DP2022-13

A S p a t i a l P a n e l D a t a A n a ly s i s o f
Fertility Rates: Unraveling Two
Myths*
Yosuke SASAKI
Takashi KAMIHIGASHIMarch 30, 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.

A Spatial Panel Data Analysis of Fertility Rates: Unraveling
Two Myths
Yosuke Sasaki∗

Takashi Kamihigashi†
2022/03/30

Abstract
This paper analyzes the relationship between total fertility rates and other socio-economic variables. The main results are against two myths: rural areas have higher total fertility rates thanurban ones, and economic inequality harms total fertility rates. The key is a spatial panel data
analysis with the lag between inception and birth. Here, spatial panel data analysis improves theefficiency of explanatory variables, and the lag produces a different result from former papers.

1

Introduction

In 2019, only 864 thousand babies were born in Japan, with a 126 million population, the number isthe lowest in the statistics, and the percentage of the total population is only 0.69%. Several academics
and journalists cite the chronically low birth rates as one of Japan’s crucial challenges. Because East
Asian countries have similar problems, and even the United States now records the lowest birth rates in
its history, the mechanism between the total fertility rates (TFRs) and other socio-economic variables
are supposed to be vital to find the solution and evaluate the policy on low total fertility rates.
Numerous people suggest that income inequality and educational costs have adverse effects on TFRs.
One may question if TFRs relate to the Gini index and the admission rates to colleges and the age cohort,and one may also be interested in the time series structure of TFRs. Moreover, several think rural areas
have higher TFRs than cities, and economic inequality affects TFRs negatively.
Several papers suggest that the factors affect TFRs, such as female employment, male and female wage, wealth transfer between the generations, parental care costs, the number of children in
one household. Moreover, in Japan, the policies to promote parental care should be evaluated. However, researchers have not estimated the relationship between the fertility rates, the Gini index, and theadmission rates after high school.
The spatial panel data analyses show that the increase of the Gini index has adverse effects on TFRs.
Also, the relationship between the population density and TFRs is the inverted U-shape. Furthermore,
real expenditure and TFRs have an inverted J-shape relationship. Then, the impulse response of thepanel vector autoregression (Sigmund & Ferstl 2019[22]) indicates that even a one-shot policy on TFRs
has favorable effects for about 20 years and adverse economic shocks have opposite effects. This paper’s
methods also validate the Easterlin hypothesis on fertility (Macunovich & Easterlin 2018[15]). This
paper shows that the negative relationship between TFRs and the population density is inverted-Ushaped, using the simple unbalanced panel data with large cross-sectional dimensions and the balanced∗†

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

1

panel data with smaller cross-sectional dimensions denser time-series ones. However, this analysis faults
that one does not consider the lag between inception and birth. With this lag, there is no statisticallysignificant relationship between population density and TFRs. The relationship between economic
inequality and TFRs is statistically false with this lag. The bivariate nonparametric regression andtime series of Moran I and Theil indices indicate the negative relationship between TFRs and colleges’
admission rates. However, the panel non-Granger causality test and the spatial panel regression do notshow a clear relationship between TFRs and the admission rates.

2

The Literature and Data Characteristics

2.1

The Literature

2.1.1

Economic and Analytical Studies on Total Fertility Rates

Several social scientists work on human fertility. However, here we focus on socio-economic approaches.
Becker (2009)[4] considers the interaction between the quantity and quality of children to explain
why education per child tends to be lower in families having more children.
Willis (1973)[23] points out that the interaction model captures an essential empirical regularity inthe cross-section relationship between fertility and measures of husband’s income and wife’s educationthat has become apparent in the emergence of a U-shaped relationship between fertility and income.
The predictions of the theoretical model of fertility demand developed in his paper fit well with thispaper’s estimation result.
Prettner et al. (2013)[19] theoretically and empirically show that if individuals put more weight on
education, they reduce fertility, increase educational investments, and hold consumption, savings, andhealth investments constant.
Lutz et al. (2006)[14] empirically find a consistent and significant negative relationship between
human fertility and population density, using fixed-effects models on the time series of 145 countries
and controlling for critical social and economic variables such as GDP per capita, infant mortality,
female labor force participation, and female literacy. In the empirical study, Kulu (2013)[12] shows thatthe desired family size in small towns and rural areas is larger than that in urban areas.
Myrskylä et al. (2009)[17] find that TFRs in highly developed countries have inverted J-shape
curves over the Human Development Index (HDI) except Japan, Canada, and South Korea. The HDI
is a summary measure of average achievement in key dimensions of human development: a long andhealthy life, being knowledgeable, and having a decent living(United Nations Development Programme’s
Human Development Report[20]). However, their finding is not robust to the redefined HDP or thedecomposition of the HDI into its subindices of education, the standard of living, and health (Harttgen& Vollmer 2014[9]).
Aldieri & Vinci (2012)[1] estimate the correlation between the female’s education and the number ofchildren in Italy and use the partner’s education to take into account the family dimension. They use azero-inflated Poisson regression and observe a negative correlation between the number of children born
and the educational level.
2.1.2

Sociological and Demographic Studies on Total Fertility Rates

As Pampel & Peters (1995)[18] review the Easterlin hypothesis, Easterlin presented his basic argumentthat swings in relative cohort size and resulting levels of relative income among cohorts of child-bearing
age produced the baby boom and then the baby bust. Initially, his discussion is concerned with describing demographic cycles (Easterlin 1968[7]). Moreover, the linkage between higher birth rates and2

adverse social-economic effects arises because large cohorts face crowding problems in three major social
institutions such as the family, education, and labor markets (Macunovich & Easterlin 2018[15]).
The second demographic transition (SDT) explains the inverted J-shape curve in sociology anddemography. Zaidi & Morgan (2017)[24] summarize the SDT as three arguments: the shift from kingchild to king-couple, the Maslowian drift, the rise of individualism, and pushback against economic
explanations.
2.1.3

Studies on Total Fertility Rates in Japan

Gini Index
Few researchers have the Gini index as the explanatory variable for total fertility rates. The panel datain this study shed light on the Gini index’s effects on TFRs.
Educational level
Some papers regress the female educational levels to TFRs.

2.2

Data

Total Fertility Rates (T F R) and the population of each cohort of every prefecture (p0to4, p5to9, p10to14,
p15to19, p20to24, p25to29, p30to34, p35to39, p40to44, p45to49, p50to54, p55to59, p60to64, p65to69,
p70to74, p75to79 and pover80) are from the National Institute of Population and Social Security Research(Japan) and its former organization. Municipal data on T F R is available online from the Statistical
Bureau of Japan.
Figure. 1 depicts the movements of total fertility rates of each prefecture in Japan.
Data on the Gini index (gini) and mean real expenditure real exp mean are estimated originallyfrom the microdata courtesy of the Statistic Bureau, Ministry of Internal Affairs and Communications.

3

Results of the Econometric Approaches

3.1

Nonparametric Analysis and the Panel Non-Granger Causality

At first, one can nonparametrically regress the bivariate relationship between the total fertility rates(TFRs) and other socio-economic variables. Figure 2 shows the relationship between T F R and pop density
and one between T F R and ad rate appear inverse corrletions.
The relationship between T F R and pop density and one between T F R and real exp mean 100 havethe panel Granger causality in both directions. These tests are based on Dumitrescu & Hurlin (2012)[6].
They note that the null of Homogeneous Non-Causality does not provide any guidance concerning thenumber or the identity of the particular panel units for which the null of non-causality is rejected.
In other words, only non-Granger causality is meaningful. However, gini and ad rate have the panel
Granger causality from T F R, and T F R has the panel non-Granger causality from gini and ad rate.
This gini and ad rate do not determine T F R in the non-Granger causality standpoint. Then, the panel
non-Granger Causality test assumes that the individuals of the panel have no cross-sectional dependence.

3.2

The Relationship Between the Total Fertility Rates in the Municipal
Level and the Population Density

Because municipal data on T F R have relatively large cross-sections (at least two thousand) and seventerms, panel data analysis is available. The estimation is based on the fixed-effects model, and the Least3

nationwide

hokkaido

aomoriiwate

miyagiakita

yamagata

fukushimaibarakitochigigumma

saitama

chiba

tokyo

kanagawa

niigatatoyamaishikawa

fukui

yamanashinaganogifu

shizuoka

aichi

mie

shiga

kyoto

osaka

hyogo

nara

wakayama

tottorishimaneokayamahiroshimayamaguchi

tokushima

kagawa

ehime

kochi

fukuokasaganagasakikumamotooitamiyazakikagoshimaokinawa

2.0
1.5
1.0

2.0
1.5
1.0

2.0
1.5
1.0

TFR

2.0
1.5
1.0

2.0
1.5
1.0

2.0
1.5
1.0

2.0
1.5
1.0
1980199020002010 1980199020002010 1980199020002010 1980199020002010 1980199020002010 1980199020002010

year

Figure 1: Total Fertility Rates in Prefectures4

1980199020002010

2.0

2.0

TFR

2.5

TFR

2.5

1.5

1.5

1.0

1.0

0

2000

4000

6000

250000

pop_density

300000350000400000

real_exp_mean_100

2.0

2.0

TFR

2.5

TFR

2.5

1.5

1.5

1.0

1.0

0.30

0.35

0.40

20

30

gini

40

ad_rateFigure 2: Nonparametric Regression Results

5

50

60

Table 1: Estimation Results: The Large Cross-Sectional Panel (n = 3755, T = 1 − 7, and N = 19235)
log(T F R) Fixed Effects Random Effects Pooling
0.2332224
0.36260635(0.0154558)(0.01481855)
Intercept******
0.6627606
0.1294545
0.08314563(0.0057179)(0.00562807)
log(pop density) (0.0654288)
***
***
***
-0.0504881-0.0140688-0.01043975(0.0056033)(0.0005074)(0.00051242)(log(pop density))2
***
***
***
2
-0.11254
0.16
0.19968
Adj.R. < 0.10 ∗ < 0.05 ∗∗ < 0.01 ∗ ∗ ∗ < 0.001
Double clustered standard deviations are in parentheses.

Square Dummy Variable as explanatory variables are few, and the omitted variables bias is severe. The
Hausman test between the fixed effects and the random effects supports this reasoning.

exp(0.6627606/2(0.0504881)) = 708.7712(1)This subsection’s relationship between the T F R and pop density is spurious because it is naturalthat we consider the lag between inception and birth.

3.3

The Spatial Autocorrelation and the Inequality Index

Moran’s I in the spatial autocorrelation of cross-sectional spatial data has an analogy to Durbin-Watsonin the serial autocorrelation of time-series data (Li et al., 2007[13]). Then, the higher Moran’s I is, thehigher the degree of spatial autocorrelation is. The spatial dependence (global spatial autocorrelation)measure of Moran’s I is represented by equation 2 (the formulation follows Rey 2004[21]):P P
n i j wij zit zjt
P 2(2)
M oran It =
s
zit
(3)
where n is the number of prefectures (i.e.n = 47), zit and zjt are the deviation of each variable from themean of each prefecture in each year (the prefecture i, j and the year t), wij are the elements of weightmatrix W (n × n) and it is equal to 1 if i and j are neighbors and 0 if they are not; s is the sum of allelements of W (spatial weights). One note that n, s and wij are invariant over time.
Figure 3 plots Moran’s I of each variable. The critical value, which rejects at the significant level
0.05 null hypothesis that there is no spatial correlation, is about 0.14 in the case of this dataset and is
added as The Critical Value(horizontal lines) in line plots of Figure 3.
According Rey (2004)[21], Figure 4 plots Moran’s I and Theil index of each variable. The Theil
index of inter-prefecutral inequalities is presented as follows(the formulation follows Cowell 2011[5]):nyit
1 X yitT heilt =log( )
n i=1 µt
µt
6

(4)

0.6
0.3

variable

0.4

variable

pop_density
real_exp_mean

0.2moran_I

moran_I

TFR

0.2
p0to4
p5to9

The Critical Valuegini

The Critical Value

col_rate

p10to14

0.1

0.0

1980

1990

2000

2010

1980

1990

year

2000

2010

year
0.6

0.5
0.5

0.4

variable

0.3

moran_I

moran_I

0.4

p15to19
p20to240.2

variable
p30to34
p35to39

0.3

p25to29

p40to44

The Critical Value

0.2

0.1

The Critical Value
0.1

0.0
1980

1990

2000

2010

1980

1990

year

2000

2010

year

0.6

0.4

variable

0.4

variablemoran_I

moran_I

p45to49
p50to54
p55to59
p60to640.2

0.3

p65to69
p70to74
p75to79
pover80

0.2

The Critical Value
The Critical Value
0.1
0.0
1980

1990

2000

2010

1980

year

1990

2000

2010

year

Figure 3: Moran’s I of Each Variable in This Study7where yit is the value of each variable in each prefecuture and each year, µt is the mean over Japanin each year and n is the number of prefectures(i.e.n = 47). Rey (2004)[21] empirically shows higher
Moran’s Is tend to be higher Theil indices. Theil indices have the drawback that they are affected by the
sample size. With higher inequality of T F R, ad rate has lower inequality over 36 years in this paper’s
dataset. Figure 4 indicates that inequality of T F R and that of ad rate have a negative relationship,
and one must consider geographical inequality to analyze the relationship of both. Next, we focus on
spatial panel regression as the model of spatial autocorrelation.

3.4

The Spatial Panel Data Analysis

Hsiao (2014)[10] suggests that panel data analysis could control the impact of omitted variables (orindividual or time heterogeneity) and generate more accurate predictions for individual outcomes thanthe cross-sectional analysis and time-series analysis.
One can apply the spatial panel data analysis to the analysis of Total Fertility Rates (TFRs) to gain
more efficiency than the static panel analysis such as the Least Squares Dummy Variable (LSDV) or the
dynamic panel analysis such as the dynamic panel Generalized Method of Moments (GMM) (Arellano& Bond, 1991[2]).
One can conduct the Maximum Likelihood (ML) (Baltagi et al. 2007[3]) and the Generalized Moments (GM) estimation (Kapoor et al. 2007[11]) to estimate the spatial panel. In this paper, the fixed
effects spatial error model is estimated because the fixed effects model is generally more appropriate than
the random effects model since adjacent spatial units’ space-time data are located in unbroken study
areas (Elhorst 2014[8]). Furthermore, as the spatial error model’s parameters are considered marginal
effects themselves (Elhorst 2014[8]), these parameters are easily interpretable. Then, other models using
the spatial weight matrix are similar to the spatial error model. This paper’s notations are based on
Millo and Piras (2012)[16], and we use their package. This paper utilizes the spatial error model tocontrol spatial autocorrelation. In the spatial error model, spatial error terms and the spatial lag terms
are considered the spatial autocorrelation terms in the error terms.
The fixed effects spatial error panel data model estimated by the ML and the GM in this paper is
as follows:y = (ιT ⊗ IN ) µ + Xβ + ε
ε = ρ (IT ⊗ WN ) ε + ν

(5)
(6)
(7)

y is the N T × 1 vector of the dependent variable (here log(T F R)), µ the vector of the time-invariant
fixed effect (not spatially autocorrelated), and X the N T × k of independent variables. Also, ιT is thevector of T × 1 ones, IN the identity matrix with N × N dimensions, IT the identity matrix with T × T
dimensions, and WN the spatial weight matrix with N × N dimensions. ε is εit ∼ IID(0, σε2 ) error
terms, and ν is νit ∼ IID(0, σν2 ). Moreover, β is the parameter vector, and ρ (|ρ| < 1) is the spatial
autocorrelation coefficient. ⊗ represents the Kronecker product.
To control age structure, variables such as age-specific proportions (from p0to4 to pover80) imply
young population proportions have negative impacts on the T F R, and this estimation results fit well the
Easterlin hypothesis that young cohort sizes have adverse effects on their fertility (Easterlin 1968[7]).
yt = (ιT ⊗ WN ) µ + Xt−1 β + ut
ut = ρ (IT ⊗ WN ) ut + εtt = 1, . . . , T

8

(8)
(9)
(10)

0.40

0.005

TFR_Theil

TFR_moran_I

0.35
0.30
0.25
0.20

0.004

0.003

0.15
1980

1990

2000

2010

1980

1990

0.27
0.26
0.25
0.24

2010

0.825

0.800

0.775

0.23
1980

1990

2000

2010

1980

1990

0.4
0.3
0.2
0.1
0.0
1980

1990

2000

2000

2010

year
real_exp_mean_100_Theilyear
real_exp_mean_100_moran_I

2000

year

pop_density_Theil

pop_density_moran_I

year

2010

0.00500.00450.00400.00350.00300.0025
1980

1990

year

2000

2010

year

0.3

gini_Theilgini_moran_I

0.00300.2
0.1
0.0

0.00250.00200.0015

−0.1
1980

1990

2000

0.0010
1980

2010

1990

0.60
0.55
0.50
0.45
0.40
1980

2000

2010

year

ad_rate_Theilad_rate_moran_I

year

0.020

0.016

0.012

0.008
1990

2000

2010

1980

year

1990

2000

year

Figure 4: Moran’s I and Theil Index of Each Variable in This Study

9

2010

Table 2: Estimation Results: Spatial Panel Maximum
Likelihood (SPML), Spatial Panel Generalized Moments(SPGM), Least Squares Dummy Variable (LSDV) and
Dynamic Panel Generalized Methods of Moments (Arellano & Bond 1991[2])(DPGMM(AB)) (n = 47, T = 35,
and N = 1645)
log(T F R)
ρSPML
0.674254(0.022147)
***

SPGM

LSDV

0.54946607
0.00082763

σν2
laglog(lag(pop density, 1))(log(lag(pop density, 1)))2
log(lag(real exp mean, 1))(log(lag(real exp mean, 1)))2
log(lag(gini, 1))log(lag(ad rate, 1))log(lag(p0to4, 1))log(lag(p5to9, 1))log(lag(p10to14, 1))log(lag(p15to19, 1))log(lag(p20to24, 1))log(lag(p25to29, 1))

DPGMM(AB)-0.0163771(0.0174095)
0.0016538(0.0013904)-5.7234362(1.7326863)***
0.2270749(0.0685412)***
0.0118918(0.0137283)-0.0360324(0.0087634)***
0.6246373(0.0315136)***
0.2359812(0.0399109)***-0.017743(0.015476)
0.0018175(0.0012316)-6.2441(1.7248)***0.24826(0.068263)***
0.014496(0.013994)-0.031813(0.0081342)***
0.67079(0.031383)***
0.21131(0.040257)***

0.0776786(0.2496147)
0.0080262(0.0190958)-8.1587635(2.0155213)***
0.3227035(0.0796065)***-0.0124552(0.0159547)-0.0291019(0.0168526).
0.6306060(0.0550940)***

0.0467749(0.0313585)

0.041251(0.031855)

0.0229333(0.0479956)-0.1808072(0.0205411)***-0.0957044(0.0138215)***-0.2683597(0.0209416)***-0.18262(0.020838)***-0.10322(0.013685)***-0.27073(0.021170)***-0.2903170(0.0307829)***-0.1192106(0.0236034)***-0.4103448(0.0330438)***

100.0257211(0.0621721)

0.353606(0.037013)***
0.485169(0.324223)-0.030917(0.025209)-5.089522(2.337559)*
0.202124(0.092200)*
0.011498(0.017563)
0.049324(0.024595)*
0.272078(0.050675)***
0.103450(0.051732)*-0.091458(0.054953).
-0.298755(0.031187)***-0.129304(0.021896)***-0.267500(0.034114)***

-0.2831767(0.0250284)***-0.0768934(0.0156915)***-0.0481813(0.0148061)**-0.28320(0.025049)***-0.083393(0.015441)***-0.044103(0.014591)**-0.3239812(0.0428268)***-0.1834299(0.0328672)***-0.0777104(0.0301375)*

log(lag(p45to49, 1))

0.0037819(0.0157401)

0.0088936(0.015535)-0.0343759(0.0254181)

log(lag(p50to54, 1))

0.0093755(0.0149187)

0.000045573(0.014862)-0.0367954(0.0203576).

-0.289960(0.045548)***-0.169882(0.030038)***-0.061538(0.031882).
-0.061216(0.034499).
-0.076422(0.027751)**

0.017104(0.011299)-0.0110085(0.0111422)-0.019842(0.023959)-0.059268(0.012431)***-0.0676346(0.0146975)***-0.0512689(0.0199666)*
0.0124371(0.0196962)-0.1367153(0.0244545)***-0.058528(0.020833)**

log(lag(p30to34, 1))log(lag(p35to39, 1))log(lag(p40to44, 1))log(lag(p55to59, 1))log(lag(p60to64, 1))log(lag(p65to69, 1))log(lag(p70to74, 1))log(lag(p75to79, 1))log(lag(pover80, 1))

0.0206318(0.0112965).
-0.0528298(0.0122143)***-0.0041745(0.0162989)-0.0069989(0.016680)

0.0247657(0.0212499)-0.1701261(0.0213852)***
0.1153499(0.0165672)***

0.016970(0.021771)-0.15362(0.021546)***
0.11265(0.015793)***-0.0489186(0.0364597)-0.032209(0.020328)-0.037297(0.027946)-0.085345(0.024770)***-0.149514(0.044055)***

0.92753

adj.R2. < 0.10 ∗ < 0.05 ∗∗ < 0.01 ∗ ∗ ∗ < 0.001
Double clustered standard deviations are in parentheses.
Table.2 does not indicate that log(pop density) and log(gini) in the model with the lag have thestatistical significance in the regression to log(T F R). This result is consistent with the fact that T F R
has panel non-Granger causality from gini in Section 3.1. In short, gini has no impact on T F R in the
viewpoint of panel data. Moreover, pop density’s effect on T F R remains unclear.

4

Conclusion

Several people think rural areas have higher T F R than cities, and economic inequality negatively correlates with T F R. However, the negative relationship between T F R and pop density and the positive
relationship between T F R and gini is false if one considers the lag between inception and birth. The
spatial panel data analysis validates the Easterlin hypothesis. ad rate has adverse effects on T F R in the11spatial panel data model. However, the panel non-Granger causality shows ad rate does not determine
T F R in the non-Granger causality standpoint.

Compliance with Ethical Standards
Funding
This study was financially supported by the JSPS KAKENHI Grant Number 20H05633.

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 of the authors.

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