A Varying Coefficient Model for Assessing the Returns to Growth to Account for Poverty and Inequality

TL;DR

This paper introduces a varying coefficient model to analyze how poverty and inequality influence the returns to economic growth, revealing significant differences across income groups and highlighting potential biases in traditional estimations.

Contribution

It develops a novel varying coefficient model that captures the heterogeneous effects of growth on different income groups, addressing limitations of mean coefficient estimates.

Findings

Returns to growth vary significantly with poverty and inequality levels.

Traditional mean coefficient estimates may be biased due to unaccounted heterogeneity.

Differences in coefficients challenge the interpretability of average effects.

Abstract

Various papers demonstrate the importance of inequality, poverty and the size of the middle class for economic growth. When explaining why these measures of the income distribution are added to the growth regression, it is often mentioned that poor people behave different which may translate to the economy as a whole. However, simply adding explanatory variables does not reflect this behavior. By a varying coefficient model we show that the returns to growth differ a lot depending on poverty and inequality. Furthermore, we investigate how these returns differ for the poorer and for the richer part of the societies. We argue that the differences in the coefficients impede, on the one hand, that the means coefficients are informative, and, on the other hand, challenge the credibility of the economic interpretation. In short, we show that, when estimating mean coefficients without accounting for poverty and inequality, the estimation is likely to suffer from a serious endogeneity bias.

Equations

$$growth=\alpha +\beta \cdot (growthdriver)+error.$$

$$y_{it}=\rho y_{i(t-l)}+x_{it}^{\prime }\beta +{\eta }_i+{\nu }_{it},t=1+l,\dots ,T,i=1,\dots ,n,$$

$$y_{it}=\rho y_{i(t-l)}+x_{it}^{\prime }{\beta }_{it}+{\eta }_i+{\nu }_{it},$$

$$y_{it}=\rho y_{i(t-l)}+x_{it}^{\prime }{\beta }_{it}+{\eta }_i+{\nu }_{it}$$

$${\beta }_{itk}={\tilde{z}}_{it}^{\prime }{\gamma }_k+a_{itk},$$

$$\left(\begin{array}{c}{\beta }_{1it}\\ {\beta }_{2it}\\ {\beta }_{3it}\end{array}\right)=\left(\begin{array}{ccc}{\tilde{z}}_{it}^{\prime }& 0& 0\\ 0& {\tilde{z}}_{it}^{\prime }& 0\\ 0& 0& {\tilde{z}}_{it}^{\prime }\end{array}\right)\left(\begin{array}{c}{\gamma }_1\\ {\gamma }_2\\ {\gamma }_3\end{array}\right)+\left(\begin{array}{c}{a}_{it1}\\ a_{it2}\\ a_{it3}\end{array}\right)\in {\mathbb{R}}^3$$

$$\left(\begin{array}{ccc}{\tilde{z}}_{it}^{\prime }& 0& 0\\ 0& {\tilde{z}}_{it}^{\prime }& 0\\ 0& 0& {\tilde{z}}_{it}^{\prime }\end{array}\right)=Z_{it}\in {\mathbb{R}}^{3\times M},\left(\begin{array}{c}{\gamma }_1\\ {\gamma }_2\\ {\gamma }_3\end{array}\right)=\gamma \in {\mathbb{R}}^M\text{mbox}and\left(\begin{array}{c}{a}_{it1}\\ a_{it2}\\ a_{it3}\end{array}\right)=a_{it}\in {\mathbb{R}}^3$$

$$E(a_{it}{a}_{js}^{\prime })$$

$$\Lambda$$

$$y_i=\rho y_{i(-l)}+{\eta }_i{\iota }_{T-l}+X_i{\beta }_i+{\nu }_i\in {\mathbb{R}}^{T-l},$$

$$y_{i(-l)}=(y_{i1},\dots ,y_{i(T-l)}{)}^{\prime }\in {\mathbb{R}}^{T-l},{\iota }_{T-l}=(1,\dots ,1{)}^{\prime }\in {\mathbb{R}}^{T-l},$$

$$X_i=\left(\begin{array}{ccc}{x}_{i(1+l)}^{\prime }& & \\ & \ddots & \\ & & x_{iT}^{\prime }\end{array}\right)\in {\mathbb{R}}^{(T-l)\times 3(T-l)},$$

$${\beta }_i=({\beta }_{i(1+l)}^{\prime },\dots ,{\beta }_{iT}^{\prime }{)}^{\prime }\in {\mathbb{R}}^{3(T-l)}\text{mbox}and{\nu }_i=({\nu }_{i(1+l)},\dots ,{\nu }_{iT}{)}^{\prime }\in {\mathbb{R}}^{T-l}.$$

$${\beta }_i=Z_i\gamma +a_i\in {\mathbb{R}}^{3(T-l)},$$

$$Z_i=(Z_{i(1+l)}^{\prime },\dots ,Z_{iT}^{\prime }{)}^{\prime }\in {\mathbb{R}}^{3(T-l)\times M}\text{mbox}and{a}_i=(a_{i(1+l)}^{\prime },\dots ,a_{iT}^{\prime }{)}^{\prime }\in {\mathbb{R}}^{3(T-l)}.$$

$$E(a_i{a}_i^{\prime })=I_{T-l}\otimes \Lambda \in {\mathbb{R}}^{3(T-l)\times 3(T-l)}$$

$$y=\rho y_{-l}+C\eta +X\beta +\nu ,$$

$$y_{-l}=(y_{1(-l)}^{\prime },\dots ,y_{n(-l)}^{\prime }{)}^{\prime }\in {\mathbb{R}}^{n(T-l)},$$

$$C=I_{T-l}\otimes \iota \in {\mathbb{R}}^{n(T-l)\times n},\eta =({\eta }_1,\dots ,{\eta }_n{)}^{\prime }\in {\mathbb{R}}^n,$$

$$X=\left(\begin{array}{ccc}{X}_1& & \\ & \ddots & \\ & & X_n\end{array}\right)\in {\mathbb{R}}^{n(T-l)\times 3n(T-l)},$$

$$\beta =({\beta }_1^{\prime },\dots ,{\beta }_n^{\prime }{)}^{\prime }\in {\mathbb{R}}^{3n(T-l)}\text{mbox}and\nu =({\nu }_1^{\prime },\dots ,{\nu }_n^{\prime }{)}^{\prime }\in {\mathbb{R}}^{n(T-l)}.$$

$$\beta =Z\gamma +a\in {\mathbb{R}}^{3n(T-l)},$$

$$Z=(Z_1^{\prime },\dots ,Z_n^{\prime }{)}^{\prime }\in {\mathbb{R}}^{3n(T-l)\times M}\text{mbox}anda=(a_1^{\prime },\dots ,a_n^{\prime }{)}^{\prime }\in {\mathbb{R}}^{3n(T-l)}.$$

$$y$$

$$W=(Z_1^{\prime }{X}_1^{\prime },\dots ,Z_n^{\prime }{X}_n^{\prime }{)}^{\prime }\in {\mathbb{R}}^{n(T-l)\times M}\text{mbox}andu=(a_1^{\prime }{X}_1^{\prime },a_2^{\prime }{X}_2^{\prime },\dots ,a_n^{\prime }{X}_n^{\prime }{)}^{\prime }+\nu \in {\mathbb{R}}^{n(T-l)}.$$

$$y=\rho y_{-l}+C\eta +W\gamma +u\in {\mathbb{R}}^{n(T-l)}$$

$$(y_{-l},C,W)\in {\mathbb{R}}^{n(T-l)\times (n+M+1)}$$

$$E\left[u{u}^{\prime }\right]$$

$$y=\rho y_{(-l)}+C\eta +\left(\begin{array}{c}{X}_1\\ 0\\ ⋮\\ 0\end{array}\right){\beta }_1+\left(\begin{array}{c}0\\ X_2\\ ⋮\\ 0\end{array}\right){\beta }_2+\dots +\left(\begin{array}{c}0\\ 0\\ ⋮\\ X_n\end{array}\right){\beta }_n+\nu \in {\mathbb{R}}^{n(T-1)}$$

$${\tilde{z}}_{it}=(1,po{v}_{i(t-3)},po{v}_{i(t-3)}^2,gin{i}_{i(t-3)},gin{i}_{i(t-3)}^2,middleclas{s}_{i(t-3)},middleclas{s}_{i(t-3)}^2{)}^{\prime }\in {\mathbb{R}}^7,$$

$$\begin{array}{rl}{\beta }_{1it}=& -2.1952^{\ast \ast \ast }+1.3705^{\ast \ast \ast }po{v}_{i(t-3)}-3.4198^{\ast \ast \ast }po{v}_{i(t-3)}^2+2.9553gin{i}_{i(t-3)}^{\ast \ast \ast }\\ & -0.8132^{\ast \ast \ast }gin{i}_{i(t-3)}^2+0.8917^{\ast \ast \ast }middleclas{s}_{i(t-3)}+2.9860^{\ast \ast \ast }middleclas{s}_{i(t-3)}^2.\end{array}$$

$$\begin{array}{rl}{\beta }_{2it}=& -0.0519^{\ast \ast }-1.2799^{\ast \ast \ast }po{v}_{i(t-3)}+3.7563^{\ast \ast \ast }po{v}_{i(t-3)}^2-3.8873^{\ast \ast \ast }gin{i}_{i(t-3)}\\ & +3.9307^{\ast \ast \ast }gin{i}_{i(t-3)}^2+5.6845^{\ast \ast \ast }middleclas{s}_{i(t-3)}-7.3372^{\ast \ast \ast }middleclas{s}_{i(t-3)}^2,\end{array}$$

$$\begin{array}{rl}{\beta }_{3it}=& -2.8512^{\ast \ast \ast }+0.6369^{\ast \ast \ast }po{v}_{i(t-3)}-0.4078^{\ast \ast \ast }po{v}_{i(t-3)}^2+0.5740^{\ast \ast \ast }gin{i}_{i(t-3)}\\ & +2.2057^{\ast \ast \ast }gin{i}_{i(t-3)}^2+5.7917^{\ast \ast \ast }middleclas{s}_{i(t-3)}-2.3057^{\ast \ast \ast }middleclas{s}_{i(t-3)}^2.\end{array}$$

$$\begin{array}{rl}{\beta }_{1it,poor}=& -0.1806^{\ast \ast \ast }+1.9801^{\ast \ast \ast }po{v}_{i(t-3)}-5.0660^{\ast \ast \ast }po{v}_{i(t-3)}^2+6.4187^{\ast \ast \ast }gin{i}_{i(t-3)}\\ & -5.5026^{\ast \ast \ast }gin{i}_{i(t-3)}^2-9.6406^{\ast \ast \ast }middleclas{s}_{i(t-3)}+13.4882^{\ast \ast \ast }middleclas{s}_{i(t-3)}^2\end{array}$$

Full text

1 Introduction

The literature shows that the variables inequality, poverty and the size of the middle class are important for economic growth. Usually the authors add the variables addressed in their special question to the growth regression and observe the effects of these variables. When explaining why certain measures of the income distribution are added, it is often explained that poor people behave differently from rich people and therefore, the economies as whole behave differently according to their level of poverty, inequality and the size of their middle class. However, adding explanatory variables does not reflect this behavior. For example, it is hard to believe that a poor economy, in which a high number of poor people cannot provide collateral and therefore, do not have access to the credit market, has the same returns to investments in physical capital as a richer economy. This paper empirically investigates the effects of measures of the income distribution on the coefficients of the drivers for economic growth. More precisely, we consider three parameters of the income distribution, namely poverty, inequality and the size of the middle class and investigate the influence of each of these variables on the coefficients of the drivers for growth proposed by Mankiw, Romer and Weil (1992). For this purpose, we formulate and apply a varying coefficient model. We focus on a panel data analysis using data from several countries over time.

The literature on growth, poverty and inequality is much too abundant to briefly discuss it here, but the explicit modeling of the impact of the latter on growth is not that frequent; see Bourguignon (2004) for a review. Generally, while the literature basically agrees on the fact that inequality affects economic growth, there is no consensus about the size or the direction of the effects. Even when concentrating on the most recent contributions you find studies arguing for the inverted U-shaped relationship (e.g., Tapia, 2015), positive effects of inequality on growth (e.g., Forbes, 2000) and negative effects (e.g., Malinen, 2013). Some argue that inequality has different effects on growth depending on the context (e.g., Halter, Oechslin and Zweimüller, 2014). On the other hand, there is a consensus that poverty affects growth negatively (e.g., Ghatak, 2015), while the size of the middle class is considered to be beneficial to economic growth (e.g., Aristotle 306 BC, Chun, Hasan, Rahman and Ulubaşoğlu, 2016; Easterly, 2001).

The channels through which poverty, inequality and the middle class influencing economic growth are diverse. This suggests that economies behave differently according to these variables. When credit market works imperfectly, lenders will demand collaterals to cope with asymmetric information. Since the poor cannot provide them, then their investment plans are likely to be wasted. Consequently, poverty and a high level of inequality could lead to low economic growth (Banerjee and Newman, 1993; Perry, Lopez and Maloney, 2006). Furthermore, high inequality and a small middle class may deter economic growth by higher socio-economic instability (Alesina and Perotti, 1996) and conflicts (Alesina, 1994; Rodrik, 1998). Poverty also influences economic growth negatively through the financial market development (Ravallion, 2010) and too much risk averse behavior of the poor (Banerjee, 2000). Sachs, McArthur, Schmidt-Traub and Kruk (2004) and Mesnard and Ravallion (2006) support the existence of a poverty trap which has obviously negative implications for economic growth.

Since economies behave differently according to poverty, inequality and the middle class, the mean returns to the drivers of economic growth may not be informative. Imagine that the population (and sample) can be divided into two groups, on composed by poor countries, and the other by rich ones, and consider the simplified growth regression

[TABLE]

In this situation it is very likely to hold that $\beta_{poor}\neq\beta_{rich}$ when studying the two groups separately. However, when pooling them, then the mean coefficient does not only reflect a theoretical parameter that is little helpful. Furthermore, there are problems when bringing the model to data. Poor countries have systematically weaker databases and therefore, the estimation of such a $\beta_{mean}$ is highly suspicious to suffer from a sample selection bias. Moreover, as the deviations from the mean coefficient are highly suspicious to move simultaneously with either the covariates, the dependent variable or both, you face an endogeneity problem in the sense that this mean-parameter cannot be estimated from model (1) with a standard OLS applied to the merged sample.

Certainly, separating the coefficient into $\beta_{poor}$ and $\beta_{rich}$ from the beginning could solve these problems. In practice however, we have many different growth drivers and a set of countries that cannot simply be separated into the groups $poor$ and $rich$. It is obvious, that then the non-modeling of $\beta$ becomes a serious endogeneity problem which typically cannot be solved, even not with more sophisticated IV methods. This motivates to estimate the growth regression with a varying coefficient model in which a “continuous transition” from poor to rich is possible. This transition is explained by the country’s individual levels of poverty, inequality and middle class in each year.

There exist also many other reasons to use a varying coefficient model for growth regression. While many authors include additional covariates like poverty and inequality additively to the growth regression, one could equally well argue that it is hard to understand why these should be essential production factors of GDP on their own; recall that the classical growth regression model is derived from the Cobb Douglas specification of a production function applied to the countries DGP. The arguments brought up in the literature why and how poverty and inequality affect growth insinuate that these covariates have an impact on the efficiency and thus the return of the production factors rather than on GDP directly. This suggests that the coefficients of the classical growth model should be modeled accordingly; it does not suggest to add these covariates additively to the model.

This article is organized as follows. Section (2) first revisits the augmented Solow growth model, deals with collecting reasonable measures of the covariates, and then discusses different estimation issues. The section concludes with formulating the desired varying coefficient model for growth regression. The results of estimation are given in section (3). We first present the results for a model where the mean growth per worker is the dependent variable. It will be seen that the coefficients change dramatically over poverty, inequality and middle size. This motivates another question which is related to the pro-poor growth discussion. If our derivatives of the income distribution affect the growth behavior of the economy, then it could be interesting to see how do the poorer and the richer parts of the countries regarding growth? We do this in an admittedly quite simple way by just looking at a growth regression with the dependent variable being either the average GDP of the lower income group (0 to 20% quantile of the income distribution) and the richest (80 to 100% quantile of the income distribution) respectively. The resulting coefficients differ dramatically. Section (4) concludes.

2 Statistical Modeling and Data Collection

2.1 Growth Regression Model and Data

We use the classic economic model for growth regression which explains the GDP production with a Cobb Douglas function specification. After some algebra one has for country $i$ in year $t$

[TABLE]

where $y_{it}$ is the logarithm of the per worker or capita GDP, $l$ some lag depth, vector $x_{it}$ contains the log of production factors (in the here considered augmented Solow-model, the physical and the human capital) and depreciation rates, $\eta_{i}$ country fixed effects, and $\nu_{it}$ a stochastic deviation with expectation zero. Sometimes time fixed effects are included but get insignificant when applying the Hodrick-Prescott filter (Hodrick and Prescott, 1997) to $y_{it}$ in order to get rid of business cycle effects.444This was also done for the time series of investment shares indicating the physical capital.

The data are based on Penn World Table 6.3 (PWT, Heston, Summers and Aten, 2009), the World Development indicators (The World Bank, 2009) and Barro and Lee (2010). The observations are obtained yearly from 1960 to 2007 for 81 countries. The depreciation rate is the sum of the depreciation rate of capital, the growth rate of productivity and the population growth (Mankiw, Romer and Weil, 1992). The sum of the depreciation rate of capital and the growth rate of productivity is approximated by 5% per year for all countries, which is added to the population growth. The logarithm of the depreciation rate is denoted by $lnn_{it}$. The saving rate of the economy is approximated by the relative investment share of the real GDP, which is denoted by $lnsk_{it}$. The human capital is calculated based on the educational attainment data from Barro and Lee (2010). Since the raw data are available every five years, interpolation splines is used to impute the missing values. We denote the logarithm of the yearly educational attainment by $lnattain_{it}$. Based on the country rating system from Heston, Summers and Aten (2009), only countries with sample quality between A and C are kept, dropping D grading samples. Furthermore, only complete time-series are incorporated for the relevant variables. This also excludes countries that were separated in a sub-period, for example Germany. Recall from our discussion above that the varying coefficient model will essentially alleviate (or even solve) the potential selection bias problem.

We are going to investigate how the factor returns $\beta$ change along the different income distributions. You could write

[TABLE]

where $\beta_{it}$ is (partly) explained by the income distribution in country $i$ at time $t$. Since estimating $\beta_{it}$ as a function of densities would be far to complex, we model some of the returns as functions of poverty, inequality and middle class, instead. Measures for these three variables are derived from the income distribution data from Sala-i-Martin (2006). His data consists of hundred data points of income distribution for 138 countries every year from 1970 to 2000, and is based on the microeconomic income surveys from Deininger and Squire (1999), while the final values are imputed, adjusted and smoothed. We use the fraction of the total population with income less than one dollar (1999 price) per day as our measurement of poverty, which is denoted by $pov1d_{it}$. The Gini coefficient is our measurement of inequality, which is denoted by $gini_{it}$. We repeated the study with the Theil index obtaining the same results. The middle class is the share of the total income that the middle sixty per cent of the population earn (cf., Easterly, 2001). More precisely, let $g_{it}(prop)$ be the cumulative income as a function of the proportion $prop$ of the total population ($pop_{it}$) of country $i$ at year $t$, i.e. the Lorenz curve times total income. Then the relative middle class is $middleclass_{it}=(g_{it}(0.8)-g_{it}(0.2))/g_{it}(1.0)$. Furthermore, as indicated at the end of Section 1, we are also interested in knowing the average incomes of the richest and the poorest twenty per cent of each country in each year. The log average GDP per capita of the poorest twenty per cent of country $i$ at year $t$ is $y_{poor,it}=ln(g_{it}(0.2)/0.2pop_{it})$. This measure was introduced by Ravallion and Chen (2003) for measuring pro-poor growth. In the same way, the log average GDP per capita of the richest twenty per cent of country $i$ at year $t$ is $y_{rich,it}=ln((g_{it}(1.0)-g_{it}(0.8))/0.2pop_{it}).$ We multiply the consumer price index (The World Bank, 2009) appropriately to these variables, so that the reference year becomes 2005 coherent to the PWT. As these series might be subject of business cycles we smooth them, too (for details cf. Koehler, Sperlich and Vortmeyer, 2011).

When combining all data, we end up with complete yearly time-series from 1970 to 2000 for the 81 countries. For our estimation we rely on the fixed effect model in the following as random effect model is unattractive given that the individual deviation $\eta_{i}$ is likely to be dependent from other determinants of economic growth. But as we face a dynamic panel model the estimates could suffer from the Nickell bias. Therefore we have calculated the size of the bias using the formulae from Phillips and Sul (2007) for various large ($>0.9$) values of $\rho$. It turns out that only for the estimates of $\rho$ the bias is larger than $3\cdot 10^{-4}$. Considering the long time series of the data, this result is not surprising. Note that we calculated the bias for equation (2); adding more regressors will reduce the bias further (see, Phillips and Sul, 2007). The Difference and the System GMM are inappropriate as $\rho$ is $>0.9$, a situation in which the former GMM performs especially bad and the latter most likely violates the necessary conditions. Alos, both GMM would suffer from too many instruments in our context, (see e.g. Roodman, 2009).

2.2 A Varying Coefficient Growth Model

As discussed, we introduce a varying coefficient model for growth regression, in which the coefficients depend on poverty, inequality and the size of the middle class, summarized in variable $z_{it}$. To the best of our knowledge the use of varying coefficient models in econometrics started in the seventies; for example Wachter (1970) for studying wage equations, and Singh, Nagar, Choudhry and Baldev (1976) allowing for coefficients with time-trends. A more generalized version, in which some of the coefficients are functions of other exogenous variables, is introduced in Amemiya (1978). Nonparametric extensions are still quite recent (for a review see Park, Mammen, Lee and Lee, 2013). Finally, Rodríguez-Poó and Soberón (2015) consider semiparametric varying coefficient models for panel data in which the drivers of the coefficients vary over both dimensions, time and subjects. That is, as there is no reason to assume that the $\beta$-coefficients are constant either over time or countries, let us rewrite (2) as

[TABLE]

with $\beta_{it}$ depending on $z_{it}$. We assume that each element of $\beta_{it}$, say $\beta_{itk}$, can be written as

[TABLE]

where $k\in\left\{1,2,3\right\}$, $a_{itk}$ a random term, and $\tilde{z}_{it}$ a spline basis evaluated at $z_{it}$. Given the data limitations it turned out that in our application it is sufficient if $\tilde{z}_{it}$ consists of an intercept, $z_{it}$, and the squared values. When stacking the $\beta$-coefficients we obtain

[TABLE]

We will assume that

[TABLE]

what just means that the correlations of the coefficients do not change over time and across individuals given $\zeta_{it}$. Stacking the time-series data of (3) you can write

[TABLE]

with

[TABLE]

Furthermore, we can stack the time-series data of (4) as

[TABLE]

with

[TABLE]

Furthermore, it follows from (7)

[TABLE]

for the random deviations from the conditional mean coefficients, and $E(\nu_{i}\nu_{i}^{\prime})=\Sigma$ for further model deviations. After stacking-time series data of (3), we additionally stack cross-sectional data

[TABLE]

with

[TABLE]

Furthermore, after stacking time-series data of (4), we additionally stack cross-sectional data

[TABLE]

with

[TABLE]

When plugging (9) into (8) you get

[TABLE]

where we used the notation

[TABLE]

The regression equation

[TABLE]

has $n+M+1$ parameters. If the matrix

[TABLE]

has full column rank, the model is identified. The following calculation shows how the unobserved heterogeneity causes uncorrelated but heteroscedastic errors if the $\nu_{it}$ are uncorrelated:

[TABLE]

Obviously, under the standard assumptions for random coefficient models, OLS gives still a consistent but inefficient estimator. In a similar context, Amemiya (1978) proposes to start with estimating the $\beta$’s from

[TABLE]

and afterward the $\gamma$, $\Lambda$ and $\Sigma$ using the $\beta$ estimates. Even with $\Sigma=\sigma^{2}I_{n(T-l)}$ his would mean to estimate $3n(T-l)+n+1$ parameters with only $n(T-l)$ data what can’t work. A feasible way to obtain an estimator is to make use of the linear structure of $E(uu^{\prime})$ and formulate an auxiliary regression in the following way. You first apply OLS to (10) to obtain consistent estimators for $\rho$, $\eta$ and $\gamma$. Then, extract the residuals from the regression and regress the squared residuals on the variables given in the linear structure of $E(uu^{\prime})$ to estimate $\sigma^{2}$ (if $\nu$ homoscedastic; extensions are obvious), and $\Lambda$. This can be done using the quadratic programming following Goldfarb and Idnani (1982, 1983). The reciprocal fitted values of this regression can be used as weights to estimate the coefficients of (10). Then you iterate until the estimated coefficients $\rho$, $\eta$ and $\gamma$ converge. This method has the problem that, when having estimated the residuals of (10) in one step, one has to find the $\lambda$’s such that the matrix $\Lambda$ has the characteristics of a covariance matrix (symmetric and positive definite). Applying OLS on the auxiliary regression does not guarantee this and may result in negative weights. Estimating the Cholesky decomposition of the matrix $\Lambda$ is not possible because of the resulting multicollinearity. Incorporating the symmetry condition is easy, and it is also possible to formulate inequality conditions for the $\lambda$’s such that $\Lambda$ fulfills some of the characteristics of a covariance matrix, e.g. to force the diagonal elements of $\Lambda$ to be positive. However, the resulting optimization procedure that calculates the $\lambda$’s itself has errors and one cannot be sure that the result of iterated least-squares combined with the iterated solution of the optimization procedure in every step converges to the desired result. Therefore, we propose to estimate the coefficients of (10) as follows:

• (1)

We estimate the coefficients of (10) in the first step using OLS,

• (2)

we extract the residuals,

• (3)

we estimate the coefficients of (10) again using least-squares with the reciprocal squared residuals as weights.

Extracting the residuals from this regression (repeat step (2)) gives weights for the next regression (repeat step (3)) and so on. We iterate this procedure, until the sum of squared differences of the coefficients from one step to the next is smaller than $0.005$. This ensures that the average squared difference from one step to another is approximately $0.00005$.

3 Results

3.1 The Effects on Economic Growth

In this subsection, we investigate the effects of poverty, inequality and the middle class on the coefficients of our growth equation. We will consider three year economic growth. Considering too short term economic growth might lead to a spurious regression problem and endogeneity, while we lose observations by considering too long term economic growth. We found three years to be reasonable, but we admit that this choice is arbitrary. More specifically, we estimate a lagged regression equation with $l=3$ and $x_{it}=(lnn_{i(t-3)},lnsk_{i(t-3)},lnattain_{i(t-3)})\in\mathbb{R}^{3}$. Furthermore,

[TABLE]

which implies $M=21$. In this case the time-series covers $31$ years, namely from the year 1973 to the year 2003.

When displaying the estimated coefficients, we report the level of significance of the coefficient ${***}$ if the p-value is almost zero, ${**}$ if the p-value is smaller than $0.01$, and ${*}$ if the p-value is smaller than $0.05$. The estimated autoregressive coefficient is $0.9318^{***}$.

The regression that explains $\beta_{1}$, which is the coefficient of $lnn$ is

[TABLE]

A graphical illustration of this is given in figure (1). The plots show the evolution of $\beta_{1}$ along $pov$ (upper left), $gini$ (upper right) and $middleclass$ (bottom left). For the variables that are hold constant in each plot, we plug in its observed averaged. Differences in inequality and the income earned by the middle class have a much larger impact on $\beta_{1}$ than differences in the poverty rate. The relationship of $gini$ and $middleclass$ to $\beta_{1}$ is similar, namely almost linear and increasing. The poverty rate has an inverted U-shaped relationship. The returns to $lnn$ are theoretically the largest in countries, where inequality is large even though the middle class earns a high fraction of the total income and a serious fraction of the total population (approximately 20%) earns below the poverty line. The boxplots (bottom right) show the estimated coefficients stratified for different country groups. The groups are Asia, Latin, sub-Saharan Africa (SSA), High Income (HI, the High Income OECD and the High Income Non-OECD) and the group of other countries (the Middle East and North Africa and the Eastern Europe). We modified the World Bank grouping appropriately for our data. The returns to $lnn$ are especially large for sub-Saharan African and HI countries. For the sub-Saharan Africa group, this result coincides with Koehler, Sperlich and Vortmeyer (2011). We also show that the $\beta_{1}$-coefficients of sub-Saharan African countries have larger variation than other countries. The coefficients of the groups Latin and Other are smaller on average and have less variation. It is interesting to see that not only for HI and sub-Saharan African but also for other countries the returns to $lnn$ can be positive. The overall distribution of the variable-coefficients seems to be in accordance with the mean-coefficients model, by which the return to $lnn$ is estimated to be $-0.00640$.

The estimated regression equation that explains the coefficient of $lnsk$ is

[TABLE]

which is plotted in figure (2). It can be observed that poverty, inequality and the middle class have a remarkable impact on the return to investment in physical capital. We observe a U-shaped relationship for the variables $pov$ and $gini$, and an inverted U-shaped relationship for $middleclass$. Therefore, the returns to investments are the highest, when poverty and inequality are either quite low or large while the middle class earns about 40% of the total income. The boxplots of figure (2) show that sub-Saharan Africa has the smallest returns to physical capital on average. This is here explained by large inequality, small middle class and large poverty. The coefficients of sub-Saharan African countries are also subject to large variation. The coefficients of Asia are small on average and show that the underlying distribution is skewed. The group Other has smaller variation and larger coefficients on average. The largest returns to physical capital are observed for Latin American countries. These countries are characterized by small poverty rates but extreme inequality und a moderate size of middle class. When considering the mean-coefficients model, the return to $lnsk$ is estimated as $0.04919$, which seems to be coherent with the our estimates.

The estimated regression equation for the coefficient of $lnattain$ is given by

[TABLE]

The boxplots in figure (3) shows that the majority of the estimated coefficients of $lnattain$ are negative. The mean-coefficients model estimates the return to $lnattain$ to be $-0.03239$, which is again negative. This counterintuitive result might be explained by a measurement problem or related to the functional misspecification of the simple linear growth model. Alternatively, Pritchett (1996) provides theories why schooling may not lead to economic growth. For example, the returns to education fall rapidly when the demand for educated labor is stagnant. We observe that $pov$, $gini$ and $middleclass$ have an almost linear and increasing relationship with $\beta_{3}$. High inequality, high poverty and a large share of income to the middle class causes a high return to school attainment. Differences in poverty have a much smaller impact on the coefficient than inequality or the middle class.

The results clearly show that coefficients can differ a lot depending on poverty, inequality and the fraction of income earned by the middle class. By considering varying coefficients, we can also see the different patterns across country groups. Certainly, we expect less bias in the coefficient estimates since we have explicitely modeled different returns of the determinants of growth.

3.2 The Effects on the Economic Growth of the Poor and the Rich

We finally investigate the growth path for the upper and the lower twenty per cent of the society in each country. Differences in the growth path of the poor and the rich naturally affect the income distribution which in turn affect growth. Therefore, this exercise will provide us insights how the income distributions evolve. We use an analogous econometric procedure as in the previous subsection, except that the dependent variables are $y_{rich,it}$ and $y_{poor,it}$ defined in Section 2. We note that these dependent variables are based on GDP’s per capita, having thus a different scaling for the dependent variable as in the previous subsection which was based on GDP per worker. We will avoid any comparison of the results based on different scalings. Instead, we focus on investigating how the income distribution measures affect the pro-poor and the pro-rich growth respectively.

The autoregressive coefficient for the poor is $0.9704^{***}$ and that for the rich is $0.9366^{***}$. This demonstrates that the series of the dependent variable is more persistent for the poor than for the rich. The evolution of the return to $lnn$ of the poorest 20 percent is

[TABLE]

and that of the richest twenty percent is given by

[TABLE]

This is graphically demonstrated in figure (4). The main drivers of the return to $lnn$ are inequality and the share of income of the middle class, while the poverty rate has hardly any influence on $\beta_{1}$. Poverty has an inverted U-shaped relationship with $\beta_{1}$ for the poor and the rich. Inequality and the share earned by the middle class have an almost linear and increasing relationship with $\beta_{1}$ for the rich. Inequality and the share of income of the middle class show an inverted U-shaped and an U-shaped relationship with $\beta_{1}$ respectively for the poor. The depreciation rate has large returns to economic growth of the rich with around 20% population living under the poverty line, large inequality and large income share of the middle class. It is similar for the pro-poor growth, except that very high inequality ($gini$ above 0.6) suppresses the return, and very low income share of the middle class ($middleclass$ below 0.35) increases it.

The evolution of the return to $lnsk$ for the poorest 20 percent of the population is

[TABLE]

and for the richest twenty percent is

[TABLE]

This is graphically demonstrated in figure (5). Poverty has smaller influence on $\beta_{2}$ than inequality and the income share of the middle class. The relationship between $\beta_{2}$ and poverty and between $\beta_{2}$ and inequality are U-shaped, while the relationship between $\beta_{2}$ and the income share of the middle class is inverted U-shaped. For both growth paths, the return to investment in physical capital is high when poverty and inequality are either very low or very high, and when the income share of the middle class is around 45%. It is interesting that when inequality is very high ($gini$ around 0.65), the return to investment in physical capital is saliently higher for the poor compared to the rich. On the other hand, when the income share of the middle class is extremely low ($middleclass$ around 0.25), the return for the poor is saliently lower compared to the rich.

The evolution of the coefficient of $lnattain$ for the poorest 20 percent of the population is

[TABLE]

and for the richest twenty percent is given by

[TABLE]

This is graphically demonstrated in figure (6). Poverty has smaller influence on $\beta_{3}$ than inequality and the income share of the middle class. Analogous to subsection (3.1), we observe that the coefficients are likely to be negative. The relationship between $\beta_{3}$ and poverty is inverted U-shaped, while they are almost linear and increasing for inequality and the income share of the middle class. The return to investment in human capital is large with high inequality and high income share of the middle class and the poverty rate being around 20% (for the poor and rich alike). Investments in human capital are slightly better for the rich in case of an extremely small or extremely large poverty rates.

The aforementioned results show that economic growth for the poor and the rich are differently influenced by the income distribution. These different economic growth paths of the poor and the rich have again implications to the income distribution.

4 Conclusion

Collecting long time-series for a large range of countries allows applying a varying coefficient model in which differences to the mean coefficient are explained by poverty, inequality and the share of income earned by the middle class. The results show that the returns exhibit a serious heterogeneity with respect to these factors. It is clear that neglecting this by estimating a simple linear model leads to inconsistent estimators; but more importantly, the researcher misses some of the maybe most important variation.

There are several reasons for considering varying coefficients. First, adding measures of the income distribution to the set of explanatory variables of the growth regression alone, does not model their impact, and it ignores the fact that the poor behave different than the rich. Furthermore, one might lose a sensible economic interpretation when simply adding additively a lot of variables to the classic growth regression. Often, the mean of the coefficients is not an informative parameter of the growth equation because of the dramatic heterogeneity of returns to production factors. Furthermore, the differences of the coefficients to their means are highly suspicious to move simultaneously with growth, which indicates an endogeneity problem. Finally, as poor countries have weaker databases and are therefore more likely to be excluded from the data, the mean coefficient might suffer from a sample selection bias when simply estimated from fixed coefficient models.

Remarkable results are that sub-Saharan African countries have highly varying and large returns to population growth, and highly varying but small returns to physical capital. Latin American countries experience highly negative returns to population growth but large positive returns to physical capital. All country groups experience negative returns to school attainment, which indicates that the variable does not take important information, such as quality of schooling into account. When expressing the coefficients as functions of poverty, inequality and the share of income earned by the middle class, then we observe that poverty has much smaller effects on the returns than the other two factors. Large inequality usually goes hand in hand with a small share earned by the middle class and vice versa. The fact that this tends to move the coefficients in opposite directions demonstrates the importance of incorporating both variables.

We also investigate the growth path of the poorest and the richest twenty percent of the society respectively. Again we observe that the returns of the growth regression are highly dependent on poverty, inequality and the share earned by the middle class. Interestingly, the returns of the two subgroups of the total population are impacted in different ways. Outstanding results are that in case of extremely high and extremely small inequality, the return to population growth is smaller for the poor than for the rich, whereas in case of an extremely small share earned by the middle class (or an extremely large share earned by the middle class), it is larger. Furthermore, in case of extremely large inequality the return to investment in physical capital is larger for the poor, but in case of an extremely small share earned by the middle class it is smaller. This shows again that small inequality and a large share earned by the middle class (or large inequality and a small share earned by the middle class) tend to force the returns in different directions. The aforementioned differences of the poor and the rich naturally affect the parameters of the income distribution, which in turn affect the growth path of the GDP per worker. This undermines again the importance of considering the income distribution when modeling the growth path.