Ly Dai
email: hunglydai@gmail.com

Economic Convergence With Safe Assets (05/2022)

Keywords: Convergence of Economic Growth, Safe Assets, Cross-Section Regression, Fixed-Effect Panel Regression. JEL Classifications: E52, F21, F41

The paper characterizes the convergence of economic growth across economies on accounting for the sovereign debts rating, a measure of the safety of government debts. The empirical analysis combines a cross-section with a panel data regression on a sample of 180 economies over 1990-2019. The evidence records that there exists the convergence of economic growth (𝛽-convergence):

an economy with lower initial per capita GDP has a higher average growth rate of per capita GDP. Moreover, on cross-section data, the sovereign debts rating amplifies the convergence process, and also exerts a positive effect on the steady state level of per capita GDP relative to that level of world leading economy. And on panel data, the convergence only applies for an economy attains a high enough sovereign debts rating (higher than a threshold of 10.8 established by the data).

This result also constitutes an inverted-U-shaped dependence pattern of economic growth on the per capita GDP. These analysis together uncover one mechanism for the convergence that a higher sovereing debts rating raises both the domestic investment and foreign capital inflows, then, stimulating the capital stock accumulation toward the steady state per capita GDP.

Introduction

The recent globalization witnesses a new phenomena on the income difference across economies: the economic growth rate experiences a reversal tendency from the divergence to the convergence across economies. While the cross-country distribution of world gross domestic product (GDP) did not shrink over 1960-1990[START_REF] Sala-I-Martin | The classical approach to convergence analysis[END_REF], the updated evidence covering up to 2020 records that the distribution has shrunk since the poor countries have grown faster than the rich ones [START_REF] Kremer | Converging to Convergence[END_REF][START_REF] Cartone | Does economic convergence hold? A spatial quantile analysis on European regions[END_REF][START_REF] Patel | The new era of unconditional convergence[END_REF].

However, one remained question is open on which factor determines the sucess of economic growth convergence across economies. And our paper aims to fill in this research gap.

The current paper investigates the convergence of economic growth by employing the sovereign debts rating, which is a proxy for the reputation of one government on the international capital market. Our analysis is based on the combination of cross-section with panel data regression on a data sample of 180 economies over the 1990-2019 period. The cross-section regression can capture the economic growth convergence implied by the theoretical model with an unique steady state, the panel-data regression can be aligned with the theoretical model with multiple steady states [START_REF] Galor | Convergence? Inferences from theoretical models[END_REF]. Therefore, the paper is consistent with both the classical approach, summarized by [START_REF] Sala-I-Martin | The classical approach to convergence analysis[END_REF] and the critique against this approach, summarized by [START_REF] Quah | Twin peaks: growth and convergence in models of distribution dynamics[END_REF].

The empirical evidence records the existence of the convergence of economic growth across economies. For an economy with a lower per capita GDP in the initial year (i.e, 1990) tends to have a lower averaged economic growth rate over the time period (i.e, 1990-2019). And the convergence of economic growth tends to be stronger on accounting for the institutional quality including the economic openness and the government effectiveness. Furthermore, the economic growth convergence depends on the sovereign debts rating. In particular, the cross-section evidence shows that the sovereign debts rating raises the likelihood of convergence as well as exerts a positive effect on the steady state level of per capita GDP. And the panel data evidence establishes that the economic growth convergence only applies for the economies with high enough sovereign debts rating, i.e, higher than a threshold of 10.8, which is uncovered by the empirical evidence. This result also constitutes an inverted-U-shaped dependence pattern of economic growth on the per capita GDP.

The paper belongs to the literature on the economic convergence, spurred by the seminal theoretical contribution of [START_REF] Romer | Increasing returns and long-run growth[END_REF] and [START_REF] Jr | On the mechanics of economic development[END_REF].

There is unconsistency on the empirical evidence on the economic growth convergence. The main reason relies on challenge to capture the steady state per capita GDP [START_REF] Sala-I-Martin | The classical approach to convergence analysis[END_REF]. On the neoclassical growth model [START_REF] Solow | A contribution to the theory of economic growth[END_REF][START_REF] Swan | Economic growth and capital accumulation[END_REF], one economy converges to the steady state, i.e, long-run equilibrium, in which the capital stock per effective unit of labor is constant. By the diminishing return to scale of capital stock, an economy with initial lower capital stock has a higher growth rate than the economy with initial huger capital stock. But this mechanism only works when different economies share the same steady state. Thus, the empirical analysis needs to hold the steady state to be constant. As [START_REF] Sala-I-Martin | The classical approach to convergence analysis[END_REF] discusses, there are two approachs to overcome this challenge, by adding the variables that proxy for the steady state or alternatively by analyzing sets of similar economies.

We follow both the first strategy by using the sovereign debts rating as proxy for the steady state per capita GDP and the second strategy by separating the group of advanced economies with the group of emerging and developing economies. Thus, our main approach is similar to [START_REF] Aghion | The effect of financial development on convergence: Theory and evidence[END_REF] who use a cross-section regression to show that the financial development amplifies the likelihood of an economy to converge to the growth rate of the world frontier.

Our evidence records a similar mechanism but with the sovereign debts rating: the rating raises the likelyhood of economic growth convergence. And for the economies which exhibits the economic convergence, the sovereign debts rating has a positive impact on the steady state level of per capita GDP.

While [START_REF] Aghion | The effect of financial development on convergence: Theory and evidence[END_REF] focuses on the cross-section analysis, we extend the cross-section to the panel data regression. Technically, the panel data captures the intra-country distribution of per capita GDP that is absent on the cross-section analysis [START_REF] Quah | Twin peaks: growth and convergence in models of distribution dynamics[END_REF]. Our panel evidence records that the convergence of economic growth only happens for the economies with a high enough value of sovereign debts rating.

Furthermore, by both of the cross-section and panel evidence, we uncover the mechanism for the impact of sovereign debts rating on the convergence of economic growth. A higher rating raises the domestic investment, then, the domestic capital stock. Thus, the sovereign debts rating also contributes on the capital accumulation process along the convergence pathway toward the steady state. This evidence provides an addition new result on the previous literature by stressing that the reputation of government is important for the convergence of economic growth.

The current paper is closely related to a recently dynamic literature on the safe assets. The safe assets can be defined as the uncertainty-insensitive assets with a constant rate of return even on the economic downturn period (Caballero et al, 2014) or alternatively as the information-insensitive assets with a rate of return unaffected by the assymetric information (Gorton, 2011). When there exists the excess demand for the safe assets as a store of wealth, an increase of safe assets supply can reduce the interest rate, then, stimulating the aggregate demand and associated economic growth rate [START_REF] Caballero | The safety trap[END_REF]). An increase of safe assets supply can also reduce the pressure of primary deficit on the inflation rate [START_REF] Brunnermeier | The fiscal theory of price level with a bubble[END_REF] or the appreciation of domestic currency [START_REF] Gopinath | Banking, trade, and the making of a dominant currency[END_REF].

Our paper sheds a refresh light on the macroeconomic impact of safe assets. In particular, the safe assets can contribute on the catching-up process of poor to the rich economies. This effect works through the role of the safety of assets on the economic growth, instead of the supply of assets as the aforementioned papers. Therefore, our evidence complements to the previous results by charactering one additonal role of the safe assets on the macroeconomy.

The paper is structured as following. After the first section on the introduction, the second section on Analysis Framework with the data sample and associted empirical model. Then, the third section shows the empirical evidence on both cross-section and panel regression, and their robustness checks. Finally, the paper closes with the fourth section on the conclusion.

Analysis Framework

Data

The data set is a cross-section sample of 180 economies over the 1990-2019 period. Each variable takes its average value over time. This approach is useful to focus on the long-run pattern of economic growth since a long time period of about 30 years is enough to absorb the short-run fluctuation over time.

The economic growth rate, denoted by (aGDPpcgrowth), is the dependent variable on our analysis. This variable captures the change of living standard of households within an economy. Moreover, it also serves as a proxy for the change of technology level, since on the neoclassical growth model [START_REF] Solow | A contribution to the theory of economic growth[END_REF][START_REF] Swan | Economic growth and capital accumulation[END_REF] implies that the GDP per capita grows with the technology progress. This variable is measured by the growth rate of real gross domestic product (GDP) per capita, on percentage. In particular, the real GDP is the GDP at constant national 2011 price on USD, which is from Penn World Tabke 10.0 (2020). And the population is from World Development Indicators updated to 2019, relased by the World Bank. Then, the real GDP per capita is the ratio of real GDP per population, on USD. Based on this data, the initial GDP per capita, denoted by (LnGDPpc1990), is the value of GDP per capita in 1990.

The sovereign debts rating, denoted by (aSovrate), plays a central role on our analysis. This variable captures the safety of sovereign debts issued in each economy. It is an index ranged from 1 to 21, with a higher index meaning a greater safety level. This index is constructed and released by the World Bank Fiscal Space Database, updated to 2019. The inputs include the ratings of sovereign debts by well-known international ageny such as the Standard & Poor, and Moody.

Other variables are explored from various resources. The trade openness, denoted by (aEopen), the ratio of exports summing imports over nominal GDP on percentage. The exports, imports and nominal GDP at current price on USD are all taken from the World Development Indicators, updated to 2019. And the government effectiveness, denoted by (aGovern), reflects the institutional quality, and is extracted from the World Government Index, constructed and released by the World Bank.

Next, the distance to world frontier, denoted by (aDistance), is the ratio of domestic productivity level over the United States one. This data is constructed and released by the Penn World Table 10.0 (2020). And the end-of-period capital stock, denoted by (LnK2019), is the logarit value of the 2019 capital stock in current USD. The investment per GDP, denoted by (aI2y), is the ratio of domestic investment per GDP on percentage. And the net total capital inflows are the ratio of negative value of current account over the GDP on percentage. All of these three variables are from the World Development Indicators. Other variables, including the initial GDP per capita, government effectiveness, distance to world frontier and end-of-period capital stock, also have a quite large standard deviaton. Therefore, the data sample offers a great variation to explore the convergence of economic growth on accounting for the sovereign debts rating.

Model

The empirical model is as following:

𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ 𝑗 =∝ + 𝛽 𝐺𝐷𝑃𝑝𝑐1990 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 𝑗 +𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 + 𝛽 𝐼𝑛𝑡𝑒𝑟 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 𝑗 * 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 + 𝛽 𝐸𝑜𝑝𝑒𝑛 𝑎𝐸𝑜𝑝𝑒𝑛 𝑗 + 𝛽 𝐺𝑜𝑣𝑒𝑟𝑛 𝑎𝐺𝑜𝑣𝑒𝑟𝑛𝑚𝑒𝑛𝑡 𝑗 + 𝛽 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑗 + 𝑢 𝑗 (1)
The focal point is the coefficients of (𝛽 𝐺𝐷𝑃𝑝𝑐1990 ), (𝛽 𝐼𝑛𝑡𝑒𝑟 ), and

(𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 ).
First, the coefficient of initial per capita GDP (𝛽 𝐺𝐷𝑃𝑝𝑐1990 ) captures the dependence of the economic growth on the initial GDP per capita. When this coefficient is significantly negative, i.e, 𝛽 𝐺𝐷𝑃𝑝𝑐1990 < 0, there exists the convergence of economic growth. Accordingly, an economy with a higher initial per capita GDP has a higher average growth rate of per capita GDP over the time period of 1990-2019.

Second, the coefficient of the interaction term between the initial per capita GDP with the sovereign debts rating (𝛽 𝐼𝑛𝑡𝑒𝑟 ) illustrates the impact of the sovereign debts rating on the convergence of economic growth rate. When this coefficient is significant, the sovereign debts rating determines the convergence rate, captured by the first derivative of the economic growth rate with respect to the initial per capita GDP.

𝛽 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 = 𝜕𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ 𝜕𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 = 𝛽 𝐺𝐷𝑃𝑝𝑐1990 + 𝛽 𝐼𝑛𝑡𝑒𝑟 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 (2)
On accounting for the sovereign debts rating, an economy can converge to the growth rate of world leading economy when its growth rate of per capita GDP depends negatively on the initial per capita GDP. This requires that (𝛽 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 < 0). In this case, the speed of convergence is amplified by the sovereign debts rating: for a higher sovereign debts rating, the tendency of economic growth convergence is stronger. Given that (𝛽 𝐺𝐷𝑃𝑝𝑐1990 < 0), this results will hold only if:

𝛽 𝐼𝑛𝑡𝑒𝑟 < 0 (3)
Third, the coefficient (𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 ) illustrates the impact of the sovereign debts rating on the long-run equilibrium or the steady state value of per capita GDP. When this coefficient is significantly positive, the sovereign debts rating exerts a positive effect on the steady state level of per capita GDP relative to that level of world leading economy.

In order to uncover this third implication, we can define variable with upper bar as the value of the leading economy, which is at the world frontier, then, is the objective for other economies to converge toward in the long run. By the equation (1), we have the value of the world leading economy as:

𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ =∝ + 𝛽 𝐺𝐷𝑃𝑝𝑐1990 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ +𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 ̅̅̅̅̅̅̅̅̅̅̅̅ + 𝛽 𝐼𝑛𝑡𝑒𝑟 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ * 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 ̅̅̅̅̅̅̅̅̅̅̅̅ + 𝛽 𝑋 𝑋 ̅ + 𝑢 ̅ (4)
Whereby, for simlicity, we denote X as the vector containing three variables including the economic openness, government effectiveness and distance to world frontier.

By taking the difference between the equation ( 1) and ( 4), we have:

𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ 𝑗 -𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ = 𝛽 𝐺𝐷𝑃𝑝𝑐1990 (𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 𝑗 - 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ) + 𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 (𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 -𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 ̅̅̅̅̅̅̅̅̅̅̅̅ ) + 𝛽 𝐼𝑛𝑡𝑒𝑟 (𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 𝑗 * 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 -𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ * 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 ̅̅̅̅̅̅̅̅̅̅̅̅ ) + 𝛽 𝑋 (𝑋 - 𝑋 ̅ ) + (𝑢 𝑗 -𝑢 ̅) (5)
By defining (𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̂𝑗 = 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 𝑗 -𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ) and

(∆𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 = 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 ̅̅̅̅̅̅̅̅̅̅̅̅ -𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 ; ∆𝑋 = 𝑋 ̅ -𝑋 𝑗 ; ∆𝑢 = 𝑢 ̅ -𝑢 𝑗 ), we rewrite this equation as:

𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ 𝑗 -𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ = 𝛽 𝐺𝐷𝑃𝑝𝑐1990 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̂𝑗 - 𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 ∆𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 + 𝛽 𝐼𝑛𝑡𝑒𝑟 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 * 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̂𝑗 - 𝛽 𝐼𝑛𝑡𝑒𝑟 ∆𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 * 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ + 𝛽 𝑋 ∆𝑋 + ∆𝑢 (6)
We assume that the convergence of economic growth rate happens at the long run equilibrium, i.e, the steady state:

𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ 𝑗 -𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ = 0
Then, we denote the steady state per capita GDP gap of an economy with the world leading economy as 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̂𝑗, * . Then, solving the equation ( 6),

i.e, 𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ 𝑗 -𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ = 0, we have:

𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̂𝑗, * = ∆𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 (𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 +𝛽 𝐼𝑛𝑡𝑒𝑟 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ )+ 𝛽 𝑋 ∆𝑋+∆𝑢 𝛽 𝐺𝐷𝑃𝑝𝑐1990 +𝛽 𝐼𝑛𝑡𝑒𝑟 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 (7)
Then, the first derivative of the steady state per capita GDP gap with respect to the sovereign debts rating is:

𝜕𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̂𝑗, * 𝜕𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 = - 𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 +𝛽 𝐼𝑛𝑡𝑒𝑟 (𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̂𝑗, * +𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ) 𝛽 𝐺𝐷𝑃𝑝𝑐1990 +𝛽 𝐼𝑛𝑡𝑒𝑟 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 (8)
When an economy converges to the growth rate of world leading economy, i.e, there exists the economic growth convergence, we have a negative term:

(𝛽 𝐺𝐷𝑃𝑝𝑐1990 + 𝛽 𝐼𝑛𝑡𝑒𝑟 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 ) ≤ 0. And we also assume that all countries lag the world leading economy in the steady state with a large distance:

(𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̂𝑗, * + 𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ < 0). Thus, only when the coefficient of sovereign debts rating is positive i.e, (𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 > 0), the value of first derivative is positive:

𝜕𝐿𝑛𝐺𝐷𝑃𝑝𝑐1990 ̂𝑗, * 𝜕𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 > 0 (9)
Other coefficients control for the role of other variables. In details, the economic openness (𝛽 𝐸𝑜𝑝𝑒𝑛 ) and government effectiveness (𝛽 𝐺𝑜𝑣𝑒𝑟𝑛𝑚𝑒𝑛𝑡 ) can have a positive impact on the growth since these variable captures the institutional quality. The distance to world frontier (𝛽 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ) can also exert a positive impact on the economic growth for the backwardness advantage (Gerchenskon, 1956):

an economy further from the world frontier can enjoy a higher economic growth by exploring the benefit from the advanced technology.

For the robustness analysis, we employ a fixed-effect panel regression.

While the cross-section regression captures the cross-country distribution of per capita GDP across economies, the panel regression focuses on the intra-country distribution of per capita GDP [START_REF] Sala-I-Martin | The classical approach to convergence analysis[END_REF][START_REF] Quah | Twin peaks: growth and convergence in models of distribution dynamics[END_REF]. Moreover, the cross-section regression reflects the empirical approach based on the theoretical growth model with one unique steady state while the panel regression follows the approach based on the model with multiple steady states [START_REF] Galor | Convergence? Inferences from theoretical models[END_REF]. Thus, each type of empirical model can reflect different aspect of economic growth convergence.

We carry out two subsequent regression equations. First, we try to detect the existence of non-linearity on the dependence pattern of economic growth on the per capita GDP. In details, the regression equation is as following:

𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ 𝑗,𝑡 = ∝ 𝑗 + 𝛽 𝐺𝐷𝑃𝑝𝑐 𝐿𝑛𝐺𝐷𝑃𝑝𝑐 𝑗,𝑡 + 𝛽 𝐺𝐷𝑃𝑝𝑐,2 (𝐿𝑛𝐺𝐷𝑃𝑝𝑐 𝑗,𝑡 ) 2 + 𝛽 𝐸𝑜𝑝𝑒𝑛 𝑎𝐸𝑜𝑝𝑒𝑛 𝑗,𝑡 + 𝛽 𝐺𝑜𝑣𝑒𝑟𝑛 𝑎𝐺𝑜𝑣𝑒𝑟𝑛𝑚𝑒𝑛𝑡 𝑗,𝑡 + 𝛽 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑗,𝑡 + 𝑢 𝑗,𝑡

Next, we investigate whether the sovereign debts rating can explain the non-linearity on the pattern of economic growth rate. The regression equation is as following:

𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ 𝑗,𝑡 = ∝ 𝑗 + 𝛽 𝐺𝐷𝑃𝑝𝑐 𝐿𝑛𝐺𝐷𝑃𝑝𝑐 𝑗,𝑡 + 𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗,𝑡 + 𝛽 𝐼𝑛𝑡𝑒𝑟 𝐿𝑛𝐺𝐷𝑃𝑝𝑐 𝑗,𝑡 * 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗,𝑡 + 𝛽 𝐸𝑜𝑝𝑒𝑛 𝑎𝐸𝑜𝑝𝑒𝑛 𝑗,𝑡 + 𝛽 𝐺𝑜𝑣𝑒𝑟𝑛 𝑎𝐺𝑜𝑣𝑒𝑟𝑛𝑚𝑒𝑛𝑡 𝑗,𝑡 + 𝛽 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑗,𝑡 + 𝑢 𝑗,𝑡 (11)

In brief, the combination of these two-step analysis can uncover the role of sovereign debts rating on the convergence of economic growth rate in the panel data sample.

Safe Assets and Economic Growth Convergence

Empirical Evidence

Table 1 shows the regression results of economic growth rate on the initial GDP per capita, sovereign debts rating and other indpendent variables. The evidence examines the impact of debts rating on the economic growth convergence, on controlling for the economic openness, government effectiveness and distance to world frontier.

The evidence records that there exists the convergence of economic growth across economies. In particular, in column 1, the coefficient of initial GDP per capita is negative. Thus, an economy with higher income per capita also has lower economic growth rate. The coefficient of the initial GDP per capita also measures the convergence rate:

𝛽 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 = 𝜕𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ 𝑗 𝜕𝐿𝑛𝐺𝐷𝑃𝑝𝑐 𝑗 = -0.381
The evidence also uncovers that the sovereign debts rating can affect the convergence of economic growth across countries. In particular, in column 2, the estimated coefficients can be captured by the following equation:

𝑎𝐺𝐷𝑃𝑝𝑐𝑔𝑟𝑜𝑤𝑡ℎ 𝑗 = 2.848 -1.202 𝐿𝑛𝐺𝐷𝑃𝑝𝑐 𝑗 + 0.264 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 -0.111𝐿𝑛𝐺𝐷𝑃𝑝𝑐 𝑗 . 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 Then, the convergence rate is dependent on the sovereign debts rating as following:

𝛽 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 = -(1.202 + 0.111 𝑎𝑆𝑜𝑣𝑟𝑎𝑡𝑒 𝑗 )
The estimated coefficient shows that an economy with a higher sovereign debts rating would have a faster convergence rate. To illustrate this finding, let's consider two economies: Singapore and Mexico. On average over 1990-2019, Singapore has an economic growth at 3.31% and associated sovereign debts rating at 20.46, while the corresponding parameters for Mexico are 1.05% and 12.23. By our empirical evidence, the convergence rate of Singapore is (-3.47) while that of Mexico is (-2,55). Thus, the difference on the convergence rate of Singapore compared with Mexico is 0.92 on its absolute value. Another important finding is the significantly positive coefficient of sovereign debts rating. In the third column, we have: (𝛽 𝑆𝑜𝑣𝑟𝑎𝑡𝑒 = 0.246 > 0).

As the equation ( 7) records, this positive coefficient proves that the sovereign debts rating exerts a positive effect on the steady state level of per capita GDP relative to that of world leading economy. Figure 1 illustrates the convergence of economic growth. With the GDP growth rate on the y-axis and the initial GDP per capita on the x-axis, the correlation line is a downward sloping line. Thus, the figure is consistent with the empirical evidence recored in Table 1 that an economy with low initial GDP per capita also has a high economic growth rate. On other words, the convergence of economic growth exists.

Next, we investigate the mechanism for the convergence of economic growth. In the neoclassical growth model [START_REF] Solow | A contribution to the theory of economic growth[END_REF][START_REF] Swan | Economic growth and capital accumulation[END_REF], the capital accumulation underlines the convergence of growth process toward the long-run equilibrium or the steady state. Based on this principle, the impact of sovereign debts rating on the capital accumulation can also underlines its role on the econonmic growth convergence. In particular, we collect three variables, including (i) the domestic capital stock at the end of time period, denoted by ( LnK2019); (ii) the domestic investment per GDP flow, denoted by (aI2y); and

(iii) the net total capital inflows per GDP, denoted by (anegCA2y). These variables are related to each other on the sense that the domestic capital stock is built up by the flows of domestic investment and foreign capital. Table 2 shows the cross-section regression results of capital stock, investment per GDP and net total capital inflows per GDP on the sovereign detbs rating. The analysis also controls for the role of economic openness, government effectiveness and distance to world frontier. In column 1, the coefficient of sovereign debts rating is significantly positive at 0.412. Thus, an increase of average sovereign debts rating by 1 point, the capital stock at the end of time period raises by 1,5 billion USD. In column 2, the coefficient of debts rating is also significantly positive at 0.503 for the domestic investment per GDP, while it is significantly negative at (-1.367) for the net total capital inflows per GDP.

Thus, for an increase of sovereign debts rating by 1 point, the domestic investment per GDP goes up by 0.5% while the foreign capital inflows per GDP goes down by 1.3%. Therefore, the evidence records that the sovereign debts rating exerts a positive impact on the domestic capital accumulation, by stimating the domestic investment rather than the foreign capital inflows. Table 3 records the regression results of economic growth convergence with the financial crisis. From column 1 to 3, the data sample covers the time period before the global crisis (1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007), while from column 4 to 6, the data sample covers the time period after the global crisis (2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018)(2019). The evidence shows that there exists the convergence of economic growth on both the time period before, in column 1, and after the crisis, in column 4. And the convergence rate tends to increase from the value at (-0.248) for the time period before the crisis to the value at (-0.415) for the time period after the crisis. Moreover, the result in column 2 and 5 shows that the sovereign debts rating also affects the convergence growth process. And the magitude of this effect raises after the crisis, captured by the increase of the coefficient of the interaction term from (-0.086) in column 2 to (-0.16) in column 5. This result holds on controlling for other independent variables in column 3 for the before-crisis period and in column 6 for the after-crisis period. In brief, the convergence of economic growth exists both before and after the global financial crisis and tends to be stronger after the crisis. Table 4 shows the regression results of economic growth rate on two different groups of economies: the advanced economies in column 1-3 and the emerging and developing economies in column 4-6. First, there exist the convergence of economic growth in both two groups, as recorded in column 1 and 4. And the convergence rate is higher in the emerging and developing economies, which is (-0.498), than in the advanced economies, which is (-1.365).

Second, the impact of sovereign debts rating on the convergence of economic growth only applies for the group of emerging and developing economies. In column 2 and 3, the coefficients of the interaction term between the initial GDP per capita and sovereign debts rating is insignificant for the group of advanced economies. And in column 5 and 6, the corresponding coefficients are significant for the group of emerging and developing economies. In brief, the convergence of economic growth is stronger in the group of emerging and developing economies than in the group of advanced economies, and the effect of sovereign debts rating only holds on the former group of economies.

Robustness Analysis

We carry out the robustness analysis of the cross-section evidence on the relationship between the safe assets and economic growth convergence. In particular, we employ a fixed-effect panel regression of the data sample covering the 1990-2019 period. The panel regression captures the time-dimension of data sample while the fixed-effect controls for the unobserved country effect.

Compared with the cross-section regression which emloys the average value of each variable over the same 1990-2019 period, the fixed-effect panel regression can characterize the fluctuation over time of the impact of safe assets on the economic growth convergence. Table 5 shows the evidence on the economic growth rate on the GDP per capita, on controlling for other independent variables. In column 1, the coefficient of GDP per capita is positive, which means that the current-period economic growth tends to be higher for a higher previous-period GDP per capita. In column 2, the coefficients of the quadratic term is also significant and negative. Thus, the pattern of economic follows a quadratic function: it is first increasing then decreasing on the previous GDP per capita. On other word, the pattern can be depicted as an inverted-U-shaped curve. As column 3 shows, this pattern holds on controlling for the economic openness, government effectiveness and distance to world frontier. In brief, the panel evidence records that the economic growth follows a quadratic pattern of initial GDP per capita.

The panel evidence differs the cross-section one on the economic growth convergence. In particular, on the panel analysis, the convergence, that a lower growth rate is associated with a higher GDP per capita, only happens when the GDP per capita is high enough. But the cross-section analysis records that the convergence happens without depending on the level of GDP per capita.

Next, we also carry out the robustness check on the role of sovereign debts rating on the economic growth convergence. Table 6 shows the panel regression results of economic growth on the sovereign debts rating and its interaction term with GDP per capita, on controlling for other independent variables. In column 1, a higher debts rating is associated with a greater economic growth rate. In column 2, the coefficient of GDP per capita is positive but that of the interaction term between the GDP per capita and the sovereign debts rating is negative. As column 3 records, this result holds on controlling for the economic openness, government effectiveness and distance to world frontier.

The empirical evidence establishes that the convergence of economic growth only happens for a high enough sovereign debts rating. This result can explain the inverted-U-shaped dependence pattern of current-period economic growth on the previous-period GDP per capita. In particular, for the low sovereign debts rating, i.e, lower than 10.8, an increase of GDP per capita is associated with higher economic growth rate. And only for the high sovereign debts rating, i.e, higher than 10.8, an increase of GDP per capita is associated with lower economic growth. Therefore, the convergence of economic growth only applies when the sovereign debts rating is high.

Table 7 shows the robusness check on the role of sovereign debts rating on the accumulation of capital stock. In column 1, a higher sovereign debts rating is associated with a huger capital stock. This increase of sovereign debts rating is also associated with more investment per GDP ratio, as shown in column 2, and more net total capital inflows per GDP, as captured in column 3. Compared with the cross-section evidence, the panel evidence is different only on the impact of sovereign debts rating on the net total capital inflows. This difference can rely on the fact that the panel regression can capture the time dimension of cross-border capital flows while the cross-section regression accounts for their cross country dimension. In brief, the cross-section evidence and associated robustness analysis by the panel data evidence jointly establishes role of the sovereign debts as a determinant of the convergence of economic growth across economies. This role works through the accumulation of capital, by stimulating both the domestic investment and foreign capital inflows.

Conclusion

The paper investigates the convergence of economic growth under the impact of the sovereign debts rating. Based on a data sample of 178 economies over 1990-2019, the empirical evidence records that the convergence exits. And its pattern depends on the sovereign debts rating: a higher rating raises both the chance and speed of convergence. This dependence pattern works through a larger capital accumulation for a bettern sovereign debts rating.

For the policy implication, the results uncover that the sovereign debts rating can be a priority for the public policy to upgrade the convergence of economic growth. This opens a new channel for the debts issuance to affect the economic growth: it can not only provide an additional finance resource for the economy but also determine the speed of growth convergence to the leading economy.

For the future research avenue, the paper can be extended by various ways.

The convergence of economic growth can depend on the context of global economy. Recently, the recovery during Covid-19 pandemic is different across economies, generating the income gap across economies. Thus, the future research can cover the impact of the pandemic on the convergence of economic growth, on accouting for the sovereign debts rating.

Figure 1 :

 1 Figure 1: Convergence of Economic Growth

  Next, we also analyze the potential change of economic growth convergence pattern by accounting for the global financial crisis in 2008, and for the development stage.

Table 1 :

 1 Descriptive Statistics

	Variables	Observations Mean Std.Dev Min	Max
	Economic Growth (aGDPpcgrowth,%)	178	2.04	1.72	-2.38 8.25
	Sovereign Debts Rating (aSovrate)	151	11.86	4.91	4.72	21
	Initial GDP per capita (LnGDPpc1990)	177	8.88	1.20	6.00 11.53

Table

1

reports the descriptive statistics of the data sample. The economic growth rate has a mean at 2.04% with a standard deviation at 1.72%. Compared with this variable, the trade openness, investment per GDP and net total capital inflows have both a higher mean and greater standard deviation. Moreover, the sovereign debts rating has a mean at 11.86 and a large standard deviation at 4.91.

Table 1 :

 1 Cross-Section Regression Results of Economic Growth Rate (aGDPpcgrowth)

	on Initial GDP per capita, Sovereign Debts Rating, Economic Openness, Government
	Effectiveness and Distance to World Technology Frontier
		(1)	(2)	(3)
	VARIABLES	aGDPpcgrowth	aGDPpcgrowth	aGDPpcgrowth
	Initial GDP per capita	-0.381***	-1.202***	-1.188***
	(LnGDPpc1990)	(0.103)	(0.163)	(0.179)
	Sovereign Debts Rating		0.264***	0.246***
	(aSovrate)		(0.0458)	(0.0665)
	LnGDPpc* aSovrate		-0.111***	-0.127***
	(Interaction)		(0.0268)	(0.0272)
	Economic Openness			0.00455**
	(aEopen)			(0.00214)
	Government Effectiveness			-0.0324

Table 2 :

 2 Cross-Section Regression Results of Capital Stock (LnK2019),

	Investment per GDP (aI2y), Net Total Capital Inflows per GDP (anegCA2y)
	on Sovereign Debts Rating, Economic Openness, Government
	Effectiveness, Distance to World Frontier
		(1)	(2)	(3)
	VARIABLES	LnK2019	aI2y	anegCA2y
	Sovereign Debts Rating	0.412***	0.503**	-1.367***
	(aSovrate)	(0.0732)	(0.231)	(0.260)
	Economic Openness	-0.0121***	0.0164*	-0.0299***
	(aEopen)	(0.00272)	(0.00858)	(0.00966)
	Government Effectiveness	-1.021**	-1.642	6.955***
	(aGovernment)	(0.392)	(1.236)	(1.391)
	Distance to World Frontier	-0.901	-7.816***	-12.08***
	(aDistance)	(0.779)	(2.456)	(2.765)
	Constant	10.69***	22.09***	26.47***
		(0.913)	(2.880)	(3.241)
	Observations	104	104	104
	R-squared	0.416	0.143	0.451
	Notes: Standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1

Table 3 :

 3 Cross-Section Regression Results of Economic Growth Rate (aGDPpcgrowth) on Initial

	GDP per Capita, Sovereign Debts Rating, Economic Openness, Government Effectiveness, Distance
	to World Frontier on Two Sub-Samples: Before Global Crisis (1990-2007) in Column 1-3, After
	Global Crisis (2009-2019) in Column 4-6
		(1)	(2)	(3)	(4)	(5)	(6)
	VARIABLES	aGDPpc	aGDPpc	aGDPpc	aGDPpc	aGDPpc	aGDPpc
		growth	growth	growth	growth	growth	growth
	Initial GDP per capita	-0.248* -1.321*** -1.199*** -0.415*** -1.320*** -1.693***
	(LnGDPpc1990)	(0.144)	(0.206)	(0.197)	(0.152)	(0.285)	(0.384)
	Sovereign Debts Rating		0.290*** 0.240***		0.270*** 0.259***
	(aSovrate)		(0.0579) (0.0789)		(0.0730) (0.0898)
	LnGDPpc*aSovrate		-0.0865** -0.113***		-0.160*** -0.218***
	(Interaction)		(0.0338) (0.0320)		(0.0415) (0.0437)
	Economic Openness			0.00428*			0.00795**

Table 4 :

 4

	Economies in Column 4-6
		(1)	(2)	(3)	(4)	(5)	(6)
	VARIABLES	aGDPpc	aGDPpc	aGDPpc	aGDPpc	aGDPpc	aGDPpc
		growth	growth	growth	growth	growth	growth
	Initial GDP per capita	-1.365*** -1.842	-3.232*** -0.498*** -1.217*** -1.025***
	(LnGDPpc1990)	(0.372)	(1.290)	(1.057)	(0.135)	(0.190)	(0.245)
	Sovereign Debts Rating		0.0522	-0.112		0.254*** 0.253***
	(aSovrate)		(0.257)	(0.203)		(0.0580) (0.0836)
	LnGDPpc*aSovrate		0.0148	0.112		-0.114*** -0.0912*
	(Interaction)		(0.191)	(0.142)		(0.0365) (0.0498)
	Economic Openness			0.00514***			-0.00174
	(aEopen)			(0.00141)			(0.00537)
	Government Effectiveness			0.250			0.0874
	(aGovernment)			(0.439)			(0.435)
	Distance to World Frontier			2.818***			1.173
	(aDistance)			(0.706)			(0.951)
	Constant	3.794*** 3.969***	2.663**	1.922*** 2.802*** 2.101***
		(0.515)	(1.414)	(1.027)	(0.156)	(0.217)	(0.786)
	Observations	36	36	36	141	107	68
	R-squared	0.284	0.300	0.698	0.089	0.286	0.313
	Notes: Standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1

Cross-Section Regression Results of Economic Growth Rate (aGDPpcgrowth) on Initial GDP per Capita, Sovereign Debts Rating, Economic Openness, Government Effectiveness, Distance to World Frontier on Two Sub-Samples: Advanced Economies in Column 1-3, Emerging and Developing

Table 5 :

 5 Panel Regression Results of Economic Growth Rate (GDPpcgrowth) on

	GDP per Capita, Economic Openness, Government Effectiveness, Distance to
	World Frontier
		(1)	(2)	(3)
	VARIABLES	GDPpcgrowth GDPpcgrowth GDPpcgrowth
	GDP per Capita	2.177***	16.15***	17.14***
	(LnGDPpc)	(0.308)	(2.524)	(4.396)
	Squared Value of LnGDPpc		-0.794***	-0.932***
	(LnGDPpc2)		(0.142)	(0.241)
	Economic Openness			0.0143**
	(Eopen)			(0.00575)
	Government Effectiveness			-1.116**
	(Government)			(0.548)
	Distance to World Frontier			5.105***
	(Distance)			(1.076)
	Constant	-17.83***	-78.02***	-79.17***
		(2.811)	(11.15)	(20.07)

Table 6 :

 6 Panel Regression Results of Economic Growth Rate (GDPpcgrowth) on

	GDP per Capita, Sovereign Detbs Rating, Economic Openness, Government
	Effectiveness, Distance to World Frontier
		(1)	(2)	(3)
	VARIABLES	GDPpcgrowth GDPpcgrowth GDPpcgrowth
	Sovereign Debts Rating	0.141***	3.921***	2.890***

Table 7 :

 7 Panel Regression Results of Capital Stock (LnK), Investment per GDP (I2y), Net Total Capital Inflows (negCA2y) on Sovereign Debts Rating, Standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1

	Economic Openness, Government Effectiveness, Distance to World Frontier
		(1)	(2)	(3)
	VARIABLES	LnK	I2y	negCA2y
	Sovereign Debts Rating	0.0326***	1.149***	0.826***
	(Sovrate)	(0.00369)	(0.0609)	(0.0805)
	Economic Openness	0.00195***	-0.00408	-0.0559***
	(Eopen)	(0.000329)	(0.00543)	(0.00742)
	Government Effectiveness	0.110***	-3.361***	-2.876***
	(Government)	(0.0362)	(0.598)	(0.797)
	Distance to World Frontier	-0.174***	-4.236***	-12.97***
	(Distance)	(0.0627)	(1.036)	(1.430)
	Constant	13.14***	13.82***	5.047***
		(0.0654)	(1.082)	(1.490)