DP2022-31

The Impact of Multi-Factor
Productivity on Income Inequality*
Takashi KAMIHIGASHI
Yosuke SASAKI

June 15, 2022

* The Discussion Papers are a series of research papers in their draft form, circulated to encourage
discussion and comment. Citation and use of such a paper should take account of its provisional character.
In some cases, a written consent of the author may be required.

The Impact of Multi-Factor Productivity on Income
Inequality
Takashi Kamihigashi ∗& Yosuke Sasaki †Abstract
Numerous empirical studies suggest that a technology change is associated with an increase in income
inequality. The Gini coefficient (or the Gini index) is commonly calculated to quantify income inequality
and analyze the relationship between inequality and other economic variables. However, the availability ofGini index data in a time series (e.g., five-year data) is sparse. Thus, it is difficult to study dynamic effects
in panel data. This study utilizes the relative share of income as an inequality measure to analyze theinteractions between cross-country income inequality and multi-factor productivity. Additional economic
variables are also considered to inform the analysis further. Using the relative share of income enables
observation of the long-term relationship dynamics between the two variables of interest because thenecessary data are available for individual countries. Panel data are also available for cross-country
factors. This study is the first to show that multi-factor productivity has a relationship with income
inequality, based on understanding the static and dynamic effects. This study defines a model withsome lags of the variable to capture the “dynamic effects.” The estimation method is the panel vector
autoregression (Sigmund & Ferstl (2019)[35]) with generalized method of moments (Blundell & Bond(1998)[4]). This method determines the multi-period structure of multi-factor productivity and income
inequality. Overall, this approach identifies the dynamic effects of multi-factor productivity on income
distribution, which is a novel finding that requires further analysis.

Keywords: Income Inequality, Muti-Factor Productivity, Cross-Country, & Panel Vector Autoregression

1

Introduction

Since the financial crisis of 2007/08, the world has experienced social and political turmoil. This turmoil
is perceived to have caused changes in income distribution, though it is argued that changes have occurredsince the 1980s, and the financial crisis exacerbated them. Because of the popularity of Piketty’s (2015)[31]’s argument that the economic system based on the concept of globalization favors the rich, uneven income
distribution is now frequently considered to be a significant societal problem.
To explain inequality in income distribution, several researchers have focused on interactions among
variables, such as the Gini index, financial development (Piketty (2015)[31], Jauch & Watzka (2016)[20]),
government spending in the economy (Jauch & Watzka (2016)[20] also use as the control variable), trade
openness (Daumal (2013)[11]) using cross-country panel data or time-series analyses of a single country.
Then, this study estimates the relationship between the top 1% income bracket, multi-factor productivity,
financial development, government spending, and trade openness. The objective of including multi-factor∗†

Center for Computational Social Science, Kobe University
Center for Computational Social Science, Kobe University

1

2 Studies on the Relationship between Income Inequality and Other Economic Variables

2

productivity is to account for technological change’s impact on income inequality/distribution in crosscountry analysis. As for technological change’s effect on income inequality/distribution, several researchers
find a positive relationship between wage dispersion and skill-biased technological change (Berman et al.
(1998)[3]). However, there are few findings on the interaction between multi-factor productivity andincome inequality/distribution in the macroeconometric literature.
Jaumotte et al. (2013)[21] discovered that increasing trade and financial globalization had separately
identifiable and opposite effects on the income distribution. Trade liberalization and export growth
were associated with lower income inequality, whereas increased financial openness was associated withhigher inequality. However, Bergh and Nilsson (2010)[2] reported that trade liberalization and economic
globalization increased income inequality. Conflicting results such as these explain the large body ofliterature focusing on financial development and the economic variables.
The primary contributions of the extant literature include the panel data analysis focusing time series
on income shares using the panel vector autoregression model (panel VAR) (Sigmund & Ferstl (2019)[35]).
Building on previous studies, this study estimates the dynamic relationship between the top 1% income
share (used to measure income inequality/distribution) and other economic variables such as multi-factor
productivity, trade openness, financial development, and government spending. The estimations of panel
VAR are conducted (see Section 4).
This study postulates that all variables are endogenous and that only past values affect the system inpanel VARs because the modeling of panel VAR requires these limitations.
Section 2 provides a review of the relevant literature. Section 2 presents definitions and sources of thevariable and brief methodologies. Section 4 details the econometrics approach and reports the study’s
main findings. These findings exemplify the dynamic interaction between relative income shares and othereconomic variables. The dynamics show that the increases in the growth rate of multi-factor productivity
positively impact the increases in the top 1% income share and worsen the income inequality.

2

Studies on the Relationship between Income Inequality and Other
Economic Variables

This paper focuses on the relationship between top 1% income share and multi-factor productivity anduses other economic variables as control variables. This section reviews papers on the interaction between
income inequality and multi-factor productivity, financial development, government spending, or trade
openness in previous literature.

2.1

Multi-Factor Productivity and Technogical Change

According to Seo and Lee (2006), [34], information and communication technology investments are positively related to productivity growth in OECD countries but not in non-OECD countries, and developed
countries have advantages over developing countries concerning the progress of digitalization. They analyzed cross-sectional and time-series data from 38 countries covering 1992 to 1996.
Few articles focus on the relationship between multi-factor productivity and income inequality directly.
Fortunately, there is much literature on skill-biased technological change (SBTC) and wage inequality.
We review the literature between SBTC and wage inequality in three parts: empirics, theory, and both.
This paragraph is the empirical part. Card and DiNardo (2002)[8] find that the evidence linking wage
inequality to SBTC is surprisingly weak. A primary candidate of an explanation for the rise in inequality
in the United States is the fall in the real value of the minimum wage, leading to a steep decline in
the influence of the minimum wage in the low-wage labor market. Berman et al. (1998)[3] claim that
SBTC was pervasive over the 1980s and 1990s, coinciding in most, if not all, developed countries. Theyshow that the factor content of SBTC in manufacturing alone implies a decrease in the proportion ofless-skilled (production) workers, about eight times that attributable to increased trade. Then their

2 Studies on the Relationship between Income Inequality and Other Economic Variables

3

calculation suggests that the effects of SBTC on relative wages are an order of magnitude larger thanthose of increased trade with the developing world, assuming that demand elasticities are approximately
the same for imports and domestic production.
In theory, Galor & Moav (2000)[15] develop a growth model with the endogenous determination ofboth technological change and the composition of the labor force in which ability-biased technologicaltransition generates patterns of wage inequality that are consistent with those observed in the UnitedStates and other advanced countries.
In theory and empirics, Krusell et al. (2000)[23] evaluate how much capital-skill complementarity
has affected the skill premium in the postwar period. They first modify the standard two-factor (capitaland labor) aggregate production that distinguishes among capital equipment, capital structures, skilledlabor, and unskilled labor. This allows for different elasticities of substitution among the factors. Using U.S. time-series data, they find that the key substitution elasticities are consistent with capital-skill
complementarity and very similar to microeconomics’ estimates. Their main finding is that with empirically plausible differences in substitution elasticities, observed factor inputs can account for most of thevariation in the skill premium. They also find that their four-factor production preserves the success ofthe standard neoclassical two–actor output: it too is consistent with the behavior of income shares andthe returns on physical capital over time. Then, they suggest that the key to narrowing inequality is
better education and training for unskilled workers because unskilled labor is competing with persistently
cheaper and better capital equipment. By improving skills, workers can use new equipment and raise
their productivity, rather than be replaced by new machines.

2.2

Financial Development

In Kappel (2010)[22], both cross-country and panel regression results show that inequality and poverty are
reduced not only through enhanced loan markets but also through better-developed stock markets. Then,in his analysis, the effect of financial development becomes relatively weak, particularly for developing
countries, and government spending only reduces income inequality in high-income countries.
According to Ghossoub & Reed (2017)[16], there is strong evidence that stock markets contributeto income inequality. Moreover, there is also evidence that public debt also contributes to inequality.
Their results indicate that financial development might contribute to income inequality more than monetary policy. Furthermore, the results suggest that stock market capitalization may significantly impact
inequality across countries.

2.3

Government Spending

Pontusson et al. (2002)[32] claim that strong unions, centralized wage bargaining, a large public sector,
and left government have muted and sometimes overcome inegalitarian tendencies in pooled cross-section
time-series analysis of wage inequality in sixteen OECD countries from 1973 to 1995.
Calderón et al. (2005)[7] find that the importance of government employment improves wage distribution.

2.4

Trade Openness

Cragg & Epelbaum (1996)[10] inform the Mexican wage debate by providing explanations for the skillbiased shift in wages, using household-level data to examine how wage and employment changes differedacross industries and occupations during the reform period.
They found that the return to occupation explains close to half of the growing wage dispersion.
Workers in the highest-paid occupations have experienced the most significant wage growth. Similar toother transition economies, the supply of managers and professionals is restricted precisely when theirskills are most required. Further evidence of the importance of labor supply elasticities in equilibrium3 Data and Methodology

4wage outcomes is that low-skill occupations, such as service workers, salespeople, and transport workers,
experienced rapid employment growth but sluggish wage growth. Rising wages in occupations requiring
more sophisticated, task-specific skills indicate that the demand for skills in Mexico has rapidly increased.
They were surprised to find that industry effects had little explanatory power, accounted for little of
the rising wage dispersion, and showed only weak patterns across industries with common characteristics.
This suggests that reform-induced rent dissipation is a less important source of wage change than theoverall demand growth for general and occupation-specific skills. Still, some of the industry patterns
are interesting. Low-skill workers have small but stable industry premia, suggesting the importanceof compensating differentials for industry conditions. In contrast, high-skill workers showed significant
wage and employment growth variations across industries. They also found that Mexico experienced
general skill intensification in the nontraded services and traded manufacturing sectors. While high-skill
employment grew at the same rate for both sectors, low-skill employment grew much more slowly in the
traded sector.
Borjas et al. (1997)[5] find that the main adverse effect of immigration and trade on U.S. native
outcomes falls on workers with less than high school education. Then the combined effects of immigrationand trade may explain half of the decline in the relative wages of high school dropouts in the 1980s and1990s.
Meschi & Vivarelli (2009)[26] find that only trade with high-income countries worsens income distribution in developing countries, through both imports and exports. However, it is found that intra–developing
countries’ trade does not decline.
Autor et al. (2013)[13] find that exposure to Chinese import competition affects local markets not justthrough manufacturing employment, which unsurprisingly is adversely affected, but also along numerous
other margins. Import shocks trigger a decline in wages primarily observed outside of the manufacturingsector, and reductions in employment and wage levels lead to a steep drop in the average earnings ofhouseholds.
Dauth et al. (2014)[12] analyze the causal impact of the rise of China and Eastern Europe on theperformance of local labor markets in Germany during the period 1988 to 2008, utilizing an instrumental
variable approach by Autor et al. (2013)[13]. They find that the rise in trade exposure has led to
substantial gains in the German economy; however, trade again produces winners and losers since workers
in import-competing industries indeed face an increased risk of job churning and lower overall employment
spells.
The main finding of Roser & Cuaresma (2016)[33]’s empirical analysis is that low-wage imports todeveloped countries tend to worsen income inequality. This finding is very robust, and the variation in the
trade variable explains a large fraction of the within-country variation in the trade variable explains a large
fraction of the within-country variation of income inequality. As opposed to most other contributions to
the literature, their analysis validates the Stolper-Samuelson theorem as a corollary of the Heckscher-Ohlinmodel; thus, their study confirms the standard prediction of trade theory.

3
3.1

Data and Methodology
Data

The source of income share data is https://wid.world. Multi-Factor Productivity is from OECD (2021),
Multifactor productivity (indicator). doi: 10.1787/a40c5025-en (Accessed on 12 November 2021). Financial development is defined as the sum of the total deposit of the financial system and the stock market
capitalization in GDP. Government spending is defined as the government expenditure in GDP. Trade
openness is also defined as the sum of imports and export in GDP. The total deposit of the financial system in GDP, the stock market capitalization in GDP, the government expenditure in GDP, and the sum ofimports and export in GDP are all from World Bank Open Data (https://data.worldbank.org). The
list of countries is Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece,3 Data and Methodology

govspend
MF P
p99p100

Tab. 1: Variables
Definition
the sum of the total deposit of the financial systemand the stock market capitalization in GDP (%)
the government expenditure in GDP (%)
multi-factor productivity (100 in 2015 in all countries)
the top 1% income sharetrade
growth or g

the sum of import and export in GDP (%)
the suffix standing for the growth rate of the variable

Variablef indev

5

SourceWorld BankWorld BankOECDWorld Inequality
Database
World Bank

Ireland, Israel, Italy, Japan, Korea, Luxembourg, Netherlands, New Zealand, Norway, Portugal, Spain,
Sweden, Switzerland, USA. We show the histograms of variables (Fig.1). These histograms in Fig.1 showthat the distributions of the growth rates of variables are nearer normal than the distributions of thelevels of variables. This is one reason why the growth rates are preferable to the levels. However, thegrowth rates of variables have fat tails compared to the normal distribution.

3.2

Methodology

3.2.1

Panel Specification Test

The first generation of the panel unit root test is different from the second generation of that in the
viewpoint of the cross-sectional dependence. There are several types of nonstationary tests used in panel
data. We apply the first generation of the panel unit root test to the unbalanced panel data. The
methods are Levin, Lin & Chu (2002)[24], Breitung (2001)[6], Im, Pesaran & Shin (2003)[19], Augmented
Dickey-Fuller based on Maddala & Wu (1999)[25] and PP based on Choi (2001)[9]. All tests include thetime trends and the intercepts. Then, the Cross-sectional Dependence (CD) tests (Pesaran (2015)[29],and Pesaran (2021)[30]) justify panel regression models and panel vector autoregression models as follows
because the CD tests cannot reject the null hypothesis of the cross-sectional independence.
3.2.2

Panel Regression

Dynamic panel regression has developed since the finding of Nickell (1981)[27] bias. Arellano & Bond
(1991)[1] overcome Nickell bias, introducing the generalized methods of moments (GMM) estimators
popularized by Hansen (1982)[18]. However, their estimation has a large bias when the coefficient of
the lag of the dependent variable is near one. In other words, the dependent variable has a unit root.
With additional moment conditions, Blundell & Bond (1998)[4] develop the system GMM estimator todeal with this bias. However, these arguments focused on single equation dynamic panel data models.
The static panel regressions, aka the fixed-effects and random-effects least square dummy variables andordinary least squares, are also estimated.
3.2.3

Panel Granger Causality Test

Granger (1969)[17] explored the relationships between certain classes of econometric models involving
feedback and the functions arising in spectral analysis. In the two-variable case, the feedback mechanism
can be broken down into two causal relations. The cross–pectrum can be considered the sum of two crossspectra, each closely connected with one of the causations.
Dumitrescu & Hurlin (2012)[14] propose a simple Granger (1969)[17] non-causality test for heterogeneous panel data models. Under the null hypothesis of homogeneous non-Causality, there is no causal

3 Data and Methodology

6findevfindev_growth
200
150

100

100
50
50
0

0
0

200

400

600

−0.5

0.0

govspend

0.5

govspend_growth
200

60
150
40
100
20

50

0

0
10

15

20

25

−0.2

−0.1

MFP

0.0

0.1

MFP_growth
125

150

count

100
100

75
50

50
25
0

0
50

75

100

125

−0.05

p99p100

0.00

0.05

p99p100_growth

100
200

75
50

100
25
0

0
0.05

0.10

0.15

0.20

−0.4

−0.2

trade

0.0

0.2

0.4

trade_growth
150

150
100
100
50

50

0

0
0

100

200

300

400

−0.2

value

Fig. 1: Histograms

0.0

0.2

4 Econometric Analysis

7

relationship for any of the cross-section units of the panel. Under the alternative, there are two subgroupsof cross-section units: one characterized by causal relationships from x to y (even though the regression
model is not necessarily the same) and another subgroup for which there is no causal relationship fromx to y. There are four main advantages. First, the standardized, average Walds statistics are simple tocompute and have a normal asymptotic distribution. Second, the original paper’s Monte Carlo simulations
show that its panel statistics lead to a substantial increase in the power of the Granger non-causality tests
even for samples with very small T and N dimensions. Third, its test statistics (based on a cross-section
average of individual Wald statistics) do not require any particular panel estimation. Finally, the test canbe easily implemented in unbalanced panels or panels with different lag orders K for each individual.
3.2.4

Panel Vector Autoregression

Sigmund & Ferstl (2019)[35] extend two general methods of moment (GMM) estimators to panel vector autoregression (PVAR) models with p lags of endogenous variables, predetermined and exogenous
variables. They first extend the first difference GMM estimator and the system GMM estimator to thisextended PVAR model.
The PVAR model has the impulse response function as the vector autoregression in time series analysis.
There are two popular impulse response functions: the orthogonalized impulse response function (OIRF)
and the generalized impulse response function (GIRF). The orderings of the dependent variables affectthe results of the OIRF; however, these orderings have no impact on the GIRF analysis.
This study estimates the simple PVAR model as follows:
yi,t =pXΦl yi,t−l + µi + ϵi,t , i = 1, 2, . . . , N, t = 1, 2, . . . , T

(1)

l=1

where m is the number of jointly determined dependent variables, N is the number of the cross-section′
dimension, T is the number of time series the dimension, yi,t = (y1,i,t , y2,i,t , . . . , ym,i,t ) is m × 1 vector of
jointly determined dependent variables, and {Φi , i = 1, 2, . . . , p} are m × m coefficient matrices.
Pesaran & Shin (1998)[28] choose to shock only one element, say its rth element, and integrate out
the effects of other shocks using the historically observed distribution of the errors. In this case, we have
GIRF (k, r, Σϵ ) = E[yi,t+k |ϵi,t,r = δr , Σϵ ] − E[yi,t+k |Σϵ ]By Setting δr =(2)p
Σϵ,r,r , we obtain the generalized impulse response function by
GIRF (k, r, Σϵ ) = Ak Σϵ (σr,r )−1/2(3)

where σr,r is the rth diagonal element of Σϵ and the m × m coefficient matrices Al can be obtained using
the following recursive relations:
Al = Φ1 Al−1 + Φ2 Al−2 + · · · + Φp Al−p , l = 1, 2, . . . ,(4)with A0 = Im (m × m identity matrix) and Al = 0 for l < 0.

4
4.1

Econometric Analysis
Panel Specification Test

Section 4.1 explores the property of the panel data in this study. Section 4.1.1 tests the existence of panel
unit root in each variable. Then, Section 4.1.2 checks the assumption of cross-sectional independence.

4 Econometric Analysis

8

Tab. 2: Panel Unit Root Test on log(p99p100)
MethodStatistic Prob. N Obs
Levin, Lin & Chu t*
0.38129 0.6485 23 846
Breitung t-stat-3.91411 0.0000 23 823
Im, Pesaran and Shin W-stat -0.8434 0.2004 23 846
ADF - Fisher Chi-square
66.4119 0.0260 23 846
PP - Fisher Chi-square
45.1714 0.5059 23 897

Tab. 3: Panel Unit Root Test on p99p100 g
MethodStatistic Prob. N Obs
Levin, Lin & Chu t*-16.1750 0.0000 23 828
Breitung t-stat-9.38374 0.0000 23 805
Im, Pesaran and Shin W-stat -17.5582 0.0000 23 828
ADF - Fisher Chi-square
388.718 0.0000 23 828
PP - Fisher Chi-square
1295.75 0.0000 23 874

4.1.1

Panel Unit Root Test

There is no package applied to the unbalanced panel data for panel unit root tests in R. We use Eviews 11
to test the nonstationarity of the unbalanced panel data. The null hypothesis of these tests is a common
unit root process for Levin, Lin & Chu (2002)[24] and Breitung (2001)[6]. In null hypotheses, Im, Pesaran& Shin (2003)[19], Augmented Dickey-Fuller based on Maddala & Wu (1999)[25] and PP based on Choi
(2001)[9] assume that the individual unit root process exists. Because the log level of each variable hasa mixed result, tests cannot reject the nonstationarity. However, the growth rate of each variable shows
the stationarity in each test.
4.1.2

Cross-sectional Dependence Test

We test the cross-sectional dependence in panels (Pesaran (2015)[29] and Pesaran (2021)[30]) to justify
panel regressions and panel vector autoregressions (PVAR) as follows. The null hypothesis is that crosssectional independence exists. The acceptances of the null in Model 3, Model 7, and Model 8 are essential
because the models of panel regressions and PVAR postulate cross-sectional independence.
Model 1:log(p99p100) ∼ 1(5)

Tab. 4: Panel Unit Root Test on log(f indev)
MethodStatistic Prob. N Obs
Levin, Lin & Chu t*
2.05385 0.9800 23 680
Breitung t-stat-0.67230 0.2507 23 657
Im, Pesaran and Shin W-stat -0.51422 0.3035 23 680
ADF - Fisher Chi-square
54.4230 0.1845 23 680
PP - Fisher Chi-square
68.2787 0.0181 23 759

4 Econometric Analysis

Tab. 5: Panel Unit Root Test on f indev g
MethodStatistic Prob. N Obs
Levin, Lin & Chu t*-7.62923 0.0000 23 664
Breitung t-stat-5.40448 0.0000 23 641
Im, Pesaran and Shin W-stat -11.1748 0.0000 23 664
ADF - Fisher Chi-square
209.502 0.0000 23 664
PP - Fisher Chi-square279.195 0.0000 23 728

Tab. 6: Panel Unit Root Test on log(govspend)
MethodStatistic Prob. N Obs
Levin, Lin & Chu t*-0.63211 0.2637 23 832
Breitung t-stat-0.44489 0.3282 23 809
Im, Pesaran and Shin W-stat -1.87622 0.0303 23 832
ADF - Fisher Chi-square
68.1486 0.0186 23 832
PP - Fisher Chi-square
35.9393 0.8568 23 882

Tab. 7: Panel Unit Root Test on govspend g
MethodStatistic Prob. N Obs
Levin, Lin & Chu t*-13.7490 0.0000 23 815
Breitung t-stat-12.0659 0.0000 23 792
Im, Pesaran and Shin W-stat -14.1224 0.0000 23 815
ADF - Fisher Chi-square275.326 0.0000 23 815
PP - Fisher Chi-square552.375 0.0000 23 859

Tab. 8: Panel Unit Root Test on log(M F P )MethodStatistic Prob. N Obs
Levin, Lin & Chu t*-3.90954 0.0000 23 672
Breitung t-stat
1.27848 0.8995 23 649
Im, Pesaran and Shin W-stat -2.14014 0.0162 23 672
ADF - Fisher Chi-square
76.3204 0.0033 23 672
PP - Fisher Chi-square
57.4062 0.1207 23 715

Tab. 9: Panel Unit Root Test on M F P g
MethodStatistic Prob. N Obs
Levin, Lin & Chu t*-12.5293 0.0000 23 671
Breitung t-stat-10.7396 0.0000 23 648
Im, Pesaran and Shin W-stat -12.7406 0.0000 23 671
ADF - Fisher Chi-square
232.559 0.0000 23 671
PP - Fisher Chi-square
447.537 0.0000 23 692

9

4 Econometric Analysis

10

Tab. 10: Panel Unit Root Test on log(trade)
MethodStatistic Prob. N Obs
Levin, Lin & Chu t*-3.46823 0.0003 23 854
Breitung t-stat-4.13334 0.0000 23 831
Im, Pesaran and Shin W-stat -2.84178 0.0022 23 854
ADF - Fisher Chi-square
68.7635 0.0164 23 854
PP - Fisher Chi-square
47.6360 0.4139 23 882

Tab. 11: Panel Unit Root Test on trade g
MethodStatistic Prob. N Obs
Levin, Lin & Chu t*-19.5763 0.0000 23 835
Breitung t-stat-17.3235 0.0000 23 812
Im, Pesaran and Shin W-stat -17.5588 0.0000 23 835
ADF - Fisher Chi-square335.193 0.0000 23 835
PP - Fisher Chi-square882.426 0.0000 23 859

Model 2:log(p99p100) ∼ lag(log(p99p100), 1)(6)log(p99p100) ∼ log(f indev) + log(govspend) + log(M F P ) + log(trade)(7)log(p99p100) ∼ lag(log(p99p199), 1) + log(f indev) + log(govspend) + log(M F P ) + log(trade)(8)

Model 3:Model 4:Model 5:
p99p100 g ∼ 1(9)p99p100 g ∼ lag(p99p100 g, 1)(10)p99p100 g ∼ f indev g + govspend g + M F P g + trade g(11)

p99p100 g ∼ lag(p99p100 g, 1) + f indev g + govspend g + M F P g + trade g(12)Model 6:
Model 7:
Model 8:

Tab. 12: Pesaran CD test for cross-sectional dependence in panels
Model 1Model 2 Model 3 Model 4
p-value < 2.2e-16 < 2.2e-16
0.5818 0.001172
Model 5Model 6 Model 7 Model 8
p-value < 2.2e-16 < 2.2e-16
0.80160.8176

4 Econometric Analysis

11

Tab. 13: Panel Regression Results (Log Level)
N = 23, T = 19 − 33, Obs = 643
GMM(AB)
GMM(BB)
LSDV-FE
LSDV-RE
OLS-5.907368-4.668240(0.662111)(0.988862)
intercept******
0.641355
0.9107513(0.0198344)
lag(log(p99p100), 1) (0.052520)******
0.056418
0.0126924
0.104204
0.113176
0.167354(0.0073970)(0.058388)(0.053222)(0.079401)
log(f indev) (0.026703)
**
*
*
**
**
-0.296568-0.0239185-0.601121-0.561398-0.206058(0.0192916)(0.135123)(0.130038)(0.204723)
log(govspend) (0.070712)
***
***
***
0.227112-0.0339899
0.990367
0.991965
0.579963(0.0160204)(0.212212)(0.213463)(0.325578)
log(M F P ) (0.117784)
*
**
***
***
*
0.083366-0.0072020
0.089698
0.056818-0.105420(0.0072473)(0.094406)(0.082888)(0.074719)
log(trade) (0.040065)**
20.50196
0.50818
0.25677
Adj.R
The cluster-robust standard errors are in the parentheses. ***, **, * denote
p-values at the 1%, 5%, 10%, level of significance, respectively.

4.2

Panel Regression

We, at first, estimate the panel regression based on the single equation model to understand the relationship between p99p100 and other variables (f indev, govspend, M F P and trade) because the cross-sectional
dependence tests in Section 4.1.2 favor the models as follows. First model has log(p99p100) as the dependent variable and log(f indev), log(govspend), log(M F P ) and log(trade) as explanatory variables.
Second model has p99p100 g as the dependent variable and f indev g, govspend g, M F P g and trade g
as the explanatory variables. Third model has the variables in second model and lag(p99p100 g, 1) as
the added explanatory variable. Ignoring the nonstationarity, we estimate the relationship between thelogarithms of economic variables in Tab.13. Because we cannot rule out the panel unit roots in thedependent and explanatory variables in Section 4.1, we use the log-based growth rate of each variable(growth ratet := log(valuet ) − log(valuet−1 )). We present the four types of the panel regression; Arellano& Bond (1991)[1] GMM estimator (AB), Blundell & Bond (1998)[4] system GMM estimator (BB), thefixed-effect estimator of the least square dummy variable (LSDV-FE), the random-effect estimator of theleast squares dummy variable (LSDV-RE) and the ordinary least square estimator (OLS). All results
here have similar coefficients and cluster-robust standard errors (Tab.14). Financial developments andmulti-factor productivities positively impact the top 1% income share, and governmental spendings negatively affect the top 1% income share. Finally, trade openness does not have a statistically significant
relationship with the top 1% income share.

4 Econometric Analysis

12

Tab. 14: Panel Regression Results (Growth Rate)
N = 23, T = 18 − 32, Obs = 613
GMM(AB)
GMM(BB)
LSDV-FE
LSDV-RE
OLS
0.00062332
0.00062332(0.00284872)(0.00251719)
intercept-0.096297-0.084107(0.063488)
lag(p99p100 g, 1) (0.063690)
0.123438
0.126586
0.120760
0.12223124
0.12223124(0.036566)(0.035943)(0.036712)(0.03757021)(0.03707941)
f indev g
***
***
***
***
***
-0.681336-0.649749-0.592580-0.60060573-0.60060573(0.164601)(0.146381)(0.170290)(0.14805137)(0.16241092)
govspend g
***
***
***
***
***
1.067471
1.045292
1.015784
0.95643111
0.95643111(0.220776)(0.185884)(0.221087)(0.21385474)(0.19648973)
MF P g
***
***
***
***
***
0.039275
0.050342
0.056217
0.05589000
0.05589000(0.051375)(0.046292)(0.048856)(0.04594620)(0.04779425)
trade g20.16794
0.19677
0.19677
Adj.R
The cluster-robust standard errors are in the parentheses. ***, **, * denote
p-values at the 1%, 5%, 10%, level of significance, respectively.

4 Econometric Analysis

13

Tab. 15: Panel Granger Causality Results (Log Level)log(p99p100)log(p99p100)(−1)-

log(f indev)
2.3051(0.02116)**log(govspend)
3.1687(0.001531)***
2.9834(0.00285)***

log(M F P )1.6182(0.1056)
3.4402(0.0005812)***
2.5783(0.009927)***

log(trade)
3.1606(0.001574)***
2.7966(0.005165)***
8.7705(< 2.2e-16)***
9.2043(< 2.2e-16)***

4.0018(6.287e-05)
log(f indev)(−1)***
5.80932.9162(6.273e-09)(0.003543)
log(govspend)(−1)******
8.51165.04053.1442(4.642e-07)(0.001666)
log(M F P )(−1) (< 2.2e-16)
***
***
***
2.17812.623
1.73261.6287(0.0294)(0.008715)(0.08317)(0.1034)log(trade)(−1)******
The p-values are in the parentheses. ***, **, * denote rejection of null hypothesisat the 1%, 5%, 10%, level of significance, respectively.
(-1) stands for the consideration of the one-lag in panel Granger Causality test.

4.3

Panel Granger Causality Test

Panel Granger Causality test (Dumitrescu & Hurlin (2012)[14]) is the extension of Granger causality intime series analysis (Granger (1969)[17]) to panel data. The null hypothesis of this test is that the nonexistence of the Granger causality is for all individuals, and the alternative is that at least one individual
has the Granger causality. Because most models reject the non-Granger causality, these results validate
the panel vector autoregression (PVAR) model as follows (section 4.4). The PVAR model postulates thatsome lags in the dependent variables determine the current dependent variables.

4.4

Generalzed Impulse Response Functions of Panel Vector
Autoregression

Fig.2 is the estimation result of the generalized impulse response functions (GIRF) in the two-lag panel
vector autoregression (PVAR) model based on the log level of the variables in Tab.1 over 20 years. Fig.3 isalso the estimation result of GIRFs in the two-lag PVAR model based on the growth rate of the variables
in Tab.1 over ten years.
Fig.2 and Fig.3 show movements of all economic variables have similar directions in the same twolag model of the PVAR. However, Fig.2 indicates the log-level PVAR model has wider 95% bootstrap
confidence intervals than the growth rate PVAR model in Fig.3. Fig.3 has more precise estimation results
than Fig.2 to forecast the dynamic impact of each economic variable. This result shows that we still needto consider the panel nonstationarity in the PVAR.
Fig.2 and Fig.3 are the estimation results of generalized impulse response functions in the PVAR
mode with the two lags. These results match the results of panel regression (Tab.14). The impacts on
p99p100 g of f indev g, M F P g are positive, the impact of govspend g is negative, and the impact oftrade g is still unclear.

-4 Econometric Analysis

14

Generalized impulse response function
GIRF and 95% confidence bands
log_findev on log_findev log_findev on log_govspend

log_findev on log_MFP

log_findev on log_p99p100

log_findev on log_trade

0.2

0.1

0.0

−0.1
log_govspend on log_findev
log_govspend on log_govspend
log_govspend on log_MFPlog_govspend on log_p99p100
log_govspend on log_trade
0.2

0.1

0.0

−0.1
log_MFP on log_findev

log_MFP on log_govspend

log_MFP on log_MFP

log_MFP on log_p99p100

log_MFP on log_trade

0.2

0.1

0.0

−0.1
log_p99p100 on log_findevlog_p99p100 on log_govspendlog_p99p100 on log_MFPlog_p99p100 on log_p99p100log_p99p100 on log_trade
0.2

0.1

0.0

−0.1
log_trade on log_findev

log_trade on log_govspend

log_trade on log_MFP

log_trade on log_p99p100

log_trade on log_trade

0.2

0.1

0.0

−0.1
5

10

15

20

5

10

15

20

5

10

15

20

5

10

15

20

5

10

15

20

steps

Fig. 2: Generalized Impulse Response Analysis of Panel Vector Autoregression (Log Level)4 Econometric Analysis

15

Generalized impulse response function
GIRF and 95% confidence bands
findev_g on findev_gfindev_g on govspend_g

findev_g on MFP_gfindev_g on p99p100_gfindev_g on trade_g

0.10

0.05

0.00

−0.05
govspend_g on findev_g govspend_g on govspend_g govspend_g on MFP_g

govspend_g on p99p100_g govspend_g on trade_g

0.10

0.05

0.00

−0.05
MFP_g on findev_g

MFP_g on govspend_g

MFP_g on MFP_g

MFP_g on p99p100_g

MFP_g on trade_g

p99p100_g on MFP_g

p99p100_g on p99p100_g

p99p100_g on trade_g

trade_g on MFP_g

trade_g on p99p100_g

trade_g on trade_g

2

2

0.10

0.05

0.00

−0.05
p99p100_g on findev_g p99p100_g on govspend_g
0.10

0.05

0.00

−0.05
trade_g on findev_g

trade_g on govspend_g

0.10

0.05

0.00

−0.05
2

4

6

8

10

2

4

6

8

10

2

4

6

8

10

4

6

8

10

4

6

8

10

steps

Fig. 3: Generalized Impulse Response Analysis of Panel Vector Autoregression (Growth Rate)

5 Conclusion

16

Tab. 16: Panel Granger Causality Results (Growth Rate)
p99p100 g
f indev g
govspend g
MF P gtrade g0.43688
6.13311.13752.6943(0.6622)(8.618e-10)(0.2553)(0.007053)
p99p100 g(−1)******
4.0556
7.28152.08442.339(5e-05)(3.302e-13)(0.03713)(0.01934)
f indev g(−1)
***
***
**
**
2.9132-1.0386
3.4668-1.5811(0.299)(0.0005266)(0.1138)
govspend g(−1) (0.003578)******
0.26927
2.32271.38533.1069(0.7877)(0.02019)(0.166)(0.00189)
M F P g(−1)*****-0.026885
0.54106
2.68835.8272(0.9786)(0.5885)(0.007181)(5.636e-09)
trade g(−1)******
The p-values are in the parentheses. ***, **, * denote rejection of null hypothesisat the 1%, 5%, 10%, level of significance, respectively(-1) stands for the consideration of the one-lag in panel Granger Causality test.

5

Conclusion

Panel regressions in Section 4.2 ignore the dynamics, and the impulse response function analysis of thepanel vector autoregressions (PVAR) in Section 4.4 neglects the effects of variables at the same time.
However, panel regressions in Section 4.2 and the PVAR model in Section 4.4 have the same result thatthe growth of the multi-factor productivity increases the growth of the top 1% income share. Then, trade
openness has no statistically significant effects on the top 1% income share, financial developments havea positive impact on the top 1% income share, and government spendings affect the top 1% income sharenegatively in Section 4.2 and 4.4.

Compliance with Ethical Standards
Funding
The JSPS KAKENHI Grant Number 15H05729 and 20H05633 financially supported this study.

Conflict of interest
The author has no conflicts of interest to declare.

Ethical approval
This article does not contain any studies with human participants performed by any authors.

5 Conclusion

17

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