An empirical analysis of the effect of increasing male wage inequality
on female marriage behavior in Japan
Shoichi SasakiGraduate School of Economics, Kobe University

This version: January 2014

ABSTRACTThis paper examines the effect of the greater inequality in the lower than in the upper
tail of the male wage distribution on female marriage behavior in Japan, using a new
analysisframework.

I

first

formulatedtheanalyticalmodel

applyingthe“median-preserving spread” method to a marriage search model and then analyzed thetheoretical hypothesis empirically, using extensive individual Japanese data from the
Employment Status Survey, which examined a large sample. The theoretical andempirical results show that the higher male wage inequality in the lower tail, which isdue to an increase in the male non-regular employment rate, has a significant effect on
the rising single-female rate. Furthermore, the increase in the “non-regular
employment” rate, regardless of the employment contract period, also has a significant
effect on the rising single-female rate, even where the increasing male wage in thelower tail is controlled for. These results suggest that policies to improve the income
security of the low-income male and to promote a shift from non-regular to regular
employment are needed to raise the female marriage rate.

Key words: higher male wage inequality in the lower tail, non-regular employment,
female marriage behavior, search model, median-preserving spread

JEL classification: J120; J310; D310


I would like to thank Kazufumi Yugami, Naoki Mitani, Yasuhide Tanaka, and seminar participantsat the Kobe University for their helpful suggestions and comments. I also thank the Kobe University
Micro-data Archive (KUMA) for the Employment Status Survey data provided.

Corresponding author; Shoichi Sasaki; Research Fellow, Graduate School of Economics, Kobe
University; 1-8-13, Maikodai, Tarumiku Kobe City, Hyogo Prefecture, Postcode 655-0046; Email:ichi2sun@fancy.ocn.ne.jp

11. Introduction
This paper examines the effect of the greater inequality in the lower tail of the malewage distribution than in the upper tail on female marriage behavior in Japan, using ananalysis framework different from previous methods.
In recent years, the declining birthrate has become a serious problem in Japan. The
tendency among the young generation to remain single or delay marriage attracts social
interest as the main cause of this phenomenon. Several studies have argued thepossibility that the tendency among young females to remain single or delay marriage isrelated to the increasing male non-regular employment rate or the lower-tail male wage
inequality (Sakai and Higuchi, 2005; Nagase, 2002; Ohta, 2007), because these
phenomena appear at almost the same period.
In fact, after the 1990s, the singlehood rate among Japanese women aged 25–44
years and the non-regular employment rate among Japanese men of the same age grouprise simultaneously over the years, as shown in Figure 1. This suggests some kind of
relationship between these phenomena. In addition, Figure 2 shows that thesingle-female rate and the greater male wage inequality in the lower tail follow thesame trends.
Therefore, it would be useful to examine the relationship between the rising
single-female rate and the increasing lower-tail male wage inequality to clarify thetendency among young women to remain single or delay marriage.
Conventionally, the mainstream economic approach to the predominant female
behavior of remaining single or delaying marriage is founded on Becker (1973). Becker
performed a cost-benefit analysis based on factors such as the declining benefits ofmarriage or the rising opportunity cost of marriage for women with the popularizationof female higher education and the reduction of the wage gap between men and women.1
Specifically, most studies examine the influence of individual attributes such as age,education qualification, income, and employment form as factors influencing thesingle-female probability.2
However, recent studies in the United States and Europe have examined
macroeconomic factors such as the increasing non-regular employment rate and malewage inequality in the lower tail with the decline in the male labor market. However,similar Japanese studies are limited. This is because statistical data suitable for such1

Oppenheimer (1997) suggests that the specialization model based on marriage gain is essentially anargument for nonmarriage, not delayed marriage.
2 Stevenson and Wolfers (2007) outline another hypothesis on the determinants of the female
marriage rate, such as cohabitation, birth control technology, household technology, change in the legal
structure of marriage, and shock to the matching function of the marriage market.

2

analyses are difficult to obtain in Japan.
In the first place, most US and European studies on the relationship between the
single-female rate and macroeconomic factors are divided into two types, independent ofeach other. The first type focuses on the decreasing number of males who can marry,
because of the decline in the labor market, while the second concerns the increasing
male wage inequality.
Both these issues can probably be analyzed by the integrated analysis framework
because they are closely related to each other. Therefore, I first modeled the relationshipbetween the greater lower-tail male wage inequality and female marriage behavior on
the “median-preserving spread.” I then applied this analysis framework, which allows
asymmetric distributional moves between the upper and lower tails, to search themarriage model in order to analyze the theoretical hypothesis. The study is based onJapanese individual data obtained from the Employment Status Survey, conductedamong a large sample.
The outline of this paper is as follows. Section 2 provides a review of the literature.
Section 3 presents the theoretical hypothesis and the estimation model. The data arethen described in Section 4. Section 5 reports the estimation results, and Section 6
concludes the paper.
2. Literature survey
Some studies have examined the relationship between the male labor market andfemale marriage behavior. Recent works that examine why females increasingly remain
single or delay marriage are divided into two types. The first type is built on the conceptthat the number of desirable men who can marry (the so-called marriageable men) is
decreasing with the growing number of low-income-earning men due to the decline in
the male labor market. The second type is based on the notion that increasing malewage inequality raises the expected value for women to remain unmarried and continue
the search for a partner.
Wood (1995), an example of the first type, suggests that the decreasing number ofregular male employees indicates an increasing number of low-income earners and asimultaneous decrease in the number of marriageable men. Thus, these phenomena
lead to higher single-female rates.
Wood (1995) examines Wilson’s (1987) hypothesis that with fewer marriageable blackmen, due to the labor market decline in urban areas, female marriage rates have
dropped. Lichter et al. (1992), Brien (1997), and Blau et al. (2000) also find that worse
male labor market conditions affect the female marriage rate negatively.
3

Oppenheimer et al. (1997) argue that career-entry uncertainties lead to delayed
marriage. In other words, those employed in stopgap jobs are less likely to marry thanare career workers because the stopgap arrangement is a sign of career immaturity,
uncertainty about the young person’s long-term prospects, and a low current
labor-market position. Sakai and Higuchi (2005), Nagase (2002), and Ohta (2007) findthat in Japan the marriage rate of persons who have worked as part-time employees(e.g., non-regular employees) in their first employment or have been unemployed islower than that of regular employees, and their marriage is delayed.
The second type of studies include Loughran (2002), Gould and Paserman (2003),
Kuo (2008), and Coughlin and Drewianka (2011). They examine the search model
hypothesis that varying the dispersion of the male wage distribution raises themarriage reservation wage for men and women postpone marriage. Thus, they indicate
a decrease in the female marriage rate.
Loughran (2002) analyzes these issues for the 22-30 female age group, using the
90%/50% and 50%/10% tile ratios of the male hourly wage for the 21–45 age group as
income inequality indices based on U.S. census data for 1970, 1980, and 1990. He foundthat higher male wage inequality in the upper tail, but not in the lower tail, indicatessignificantly delayed female marriage.
Gould and Paserman (2003) examine the same issues for the 21–30 female age group,
using the standard deviation of weekly wages of full-time male workers of the 16–
64-age group. They used the same data that Loughran had employed as the main
inequality indexes, controlling for city fixed effects. Gould and Paserman argue that
25% of the 20-year decline in the female marriage rate, from 1970 through 1990, can be
explained by increasing male wage inequality.
In Taiwan, where the singlehood rate is rising rapidly, Kuo (2008) examined the
Manpower Utilization Survey data on women in the 22–30 age group for the 1980–2005
period, controlling for geographic and linear-time fixed effects. Kuo concludes that
whereas the increasing male wage inequality has a negative effect on the female
marriage rate, the effect is neither large nor significant. Kuo also shows, by estimating
the 90%/50% and 50%/10% ratios simultaneously without controlling for geographic andlinear-time fixed effects, that higher male inequality has a greater marginal effect in
the lower tail than in the upper tail. This result is different from findings based on US
data.
Coughlin and Drewianka (2011) estimate the effect of rising male wage inequality on
the female marriage hazard and conclude that inequality continues to encourage young
women to delay marriage.
4

Researchers have rarely employed individual data to conduct country-level analysis
in Japan because such data are difficult to obtain. Tachibanaki and Kimura (2008),
among others, have used aggregate data based on official government statistics insteadof individual data, but they performed simple empirical analysis without controlling forvarious factors.
Recently, Yugami and Sasaki (2011) conducted an analysis based on the Japanese
General Social Surveys, which provide individual data. However, their estimation
neither uses the 90%/50% and 50%/10% ratios, because of data limitations, nor provides
a clear result on the effect of the increasing non-regular employment rate.
However, not all previous studies have employed the analysis framework thatconnects both types of study. Besides, the mechanisms of the rising single-female rate
differ according to the cause—fewer marriageable men or increasing male wage
inequality. The first focuses on female behavioral responses to the lower tail of the malewage distribution, while the second considers the upper tail. Furthermore, the firstmechanism has not been clarified in detail because of the absence of a micro-foundation
analysis such as the search model.
Overcoming the limitations of previous methods, the present analysis framework
helps explain how the female reservation wage and marriage probability respond to theasymmetric male wage distribution in Japan. This phenomenon—higher inequality inthe lower tail than in the upper tail—is due to the rise in the male non-regular
employment rate with the decline in the male labor market.
I

first

formulateanintegrated

analysisframeworkby

applyingthe“median-preserving spread,” instead of the “mean-preserving spread,” to the marriage
search model. The theoretical hypothesis thus derived is based on more realistic wage
inequality changes in Japan than in previous studies. This analysis framework has anadvantage in that it explains phenomena that consistently happen in the youth laborand marriage markets.
I also analyze the theoretical hypothesis empirically, using anonymous Japaneseindividual data from the Employment Status Survey, conducted among a large sample.
3. Theoretical hypothesis and estimation modelThis paper constructs a theoretical model based on Loughran (2002), Gould andPaserman (2003), and Kuo (2008). However, it differs from previous work on one point:
the single-female rate in Japan. I apply a relatively new analysis concept, themedian-preserving spread, to a marriage search model to examine the effect of therecent increase in lower-tail male wage inequality. The asymmetry between the lower
5and upper tails is due to the rise in the male non-regular employment rate. The
theoretical hypothesis derived is consistent with the increase in the male non-regular
employment rate and the higher male inequality in the lower tail.

3.1. Search model
I outline a simplified form of the previous analytical framework: the Loughran (2002)
model. The marriage market is supposedly characterized by imperfect information.
Women do not know the income level of each individual from whom they receive amarriage proposal, but they know the characteristics of the whole male income
distribution. In such a marriage market, women consider the reservation male wage,that is, the lowest wage level at which they would be willing to accept a marriage
proposal. Each female receives a marriage proposal with a certain offer rate and makes
a dichotomous decision, to marry or not. Women reject a proposal if the man’s wage isless than the reservation level and continue to search for a marriage partner. They stop
the search when they accept someone’s marriage proposal.
Therefore, females marry when the present value of the marriage is equal to thepresent value of continuing to search for their marriage partner. This condition isformulated as follows:𝜆 ∞𝑤 ∗ = 𝑏 − 𝑐 + ∫ (𝑤 − 𝑤 ∗) 𝑑𝐹(𝑤)𝑟 𝑤∗（1）
where 𝐹 is the distribution function of male wage W with mean μ and variance σ,which are conditional on individual attributes such as age and education. λ is the offer
rate that each female receives, which depends on marriage market conditions, such as
the sex ratio. 𝑏𝑡 is the utility that a woman gains when she remains single, 𝑐𝑡 is thecost of searching for a marriage partner, 𝑤𝑡 is the present value of the offer of marriage
at time t (considering the probability of divorce),3 𝑤 ∗ represents the female reservation
wage for men, and r is the discount rate.
The probability of accepting an offer of marriage (hazard rate of the marriage), p,
which is inversely proportional to the reservation wage, is defined as follows:𝑝 = 𝑞[1 − 𝐹(𝑤 ∗ )](2)
In this situation, the average search duration for women is 1⁄ 𝑝 (inverse of thehazard rate of marriage). In other words, the higher the reservation level, the lower the
probability p that a woman will accept the marriage offer. Likewise, the longer the3

I omit the additional character t, representing time, to simplify the expression.

6

duration of search, the higher the female first-marriage age and singlehood rate.
Based on equation (1), let us consider how a change in the male mean wage would
influence the female reservation wage for men. Here, from Mortensen (1986), thefollowing relationship holds.
0≤

𝜕𝑤 ∗
𝑝
=
≤1
𝜕𝜇
𝑟+𝑝
(3)

Thus, an increase in the mean of the male wage distribution raises the female
reservation wage by an amount less than the increase in the mean.
Moreover, a higher mean of the male wage distribution affects the probability ofaccepting an offer of marriage as follows:𝜕𝑝𝑝= 𝜆𝐹 ′ (𝑤 ∗ ) [1 −
]>0
𝜕𝜇
𝑝+𝑟
(4)
In other words, the higher the acceptance probability for an offer of marriage, the lower
the single-female rate will be.
Furthermore, Mortensen (1986) applies to the search model the concept of themean-preserving spread, which Rothschild and Stiglitz (1970) introduce to analyze themarginal effect of a risk increase. Mortensen derives the marginal effect of an increasein the variance σ with the mean μ of male wage remaining unchanged. With theassumption of risk-neutral agents, the following relationship holds:𝑤∗𝜕𝑤 ∗ 𝜆 ∫0 𝐹𝜎 (𝑤 ∗ , 𝜎)
=
>0
𝜕𝜎
𝑟+𝑝
(5)
where 𝐹𝜎 (𝑤 ∗, 𝜎) represents the partial differentiation of the distribution function aboutthevariance

𝜎

and,

from

the

definition

of

the

mean-preserving

spread,𝑤∗∫0 𝐹𝜎 (𝑤 ∗, 𝜎) 𝑑𝑤 > 0 holds. Thus, an increase in the dispersion of male wages indicates ahigher female reservation wage for men.
Then, a change in the acceptance probability for the marriage proposal andtermination of search by varying the dispersion of the male wage distribution is
described as follows:𝜕𝑝𝜕𝑤 ∗= −𝜆𝐹 ′ (𝑤 ∗ )− 𝜆𝐹𝜎 (𝑤 ∗ , 𝜎) < 0
𝜕𝜎
𝜕𝜎
(6)
Therefore, varying the dispersion of the reservation wage in the upper tail raises thereservation wage per se and results in a lower acceptance probability for the marriage
7

proposal. However, the actual movement of the increasing wage inequality is commonly
more asymmetric than symmetric around the mean wage. In that case, the acceptanceprobability of the marriage proposal will not be captured strictly according to (6)(Loughran, 2002).

3.2. Implication of a rise in the non-regular employment rate
I will now explain, using the search model, how increasing male wage inequality
affects the single-female rate. This interpretation differs from that of previous studies.
Non-regular employees are generally low-wage earners. Therefore, the larger thenumber of such employees, the lower the mean male wage in the marriage market, andthe greater the dispersion of the wage distribution in the lower tail. Thus, the change inthe male wage distribution is asymmetric between the upper and lower tails. However,
the mean-preserving spread applies only when the wage distribution spreads
symmetrically around the mean wage, because it holds the mean constant. Actually,
Loughran (2002) mentions that equation (5) does not hold for non-mean-preserving
spreads, for example, a higher inequality in the lower than the upper tail.
To avoid this problem, this study developed the median-preserving spread as ananalysis framework to allow for asymmetric distribution changes between the upperand lower tails.4 Aldashev (2010) applies the median-preserving spread to the job
search model.
Here, I apply this method for the first time to a marriage search model5 and derive atheoretical hypothesis on the effect on the female reservation wage for men in the caseof median changes and increasing inequality in the upper or lower tails.
3.2.1. Deriving the effect of median changeFirst, I derive the effect of a median change on the reservation wage. The reservation
wage is the same as previously described (1).
𝜆 ∞𝑤 ∗ = 𝑏 − 𝑐 + ∫ (𝑤 − 𝑤 ∗) 𝑑𝐹(𝑤)𝑟 𝑤∗(7)
For an increase in the median of the wage distribution, 𝐹(𝑥), to just 𝜃 , the4Labor economic studies using the median-preserving spread in a search model include M𝑜̈ ller andAldashev (2007) and Aldashev (2010).
5

As far as I know, no previous studies have applied the median-preserving spread to a marriage

search model.

8reservation wage 𝑤 ∗ can be described as follows:
𝜆 ∞[𝑤 − 𝑤 ∗(𝜃)] 𝑑𝐹(𝑤 − 𝜃)𝑤 ∗ (𝜃) = 𝑏 − 𝑐 + ∫𝑟 𝑤∗ (𝜃)(8)
Subtracting (1) from (8) yields∞∞𝑤 ∗ (𝜃)𝑤∗

𝜆(𝑤 ∗ (𝜃) − 𝑤 ∗ ) = [ ∫ (𝑤 − 𝑤 ∗ (𝜃))𝑑𝐹(𝑤 − 𝜃) − ∫ (𝑤 − 𝑤 ∗)𝑑𝐹(𝑤)]𝑟(9)Integrating the first and second term on the right side of (9) respectively gives𝑤 ∗ (𝜃)∞∫ (𝑤 − 𝑤 ∗ (𝜃))𝑑𝐹(𝑤 − 𝜃) = 𝐸(𝑤) + 𝜃 − 𝑤 ∗ (𝜃) + ∫ 𝐹(𝑤 − 𝜃)𝑑𝑤
𝑤 ∗ (𝜃)

0

(10)
𝑤∗

∞

∫ (𝑤 − 𝑤 ∗ )𝑑𝐹(𝑤) = 𝐸(𝑤) − 𝑤 ∗ (𝜃) + ∫ 𝐹(𝑤 − 𝜃)𝑑𝑤
𝑤∗

0

(11)
Partially differentiating (8) with 𝜃 using (10) and (11),𝜕𝑤 ∗ 𝜆𝜕𝑤 ∗ (𝜃)= [1 −(1 − 𝐹(𝑤 ∗ )) − 𝐹(𝑤 ∗ )]
𝜕𝜃
𝑟
𝜕𝜃
(12)
Therefore, the following inequality holds:𝜕𝑤 ∗𝜆(1−𝐹(𝑤 ∗ ))𝑝0≤ 𝜕𝜃 = 𝑟+𝜆(1−𝐹(𝑤 ∗ )) = 𝑟+𝑃 ≤1(13)Thus, a higher male median wage increases the female reservation wage by an amountless than the increase in the median.
Moreover, how the increase in the median of the male wage distribution affects theacceptance probability of a marriage offer is described as follows:𝜕𝑝 𝐹 ′𝑝= ∗ [1 −
]>0
𝜕𝜃 𝑤
𝑝+𝑟
(14)
In other words, the higher the acceptance probability of a marriage offer, the lower the
single-female rate will be.

93.2.2. The effect of a greater wage inequality in the upper or lower tailWhen the female reservation wage is higher than the median of the male wage
distribution, a greater wage inequality in the lower tail does not affect the reservation
wage for women who adopt suitable search strategies. Therefore, the reservation wage
is assumed to be lower than the median, and the expected value of accepting an offer ofmarriage above the reservation wage male wage is described as follows:∞Λ = ∫ (𝑤̅ − 𝑤 ∗ )𝑑𝐹(𝑤)𝑤∗(15)
where 𝑤
̅ is the median, and 𝑤 ∗ is the reservation wage, 𝜎𝐿 is the dispersion in thelower tail, and 𝜎𝑈 is the dispersion in the upper tail.
It is possible to rewrite (15) as̅𝑤∞

Λ = ∫(𝑤 − 𝑤 ∗ )𝑑𝐹(𝑤) + ∫ (𝑤 − 𝑤 ∗ )𝑑𝐹(𝑤)
𝑤∗

̅
𝑤

(16)
where the first bracket in the left-hand side is̅
𝑤∫(𝑤 − 𝑤

∗ )𝑑𝐹(𝑤)

= (𝑤
̅ −𝑤

̅
𝑤

̅
𝑤

𝑤∗

𝑤∗

𝑤
̅ − 𝑤∗̅) − ∫ 𝐹(𝑤)𝑑𝑤 =− ∫ 𝐹(𝑤)𝑑𝑤
2∗ )𝐹(𝑤𝑤∗

and the second bracket in the left-hand side is∞∫ (𝑤 − 𝑤∞∗ )𝑑𝐹(𝑤)

̅
𝑤

∞

∞
∗

= ∫ 𝑤𝑑𝐹(𝑤) − ∫ 𝑤 𝑑𝐹(𝑤) = ∫ 𝑤𝑑𝐹(𝑤) −
̅
𝑤

̅
𝑤

̅
𝑤

𝑤∗
2

Thus, the following holds:
̅
𝑤

∞

𝑤∗

̅
𝑤

𝑤
̅
Λ = − ∫ 𝐹(𝑤)𝑑𝑤 + ∫ 𝑤𝑑𝐹(𝑤) − 𝑤 ∗2(17)
Since the increase in the truncated mean raises the expected value of the upper tail,𝜕∞∫̅ 𝑤𝑑𝐹(𝑤) > 0.
̅ 𝑤
𝜕𝑤

Additionally, an increase in the upper-tail dispersion raises the expected value of theupper tail:

10

∞

𝜕
∫ 𝑤𝑑𝐹(𝑤) > 0𝜕𝜎𝑈̅𝑤

Thus, from (16), the following holds:
𝜕Λ
>0
𝜕𝜎𝑈
(18)
On the other hand, with an increase in the lower- tail dispersion,
𝜕𝐹(𝑤)
𝜕𝜎𝐿

> 0 holds, so that the following holds from (17):
̅
𝑤

𝜕Λ
𝜕
=− ∫
𝐹(𝑤)𝑑𝑤 < 0
𝜕𝜎𝐿
𝜕𝜎𝐿
𝑤∗

(19)
From (18) and (19), the following are derived:
𝜕𝑤 ∗
>0
𝜕𝜎U
(20)
𝜕𝑤 ∗
<0
𝜕𝜎𝐿
(21)
The dispersion in the upper tail of male wage distribution has a marginal effect on theprobability of acceptance of a marriage offer as described below:
𝜕𝑝𝜕𝑤 ∗= −𝜆𝐹 ′ (𝑤 ∗)− 𝜆𝐹𝜎𝑈 (𝑤 ∗, 𝜎𝑈 ) < 0
𝜕𝜎𝑈
𝜕𝜎𝑈
(22)
Here, 𝐹𝜎𝑈

(𝑤 ∗, 𝜎𝑈 ) in the second term of the right-hand side is a partial

differentiation of the distribution function of the dispersion (𝜎𝑈 ) in the upper tail. Thus,
it is zero if the reservation wage is lower than the median, as assumed above. Therefore,
the greater dispersion in the upper tail raises the reservation male wage (in the firstterm of right-hand side) and the single-female rate.
On the other hand, the probability of acceptance of an offer of marriage, expressed as
the size of the dispersion (𝜎𝐿 ) in the lower tail, is described as follows:𝜕𝑝𝜕𝑤 ∗= −𝜆𝐹 ′ (𝑤 ∗ )− 𝜆𝐹𝜎𝐿 (𝑤 ∗, 𝜎𝐿 )
𝜕𝜎𝐿
𝜕𝜎𝐿
(23)
11

The right-hand side indicates the difference between the effect of the decline in thefemale reservation wage due to the higher male wage inequality in the lower tail (thefirst term) and the effect of the variance of the male wage distribution below the female
reservation wage (the second term). However, a strong possibility exists that thedispersion of the male wage distribution in the lower tail is so large that the latter effectis dominant, considering the increasing number of low-income non-regular employees
among young Japanese men. Therefore, when the latter effect exceeds the reduction
effect of the female reservation wage, the acceptance probability of a marriage offer
drops, and the single-female rate rises. This point needs to be clarified in positive
analysis.

3.3. Estimation modelI examine the theoretical hypothesis mentioned above, using a probit model in which
the dependent variable is whether the female is single or not. The estimation model isshown in (24):𝑦𝑖∗ = 𝑥𝑖 𝛼 + 𝐼𝛽 + 𝑀𝛾 + 𝜀𝑖𝑦𝑖 = 1 𝑖𝑓 𝑦𝑖∗ > 0,

and 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒(24)

where 𝑖 represents the 𝑖th female.
The explanatory variables are female individual attributes 𝑥𝑖 , such as age group andeducation background, male wage inequality indices or labor market condition indices I,
such as the 90%/50% tile ratio, the 50%/10% tile ratio, the non-regular employment rate,
and the unemployed rate, which I mostly focus on. I also use the control variable 𝑀,such as the sex ratio, the female habitation dummy variable in large urban areas, as theproxy variable for the marriage markets,6,7 where 𝜀𝑖 is the error term. The sex ratio isthe male-to-female ratio in the marriage market by age group and year. The higher theratio, the greater the advantage for women in the marriage market and, therefore, thelower the single-female probability. The single-female probability is probably higher in

6

However, some previous studies include other employment conditions such as female regular

employment, non-regular employment, and unemployment as explanatory variables, I have not added
these variables in this paper because many females retire or change their status to non-regular
employment on marriage or childbirth, causing endogeneity between the employment condition andmarriage behavior. I have not included female wage, either, as the explanatory variable for a similarreason.
7 I have not included the year dummy variable because the regular or non-regular employment rate

and the year dummy show a certain amount of correlation (correlation coefficient = 0.5–0.7).

12large urban areas than in other areas because the social norm of marriage in large
urban areas is not as strong as in other areas. In addition, the single-female probability
would be lower, and the female age group therefore higher.
4. DataIn this paper, I use official Japanese statistics (anonymous individual data) from the
Employment Status Survey for 1992, 1997, and 2002, 8 obtained from the Kobe
University Micro-data Archive in Japan.
Considering that the data are anonymous, information on prefectures andhouseholds (with more than nine members) is kept secret, and age and wage
information is not indicated in actual numbers but as class values, to avoid
identification of an individual or household. However, I actually use resampled data,
which comprise 80% of the entire sample. Thus, the sample size is approximately
2,370,000 over three years.
In this paper, I restrict the sample to the 25–44 female age group because women
younger than 25 years include students and approximately 90% of women are marriedby age 44.9 Finally, the sample for analysis includes 367,690 women.
It is assumed that the potential marriage partners are males of the same age groupand year. Marriage market segmentation by education background is not assumed.10
In this paper, I use an hourly, and not yearly, wage because the cyclical change inlabor supply influences yearly wage inequality (Loughran, 2002). I calculate thelogarithm median wage and the wage inequality indices using an hourly wage. I also
calculate the male employment index and sex ratio in the marriage market for each age

8Currently, data are available only for these three years.

9 Because the 2002 questionnaire does not distinguish between divorcees or widows and single women,

it must be noted that the single-female rate for 2002, unlike in other years, includes divorcees andwidows.
10

I assume that the women’s potential marriage partners belong to the same 5-year age group

because the mean age difference between man and woman at the first marriage was 2.6 years in 1992,
2.4 years in 1997, and 1.7 years in 2002. This gap has been gradually reducing according to the Annual
Population and Social Security Surveys carried out by the National Institute of Population and Social
Security Research. In addition, I also assume, following Gould and Paserman (2003), that the potential
marriage partners are from all educational backgrounds, because changes in wage inequality may
affect female education choice. These are the reasons I set the marriage market the way I did in thispaper.

13

group and year.11
The descriptive statistics of the variables used in this analysis are shown in Table 1.
Figure 3 shows the change in the kernel distribution of the 25–44 male age group’s
logarithmic hourly wage for 3 years (unwaged persons are excluded).
The overall the wage distribution seems to be more skewed to the upper side in 1997
than in 1992. It then shifted, overall, to the lower side with a slightly higher lower-tailinequality in 2002 than in 1992.
5. Estimation results and discussion
5.1. Baseline estimation results
I first compare the estimation results on the effect of male wage inequality indiceswhen the mean male wage and median male wage are controlled for.
5.1.1. Male inequality indices with mean wage control
Table 2 shows the baseline estimation results when the mean male wage iscontrolled for. I use the 90%/50%, 50%/10%, and 90%/10% tile ratios as the wage
inequality indices in the upper tail, in the lower tail, and for the whole distribution,
respectively.
The coefficient of the mean male wage does not have a statistically significant effect,
but the sign is negative in all cases for the single-female probability. Thus, the result isby and large consistent with the theoretical hypothesis.
The coefficient of the 50%/10% tile ratio, the index being considered here, has asignificant positive effect at the 1% level when it is used as an explanatory variable,
either alone or in combination with the 90%/50% tile ratio. Thus, the result is consistent
with the theoretical hypothesis. The coefficient of the 90%/10% tile ratio has asignificant positive effect at the 1% level. On the other hand, the coefficient of the
90%/50% tile ratio does not have a significant effect.

11

I use the median of each wage grade as the personal wage and multiply it 1.25 times for thehigh-end yearly wage group, more than 10 million yen. Furthermore, the real wage is based on the2002 consumer price index. From 2002, however, each wage group in the Employment Status Survey
questionnaire is further subdivided. Therefore, I standardize the wage groups according to those of
1992 and 1997, following Shinozaki (2001), Kambayashi (2010a), and Yugami and Sasaki (2011), toremove the influence of class interval changes on the setup conditions of wage inequality indices. In
addition, I define the non-regular employment rate, following Ohta (2007), in terms of all employees
and the unemployment rate as a proportion of the population.

14

The marginal effect is the increase in the single-female probability (byapproximately 1.7%) for every 1% rise in the 50%/10% tile ratio of the male wage in themarriage market.
Among the control variables, the dummies for being above a college graduate and forfemale habitation in large urban areas have a consistently significant positive effect at
the 1% level. Thus, the results are not consistent with the theoretical hypothesis
derived by the mean-preserving spread but are consistent with the theoretical
hypothesis derived by the median-preserving spread.
5.1.2. Male inequality indices with median wage control
Table 3 shows the baseline estimation result, controlling for the median male wage.
The median male wage does not have a statistically significant effect, but the sign isnegative in all cases for the single-female probability. Thus, the result is by and large
consistent with the theoretical hypothesis.
In this estimation, I use the 90%/50%, 50%/10%, and 90%/10% tile ratios. The
50%/10% tile ratio, the inequality index being considered here, also has a significant
positive effect at the 1% level when it is used as explanatory variable, either alone or in
combination with the 90%/50% tile ratio. Thus, the result is consistent with theoretical
hypothesis. The coefficient of the 90%/10% tile ratio also has a significant positive effectat the 1% level.
The marginal effect is the increase in the single-female probability (byapproximately 1.7%) with a 1% rise in the 50%/10% tile ratio of the male wage in themarriage market when the median wage is controlled for. As with the median wage, thisresult is consistent with the theoretical hypothesis derived by the median-preserving
spread.
5.1.3. Estimation results on male employment indices
Table 4 shows the estimation results on male employment condition indices when themean or median male wage is controlled for. The marginal effect of median wage control
is the increase in single-female probability (by approximately 0.02% and 0.03%,
respectively) per 1% rise in the male non-regular employment rate and the unemployedrate in the marriage market. The coefficients of both the non-regular employment rate
and the unemployed rate have a significant positive effect at the 1% level in both cases.
These results are consistent with those of previous studies.
From the above results, it is confirmed that the deterioration of male employment
indices have a significant positive effect on the single-female rate.
15

Considering the nature of data in this paper, however, both results are similar
because the mean wage and median wage are exactly the same. It is of enormous
significance that I obtained a result that is consistent with the theoretical hypothesis byusing an estimation model that can capture asymmetric changes in male wage
inequality around the median wage more exactly than previous methods could.

5.2. Contributory factors of single-female rate changes
Using the estimation results on the marginal effect of male wage inequality andemployment condition indices, I try to predict the magnitude of change in the
single-female rate from 1992 to 2002. Table 5 shows how much of the change in the
single-female rate can be explained by the male wage inequality trend according to theestimated marginal effects in Tables 3 and 4.
The single-female rate of the 25-44-age group has risen 15.43% in these 10 years. In
this period, the average changes in the 90%/50% and 50%/10% tile ratios of the malewage in the same age group are a 5.5% decrease and a 9.7% increase, respectively.
In addition, the marginal effect coefficients are -0.030 and 1.690, respectively, when
the above indices are simultaneously used as explanatory variables. Thus, the predictedchanges are 0.2% and 16.4%, respectively, based on the average changes multiplied by
the coefficients of marginal effect.
From these results, the changes in the male 90%/50% and 50%/10% tile ratios
explain 1.1% and 106.3%, respectively, of the rise in the single-female rate. In otherwords, the 50%/10% tile ratio, representing the wage inequality size in the lower tail,has a significant explanatory power for the rise in the single-female rate. Furthermore,with a similar calculation, the male non-regular employment rate can predict 53.3% of
the rise in the single-female rate. This index has a significant explanatory power, too.

5.3. Estimation results based on married male data
Gould and Paserman (2003) believe that, when a large number of single men tend tobe low-income non-regular employees, male wage inequality increases in the lower tail.
To control for this reverse causality, I estimate the model using only married maledata.12 Table 6 shows the estimation results.
The coefficient of the 50%/10% tile ratio has a significant positive effect at the 1%
level when it is used as an explanatory variable, either alone or in combination with the
90%/50% tile ratio. The marginal effect is the increase in single-female probability (by12An analysis based only on married male data has merits in that not only is the influence of reverse
causality removed but the influence of the inequality generated by changes in the returns to skill,ability, etc., on the single-female rate is also examined.

16

approximately 1.0%) per 1% rise in the 50%/10% tile ratio. In addition, the coefficientsof the non-regular employment rate and unemployed rate have significant positiveeffects at the 1% level, similar to the results in Table 4. These findings confirm therobustness of the results in Tables 3 and 4.

5.4. Estimation results based on wage inequality and employment indicesThis study assumes that a high male non-regular employment rate is reflected as anincrease in male wage inequality in the lower tail. I therefore consider the male wage
inequality indices as explanatory variables.
The results confirm that the greater male wage inequality in the lower tail has asignificant positive effect on the single-female rate. In addition, I also use theemployment condition indices as explanatory variables, separately, and estimate theeffects. The estimation results confirm that the non-regular employment rate also has asignificant positive effect on the single-female rate.
Then, how does the result turn out if both wage inequality and employment conditionindices are simultaneously used as explanatory variables? According to the theoretical
hypothesis, the marginal effect of wage inequality indices is considered significant. On
the other hand, the significance and marginal effect of the employment conditionindices are absorbed by the wage inequality indices and become negligible.
Table 7 (columns 1 and 2) shows the estimation results based on the 90%/50% and50%/10% tile ratios as well as the male non-regular employment or unemployed rate.
The coefficient of the 90%/50% tile ratio is negative, but does not have a statisticallysignificant effect on the single-female rate. The coefficients of the 50%/10% tile ratio andthe non-regular employment rate together have a significant positive effect on the
single-female rate. However, the 50%/10% tile ratio does not have a statisticallysignificant effect in combination with the unemployed rate, although the sign is positive.
Thus, I can confirm that the result is almost robust, although the marginal effect of the
50%/10% tile ratio, independent of the non-regular employment rate, is one-half lower
when considered along with the non-regular employment rate.
On the other hand, the male non-regular employment rate and the unemployed rate
have a significant positive effect at the 1% level on the single-female rate. In neithercase does the marginal effect become as small as the result in Table 4. In other words, I
obtained a result different from the hypothesis.
How can this result be interpreted? The rise in the non-regular employment rate isbelieved to reflect the greater wage inequality in the lower tail, which might in turn be
related to just one point. An employment condition index such as the non-regular17

employment rate may capture the uncertainty effects of employment that the wage
inequality index cannot capture.
In other words, it is suggested that women predict the long-term increase in
low-income, unstable employment in the marriage market considering not only thegreater male inequality in the lower tail but also the rise in the non-regular
employment rate based on the designation. Therefore, they postpone marriage. This isalso consistent with Oppenheimer’s argument that job uncertainty delays marriage.
Incidentally, it is believed companies could manage human resources—develop
ability, for example—according to the designation. Furthermore, companies tend to hire
non-regular employees by designation and treat them as a buffer for labor adjustment
because lay-off of regular employees costs much more in Japan than does that of
non-regular employees.
I now further discuss non-regular employment, a common form of work for young
men and a social issue recently in Japan. “Regular employment” and “non-regular
employment” are “name at work (designation)” types in the Employment Status Survey.
Such a classification method is peculiar to Japan. It is uncommon in the United Statesand in European countries for classification of employment status. In addition, thesurvey also classifies positions according to the employment agreement. In other words,
workers with an agreed employment term of more than one year are classified aspermanent employees, and those with a term of less than one year are grouped as
temporary employees.
Considering the name-at-work classifications and employment status based on theagreed term, we can classify employment positions into four categories: permanent
regularemployment,

employment,

andtemporary

temporary

regularemployment,

permanent

non-regular

non-regular

employment.

Therefore,

non-regular

employment is classified into two categories: permanent non-regular employment andtemporary non-regular employment.
In Japan, permanent non-regular employees have increased in recent years(Kambayashi, 2010b). It is recognized that permanent non-regular employment is often
renewed many times, so that the actual duration of service is relatively long, though at
low wages (Koyo̅ no Arikata ni Kansuru Kenkyukai [Meeting to study systems of
employment] 2009). However, non-regular employees’ contracts, even if permanent,
have increasingly not been renewed recently with the deteriorating labor market
conditions.
Do the increasing permanent and temporary non-regular employment rates havedifferent effects on the single-female rate? I show the estimation results using the18

indices in Table 7 (columns 3 and 4). The results show that both indices have asignificant positive effect at the 5% level on the single-female rate. In addition, themarginal effect of permanent non-regular employment is slightly larger than that of
temporary non-regular employment. Therefore, even if the name “non-regular” is thesame, the marginal effect of increasing male permanent employment is larger than thatof male temporary employment when the effects of increasing inequality in the lower
tail are controlled for. Thus, the possibility that the increasing male permanent
non-regular employment rate will accelerate the rising single-female rate is a concern.
6. Conclusion
Clarifying the tendency among young women to remain single or delay marriage,
putatively the main cause of the declining birthrate in Japan, is a socially urgent need.
Some recent studies in the United States and other countries have applied theimplications of the search model to marriage behavior. These studies, based on themean-preserving spread, have concluded that women might delay marriage in responseto the increasing inequality in the upper tail of the male wage distribution.
However, the phenomenon of the greater inequality in the lower tail than the upper
tail of the male wage distribution in Japan is due to the rising male non-regular
employment rate. Therefore, the previous analysis framework could not strictly analyzea phenomenon such as this. Besides, a sophisticated study with Japanese individual
data is difficult to carry out because of data availability limitations.
In this paper, I model the relationship between asymmetric change in the male wage
distribution due to rising male non-regular employment rates and female marriage
behavior, using the “median-preserving spread” framework. This method allows us toanalyze, for the first time, the effect of asymmetric inequality change between the upperand lower tails of the wage distribution on the marriage search model, assuming thatthe rising non-regular employment rate reflects increasing male wage inequality in thelower tail. I then examine the theoretical hypothesis by empirical analysis, using a large
Japanese individual dataset.
The results show that the increasing male inequality in the lower tail has asignificant positive effect on the single-female rate when female education background,age group, sex ratio, and male median wage are controlled for. Thus, the results are
consistent with the theoretical hypothesis. Besides, the contribution to the rising
single-female rate is considerably large.
In other words, while the bottom of the distribution dropped below the low income
level in Japan, the inequality in the lower tail has increased since the 1990s. The
19

increasing male inequality in the lower tail is proposed as one of the significant factorsof the rising single-female rate.
However, the increasing wage inequality in the lower tail and the rising non-regular
employment rate, when simultaneously used as explanatory variables, show asignificant positive effect on the single-female rate. Non-regular employment might beinterpreted as the long-term job uncertainty that Oppenheimer deliberates on. From
this result, it is suggested that women predict a long-term rise in lower income,unstable male employment in the marriage market by considering, according to thedesignation, not only the increasing male inequality in the lower tail but also the rising
non-regular employment rate. Therefore, they delay marriage.
The policy implication derived from the results of positive analysis is that raising thelow male wage level and supporting a shift from non-regular to regular employment byimproving the labor market environment would play an important role in increasing thefemale marriage rate in Japan.
To that end, an ability development program, employment training improvement,
and the establishment of an evaluation system for occupation ability are necessary.
Improving income security is an additional requirement. However, the conditions andrequirements are different for permanent versus temporary non-regular employees.
Thus, the remedial measures implemented should be appropriate to each category.
Nevertheless, I could not precisely establish the superiority of the new analysis
framework over the previous methods in some analyses because of data limitations. The
significant academic contribution of this study is that I rigorously analyzed, not only
normatively but also positively, the remaining-single or delaying-marriage mechanisms
in Japan. To do so, I applied the “median-preserving spread,” a new analysis framework,
to a marriage search model and obtained results different from those of previous studies
such as Loughran (2002) and Gould and Paserman (2003). Employing the search model,
I found, theoretically and empirically, that women might delay marriage in response toincreasing inequality in the lower tail of the male wage distribution.
How did these influences affect the so-called lost-decade generation born in the
1980s, which faces job scarcity? To examine this question, the researcher would need toaccumulate substantial data. This remains a future challenge.
In addition, because of data limitations I could not conduct a fixed effect model
analysis for residence area, which some conventional studies have done. This
examination is also a future challenge and a prospect for this research horizon.

20

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23

Fig.1. Trends of the single-female rate and the male non-regular employment rate.

Fig.2. Trends of the single-female rate and the male non-regular employment rate.

24

Fig.3. Changes in the kernel distribution of male logarithmic hourly wage (Data source:
Employment Status Surveys of 1992, 1997, and 2002).

25

Table 1: The descriptive statistics of the variables
Women of ages 25–44 years

Variable

Obs.

Mean

Std. Dev.

Min.

Max.

Female age-

25–29 years old

367690

0.238

0.426

0

1

group dummy

30–34 years old

367690

0.235

0.424

0

1

35–39 years old

367690

0.243

0.429

0

1

40–44 years old

367690

0.283

0.451

0

1

FemaleBelow junior high graduate

367690

0.075

0.264

0

1

education

High school graduate

367690

0.539

0.498

0

1

background

Graduate from vocational school

367690

0.272

0.445

0

1

dummy

Above college graduate

367690

0.110

0.313

0

1

Female single probability

367690

0.219

0.414

0

1

Male

Mean (logarithmic)

367690

7.452

0.177

7.128

7.735

logarithmic

Median (logarithmic)

367690

7.497

0.188

7.148

7.823

hourly90%/10% tile ratio

367690

1.300

0.163

1.043

1.540

in marriage

90%/50% tile ratio

367690

0.599

0.073

0.462

0.698

market50%/10% tile ratio

367690

0.701

0.102

0.570

0.842

Male

Regular employment rate (%)

36769096.2382.626

88.65098.270employment

Non-regular employment rate (%)

367690

3.762

2.626

1.730

11.350condition in

Permanent employment rate (%)

36769090.4893.415

81.62694.202marriage

Temporary employment rate (%)

367690

5.069

1.352

3.588

8.503

marketUnemployment rate (%)

367690

4.369

2.146

2.185

9.678

Sex ratio in marriage market

36769090.8272.414

88.602

94.486

Female habitation dummy in large urban areas

367690

0.322

0.467

0

1

Year dummy

1992

367690

0.372

0.483

0

1

1997

367690

0.331

0.471

0

1

2002

367690

0.297

0.457

0

1

wage

Note: The regular/non-regular employment rate is the ratio of regular/non-regular employees to all
employees.

26

Table 2: The marginal effects of various male inequality indices(Baseline estimate result 1)
Male wage inequality indices (tile ratios)
90%/10%90%/50%50%/10%90%/50%+50%/10%

Log male mean wage

-0.011-0.275-0.008-0.066(0.184)(0.216)(0.154)(0.140)

Log male wage

0.802

0.367

1.743

0.084

Inequality(0.257)(0.465)(0.302)--

Indices(Number of obs.
Pseudo R2

***

-

)

(

-

)

(

***

(0.374)
1.725

-

)

(0.334)367690367690367690367690

0.153

0.149

0.156

0.156

***

Notes:
1. Men in the marriage market were considered with controls for age group, year dummy, and mean
wage.
2. Dependent variable: single female = 1, others = 0.
3. Other explanatory variables are female education background dummy, female age-group dummy,
sex ratio in marriage market, and female habitation dummy in large urban areas.
4. Within the second-step parentheses are standard errors modified by clustering based on age group,
education background, and year dummy.
5. ***, **, and * indicate 1%, 5%, and 10% levels of significance, respectively.

27

Table 3: Marginal effects of various male wage inequality indices(Baseline estimate result 2)
Male wage inequality indices (tile ratios)
90%/10%90%/50%50%/10%90%/50%+50%/10%

Log male median wage

0.024

-0.377-0.120-0.130(0.194)(0.257)(0.164)(0.189)Log male wage

0.838

0.118

1.687

-0.030Inequality(0.300)(0.581)(0.289)--

Indices(Number of obs.
Pseudo R2

***

-

)

(

-

)

(

***

(0.447)
1.690

-

)

(0.298)367690367690367690367690

0.153

0.149

0.156

0.156

***

Notes:
1. Men in the marriage market are considered with controls for age group, year dummy, and mean
wage.
2. Dependent variable: single female = 1, others = 0.
3. Other explanatory variables are female education background dummy, female age-group dummy,
sex ratio in marriage market, and female habitation dummy in large urban areas.
4. Within the second-step parentheses are standard errors modified by clustering based on age group,
education background, and year dummy.
5. ***, **, and * indicate 1%, 5%, and 10% levels of significance, respectively.

28

Table 4: Marginal effects of various male employment indices(Baseline estimate result 3)
Non-regular employment rate

Unemployment rate

Log maleLog maleLog maleLog maleMean wage

Median wage

Mean wage

Median wage

Log male mean or

0.049

0.089

-0.057-0.009Median wage

(0.156)(0.184)(0.117)(0.136)

Male employment

0.020

0.021

0.029

0.030

Condition indices(0.007)

Number of obs.

367690367690367690367690

0.157

0.157

0.160

0.160

Pseudo R2***(0.007)***(0.007)***(0.007)***

Notes:
1. Men in the marriage market are considered with controls for age group, year dummy, and mean
wage.
2. Dependent variable: single female = 1, others = 0.
3. Other explanatory variables are female education background dummy, female age-group dummy,
sex ratio in marriage market, and female habitation dummy in large urban areas.
4. Within the second-step parentheses are standard errors modified by clustering based on age group,
education background, and year dummy.
5. ***, **, and * indicate 1%, 5%, and 10% levels of significance, respectively.

29

Table 5: Estimation results of contributions to change in single-female rate
Average

Marginal

PredictedchangeeffectchangeContribution△Single-female rate

0.15428△Male 90%/50% tile ratio-0.055-0.030

0.002

1.1%

△Male 50%/10% tile ratio

0.097

1.690

0.164

106.3%△Single-female ratio

0.15428△Male non-regular employment rate

3.917

0.021

0.082

53.3%

30

Table 6: Marginal effects of male wage inequality and employment condition indicesbased only on married male data on single-female rate
Male wage inequality indices (tile ratios)

Employment conditionindices90%/10%90%/50%50%/10%90%/50%+Non-regular

Unemploy

50%/10%

Employment

ment rate

Rate
Male median

0.105-0.389-0.128-0.182-0.077-0.010Wage(0.256)(0.274)(0.228)(0.251)(0.143)(0.104)

Male wage

1.191-1.3671.101-0.283

0.050

0.107

Inequality(0.352) ***(0.409)(0.204) ***(0.508)(0.012) ***(0.017) ***

-

-

-

0.939

-

-

Indices
(

-

)

(

-

)

(

-

)

(0.265) ***

(

-

)

(

-

)

Number of obs.

367690367690367690367690367690367690Pseudo R2

0.150

0.153

0.155

0.156

0.159

0.162

Notes:
1. Men in the marriage market are considered with controls for age group, year dummy, and mean
wage.
2. Dependent variable: female – single = 1, others = 0.
3. Other explanatory variables are female education background dummy, female age-group dummy,
sex ratio in marriage market, and female habitation dummy in large urban areas.
4. Within the second-step parentheses are standard errors modified by clustering based on age group,
education background, and year dummy.
5. ***, **, and * indicate 1%, 5%, and 10% levels of significance, respectively.

31

Table 7: Estimation results based on simultaneous use of male wage inequality andemployment indices
Male employment indices

Male median wage

90%/50% tile ratio

50%/10% tile ratio

Non-regular

Unemployment

Permanent

andTemporary

Employment rate

Rate

Non-regular

Non-regular

Employment rate

Employment rate

0.018

-0.099-0.052-0.020(0.159)(0.127)(0.160)(0.158)-0.234-0.316-0.261-0.211(0.314)(0.288)(0.298)(0.329)

0.869

0.450

0.772

0.976

(0.427)(0.374)

0.026

0.032

(0.385)****(0.396)

Male employment

0.015

condition indices(0.007)

Number of obs.

367690367690367690367690

0.158

0.160

0.159

0.158

Pseudo R2**(0.008)***(0.014)

and

**

0.029
**

(0.014)**

Notes:
1. Men in the marriage market are considered with controls for age group, year dummy, and mean
wage.
2. Dependent variable: single female = 1, others = 0.
3. Other explanatory variables are female education background dummy, female age-group dummy,
sex ratio in marriage market, and female habitation dummy in large urban areas.
4. Within the second-step parentheses are standard errors modified by clustering based on age group,
education background, and year dummy.
5. ***, **, and * indicate 1%, 5%, and 10% levels of significance, respectively.

32