.0241 and 0.0357, respectively. The Cobb-Douglas production frontier, translog frontier, and restricted translog frontier give relative higher estimates of [[??].sup.2.sub.u] and lower [[??].sup.2.sub.v], where ([[??].sup.2.sub.u], [[??].sup.2.sub.v]) = (2.6714, .0217), ([[??].sup.2.sub.u], [[??].sup.2.sub.v]) = (2.8809, .0194),"and ([[??].sup.2.sub.u], [[??].sup.2.sub.v]) = (3.2214,".0209), respectively. All estimates suggest a relatively small magnitude for the random noise relative to that of the efficiency term. With an estimated [??] of -0.1383, our semiparmetric model implies that increased EF will increase technical efficiency ([TE.sub.it]). The Cobb-Douglas, translog, and restricted translog models suggest an estimated [??] of -0.1156 and -0.0519 and -0.0404, respectively, confirming our sempiparametric estimate (with the same sign, although of different magnitudes).

The restricted translog production function assumes that the coefficients of [Ln.sub.2]([H.sub.it]), [Ln.sup.2]([L.sub.it]) [Ln.sub.2]([K.sub.it]), Ln([L.sub.it])Ln([K.sub.it]), Ln([L.sub.it])Ln([H.sub.it]), and Ln([H.sub.it])Ln([K.sub.it]) are jointly equal to zero. To check its validity empirically, we perform a likelihood ratio test for the null of [[beta].sub.2H] = [[beta].sub.2L] - [[beta].sub.2K = [[beta].sub.HL] = [[beta].sub.HK] = [[beta].sub.LK] = 0 which gives us a test statistic of 0.0943, smaller than the critical value of 12.592 at the 5% significance level. Since we cannot reject the null hypothesis and the restricted translog is also nested in our semiparametric model, we focus on Cobb-Douglas and restricted translog models as parametric counterparts in the following analysis.

Since our semiparametric estimators are smooth functions of the environmental variable EF, we plot our coefficients [mathematical expression not reproducible] and their 95% confidence intevals, which are based on the asymptotic results in Yao et al. (Forthcoming), against EF of the world index in Figure 1. For comparison purposes, we superimpose the parametric counterpart estimates. As expected, our smooth coefficients are mostly positive and nonlinear across the entire range of the EF index, and exhibit different patterns.

The upper-left panel in Figure 1 presents the neutral effects of EF [??] (EF) against its parametric counterparts. The neutral effects of EF on production from our semiparametric estimates are decreasing with economic freedom, but much smaller than those from parametric estimates. The upper-right panel in Figure 1 describes the output elasticity of human capital (or marginal product of human capital). It shows a decreasing pattern in general but also suggests improved productivities from human capital when EF is above 7.5 (approximately Japan in 2010). The 95% confidence bounds of sempiparametric estimates contain most of its parametric counterparts for the output elasticity of human capital, but it is not the case for the neutral effect. It suggests that parametric models yield fairly reasonable estimates for the output elasticity of human capital, but q[u.sub.it]e likely misspecify the neutral effect.

The lower-left panel in Figure 1 plots the output elasticity of labor (or marginal product of labor). It demonstrates that the output elasticity of labor decreases as the EFW index rises. However, output elasticity of labor decreases at a slower rate once EF reaches 6.0 (China in 2010). Our semiparametric estimates are generally much lower than the parametric estimates. The output elasticity of capital in the lower-right panel exhibits an increasing pattern in general but at a lower rate when EF reaches 6.0. Our results suggest higher output elasticity of capital under semiparametric specification than parametric counterparts. In both cases, the confidence intervals of our semiparametric estimates do not contain the parametric counterparts, suggesting the parametric models are likely misspecified. The returns to scale estimates suggested by our semiparametric model at 10th, 50th, and 90th percentiles are 0.9477, 1.1258, and 1.9209, respectively, with a mean of 1.2972. In comparison, Cobb-Douglas and restricted translog models indicate returns to scale estimates of 1.5274 and 1.3367, respectively.

Discrepancies between semiparametric and parametric estimates imply that the parametric approaches, which are very much likely to suffer from misspecification, overestimate the neutral effect of economic freedom, output elasticity of labor and underestimate output elasticity of capital, but deliver reasonable estimates of output elasticity of human capital.

Given the pattern exhibited in Figure 1 and the fact that Ln(K)'s magnitude is much larger than other inputs, we expect that EF shifts the production frontier upward. For illustration, let us consider the case of China. In 2010, China had a chain-linked EF score of 6.07. Our semiparametric model suggests that if China had the EF level of the United States (7.76), its production frontier would be shifted upward by $3,914,667 millions of 2005 U.S. dollars or 20.83% higher than its estimated 2010 outputs on the frontier, ceteris paribus. In comparison, the Cobb-Douglas suggests that the same increase in EF will have a much smaller impact on the production frontier, with an upward shift of $56,328 millions of 2005 U.S. dollars (or 0.28% higher than the estimated 2010 production frontier). The restricted translog model indicates a larger upward shift of $4,971,707 millions of 2005 U.S. dollars (or 32.44% higher) on the frontier.

Figure 2 presents kernel density estimates of composite error and technical efficiency for all three models. The left panel indicates that composite errors for all three models cluster around -0.5 which suggests majority of observations are not fully technically efficient on average (E([[epsilon].sub.it]|[EF.sub.it]) < 0) and operates below production frontier. Moreover, the kernel density of composite error for semiparametric estimates is taller and more tightly centered around -0.5, suggesting smaller composite error in absolute values and higher technical efficiency on average under semiparametric estimates than those of parametric counterparts, whose densities exhibit a fairly obvious negative skewness with larger magnitudes in absolute value.

The right panel of Figure 2 presents density estimates of technical efficiency for semiparametric and parametric estimates. The results are consistent with what is suggested in the left panel. Compared with parametric counterparts, the density of semiparametric technical efficiency is more tightly centered at a higher level, suggesting higher efficiencies under the semiparametric approach. The average and median technical efficiency are 0.6955 and 0.7046 for the semiparametric model, at least 20% higher than those from the parametric models, which are 0.5009 and 0.4744 for Cobb-Douglas, and 0.5699 and 0.5625 for translog. The results suggest that technical efficiencies are likely to be mismeasured in parametric models. Our results are consistent with Adkins, Moomaw, and Savvides (2002) and Klein and Luu (2003) in the sense that the majority of observations are technically inefficient. However, our estimates of technical efficiency are smaller in absolute values. A direct comparison between our and their estimates may not be appropriate since they incorporate many other variables such as political rights and civil liberty in the estimation of technical efficiency. However, we conjecture that their estimates can be misleading, since they assume constant output elasticities which can either overestimate or underestimate the impact of EF on the elasticities as demonstrated in Figure 1.

In Equation (1), EF shifts production frontier directly via [alpha]([EF.sub.it]) and indirectly through the output elasticities of input variables, [[beta].sub.H]([EF.sub.it]), [[beta].sub.L]([EF.sub.it]), and [[beta].sub.K]([EF.sub.it]). Economic freedom also affects efficiency through g([EF.sub.it];[eta]). To give the audience a more vivid picture, we follow Yao et al. (Forthcoming) to obtain partial effect of EF on technical efficiency change from [partial derivative] ln [[??].sub.it]/[partial derivative][LC.sub.it] = [eta] {ln [[??].sub.it] }, where In [[??].sub.it] = -[u.sub.i] exp ([eta][LC.sub.it]) and [[??].sub.it] = exp (-[u.sub.it]). Note that for [[??].sub.it], we use the actual estimated [u.sub.it], with [u.sub.i] replaced by estimated mean value and [eta] by its estimate. We did not use the Jondrow, Knox, and Schmidt (1982) estimate [TE.sub.it] discussed before, since it depends on the unknown composite error eit term, whose estimate generates a fair amount of random fluctuations in the graphs. Thus, we plot the partial effect of EF on the efficiency changes for all three estimates against EF in the left panel of Figure 5. The partial effects are found to be positive for all three but decrease with higher levels of economic freedom. This suggests that technical efficiency increases with more economic freedom, but at a decreasing rate.

If one believes that the smooth varying coefficients should take a parametric form, then one naturally would like to perform a model specification test. If the parametric functional form is not rejected, then one could use it for simplicity. We perform a model specification test proposed by Yao et al. (Forthcoming) to test parametric models against our semiparametric model. Following the wild bootstrap procedure described in Yao et al. (Forthcoming), the test statistics [[??].sub.n] (7) for the null hypotheses of Cobb-Douglas and translog model specifications are 343.62 and and 142.7, respectively, and the boot-strapped p values are zero under both scenarios, strongly rejecting the parametric models. Consequently, to address the effects of EF on technical efficiency, the two popular parametric frontier models, being too restrictive in their functional forms, are likely misspecified.

V. ROBUSTNESS TESTING

To probe the robustness of the empirical results from Section IV, we employ the index of economic freedom (IEF) from the Heritage Foundation (Miller and Kim 2017) as our alternative measurement of EF and replicate the analysis above. While the IEF is based on the same philosophical foundation as the EFW, it covers slightly different data and is scaled from 0 to 100 as compared to the 0-10 of the EFW. The IEF covers variables in ten major areas, including business freedom, trade freedom, monetary freedom, government size and spending, fiscal freedom, property rights, investment freedom, financial freedom, freedom from corruption, and labor freedom. We match the IEF with our dataset to improve comparability with our earlier results. As of 2010, Singapore exhibits the highest IEF score of 86.1 whereas Zimbabwe has the lowest score of 21.4. Both measurements of EF are highly correlated, with a correlation of .8364. The IEF scores as of 2010 are listed in Table 1 and summary statistics are presented in Table 2.

The smooth coefficient estimates using IEF as the "environmental" variable are presented in Table 3 Panel B. The smooth coefficient estimates of output elasticities have similar magnitudes and are generally comparable with the estimates obtained from using EFW as the environmental variable. For the parameter estimate [??], it provides estimates at a smaller magnitude, where [mathematical expression not reproducible] = (.2834, .0217,-.0056). With an estimated [??] of -.0056 using an alternative measurement of economic freedom, it further confirms that higher levels of EF increase technical efficiency. The maximum likelihood estimation results for parametric models are tabulated in Table 4 Panel B. Comparing with the smooth coefficient estimates using IEF, the Cobb-Douglas, translog, and restricted translog models provide much larger estimates of [[??].sup.2.sub.u], similar estimates of [[??].sup.2.sub.v], and lower estimates of [??], where [mathematical expression not reproducible] = (8.3085, .0214,-.2292), [mathematical expression not reproducible] =(4.1121, .0209,-.1855), and [mathematical expression not reproducible] = (6.7571, .0212, -.2165), respectively.

We also plot smooth coefficients using IEF as environmental variable and their 95% confidence bounds in Figure 3 to facilitate comparison. Parametric estimates from the Cobb-Douglas and restricted translog models are also superimposed. As expected, the smooth coefficients using IEF as the environmental variable demonstrated similar patters as smooth coefficients generated using EFW as the environmental variable.

The neutral effects of EF presented in the upper-left panel of Figure 3 show a decreasing pattern as EF increases. The neutral effects on the production are much smaller than those from its parametric counterparts. The smooth coefficient estimate of output elasticity of human capital described in the upper-right panel of Figure 3 indicates a decreasing pattern as EF increases but it starts to increase when IEF score exceeds 70. It is consistent with the observation of output elasticity of human capital obtained when using EFW as the environment variable, which starts to increase when EF exceeds 7.5. A large portion of parametric estimates also fall into the 95% confidence interval, suggesting reasonable estimates of output elasticity of human capital from parametric models.

The output elasticity of labor is presented in the lower-left panel of Figure 3 while output elasticity of capital is presented in the lower-right panel. Smooth coefficient function of output elasticity of labor decreases in general with economic freedom. Parametric models suggest higher output elasticity of labor as a fairly large portion of parametric estimates is above the upper bound of the confidence interval. The output elasticity of capital shows an increasing pattern as EF increases but at a slower rate once EF reached 60. This is also consistent with the observation made using EFW as the environmental variable, where the output elasticity of capital increases at a much slower rate once EF reaches the threshold of 6.0. Parametric results indicate lower estimates of output elasticity of capital than our semiparameric estimate.

We also present the kernel density estimates of the composite error and technical efficiency using IEF as EF in Figure 4. The results are similar to those indicated using EFW as the environmental variable. Composite errors from both semiparametric and parametric models clusters around -0.45 but semiparametric estimates are more tightly centered around -0.45. This indicates smaller composite error and higher technical efficiency on average for our semiparametric estimates. The kernel density of technical efficiency for semiparametric model is also more tightly centered at a higher level, suggesting higher efficiencies under the semiparametric specification. We also plot the efficiency changes for all three models in the right panel of Figure 5 using IEF as the environmental variable. We observe the same decreasing pattern for the partial effect of EF as before, although the partial effect for the semiparametric model has a much smaller magnitude due to a reduced estimate of rj.

VI. CONCLUSION

In this paper, we apply a multistep semiparametric smooth coefficient stochastic production frontier estimator proposed by Yao et al. (Forthcoming) to investigate the effects of EF on production frontier and technical efficiency. We contrast the semiparametric estimate with Cobb-Douglas and translog. Allowing the output elasticities and technical efficiency to depend on the environmental variable, economic freedom, we observe significant variation on output elasticities. A model specification test indicates that both Cobb-Douglas and translog production frontiers are likely to be misspecified.

Our results add to the literature on EF and growth in three ways. First, our results highlight the importance of flexible modeling as parametric estimates of the marginal productivity of inputs are shown to be fairly restrictive. Parametric approaches overestimate the neutral effect of EF and the output elasticity of labor, and underestimate the output elasticity of capital, relative to our semiparametric estimates. Second, our empirical average efficiency estimates are at least 20% higher than those obtained from the parametric counterparts. Third, we find that the output elasticities of labor, human capital, and physical capital vary with the level of economic freedom. In this way our results are similar to that of Hall, Sobel, and Crowley (2010), but more precise. Our results suggest that the output elasticities are mostly positive. Interestingly, increased EF generally lowers the output elasticities or marginal products of human capital and labor but leads to improvements in the marginal product of capital.

Like the previous literature we find that EF shifts the semiparametric stochastic production frontier upward and reduces technical inefficiency. Thus, higher EF is favored for a country to improve its efficiency in general. However, we discover that EF positively impacts the frontier mainly through the nonneutral effect on the output elasticity of capital. Thus, on the policy front, countries with a higher capital to labor ratio are especially advised to improve its economic freedom, if the policy goal is to be as efficient as possible with its inputs.

ABBREVIATIONS

EF: Economic Freedom

EFW: Economic Freedom of the World

GDP: Gross Domestic Product

IEF: Index of Economic Freedom

PWT: Penn World Table

SF: Stochastic Frontier

doi: 10.1111/ecin.12548

Online Early publication January 18, 2018

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FAN ZHANG, JOSHUA HALL (ID) and FENG YAO

Zhang: Assistant Professor, Department of Business Management, Ripon College, Ripon, WI 54971. Phone 920748-8198, Fax 920-748-7243, E-mail zhangf@ripon.edu

Hall: Associate Professor, Department of Economics, West Virginia University, Morgantown, WV 26506. Phone 304-293-7870, Fax 304-293-5652, E-mail joshua.hall @ mail .wvu.edu

Yao: Associate Professor, Department of Economics, West Virginia University, Morgantown, WV 26506. Phone 304-293-7867, Fax 304-293-5652, E-mail feng.yao@mail.wvu.edu

(1.) Some well-cited papers in this literature include Dawson (1998), Gwartney, Lawson, and Holcombe (1999), Heckelman and Stroup (2000), Carlsson and Lundstrom (2002), Dawson (2003), De Haan, Lundstrom, and Sturm (2006), and Justesen (2008).

(2.) In this respect, our work is similar to that of Hall, Sobel, and Crowley (2010), who find in a cross-section that the marginal product of labor and capital varies with the institutional quality of a country.

(3.) Throughout the paper we will use output elasticity and marginal product interchangeably.

(4.) Our empirical approach, however, is different from theirs and thus our results are not directly comparable.

(5.) Yao et al. (2017) propose a four-step semiparametric estimator, establish their consistency and asymptotic normality, and carry out a comprehensive Monte Carlo study.

(6.) We look at only 5-year increments for two reasons. First, institutions change slowly and thus 5-year increments is standard in the economic freedom/growth literature. Second, the EFW index only reports at 5-year intervals from 1990 to 2000.

(7.) Yao et al. (2017) propose a modified test for parametric frontier specification based on Li et al. (2002) and Li and Racine (2010) for panel data framework and establish asymptotic properties for the test statistics [[??].sub.n]. Please refer to Yao et al. (2017) for details.

Caption: FIGURE 1 Plots of Smooth Coefficient Frontiers against EFW

Caption: FIGURE 2 Kernel Density of Composite Error and Technical Efficiency EFW

Caption: FIGURE 3 Plots of Smooth Coefficient Frontiers against Heritage Index

Caption: FIGURE 4 Kernel Density of Composite Error and Technical Efficiency Heritage Index

Caption: FIGURE 5 Partial Effect of Economic Freedom on Technical Efficiency

TABLE 1 List of Countries and Economic Freedom as of 2010

Country          EFW    IEF    Country           EFW    IEF

Albania          7.51   66.0   Denmark           7.75   77.9
Argentina        5.72   51.2   Dominican Rep.    7.06   60.3
Australia        8.10   82.6   Ecuador           5.70   49.3
Austria          7.53   71.6   Egypt             6.79   59.0
Bahrain          7.69   76.3   El Salvador       7.43   69.9
Bangladesh       6.52   51.1   Estonia           7.76   74.7
Barbados         6.64   68.3   Fiji              6.33   60.3
Belgium          7.47   70.1   Finland           7.73   73.8
Belize           6.45   61.5   France            7.43   64.2
Benin            5.81   55.4   Gabon             5.50   55.4
Bolivia          6.39   49.4   Germany           7.53   71.1
Botswana         7.22   70.3   Ghana             6.86   60.2
Brazil           6.56   55.6   Greece            6.75   62.7
Bulgaria         7.22   62.3   Guatemala         7.24   61.0
Burundi          5.02   47.5   Honduras          7.04   58.3
Cameroon         6.56   52.3   Hong Kong         8.85   89.7
Canada           8.05   80.4   Hungary           7.32   66.1
Central Afr.     5.72   48.4   Iceland           6.41   73.7
Chile            7.94   77.2   India             6.59   53.8
China            6.07   51.0   Indonesia         7.05   55.5
Colombia         6.33   65.5   Iran              6.46   43.4
Congo, Dem.      5.61   41.4   Ireland           7.75   81.3
Congo, Rep.      4.95   43.2   Israel            7.60   67.7
Costa Rica       7.41   65.9   Italy             7.11   62.7
Cote d'Ivoire    5.85   54.1   Jamaica           7.12   65.5
Croatia          6.97   59.2   Japan             7.51   72.9
Cyprus           7.65   70.9   Jordan            7.75   66.1
Czech Rep.       7.35   69.8   Kenya             7.07   57.5

Country          EFW    IEF    Country           EFW    IEF

Korea, South     7.28   69.9   Senegal           5.78   54.6
Kuwait           7.40   67.7   Sierra Leone      6.91   47.9
Latvia           7.01   66.2   Singapore         8.53   86.1
Lithuania        6.99   70.3   Slovak Rep        7.44   69.7
Luxembourg       7.59   75.4   Slovenia          6.55   64.7
Malawi           5.97   54.1   South Africa      6.87   62.8
Malaysia         7.00   64.8   Spain             7.26   69.6
Mali             5.97   55.6   Sri Lanka         6.26   54.6
Malta            7.61   67.2   Sweden            7.61   72.4
Mauritius        7.92   76.3   Switzerland       8.23   81.1
Mexico           6.69   68.3   Syria             5.73   49.4
Morocco          6.45   59.2   Taiwan            7.74   70.4
Namibia          6.51   62.2   Tanzania          6.53   58.3
Nepal            5.92   52.7   Thailand          6.60   64.1
Netherlands      7.58   75.0   Togo              5.62   47.1
New Zealand      8.10   82.1   Trinidad & Tob.   6.95   65.7
Niger            5.93   52.9   Tunisia           6.06   58.9
Norway           7.38   69.4   Turkey            6.52   63.8
Pakistan         5.98   55.2   Uganda            7.56   62.2
Panama           7.25   64.8   Ukraine           5.87   46.4
Paraguay         6.62   61.3   United Kingdom    7.90   76.5
Peru             7.51   67.6   United States     7.76   78.0
Philippines      7.09   56.3   Uruguay           7.29   69.8
Poland           7.09   63.2   Venezuela         3.84   37.1
Portugal         7.05   64.4   Zambia            7.68   58.0
Romania          7.14   64.2   Zimbabwe          4.51   21.4
Russia           6.34   50.3
Rwanda           7.20   59.1

TABLE 2
Summary Statistics

Symbol   Variable Description              Mean      Std.

Ln(Y)    Log of output-side real GDP      11.3531   1.9074
Ln(H)    Log of human capital             0.8947    0.2442
Ln(L)    Log of labor force               1.5453    1.6410
Ln(K)    Log of capital stock             12.2795   2.0745
EFW      Economic freedom of the world    6.6790    1.0913
IEF      Index of economic freedom        61.8722   10.5502

Symbol    Min.      Max.     Bandwidth

Ln(Y)    7.0998    16.3799      -Ln(H)
0.1397    1.2864       -Ln(L)
-2.8134   6.6611       -Ln(K)
6.9201    17.5224      -EFW
2.9707    9.1509     0.7891
IEF       21.4      8.97      0.6688

TABLE 3 Estimation of the Semiparametric Smooth
Coefficient Stochastic Frontier

Panel A: Economic Freedom of the World Index

MLE                  Estimates   Standard error
[[??].sup.2.sub.u]    2.0241         0.4566
[[??].sup.2.sub.v]    0.0357         0.0037
[??]                  -0.1383        0.0132
Smooth coefficient     Mean      10th percentile
[??](EF)              2.0446         1.0160
[[??].sub.H](EF)      0.3374         -0.0307
[[??].sub.L](EF)      0.2184         0.1557
[[??].sub.K](EF)      0.7413         0.5922

Panel B: Index of Economic Freedom

MLE                  Estimates   Standard error
[[??].sup.2.sub.u]    0.2834         0.0373
[[??].sup.2.sub.v]    0.0333         0.0027
[??]                  -0.0056        0.0017
Smooth coefficient     Mean      10th percentile
[??](EF)              2.0068         1.4956
[[??].sub.H](EF)      0.4299         0.1771
[[??].sub.L](EF)      0.2340         0.1767
[[??].sub.K](EF)      0.7344         0.6391

Panel A: Economic Freedom of the World Index

MLE
[[??].sup.2.sub.u]
[[??].sup.2.sub.v]
[??]
Smooth coefficient   Median   90th percentile
[??](EF)             2.0949       2.9463
[[??].sub.H](EF)     0.1654       0.9552
[[??].sub.L](EF)     0.1911       0.3418
[[??].sub.K](EF)     0.7614       0.8467

Panel B: Index of Economic Freedom

MLE
[[??].sup.2.sub.u]
[[??].sup.2.sub.v]
[??]
Smooth coefficient   Median   90th percentile
[??](EF)             1.9827       2.5315
[[??].sub.H](EF)     0.2922       0.6694
[[??].sub.L](EF)     0.2024       0.3712
[[??].sub.K](EF)     0.7606       0.7821

TABLE 4 Maximum Likelihood Estimation of Parametric Benchmark Models

                      Cobb-Douglas Model      Translog Model

Variables            Coefficient   Prob.   Coefficient   Prob.

Panel A: Economic Freedom of the World Index

Constant                 5.3703    .0000       7.7710    .0000
EF                       -.3178    .0129       -.1716    .3067
E[F.sup.2]                .0231    .0080        .0006    .9490
Ln(H)                     .6076    .0000       4.5700    .0000
Ln(L)                     .3836    .0000       1.2026    .0000
Ln(K)                     .5362    .0000       -.4002    .1029
[Ln.sup.2](H)                --     --         1.8076    .0012
[Ln.sup.2](L)                --     --          .0830    .0005
[Ln.sup.2](K)                --     --          .0731    .0000
EFLn(H)                      --     --          .1381    .1480
EFLn(L)                      --     --          .0216    .3247
EFLn(K)                      --     --          .0051    .7832
Ln(H)Ln(L)                   --     --          .6310    .0000
Ln(H)Ln(K)                   --     --         -.7418    .0000
Ln(L)Ln(K)                   --     --         -.1442    .0000
[[??].sup.2.sub.u]       2.6714    .0234       2.8809    .0539
[[??].sup.2.sub.v]        .0217    .0000        .0194    .0000
[??]                     -.1156    .0013       -.0519    .1814
Log-likelihood                359.065               390.614
Observations                    440                   440

Panel B: Index of Economic Freedom

Constant                 5.8476    .0000       4.7966    .0000
EF                       -.6265    .0000       -.4942    .0072
E[F.sup.2]                .0425    .0000        .0338    .0164
Ln(H)                     .5589    .0000       2.9839    .0045
Ln(L)                     .3178    .0000        .4311    .1707
Ln(K)                     .5995    .0000        .4715    .0937
[Ln.sup.2](H)                --     --         1.6340    .0027
[Ln.sup.2](L)                --     --          .0398    .1067
[Ln.sup.2](K)                --     --          .0159    .3942
EFLn(H)                      --     --         -.2100    .0523
EFLn(L)                      --     --         -.0121    .6699
EFLn(K)                      --     --          .0184    .3723
Ln(H)Ln(L)                   --     --          .4007    .0111
Ln(H)Ln(K)                   --     --         -.3676    .0169
Ln(L)Ln(K)                   --     --         -.0402    .2791
[[??].sup.2.sub.u]       8.3085    .0227       4.1121    .1197
[[??].sup.2.sub.v]        .0214    .0000        .0209    .0000
[??]                     -.2292    .0000       -.1855    .0003
Log-likelihood                367.893               380.418
Observations                    440                   440

                           Restricted
                          Translog Mode

Variables            Coefficient   Prob.

Panel A: Economic Freedom of the World Index

Constant                 7.9271    .0000
EF                       -.6367    .0000
E[F.sup.2]                .0100    .3059
Ln(H)                     .6522    .2167
Ln(L)                     .4167    .0002
Ln(K)                     .2678    .0010
[Ln.sup.2](H)                --     -[Ln.sup.2](L)
--     -[Ln.sup.2](K)
--     -EFLn(H)
-.0368    .6378
EFLn(L)                  -.0117    .4696
EFLn(K)                   .0443    .0005
Ln(H)Ln(L)                   --     -Ln(H)Ln(K)
--     -Ln(L)Ln(K)
--     -[[??].sup.2.sub.u]
3.2214    .0170
[[??].sup.2.sub.v]        .0209    .0000
[??]                     -.0404    .0000
Log-likelihood                372.628
Observations                    440

Panel B: Index of Economic Freedom

Constant                 5.8135    .0000
EF                       -.6271    .0000
E[F.sup.2]                .0431    .0024
Ln(H)                    1.8180    .0028
Ln(L)                     .2154    .2017
Ln(K)                     .5183    .0000
[Ln.sup.2](H)                --     -[Ln.sup.2](L)
--     -[Ln.sup.2](K)
--     -EFLn(H)
-.1988    .0354
EFLn(L)                  -.0215    .6194
EFLn(K)                   .0135    .4237
Ln(H)Ln(L)                   --     -Ln(H)Ln(K)
--     -Ln(L)Ln(K)
--     -[[??].sup.2.sub.u]
6.7571    .0848
[[??].sup.2.sub.v]        .0212    .0000
[??]                     -.2165    .0000
Log-likelihood                372.176
Observations                    440