Trends in Income Inequality: Evidence from Developing and Developed Countries

We use a novel approach to answer questions like: is there a secular trend and shift in inequality across developing and/or developed countries? Do the developing countries converge the inequality level of developed countries? Unlike some recent studies that rely on casual observation of the time series, we conduct recent econometric techniques that allow us to check if there are statistically significant trends and whether these trends are deterministic or stochastic. We also test the inequality gap between developed countries and different developing country groups using convergence test. Using data of 34 countries over the 1960–2020 period, we find that inequality has been either increasing or stable in developed countries; in developing countries, both improvements and deterioration in inequality are observable. We show also that few developing countries reduce the inequality gap with developed countries. These heterogeneous trends suggest that the inequality in the last half-century is driven by national rather than global contexts. Our analysis of the Official Development Assistance and inequality nexus supports this inference.


Introduction
Income inequality has become a growing concern for policymakers in the last few decades. In 2015, the United Nations established the Sustainable Development Goals (SDGs) including "reduce inequality within and among countries" by 2030. A growing strand of literature recently suggests that income inequality has increased around the world since the 1980s. Piketty (2014) and Piketty and Zucman (2014) show that income inequality, especially in developed countries, has increased since the beginning of the 1980s and they attribute this phenomenon to the sustainable increase of the capital-to-income ratio. Furthermore, they expect that the increase in inequality will continue in this century. Ravallion 1 3 (2014) shows that the within inequality for the developing world as a whole has been slowly rising in the 1990s and thereafter declined slightly. Roine and Waldenström (2011) and Alvaredo et al. (2017a;b) suggest that income inequality has increased rapidly since 1980 across most countries, but at different speeds. These findings lead to the following questions: is there a secular trend and shift in income inequality across developing and/ or developed countries in the last six decades? Do the developing countries converge the inequality level of developed countries?
The aim of this paper is to provide statistical answers of these questions. Notably, the literature extensively discusses the trend of inequality across developed countries; however, there is a lack of statistical evidence. Additionally, the behaviour of inequality across developing countries, and particularly relative to developed countries, remains underexplored. This is important to assess any inequality gap between these two groups of countries.
Our first contribution is to estimate the trend of income inequality, in order to evaluate inequality within countries. To the best of our knowledge, this is the first study to estimate the inequality trend directly. Most analyses of the inequality trend rely on casual observation of inequality over time, or they compare inequality averages and distributions across time intervals (see, inter alia, Sala-i-Martin, 2006;Piketty & Zucman, 2014;Ravallion, 2014;Alvaredo et al., 2017aAlvaredo et al., , 2017b. This is surprising because estimating the inequality slope is essential to check if there is a statistically significant trend and whether this trend is deterministic or stochastic, as the effect of shocks on income inequality will have permanent effects with the former, but only temporary effects with the latter. The behaviour of inequality becomes less homogenous after the 1970s thereby more difficult to observe by a cursory look at data (Roine & Waldenström, 2011). Additionally, it is important to test whether there is a break in inequality trend in 1980s, as suggested by the literature (e.g., Piketty, 2014). 1 Furthermore, understanding the nature, the trend and other time series characteristics of the underlying series, should improve our ability to forecast its movements (see Arezki et al., 2014b;Harvey et al., 2017).
According to the SDGs, income inequality across countries may have been reduced due to the growth of Official Development Assistance (ODA) and financial flows to the least developed countries in the last few years. Sala-i-Martin (2006) also suggests that inequality across countries has decreased because of the rapid growth rate of some of the largest, yet poorest, countries such as China, India, and several other countries across Asia. Our second contribution is, therefore, to test the inequality gap between developing and developed countries using two approaches. Particularly, we assess the trend of income inequality of developing countries relative to developed countries. This allows us to test for changes in inequality across countries. We also examine the convergence of inequality using the log t regression test and the club clustering algorithm, which allows us to find countries that share the same transformational route to achieve the steady state without pre classification or grouping of countries.
To estimate the trend in inequality, we use the Gini index, which is the most widely used measure of inequality in the literature. We base our selection of countries on being able 1 Roine and Waldenström (2011) examine if there are common breaks in income inequality across 18 countries, most of them are developed countries, over . Thus, they use methods allowing for multiple endogenous breaks. However, we use the unit root test allowing for an exogenous break to test the proposed break in the 1980s. Additionally, we use also unit root tests allowing for a single endogenous break as our main aim is to estimate the trend per se rather than the breaks in trend thus we need enough observations before and after the break. Kellard and Wohar (2006) and Ghoshray (2011) use unit root tests allowing for up to two structural breaks to estimate the long-run trend of commodity prices in the last century. Thus, we allow for a single break because we use data for the last half century.
to obtain data for the last six decades. This was determined by the desire to have a mix of developed and developing countries in our sample, but the necessity to have a dataset that is long enough to allow for the testing of structural breaks. This gives us a sample of 34 developed and developing countries 2 over the period 1960-2020. 3 The developing countries in our sample include, mainly, Asian and Latin American countries. This is helpful for our analysis, as these two regions experienced different changes in inequality in recent decades. Sala-i-Martin (2006), for example, shows that inequality decreased in Asia and increased in Latin America over the period 1970-2000. Figure 1 shows the Gini coefficient (in logs) for the developed and developing countries in our sample. The Gini coefficient for some countries, such as the United States and Hong Kong, trends upward. For other countries, such as Brazil, Iran, Mexico and Venezuela, it is trending downward. In some cases, such as the United Kingdom, Hungary, Sierra Leone and Argentina, we can see a break in trend. Therefore, the figure provides initial evidence suggesting different patterns of inequality across countries.
To analyse these inequality patterns statistically, we use time series analysis rather than panel analysis, in order to understand the trend in inequality for each country. We employ unit root tests allowing for breaks to check whether the Gini index is difference stationary (DS) or trend stationary (TS) with a structural break. 4 If the Gini index exhibits unit root behaviour, then it is said to contain a single stochastic trend and thus we estimate this trend using a DS model. If, however, the underlying Gini index is found to be stationary with a structural break, then the index is considered to be trend stationary and we estimate the two trends, before and after the break, using a TS model.
The results show that inequality is well represented by a single stochastic trend for 21 developed and developing countries, as we cannot reject a unit root after allowing for endogenous breaks in the slope and intercept. The estimated trend for these countries suggests that inequality exhibits positive or trendless behaviour for the developed countries amongst the 21, but a mix of positive, negative and trendless behaviour for the developing countries. For the remaining 13 countries, three developed and 10 developing, we find 2 Our sample includes 19 developing countries (Argentina, Bangladesh, Brazil, Chile, Colombia, Costa Rica, Hong Kong, Indonesia, Iran, Madagascar, Mexico, Malaysia, Pakistan, Panama, Philippines, Sierra Leone, Thailand, Taiwan and Venezuela) and 15 developed countries (Australia, Canada, Germany, Finland, France Hungary, Italy, Japan, Korea, Norway, Puerto Rico, Portugal, Sweden, the United Kingdom and the United States). 3 Islam and Madsen (2015) backdate the Gini coefficients using labour's income share, which is not available for most developing countries. Additionally, our aim is to compare the behaviour of inequality in the last six decades across developing and developed countries. Therefore, we use Gini index from the Standardized World Income Inequality Database (SWIID) as this dataset improves greatly the comparability of inequality across countries by standardizing the consumption and wage income (Delis et al., 2013;Solt, 2016). The data are available from 1960. 4 Recently, some empirical studies have examined the persistence of inequality, with mixed findings. Islam and Madsen (2015), for instance, test the persistence of Gini coefficients and the top 10% income shares for a sample of 21 OECD countries over the period 1870-2011 using the Carrion-i- Silvestre et al. (2005) panel stationarity test, which allows for multiple structural breaks. They find that the increasing inequality after 1980 is driven by a deterministic trend, as suggested by Piketty (2014). Christopoulos and McAdam (2017) provide opposite evidence. Using the Gini index, they test the persistence of inequality for a panel of 47 OECD and non-OECD countries from 1975 to 2012. To do so, they introduce a new panel unit root test to address unknown structural breaks. Their results suggest that inequality measures contain a unit root. They conclude that inequality measures are exceptionally persistent, if not strictly a unit root, which implies that shocks to income inequality have permanent, or, at least, very long-lasting, effects. Sanso-Navarro and Vera-Cabello (2020) find that Gini coefficient has alternated between stationary and nonstationary regimes over the period 1960-2017. that the trend is represented by two slopes, as they experienced a break in inequality trend around the late 1980s and early 1990s. Unlike some recent work that suggests the inequality tends to increase after the secular break in the 1980s, the observed heterogonous breaks in some countries seem to be caused by domestic economic and political changes. After the break, inequality has decreased for some of them, but increased or remained more or   less constant for others. To clarify the picture of the trend over the whole sample period, we calculate measures of the prevalence of a trend i.e. the trend lasts for more than 50% of the sample period. We find that income inequality has been either increasing or stable in developed countries, whilst the evolution of inequality is more heterogeneous for developing countries, as different countries have experienced one or other of higher, lower or unchanged inequality. These findings indicate that the inequality gap varies between developing and developed countries. The results of the relative Gini coefficients of developing countries confirm this inference. Furthermore, we find that developing countries form two clubs. The first club includes mainly Latin American and Caribbean countries whilst the second includes Asian countries, alongside the European countries.
Overall, there is no consistent secular pattern in inequality, especially among developing countries, at least in the last six decades. These heterogeneous patterns suggest that income inequality is shaped by domestic economic and political contexts thereby domestic policies such as taxation are more important than international policies and foreign aid, such as ODA, in reducing inequality (see Sumner, 2012;Alvaredo et al., 2017b). We analyse the nexus between ODA and inequality at the end and find no evidence of the role of ODA reducing the inequality gap between developing countries and developed countries.

Literature Review
A growing strand of literature discusses the changes in income inequality in the last few decades. For example, Atkinson (2003) studies a sample of nine OECD countries over the period 1945-2001 and finds that inequality increased in most countries. However, the magnitude of this increase varies across countries. Using data of China, France, the United Kingdom and the United States over the period 1978-2014, Alvaredo et al. (2017a) also show that inequality, measured by top income and wealth shares, increased in most countries in recent decades. They also find that the magnitude of this change differs across countries. This heterogeneity illustrates the role of national policies such as tax and social transfer policy in shaping inequality (Alvaredo et al., 2017a, b).
This heterogeneity across countries enhances some studies to explore some common patterns and features of inequality across subsamples. Piketty and Saez (2006) and Atkinson et al. (2007), for instance, provide evidence about homogenous negative trend in developed countries until the 1970s then the inequality behaviour becomes less homogenous. Particularly, they observe a noticeable difference in inequality behaviour between Anglo-Saxon countries and other developed countries such as continental European countries and Japan. Other studies distinguish between Anglo-Saxon, continental European and Nordic countries. For example, Atkinson et al. (2011) and Roine and Waldenström (2011) find difference in inequality trend between Anglo-Saxon group and continental European group. They show also that Nordic countries follow the pattern of Anglo-Saxon countries. In contrast, Morelli et al. (2015) reveal that English-speaking countries have very fast growth in income inequality whilst Nordic countries experience no or modest increases in top income shares in recent decades. Overall, these studies provide evidence of "great U-turn" of income inequality i.e. they show that most developed countries have downwrd trend in inequality during the first three quarters of the twentieth century, however inequality increases at different speeds across countries since the 1970s. Addtionally, there seems to be a consensus among economists that the ineqaulity of Anglo-Saxon countries grows faster than the remaning developed countries.
Remarkably, the literature focuses mainly on developed countries 5 because of the rapid growth of inequality in these countries since the 1970s and the data limitation of developing countries. Therefore, the picture about the income inequality in developing world is still ambiguous comparing with developed countries (Morelli et al., 2015). The scarce evidence from developing countries also shows the heterogeneity of inequality pattern across them. For example, Atkinson et al. (2011) use data of five developing countries, China, India, Singapore, Indonesia and Argentina, and find substantial heterogeneity across these countries as their inequality display a U or L shape pattern. Roine and Waldenström (2011) study four groups and one of them is Asian countries which includes Japan and two developing countries, India and Singapore. They find that inequality falls in India until 1980 and increases after that. Singapore as well experienced a U-shape pattern, however the break happened after the Asian crisis in 1997. Morelli et al. (2015) include three emerging countries, China, India, and South Africa, in their analysis and find that the inequality trends of these countries seem close to those in the English-speaking countries. The trend changes in the early 1980s in India and in the late 1980s in China and South Africa and these three countries experience increase in inequality after the 1990s. Using the averages of six developing regions, Ravallion (2014) also provides evidence about the mixed patterns of income inequality in developing world during the period 1980-2010. The study shows that the inequality has downward trend since 1990s in three regions, Latin America and the Caribbean, Eastern Europe and Central Asia and the Middle East and North Africa. On the other hand, the inequality has upward trend in both South Asia and East Asia and no clear trend in Sub-Saharan Africa. For the developing world as a whole, the paper reports that the inequality has been slowly growing until the 1990s and thereafter declined slightly.
In this study, we focus on developing countries in our analysis. We study the pattern of income inequality of developing countries and compare it with developed countries. To do so, we first estimate the trend of developing and developed countries using either TS model for stationary series or DS model for non-stationary series. This is important as estimating the trend using TS model for non-stationary series can lead bias results i.e. we may conclude that the trend is significant when it is actually not (see Kim et al., 2003). Additionally, using a DS instead of a TS model to estimate the trend is inefficient if inequality is generated by a stationary process because DS model has low power comparing with TS model (see Ghoshray, 2011). Furthermore, we use two approaches to test the convergence of income inequality. The first approach allows us to check if developing countries converge the inequality level of developed countries, whilst the second is useful to find the convergence clubs without pre classification or grouping of countries. This is important as the literature finds heterogenous patterns of inequality across countries which illustrates the difficulty to group them according to inequality behaviour. 6

Data
Income inequality is captured by the Gini index, which is the most widely used measure of inequality in the empirical literature (e.g., Beck et al., 2007;Christopoulos & McAdam, 2017;Delis et al., 2013;Dollar & Kraay, 2002;Islam & Madsen, 2015). The data are obtained from the Standardized World Income Inequality Database (SWIID) (Solt, 2016). SWIID provides the most comprehensive database on the Gini index and it is currently the best suited data set to perform a cross-national study on income inequality, as it standardises consumption and wage income (see Solt, 2016;Christopoulos & McAdam, 2017). Therefore, this data set serves our purpose, comparing the inequality behaviour across countries, well because it improves greatly the comparability of the inequality among countries (Delis et al., 2013). The Gini coefficient is derived from the Lorenz curve and ranges between 0 (perfect equality) and 100 (perfect inequality). In the SWIID data set, data on the Gini index are available for 192 countries over the period 1960-2020/21. We use the post-tax Gini index to control for the impact of fiscal policy on inequality.
To estimate the trend whilst allowing for a break in trend, we need a long period, so we select countries with at least 50 observations and no in-sample Not Available (NAs). This gives us an unbalanced data set for 34 countries, developing 19 countries and 15 developed countries, from 1960 to 2020.

Methodology
In this section, we discuss DS and TS models which we use to assess the inequality within countries. Then, we present β-convergence approach that we apply to evaluate the inequality among countries.

Estimating Income Inequality Trend
To analyse the trend of inequality, we use the methodology commonly employed to test for the trend of primary commodity prices relative to manufacturing goods (i.e. the Prebisch-Singer hypothesis; see for example Kellard & Wohar, 2006;Ghoshray, 2011;Arezki et al., 2014a). The first step is to consider the underlying nature of the Gini coefficient. It could be trend stationary (TS) or difference stationary (DS). 7 If the underlying Gini 7 Some studies highlight that conventional unit root tests are potentially unreliable for bounded variables such as the Gini index, with values between 0 and 100 (Cavaliere & Xu, 2014). However, the adjustment path of the Gini index may be protracted, even if it is ultimately mean reverting (Christopoulos & McAdam, 2017). Furthermore, many bounded series such as nominal interest rates, cannot be strongly negative, and unemployment rate, roughly a percentage, is often treated as possessing a unit root. We follow other studies such as Islam and Madsen (2015) by using the logarithm of the Gini index; however, to check for robustness, we have followed the suggestion of Wallis (1987) and also conducted all of the analyses using a logis-1 3 coefficient series are found to be trend stationary, we test the trend by estimating the following log-linear time trend model: (1) Gini t = + t + t

= ln
Gini t 1−Gini t . Results from the logistic transformation are quantitatively very similar and qualitatively identical to the results obtained from the logarithm series. These results are available by request.

Footnote 7 (Continued)
where Gini t is the logarithm of Gini coefficient, t is a linear trend and the random variable t is stationary with mean zero. The focal point of interest is the coefficient which represents the growth rate of inequality. If > 0 then it indicates upward slope of Gini coefficient, i.e. the inequality increases over time, otherwise, for < 0, we conclude that inequality has downward slope; thereby it tends to decrease over time. If Gini coefficient exhibits a unit root behaviour, then we adopt the model as estimating the trend stationary model given by Eq.
(1) will generate misleading results about trend, i.e. we may conclude that the trend is significant when it is actually not. 8 More specifically, we cope with this issue by estimating the stochastic trend using the following difference stationary model: where t is a stationary and invertible error process. As aforementioned, if is positive (negative) and statistically significant then it indicates upward (downward) slope of Gini coefficient. Importantly, if Gini coefficient is a trend stationary process but is treated as a difference stationary process, then estimating the trend using Eq. (2) is inefficient, lacking power relative to those estimated from Eq. (1) (see Ghoshray, 2011). Finally, the standard errors of both models are heteroskedasticity and autocorrelation robust by using Newey-West standard errors. Perron (1989) shows that if a structural break is ignored, the power of the unit root test is lowered; thus he suggests a unit root test allowing for a structural break. His paper, however, was criticised for the fact that he assumed that the date of the structural break is known, i.e. exogenous. Stock and Watson (1988a, b) and Christiano (1992) criticise this test as an exogenously chosen break date may lead to false inferences. Consequently, some researchers, such as Zivot and Andrews (1992) and Perron and Vogelsang (1992a, b), have developed a unit root test that allows for an unknown break to be determined endogenously from the data. However, these tests have the limitation that the critical values are derived while assuming no break under the null hypothesis. Nunes et al. (1997) illustrate that this assumption leads to size distortions in the presence of a unit root with structural breaks. As a result, this test may tend to suggest evidence of stationarity with a break (Lee & Strazicich, 2003). Lee and Strazicich (2013) propose a one break minimum Lagrange Multiplier (LM) unit root test with alternative hypothesis unambiguously implies the series is trend stationary; thus it is unaffected by a break under the null. Additionally, they illustrate that this test is free of size distortions and spurious rejections in the presence of a unit root with a break, as it employs a different detrending method (see also Lee et al., 2006) and tends to estimate the break point correctly. 9 Therefore, we employ the Lee and Strazicich (2013) test that allows for a single structural break. 10 To briefly describe the Lee and Strazicich (2013) method, consider the following data generating process (DGP): where Gini t is Gini index (in log) and X t denotes the changes in level and trend as follow and TB refers to breakpoint. As mentioned before, this test allows for a structural break under both the null ( H 0 ∶ ∅ = 1 ) and alternative ( H A ∶ ∅ < 1 ) hypotheses. Note that the critical values depend on the break fraction, λ = TB∕T where T is the total number of observations.
The statistic of LM test can be estimated using following regression: where , t = 2, 3, … , T ; ∅ are coefficients on the regression of ΔGini t on ΔX t ; is given by Gini 1 − X 1 ∅ . Gini 1 and X 1 are the first observations of the Gini t and X t sequences respectively. The lagged terms, are added to correct for serial correlation. The appropriate lag length, p, is selected using the general to specific method (GTOS). The LM test statistics are given by the τ statistic testing the null hypothesis H 0 ∶ ( = 1 ). The LM unit root test determines the break points endogenously by utilising a grid search. To eliminate endpoints trimming of the infimum ( inf ) is made at 10%. The test determines the breakpoints where the test statistic is minimised. The LM test is given as LM τ = inf̂ (λ) where λ is the break fraction as mention above.
After determining whether the Gini coefficient is trend stationary (TS) with a structural break or difference stationary (DS), we estimate the deterministic or stochastic trend. For TS Gini coefficients, we test the shift of inequality slope. To do so, we follow Arezki et al. (2104a) by considering piecewise regressions. Specifically, we estimate Eq. (1) before, regime 1, and after, regime 2, the breakpoint and we test if the difference between two regimes is statistically significant. For DS Gini coefficients, we estimate Eq. (2) for the whole period.

Testing the Convergence
According to Table 1, income inequality is higher in developing countries comparing with developed countries. Thus, the next step in our analysis is to test the trend of Gini coefficients of the 19 developing countries relative to the unweighted averaged Gini coefficient of the developed countries. To do so, we test the stationarity of the relative Gini coefficients of developing countries using the aforementioned unit root tests i.e. unit root tests without and with one break. If we reject the null hypothesis of unit root then country may converge the inequality level of developed countries. 11 However, this approach requires to group the samples in advance based on prior information or assumption e.g. this study tests the inequality behaviour of developing countries relative to developed countries. Additionally, it depends on the stationarity properties of the series. Therefore, we also use the log t regression and the club clustering algorithm suggested by Phillips and Sul (2007) to test the convergence hypothesis. Phillips and Sul methodology is useful not only to test the convergence for the whole sample but also to cluster countries into clubs with similar transition paths. This methodology avoids ex-ante sample separation by using data-driven algorithm for identifying convergence clusters. In addition, this methodology is robust to the stationarity properties of the series. According to this methodology, our inequality index, Gini it , can be decomposed to systematic elements ( h it ) and transitory elements ( g it ) based on the nonlinear time-varying factor model as follows: To separate common components from idiosyncratic components, Eq. (6) can be restructured as: where ∅ it is time-varying idiosyncratic element, which is used to measure the distance between Gini it and u t . u t represents a single element that is common to all countries, which reflects some determined or random trend behaviours.
If the inequality of all countries tends to the same value in the long run, then Gini it needs to meet the following convergence condition: this condition is equivalent to lim t→∞ ∅ it = ∅ for any i.
Assume that the idiosyncratic element, ∅ it , can be written as follows: where L(t)t denotes a slowly varying function and is a decay rate. We set L(t) = log(t) as the results of Monte Carlo simulation of Phillips and Sul (2007) show that L(t) = log(t) can produces the least size distortion and the best test power.
According to Eq. (9), we can test the convergence of Gini it by the following null hypothesis: If the null hypothesis is valid, then it indicates that the income inequality of all countries in our sample is converged; otherwise it indicates that the inequality in the part of our sample is not converged.
To estimate ∅ , Phillips and Sul (2007) removed the common elements in Eq. (7) as follow: where h it is called the relative transition parameter as it the transition path of country i relative to the panel average at time t. The cross-sectional mean of h it is 1, and the crosssectional variance h it meets the following condition: The log t regression model is used to implement the hypothesis test: . r is the initiating sample fraction. 12 We reject the null hypothesis of convergence at 5% level of significance if the value of t-statistic is below − 1.65 which indicates the divergence of the whole sample. In this analysis, we use the Hodrick and Prescott (1997) filter, with smoothing parameter of 400 (see Correia et al., 1992 andCooley &Ohanian, 1991), to extract the trend component and to remove the effects of business cycle. The rejection of convergence hypothesis does not rule out the possibility of the convergence as the convergence may still exist in the subgroups i.e. clubs. Phillips and Sul (2007) developed a data-driven algorithm to explore the existence of club convergence. The procedure includes the following five steps:

Step 1: Sorting
Sorting the countries decreasingly according to the value of their inequality index in the last period.

6
Step 2: The formation of core group.
Find the first k countries such that the t-test statistic of the log t regression t k is larger than -1.65 for the subgroup with individuals {k, k + 1} . If there is no k that satisfies this condition, then quit the algorithm i.e. there is no convergence subgroup exists. Otherwise, if there is k that meets the condition, t k > −1.65 , then continue to perform the log t-test for the group with until that this condition fails. Then select j * that yields the largest t-test statistic, then countries {k, k + 1, … , k + j * }, form a core group.

Step 3 Sieving individuals
Create a complementary group, G c j * , which includes all remining countries i.e. those are not included in the core group. Add one country from this group to the core group at each time and perform the log t-test. If the t-test statistic t b >c * = 0 , then include this country to the club candidate group. Then, perform the log t-test for this club candidate group using the abovementioned steps, if the t-test statistic � t b > −1.65 , then we can obtain the initial convergence club; if not, increase the critical value c * and repeat the above step until � t b > −1.65.

Step 4 Recursion and stopping rule
Create a subgroup for the remining countries which do not belong to the first convergence club. Perform the log t-test for this subgroup. If the t statistic � t b > −1.65 , then this subgroup forms another convergence club; otherwise, repeat the steps 1-3 for this subgroup.
Steps 1-4 can obtain the initial clubs. To avoid splitting a convergent group into several small convergent subgroups, Schnurbus et al. (2017) suggest that the log t-test is performed for all pairs of the subsequent initial clubs, if the corresponding t statistic satisfies the convergence hypothesis, i.e. is greater than − 1.65, merge them to form the new clubs. Repeat this procedure until no clubs can be merged. This step leads to final convergence clubs.

Results and Discussion
The results of the unit root testing procedures without break are presented in Table 2. Tables 3 and 4 report the results of the unit root tests with exogenous and endogenous breaks, respectively. Tables 5 and 6 display the results of estimating the trend using TS and DS models, respectively. Table 7 shows the prevalent trend over the sample period. The unit root results of the relative Gini coefficients of developing countries are presented in Tables 8 and 9. Tables 10 and 11 display the results of the log t regression and the club clustering algorithm. Table 12 shows the results of β-convergence test. Finally, we discuss the relationship between ODA and inequality in developing countries in the last half century, Table 13.

Conventional Tests with No Breaks
As a prelude to conduct unit root tests allowing for a structural break, we employ three conventional no-break unit root tests. Table 2 presents our results. The three tests indicate that for all countries the unit root hypothesis cannot be rejected at 5% significance level. These preliminary results support the findings of Christopoulos and McAdam (2017) that the inequality measure has a unit root and thus the shocks to inequality are permanent. However, we should consider that the non-rejection of the null hypothesis of a unit root could be misleading because these tests ignore the possibility of a structural break in either the intercept or trend. We address this by employing unit root tests allowing for single break in trend and intercept.

Exogenous Break Unit Root Test
While we are concerned primarily with testing the unit root allowing for an endogenously determined break, it is useful first to test the proposed break by Piketty (2014) and other studies, as coming in the 1980s. Therefore, we use the unit root test allowing for an exogenous break suggested by Perron (1989) and we select 1980, 1985 and 1990 as breakpoints in intercept and slope. We select these three years as the literature, such as Islam and Madsen (2015) Piketty (2014) and Roine and Waldenström (2011), suggests that there is a shift in inequality in the 1980s because the end of the post WWII equalization. Our results are  Table 3. We find evidence of a shift in inequality trends in Costa Rica, Finland, Italy and Sierra Leone in 1980, in Brazil, Chile and Korea in 1985and in Argentina, Indonesia, Korea, Pakistan and Venezuela in 1990. Overall, we find only limited evidence for the proposed breaks in the 1980s. However, we assume with this test that the breakpoint is the same across countries and it is known, which could lead to false inferences, as discussed above. Thus, the next step is to test for a unit root allowing for endogenous break in slope and intercept.

Endogenous Breakpoint Unit Root Tests
We employ three unit root tests, developed by Zivot and Andrews (1992), Perron andVogelsang (1992a, 1992b) and Lee and Strazicich (2013), which allow for one endogenous break in both intercept and slope. These three tests may show different break dates as the dates are determined endogenously i.e. they are estimated rather than selected a priori (see Ghoshray & Johnson, 2010;Maslyuk & Smyth, 2008). Therefore, we follow other studies such as Ghoshray and Johnson (2010) by using the test proposed by Lee and Strazicich as benchmark because it allows for a break under the null. Additionally, this test employs a different detrending method and tends to estimate the break point correctly. Importantly, we should be careful not to interpret these tests as tests for structural breaks (see Kellard & Wohar, 2006;Ghoshray, 2011). Table 4 reports the results from applying these three tests on the logarithm of Gini index of the 34 countries. According to the Zivot-Andrews and Perron-Vogelsang tests we are unable, for one of these two tests at least, to reject the null of the unit root at any conventional levels of statistical significance for 24 countries. The benchmark test, Lee and Strazicich, shows that we are unable to reject the null of the unit root at any conventional levels of statistical significance for 17 countries. For the remaining 17 countries, the null hypothesis of a unit root is rejected, in favour of the alternative that the process is trend stationary with a single structural break. However, the null hypothesis Table 9 Unit root tests with a single structural break of developing countries relative to developed countries *** , ** and * denote significance at the 1%, 5% and 10% levels, respectively. The numbers within parentheses denote the lag length. TB denotes the break date. Critical value at 1%, 5% and 10% for Zivot-Andrews   of a unit root is rejected at the 5% significance level only 13 countries. For all countries where the null is rejected using the Zivot-Andrews and/or Perron-Vogelsang tests, we also reject the null using the Lee-Strazicich test, except for Canada and Puerto Rico. As noted, the breakpoints could vary across tests, however the significant breakpoints seem consistent across the three tests. Although the exact timing of the breakpoints differs across countries, most of them occur between the late 1980s and early 1990s as suggested by, for example, Sala-i-Martin (2006) and Piketty (2014). Piketty (2014) suggests that since the beginning of the 1980s, developed countries experienced an increase in inequality-and that this will continue in this century. However, our findings show that only three out of 15 developed countries experienced a shift in inequality trend at the 5% significance level. Moreover, 10 out of 19 developing countries also experienced a shift in inequality trend. Clearly, given that the null hypothesis of unit root cannot be rejected using conventional no-break unit root tests, allowing for the possibility of one structural break under the alternative hypothesis greatly affects the conclusions to be drawn from the unit root tests.
To summarise, we cannot reject the unit root null for 21 countries at the 5% significance level. Therefore, the trend of these countries is represented by a single trend, which we shall estimate using a DS model. For the remaining 13 countries, the null hypothesis of a unit root is rejected in favour of the alternative, thus the process is trend stationary with a single structural break. This implies that inequality has two trends, before and after the break date, which we shall estimate using a TS model. The results suggest also that shocks to income inequality for these 13 countries have only transitory effects, but permanent effects for the other 21 countries.

Estimating the Inequality Trend
After determining which series exhibit trend stationary behaviour, allowing for a shift in intercept and trend, we can estimate the trend with an appropriate model. 13 countries are found to exhibit trend stationary behaviour with a single structural break, so we consider piecewise regressions to estimate and compare the slopes before and after the break. 13 To do so, we fit a linear trend model using Eq. (1) before, regime 1, and after, regime 2, the break date. The results are summarised in Table 5. ̂ 1 and ̂ 2 represent the estimated slope for regime 1 and regime 2, respectively. The values in brackets are the p-values for the corresponding parameters. We compare ̂ 1 and ̂ 2 using Wald test to test if the difference between the two slopes is statistically significant, see the last column in Table 5. For the remaining 21 countries that we cannot reject unit root after allowing for a structural break, we estimate the trend using Eq.
(2). The results are presented in Table 6. Table 5 shows that almost all slopes before the break, regime 1, are statistically significant and most of them are positive. Particularly, ̂ 1 is positive for all developed countries except Hungary and positive for eight out of 10 developing countries. The second column in Table 5 reports the slope after the break date, regime 2. Most slopes are statistically significant. ̂ 2 is positive and statistically significant for Hungary, negative for Portugal and insignificant for Korea. This slop is positive and statistically significant for only two out of 10 developing countries. The final step is to test the difference between slopes before and after the break. The last column in Table 5 shows that all TS countries in our sample, except Bangladesh and Korea, experienced a statistically significant change in the slope of the inequality trend. The shift in the inequality slope indicates that inequality tends to increase in two countries after the break, because the slope either switches from negative to positive (Hungary) or becomes steeper upward after the break (Indonesia). In contrast, inequality tends to decrease after the break in eight countries, as the slope switches from positive to negative (Argentina, Brazil, Chile, Colombia, Panama, Portugal and Thailand), becomes steeper downward (Venezuela) or flatter upward (Pakistan).
We find that the difference between the two slopes is positive and statistically significant in both Hungary and Indonesia. This means that the inequality tends to increase after the break. Therefore, we attempt to identify the potential drivers of those breaks through matching them to historical events. The breakpoint of Hungary represents the collapse of the communist system. Indonesia experienced a drastic increase in inequality after the Asian Financial Crisis in the late 1990s. Ravallion and Lokshin (2007), for example, studied the lasting effect of this crisis on poverty in Indonesia and found that the 1998 crisis can explain a large share, possibly half, of the poverty count in 2002 (see also Sala-i-Martin, 2006). 14 This confirms that shocks to income inequality for these TS countries are long-lasting/permanent. Overall, only few countries with exceptional circumstances have experienced an increase in income disparity trend post 1980s and 1990s. Some countries such as Argentina, Brazil, Chile, Colombia, Panama, Portugal and Venezuela, on the other hand, exhibit remarkable success in reducing inequality after the break. Table 6 presents the results of DS models for the remaining 21 countries, 12 developed and nine developing. We cannot reject the unit root hypothesis after allowing for an endogenous break in slope and intercept for these 21 countries. We find a statistically significant slope only for 10 countries, five positive and five negative. The results indicate that seven developing countries out of nine have a significant trend, two upward, Hong Kong and Madagascar, and five downward, Iran, Malaysia, Mexico, Philippines and Sierra Leone. The trend for developed countries is either significant positive, Japan, Puerto Rico and the United States, or insignificant, Australia, Canada, Finland, France, Germany, Italy, Norway, Sweden and the United Kingdom. These results support those for TS Gini coefficients presented in Table 5 that there is no secular pattern of inequality in last 50 years.
The results of the DS model of these 21 countries, along with the results of TS models of the remaining 13 countries, provide mixed evidence about within-country inequality. Inequality has decreased in most Latin American and Caribbean countries. The TS model shows that the difference between inequality trends before and after the break is negative and significant for Argentina, Brazil, Chile, Colombia, Panama and Venezuela (see Table 5), whilst the DS model reports that the trend is negative for Mexico, insignificant for Costa Rica and positive for Puerto Rico (see Table 6). The results of Asian countries seem more heterogenous. Half of Asian countries show similar success, Iran, Malaysia, Pakistan, Philippines, Sierra Leone and Thailand. The trend of Indonesia shifts toward higher inequality after the break (see Table 5), whilst Bangladesh, Hong Kong and Japan have a positive significant trend over the whole period (see Tables 5 and 6). The trends of the remining Asian countries, Korea and Taiwan, suggest that inequality has remained more or less constant. 15 These findings differ from the results of Sala-i-Martin (2006), which show that inequality decreased in Asia, with varying results for Latin America. The results of remining developed countries are mixed. Inequality does not show a significant slope in Australia, Canada, Finland, France, Germany, Italy, Norway, Sweden and the United Kingdom and it increases in Hungary and United States and decreases only in Portugal after the break. Finally, this sample includes only two Sub-Saharan African countries, Madagascar and Sierra Leone. Table 6 displays a positive trend for the former and negative for the latter.
To clarify the picture of the trend over the whole period, we calculate the measure of the prevalence of a trend for each country over the whole sample period. This is important because some countries have different slopes before and after the break. To do so, we follow Kellard and Wohar (2006) and Ghoshray (2011) by constructing relative measures of a prevalence of a trend. More specially, we measure the prevalence of a negative trend for each country as follows: where (−) represents the ratio of the number of years that a statistically significant negative trend exists, λ( −), to the total number of years, N. In the same way we measure the prevalence of positive, (+) , and trendless, (.) , behaviour. For example, the breakpoint for Argentina is 1992 thus we estimate regime 1 for the period 1961-1992 (32 years) and regime 2 for the period 1993-2019 (27 years). We find a statistically significant positive slope for regime 1, therefore λ( +) = 32 thereby (+)=32/59 = 0.54 and a statistically significant negative slope for regime 2, thus λ(-) = 27 thereby (−)=27/59 = 0.46. Table 7 shows the results for all 34 countries. If we define the prevalent trend as the trend lasting for more than 50% of the sample period, then the results show that the prevalent trend is negative for nine countries, positive for 13 and trendless for 12 countries. Furthermore, we can see that the prevalent trend varies across developing countries, as eight countries display a negative trend, eight countries have a positive trend and the remaining (−) = (−) N three countries have insignificant trend. The prevalent trend for developed countries is more consistent, with either positive or insignificant trends for more than 50% of the sample period. Finally, most developing countries experienced high inequality compared with developed countries (see the summary statistics presented in Table 1). However, our findings also show that, within developing countries, the inequality picture is mixed. In particular, inequality tends to decrease in some countries like Brazil, Iran, Mexico, Colombia and Venezuela while it tends to increase or remain more or less constant in other countries, such as Hong Kong, Madagascar, Indonesia and Taiwan. These mixed trends indicate that only few developing countries may reduce the inequality gap with developed countries. We further investigate this point in next section.

The Inequality Trend for Developing Countries Relative to Developed Countries
We test the trend of Gini coefficients of the 19 developing countries relative to the unweighted averaged Gini coefficient of the developed countries. The developed countries have lower inequality compared with developing countries (see Table 1). Thus, testing the stationarity of the relative Gini coefficients of developing countries demonstrates the change in inequality gap with developed countries. More specifically, the rejection of null hypothesis of a unit root of the relative Gini coefficients of developing countries suggests that these countries may converge to the inequality level of developed countries. Tables 8 and 9 present the unit root test results without and with a structural break, respectively, for the relative Gini coefficients of the 19 developing countries. Table 8 reports that relative Gini coefficients contain unit roots; we cannot reject null hypothesis for one country, Bangladesh, but only with the ADF test. The results of the Lee-Strazicich test presented in Table 9 show that we reject the null hypothesis of a unit root, in favour of a trend stationary alternative, with structural break in slope and intercept, for four countries.
The overall findings of the 19 developing countries provide mixed evidence about the convergence of inequality between developing and developed countries. However, we should interpret these results with caution as we allow only for one break due to data limitation. Using unit root with more than one break may lead to rejection of null hypothesis of unit root for more countries, therefore more evidence to support the convergence. 16 In addition, this approach needs to group the samples in advance based on prior information or assumption e.g. the quality of institutions, geographical locations or development level (like our case), which could lead to fail to identify the potential club convergence. In the next step, we address these two issues by using the log t regression and the club clustering algorithm suggested by Phillips and Sul (2007) to test the convergence hypothesis.

Robustness Check
In this section, we use Phillips and Sul (2007) methodology to test convergence hypothesis. Using this approach allows us to extend our sample by including more countries with less observations as there is no need to apply unit root tests or estimate the trend of different regimes, which requires sufficient observations. However, we add only countries with relatively similar number of observations as the approach needs balance sample. Thus, we include countries which have more than 45 observations in this exercise. This increases our sample from 34 to 44 countries. 17 Table 10 presents the results of the log t regression for the whole sample. We reject the convergence as t-statistic, − 2.824, is less than − 1.65. Therefore, the next step is to check the possibility of club clustering. Table 11 shows that we have two convergent clubs. The first club is comprised of 16 countries whilst the second club is made up of 28 countries. Interestingly, the first club includes mainly Latin American and Caribbean countries whilst most countries in the second club are European and Asian. Notably, most developed countries clustered in club 2, 16 out of 19, while the developing countries equally distributed between these two clubs, 13 in club 1 and 12 in club 2. This confirms the observed heterogonous behaviour of income inequality across developing countries. Recall the results of DS and TS models presented in Tables 5 and 6 which suggest that Latin American and Caribbean countries have homogenous inequality behaviour, the findings of club clustering algorithm confirm these results as most Latin American and Caribbean countries are in the same club thereby they share a common equilibrium.
We further investigate whether the β-convergence exists in these clubs. The β-convergence tests the nexus between the growth rate and the initial level (Bai et al., 2019;Barro & Sala-i-Martin, 1992). If the coefficient of initial level is negative and statistically significant then the convergence exists. Therefore, we regress growth rate of Gini coefficient to its initial level for each club and presents the results in Table 12. It can be seen that the coefficient of both clubs is significantly negative, consequently the β-convergence exists in each club.

The Relationship Between ODA and Inequality
Our findings show heterogonous behaviour of income inequality across countries in our sample, especially developing countries. We have found that, in the last half century, inequality tends to decrease in 13 countries; Argentina, Brazil, Chile, Colombia, Iran, Malaysia, Mexico, Pakistan, Panama, Philippines, Sierra Leone, Thailand and Venezuela. These heterogonous patterns and breaks observed across countries could be attributed to drastic 17 We use linear interpolation to balance our sample thus we use countries with at least 45 observations to minimize the impact of interpolation. domestic economic and political changes. This implies that reducing inequality in developing nations requires more attention to institutional and political reforms such as improving the educational equality and applying more progressive tax system rather than global policies (see Sumner, 2012;Alvaredo et al., 2017b). To clarify this point, we assess the relationship between income inequality and Official Development Assistance (ODA). ODA is one of the main global policies in the last half century aimed at reducing poverty in the developing world through flows of resources from developed countries to developing countries. We use data for ODA from the World Bank's World Development Indicators (WDI) over the period 1960-2020, to assess the association between ODA and inequality in developing countries. Table 13 presents the results of ODA analysis. The correlation between ODA and inequality is mixed. The correlation coefficient is negative and statistically significant in five countries, however our trend findings show that the inequality decreases only in two of them. Interestingly, the correlation coefficient is positive and statistically significant in five countries. Overall, the inequality decreases in 13 countries and ODA has negative correlation with inequality only in two of them, Pakistan and Sierra Leone. Indeed, most of these 13 countries, especially Latin American, received less ODA in the last half century compared with other developing countries in our sample (see the second column in Table 13). This suggests that ODA is not the cause of the observed decrease in inequality in these countries. One possible approach to assess the relationship between ODA and inequality is the cointegration. However, this approach is not appropriate as we find evidence in favour of stationary Gini index after allowing for a break in trend for several developing countries. Thus, we use a stationary VAR analysis based on the first differences of Gini index for countries that we cannot reject a unit root after allowing for a break, along with the detrended Gini index for countries which we found the index stationary around broken trend function, i.e. the residuals from estimation of (3). ODA, on the other hand, exhibits unit root behaviour for all countries except Pakistan and Venezuela, see Appendix 1. Therefore, we use the first differences for all countries except Pakistan and Venezuela. To select the lag for stationary VAR(p), we follow Lütkepohl (2005) and Harvey et al. (2017) through comparing the results of the Schwarz, Akaike and Hannan-Quinn Information Criterion. If the ICs agree then that lag length is selected. Otherwise, we select the lag length according to the IC that shows the most evidence of Granger causality.
The results of stationary VAR suggest that ODA appears to Granger cause income inequality only in five countries, see the last column in Table 13. This weak nexus between ODA and inequality supports other studies shows the importance of domestic policies in order to achieve the 10th SDG by 2030.

Conclusion
This paper examines trends in income inequality over the last six decades. We use Gini coefficient data for a sample of 34 countries, 15 developed and 19 developing, over 1960-2020. To estimate the trend in inequality, we use unit root tests that allow for breaks. This allows us to determine whether the inequality measures are trend stationary with a structural break, or first difference stationary. Given these unit root results, we estimate the trend either using a trend stationary model, before and after the break, or a difference stationary model. Additionally, we compare the slopes before and after the break to check the direction and significance of the shift in trend. We evaluate also the inequality gap between developing and developed countries by conducting convergence test on the Gini coefficients of developing countries relative to developed countries. To the best of our knowledge, this is the first comparative study of income inequality trend of developing and developed countries. The literature focuses on inequality trend of developed countries. However, the trend of inequality of developing countries, which have higher income inequality comparing with developed countries, is still underexplored. Understanding the behaviour of inequality of these countries is important to reduce the inequality within and among countries which is one of the Sustainable Development Goals. Our results not only confirm earlier findings but also offer new insights. We find that several countries experience a break in income inequality in the 1980s and inequality tends to increase in most developed countries as suggested by literature. However, we show that the trend of income inequality is more heterogenous in developing countries as inequality decreases in some of them. We test whether Official Development Assistance (ODA) causes this decline in inequality.
More specifically, we find that inequality exhibits different behaviours, especially across developing countries. The inequality trend is well represented by a single slope for 21 developing and developed countries, but by two slopes for the remaining 13 countries, which experienced a break in inequality trend around the late 1980s and early 1990s. The observed breaks in inequality trends could be attributed to drastic domestic economic and political changes. After the break, inequality decreased for some countries, increased for others and remained more or less constant for the remaining countries. The prevalent behaviour over the whole sample period is positive or trendless for developed countries, whilst it is much more heterogeneous for developing countries: positive, negative or trendless. Developing countries also show mixed performances regarding the inequality gap with developed countries, as their Gini coefficients exhibit different convergence behaviours comparing with developed countries. We also use club convergence approach and find that our sample form two convergence clubs. The first club mainly includes Latin American and Caribbean countries and the second club is mainly composed of European and Asian countries.
Reducing income inequality within and across countries is one of the Sustainable Development Goals of the United Nations that they aim to achieve by 2030. The observed mixed patterns of inequality across developing countries in the last six decades indicate that the key drivers of inequality seem to be domestic rather than global. Therefore, our final thought is that achieving this goal requires more attention to reform the domestic policies such as taxation and redistribution policies, rather than the global policies suggested by United Nations such as ODA and financial flows. Our analysis to the nexus between ODA and income inequality confirms this inference. Particularly, countries with downward inequality trend, mainly Latin American, received less ODA in the last half century and ODA has a weak relationship with inequality of these countries. The observed decline in inequality in these countries could be attributed to domestic policies such as improvements in social protection systems (e.g. the broad-based pensions of Brazil and Chile and the universal health systems of Brazil and Colombia), the growth of wages and expanding access to education.
Finally, there are some limitations which could be considered in the future research. We focus on small sample of countries because of data availability. Thus, it will be interesting to study the inequality trend of other countries, particularly Sub-Saharan African countries. Additionally, we use Gini index as a measure of income inequality because it covers the entire spectrum of the income distribution which allows us to investigate the trend of income disparity across different cohorts. However, using other measures of income inequality such as top ten percent income share or income share held by lowest ten percent is important to test the trend of the wealthy and poor and to explore which cohort leads the shift in the overall inequality index.