Theoretical framework

The terminology ‘kinetics’ is borrowed from chemistry, where it refers to the response of the rate of a chemical reaction to the concentration/availability of substrates; here it describes how first marriage rate responds to the availability of preferred mates. If the marriage squeeze hypothesis is true, marriage kinetics can be derived by analysing the flow of singles in the context of mortality and migration.

Figure 1 shows the flow of singles (men or women) within a short time interval Δt for a region open to migration. Here, Δt is a short time interval within an intercensal period; P is the population size of a cohort in the region at the beginning of Δt; S ₁ is the proportion single in P at the beginning of Δt; ΔM is the net migration (in-migrations minus out-migrations) during Δt (for simplicity, migration is assumed to occur at the beginning of Δt); S ₁+δS is the proportion single in the net migration population ΔM; μ is the mean mortality rate during Δt (it is assumed that there is no mortality differential by marital status); η is the mean first marriage rate during Δt; and S ₂ is the proportion single in the population (including migrants) at the end of Δt.

Fig. 1

The flow of singles (men or women) within a short time interval Δt for a region open to migration. See text for definition of abbreviations.

Based on the balance of singles, it follows that:

$$\eta ={{PS_{1} {\plus}\Delta M \cdot \left( {S_{1} {\plus}\delta S} \right){\minus}\left[ {PS_{1} {\plus}\Delta M \cdot \left( {S_{1} {\plus}\delta S} \right)} \right] \cdot \mu \Delta t{\minus}\left( {P{\plus}\Delta M} \right) \cdot \left( {1{\minus}\mu \Delta t} \right) \cdot S_{2} } \over {\left[ {PS_{1} {\plus}\Delta M \cdot \left( {S_{1} {\plus}\delta S} \right)} \right] \cdot \Delta t}}$$

The above equation reflects multiple decrements in singles: marriage, mortality and migration. If the marriage squeeze hypothesis is true, a lower availability of preferred mates brings a lower marriage rate. Furthermore, if the relationship is linear, it follows that:

$$\eta ={{PS_{1} {\plus}\Delta M \cdot \left( {S_{1} {\plus}\delta S} \right){\minus}\left[ {PS_{1} {\plus}\Delta M \cdot \left( {S_{1} {\plus}\delta S} \right)} \right] \cdot \mu \Delta t{\minus}\left( {P{\plus}\Delta M} \right) \cdot \left( {1{\minus}\mu \Delta t} \right) \cdot S_{2} } \over {\left[ {PS_{1} {\plus}\Delta M \cdot \left( {S_{1} {\plus}\delta S} \right)} \right] \cdot \Delta t}}=k\overline{A} $$

where, $$\overline{A} $$ is the mean availability of preferred mates during Δt and k is a constant coefficient. Under the assumption that $${{\Delta M} \mathord{\left/ {\vphantom {{\Delta M} P}} \right. \kern-\nulldelimiterspace} P}\to0$$ , $$S_{2} {\minus}S_{1} =\Delta S\to0$$ and $$\overline{A} \to A$$ when $$\Delta t\to0$$ , it follows that:

$$\eqalignno{ \mathop{{\lim }}\limits_{{\Delta t\to0}} k\overline{A} = \ & kA=\mathop{{\lim }}\limits_{{\Delta t\to0}} \eta \cr {\rm } & =\mathop{{\lim }}\limits_{{\Delta t\to0}} {\displaystyle{{\minus}P \cdot \left( {S_{2} {\minus}S_{1} } \right){\minus}\Delta M \cdot \left( {S_{2} {\minus}S_{1} } \right){\plus}\Delta M \cdot \delta S\atop{\plus}\left( {P{\plus}\Delta M} \right) \cdot \mu \Delta t \cdot \left( {S_{2} {\minus}S_{1} } \right){\minus}\Delta M \cdot \delta S \cdot \mu \Delta t} \over {\left[ {PS_{1} {\plus}\Delta M \cdot \left( {S_{1} {\plus}\delta S} \right)} \right] \cdot \Delta t}} \cr {\rm } & {\rm }={{{\minus}dS_{1} } \over {S_{1} dt}}{\minus}0{\plus}{{dM} \over {Pdt}} \cdot {{\delta S} \over {S_{1} }}{\plus}0{\minus}0 $$

Thus, the differential form of marriage kinetics in the context of mortality and migration is:

(1) $${{{\minus}dS} \over {Sdt}}{\plus}{{dM} \over {Pdt}} \cdot {{\delta S} \over S}=kA$$

Equation (1) differentiates between nominal and actual first marriage rates. The term −dS/Sdt is the nominal first marriage rate or force of nuptiality (Hajnal, Reference Hajnal1953); over an intercensal period, the mean value of it: ${{\mathop{\int}\nolimits_{t_{1} }^{t_{2} } {\left( {{\minus}{{d\ln S} \over {dt}}} \right)dt} } \mathord{\left\big/ {\vphantom {{\mathop{\int}\limits_{t_{1} }^{t_{2} } {\left( {{\minus}{{d\ln S} \over {dt}}} \right)dt} } {\left( {t_{2} {\minus}t_{1} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {t_{2} {\minus}t_{1} } \right)}}={\minus}{{\ln S_{2} {\minus}\ln S_{1} } \over {t_{2} {\minus}t_{1} }}$

depends only on the proportions single at the two ends (i.e. starting and ending censuses) of the period. The term dM/Pdt is the instantaneous net migration rate, (dM/Pdt) (δS/S) is the migration-based adjustment to the nominal first marriage rate, and ${{{\minus}dS} \over {Sdt}}{\plus}{{dM} \over {Pdt}} \cdot {{\delta S} \over S}$ is thus a representation of the adjusted or actual first marriage rate. Evidently, the nominal first marriage rate will be lower than the actual rate if the net migration population is composed of single immigrants; in extreme cases, the nominal rate could be zero or even negative, although the actual rate will always be positive.

The differential form of the kinetics equation considers instantaneous force of nuptiality. For application to census data, it is integrated to get its integral forms. To do so, two assumptions are made. Firstly, migration rate is constant during an intercensal period. As $\overline{r} =\overline{b} {\minus}\overline{\mu } {\plus}\overline{\tau } $ and for a cohort $$\overline{b} =0$$ (Preston et al., Reference Preston, Heuveline and Guillot2001), it follows that $$\overline{\tau } =\overline{r} {\plus}\overline{\mu } $$ , i.e. mean net migration rate is the sum of the mean growth rate and mortality rate over an intercensal period; here, $$\overline{r} $$ is the mean growth rate, $$\overline{b} $$ is the mean birth rate, $$\overline{\mu } $$ is the mean mortality rate, and $$\overline{\tau } $$ is mean net migration rate. Secondly, mate availability A is a linear function of t, i.e. A_t =A ₀+at.

The integration is illustrated for two typical situations of migration. If migrants are composed of singles only, δS/S=(1−S)/S. When a logistic curve is assumed for decrement of singles, ln[(1−S)/S] is a linear function of time/age. Let:

$$\ln \left[ {{{\left( {1{\minus}S} \right)} \mathord{\left/ {\vphantom {{\left( {1{\minus}S} \right)} S}} \right. \kern-\nulldelimiterspace} S}} \right]=u{\plus}vt,$$

i.e.

$${{\left( {1{\minus}S} \right)} \mathord{\left/ {\vphantom {{\left( {1{\minus}S} \right)} S}} \right. \kern-\nulldelimiterspace} S}=e^{{u{\plus}vt}} \Rightarrow v={{\left( {\ln {{1{\minus}S_{2} } \over {S_{2} }}{\minus}\ln {{1{\minus}S_{1} } \over {S_{1} }}} \right)} \mathord{\left/ {\vphantom {{\left( {\ln {{1{\minus}S_{2} } \over {S_{2} }}{\minus}\ln {{1{\minus}S_{1} } \over {S_{1} }}} \right)} {\left( {t_{2} {\minus}t_{1} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {t_{2} {\minus}t_{1} } \right)}}.$$

It follows that:

(2) $$\eqalignno{ & {\rm }\mathop{\int}\limits_{t_{1} }^{t_{2} } {\left( {{\minus}{{d\ln S} \over {dt}}{\plus}{{dM} \over {Pdt}} \cdot {{\delta S} \over S}} \right)} dt=\mathop{\int}\nolimits_{t_{1} }^{t_{2} } {kAdt=\mathop{\int}\nolimits_{t_{1} }^{t_{2} } {k \cdot \left( {A_{0} {\plus}at} \right)} } dt \cr & \Rightarrow{\minus}\left( {\ln S_{2} {\minus}\ln S_{1} } \right){\plus}\left( {\overline{r} {\plus}\overline{\mu } } \right) \cdot \mathop{\int}\nolimits_{t_{1} }^{t_{2} } {e^{{u{\plus}vt}} } dt=kA_{0} \cdot \left( {t_{2} {\minus}t_{1} } \right){\plus}{1 \over 2}ka \cdot \left( {t_{2}^{2} {\minus}t_{1}^{2} } \right){\plus}C \cr & \Rightarrow{\minus}{{\ln S_{2} {\minus}\ln S_{1} } \over {t_{2} {\minus}t_{1} }}{\plus}\left( {\overline{r} {\plus}\overline{\mu } } \right) \cdot {{\left( {{{1{\minus}S_{2} } \over {S_{2} }}{\minus}{{1{\minus}S_{1} } \over {S_{1} }}} \right)} \mathord{\left/ {\vphantom {{\left( {{{1{\minus}S_{2} } \over {S_{2} }}{\minus}{{1{\minus}S_{1} } \over {S_{1} }}} \right)} {\left( {\ln {{1{\minus}S_{2} } \over {S_{2} }}{\minus}\ln {{1{\minus}S_{1} } \over {S_{1} }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\ln {{1{\minus}S_{2} } \over {S_{2} }}{\minus}\ln {{1{\minus}S_{1} } \over {S_{1} }}} \right)}}\cr &=k \cdot {{A_{t_{1} } {\plus}A_{t_{2} } } \over 2}{\plus}{C \over {t_{2} {\minus}t_{1} }}{\rm } $$

If migrants are composed of the married only, δS/S=(0−S)/S=−1. It follows that:

(3) $$\eqalignno{ & {\rm }{\minus}\left( {\ln S_{2} {\minus}S_{1} } \right){\plus}\left( {\overline{r} {\plus}\overline{\mu } } \right) \cdot \mathop{\int}\limits_{t_{1} }^{t_{2} } {{{0{\minus}S} \over S}} dt=kA_{0} \cdot \left( {t_{2} {\minus}t_{1} } \right){\plus}{1 \over 2}ka \cdot \left( {t_{2}^{2} {\minus}t_{1}^{2} } \right){\plus}C \cr & \Rightarrow{\minus}{{\ln S_{2} {\minus}\ln S_{1} } \over {t_{2} {\minus}t_{1} }}{\minus}\left( {\overline{r} {\plus}\overline{\mu } } \right)=k \cdot {{A_{t_{1} } {\plus}A_{t_{2} } } \over 2}{\plus}{C \over {t_{2} {\minus}t_{1} }}{\rm } $$

Thus, the generalized differential form leads to various integral forms according to the marital status composition of migrants and/or empirical representation of availability. Theoretically, actual first marriage rate can also be computed by dividing total marriages by total person-years of a cohort between two censuses. However, for a cohort aged 15–19 in a census year, the count of marriages in each age from 16 to 20 years in the next year may not be available (e.g. in Japan’s 1920–1940 vital statistics); in such cases, Eqns (2) and (3) have the merit of needing only statistics on marriages in each census year, but not statistics between census years. It is worth noting that if t ₂−t ₁ is shorter, the assumptions in the integration (e.g. constant migration rate) over an intercensal period are easier to satisfy. In this sense, census data with a 5-year intercensal period (e.g. Japan and South Korea) are preferred to those with a 10-year intercensal period (e.g. Malaysia and Taiwan) in testing kinetics equations.

Through Eqns 2 and 3, the theoretical formulation can be linked to the census data via a linear regression analysis to test whether and how lower mean mate availability predicts a lower adjusted or actual mean first marriage rate in a cohort over an intercensal period.

Census data from Japan

The data used for regression analysis come from: (1) the Population Census of Japan, 1920–1940 (Statistics Bureau, 2011); and (2) Vital Statistics, 1920–1940 (Tōkei-Kyoku, 1920–1940). These data are generally in both Japanese and English, but the original Japanese version is more comprehensive and is thus used in this study. The choice of 1920–1940 census data is explained later.

The Japanese government has conducted a population census every five years since 1920, with a break in 1945 due to WWII. The census counts persons that live in local areas over 6 months, by age group (0–14, 15–19, 20–24, etc.), sex and marital status at both national and prefecture (the first administrative division level in Japan; Wikipedia Contributors, 2015) levels. Based on such data, period female SMAM can be calculated at the national level (Retherford et al., Reference Retherford, Ogawa and Matsukura2001). Until 1925, female SMAM was close to 21 years, the upper limit of SMAM typical of a non-European marriage pattern; in 1940, female SMAM was already more than 23 years, the lower limit typical of a European pattern (Hajnal, Reference Hajnal1965). From 1920 to 1940, the proportion of single women at age 40–44 levelled off at 2%, far less than the lower limit level of 10% in a European marriage pattern (Hajnal, Reference Hajnal1965). Thus, there was a trend towards later but not less marriage (for a discussion of the development of marriage that is not only later but also less in East Asia in recent decades, see Retherford et al. (Reference Retherford, Ogawa and Matsukura2001), Chen & Chen (Reference Chen and Chen2014) and Raymo et al. (Reference Raymo, Park, Xie and Yeung2015)); in Hajnal’s terms, it might also be said that the Japanese marriage pattern shifted from a non-European one to a semi-European one from 1920 to 1940.

It makes no sense to do regression analysis at the national level, as only one value of mate availability and one first marriage rate can be arranged from each census. However, it is feasible to conduct analysis with census data from 47 prefectures. Three life-course stages (15–19→20–24, i.e. from age 15–19 to age 20–24; 20–24→25–29; 25–29→30–34) are analysed in the study; these covered more than 90% of marriages in single women during the period 1920–1940 (Statistics Bureau, 2011). Given that there are 47 prefectures, and for each prefecture four intercensal periods and three life-course stages are considered, there are 47×4×3=564 data points for regression analysis; each point corresponds to one value of mean mate availability and one mean first marriage rate in a cohort over an intercensal period.

Assignment for variables in the regression analysis

Firstly, a measure of mate availability is introduced. Lampard (Reference Lampard1993) indicated that although theoretically inferior to other more complicated measures of mate availability like iterated availability ratio (IAR), sex ratio may in practice be a better predictor of marriage rate. Thus, mate availability for single women in an age group and a given census can be represented empirically as follows (subscript e refers to empirical):

(4) $$\eqalignno{ A_{e} \,{\rm =}\, & {1 \over 2} \cdot {{{\rm number}\,{\rm of}\,{\rm single}\,{\rm men}\,{\rm in}\,{\rm the}\,{\rm same}\,{\rm age}\,{\rm group}} \over {{\rm number}\,{\rm of}\,{\rm single}\,{\rm women}\,{\rm in}\,{\rm the}\,{\rm age}\,{\rm group}}}{\plus} \cr & {1 \over 2} \cdot {{{\rm number}\,{\rm of}\,{\rm single}\,{\rm men}\,{\rm in}\,{\rm the}\,{\rm next}\,{\rm age}\,{\rm group}} \over {{\rm number}\,{\rm of}\,{\rm single}\,{\rm women}\,{\rm in}\,{\rm the}\,{\rm age}\,{\rm group}}} $$

Mate availability, as defined in Eqn (4), is a reasonable representation of the availability of mates with a preferred age difference. The survey on age preference nearest to the period 1920–1940 was the 1982 National Fertility Survey (National Institute of Population and Social Security Research, 1982), which indicated that the preferred age difference among single women roughly varied from zero to five years, with an average at about three years. The cross-country studies on mate selection preferences by Buss (Reference Buss1989) and Kenrick and Keefe (Reference Kenrick and Keefe1992) also indicated an age difference preference by females within this range across the countries surveyed (underdeveloped, developing and developed). Thus, it is reasonable to represent single females’ preferred mate set as single males in two adjacent age groups and preferred age difference as the mid-point between zero (mates in the same age group) and five years (mates in the next age group) (see also Torabi & Baschieri, Reference Torabi and Baschieri2010). Mate availability calculated in this way declined with time in the pre-WWII period but increased with time in the post-war decades after a major loss of marriageable men in the war; thus, it is better to use the pre-war data to test the marriage squeeze hypothesis, which concerns the response of marriage behaviour to a decline in mate availability, i.e. a relative, not absolute, shortage of mates, with time.

Secondly, for a cohort, the adjusted intercensal first marriage rate is obtained by adding a migration-based adjustment to the nominal marriage rate according to Eqns (2) or (3). As discussed in the theoretical formulation, the nominal mean first marriage rate over an intercensal period is −(lnS ₂−lnS ₁)/(t ₂−t ₁). Here, the time interval is the intercensal period (five years) and the two proportions correspond to a cohort at the two ends of an intercensal period; for example, women aged 15–19 in 1920 were 20–24 years old in 1925 and two proportions can then be found.

In adjusting nominal rate, the intercensal growth rate is assumed to be constant and calculated as:

$$\bar{r}\,\,{\rm =(ln}\,P_{{\rm 2}} {\rm {-}ln}\,P_{{\rm 1}} {\rm )/(}t_{{\rm 2}} {\rm {-}}t_{{\rm 1}} {\rm ),}$$

i.e. the mean growth rate over a full intercensal period. Mortality rate is also assumed to be constant over an intercensal period and can be computed using two methods. The first method divides total deaths by total person-years during an intercensal period; this method needs annual statistics of deaths at each age (note: the females aged 15–19 in 1920 were aged 16–20 in 1921). The second method is to compute the mean value of two rates at the beginning and end of an intercensal period: for a cohort, if the mortality rate was μ ₁ in 1920 (calculated by counts of deaths in each prefecture; Tōkei-Kyoku, 1920–1940) and μ ₂ in 1925, the intercensal mortality rate is then calculated as $$\bar{\mu }{\rm =(}\mu _{{_{{\rm 1}} }} {\rm {\plus}}\mu _{{\rm 2}} {\rm )/2}$$ ; generally, μ ₁ was close to μ ₂. The first method is the one used in the current study. However, intercensal mortality rates computed by the two methods are almost the same; for example, the coefficients of correlation between mortality rates at the life-course stage 15–19→20–24 calculated by the two methods are 0.943, 0.986, 0.984 and 0.982 in the four intercensal periods, respectively. When annual age-specific vital statistics are not available, the second method can be used, as it needs only age- or age-group-specific vital statistics of a cohort in each census year, but not vital statistics in the years between the two censuses.

Intercensal migration rate, i.e. mean migration rate over a full intercensal period, is computed by adding intercensal growth and mortality rates together (the results are available from the author). In general, prefecture-specific migration rates were in the interval from −40‰ to 40‰ (note: the unit of migration rate is the same as that of growth rate, year⁻¹). Most areas had a negative migration rate, i.e. out-migration; however, metropolitan areas like Tokyo, the capital of Japan, and Osaka, historically the centre of commerce in Japan, had a positive migration rate, i.e. in-migration (for a review of migration to economic centres, see Yap, Reference Yap1977). This pattern arose because in that period single girls migrated from rural areas to large cities, e.g. Osaka and Tokyo, for temporal/seasonal work before marriage (note: single women after age 25 or married women were generally not favoured by employers) and married women migrated for union with their husbands (Nojiri, Reference Nojiri1949; Taeuber, Reference Taeuber1958). Among men, single non-heir sons migrated to large industry centres for a better subsistence chance and married men migrated for temporary or seasonal work. There was a trend towards universal emigration from 1935 to 1940, as more and more men and women migrated to other countries for military services or land opportunities during war-time (Taeuber, Reference Taeuber1958). In the 1940 census, overseas soldiers were still included in their original hometown populations, although overseas landers were excluded (Tōkei-Kyoku, 1920–1940).

Using intercensal migration rate, nominal first marriage rate is adjusted using Eqn (2) in the case of the life-course stage 15–19→20–24, in that migrants at this stage were almost universally single women. As for the stage 20–24→25–29, the nominal rate is adjusted using Eqn (3), in that regardless of migration within Japan or migration to other countries, most (75% on average) migrants at this stage were married women (see above discussion on sex, age and spatial patterns of migration; for more details, see Taeuber, Reference Taeuber1958). In general, the ratios of the migration adjustment-induced change in first marriage rate to nominal first marriage rate were in the interval from −10% to 10%; however, they were evidently bigger in the metropolitan areas of Tokyo and Osaka (details available from the author). Table 1 lists Pearson’s coefficients of correlation between the nominal and adjusted/actual marriage rates at the stages 15–19→20–24 and 20–24→25–29 in the four intercensal periods. With migration rates in the current study, there was a high correlation between nominal and adjusted first marriage rate. A simulation with two times the migration rate in this study (i.e. a level at 80‰) indicates that, under such a condition, the correlation coefficient declines quickly from 0.954 to 0.819 at the stage 15–19→20–24 during the period 1920–25; a simulation with three times the migration rate (i.e. a level at 120‰) produces a lower correlation (Table 1). No adjustment is made to the nominal first marriage rate in the case of stage 25–29→30–34 because migration at this stage can be disregarded (Taeuber, Reference Taeuber1958; Nakagawa, Reference Nakagawa2010).

Table 1

Pearson’s coefficients of correlation between nominal and adjusted first marriage rates among female cohorts in four intercensal periods from 1920 to 1940 in Japan

^a Actual migration rate.

^b Simulated situation at two times the actual migration rates.

^c Simulated situation at three times the actual migration rates.

For each test, degree of freedom=45 and p<0.001.

The adjusted rates indicate that in each prefecture there was an evident secular decline in first marriage rate at the life-course stage 15–19→20–24 from 1920 to 1940, and thus a steady marriage postponement during the period (prefecture details are available from the author; for the overall trend, see Table 2). By contrast, in almost every prefecture first marriage rate at the stages 20–24→25–29 and 25–29→30–34 only declined slightly from 1920 to 1925, but remained almost constant from 1925 to 1940 (for the overall trend, see Table 2). Thus, the transition to later marriage during the period 1920–1940 was mainly driven by a decline in first marriage rate at the stage 15–19→20–24.

Table 2

Regression coefficients from models with intercensal first marriage rate as the response variable

^a Fixed effects model: intercensal first marriage rate is corrected using Eqn (4); both prefecture and intercensal period fixed effects are included.

^b Fixed effects model: intercensal first marriage rate is corrected using Eqn (4); both prefecture and intercensal period fixed effects are included; 1935–1940 intercensal period is not considered.

^c Random effects model: first marriage rate is corrected using Eqn (4); intercensal period dummy variable is included.

^d Fixed effects model: intercensal first marriage rate is not corrected; both prefecture and intercensal period fixed effects are included.

^e Regression coefficient of first marriage rate on the intercensal period 1920–1925 is treated as ‘intercept’ (not given by model but computed manually), which is the mean of individual intercepts for prefectures.

^f Regression coefficients of first marriage rate on the intercensal periods 1925–1930, 1930–1935 and 1935–1940 are given by the model as the difference from that for the period 1920–1925.

^g Regression coefficient of first marriage rate on mate availability.

^h The intercept is given by model.

R ² values for the three life-course stages are: Model 1, 0.957, 0.469 and 0.761; Model 2, 0.947, 0.693 and 0.794; Model 3, 0.943, 0.401 and 0.706; and Model 4, 0.960, 0.539 and 0.761.

***p<0.001.

Panel regression model

The primary analysis is a two-way fixed effects model. Prefectures are taken as panels. Given that there were only four intercensal periods from 1920 to 1940, intercensal period is included as the second fixed effect, in the form of a dummy variable (Wooldridge, Reference Wooldridge2012). Heteroskedasticity is taken into account by computing robust covariance matrices of coefficients (Zeileis & Hothorn, Reference Zeileis and Hothorn2002). The modelling platform is the statistical package ‘plm’ (Croissant & Millo, Reference Croissant and Millo2008) in R 3.1.2 (R Core Team, 2014). The model structure is:

$$y_{{_{{ij}} }} {\rm =}\upsilon _{i} {\plus}\theta _{j} {\plus}\beta x_{{i_{j} }} {\plus}{\varepsilon}_{{ij}} $$

where y _ij is the adjusted intercensal first marriage rate in prefecture i (=1, 2, …, 47) and intercensal period j (e.g. 1920–1925); x _ij is the intercensal mate availability in prefecture i and intercensal period j; β is the regression coefficient; υ _i is the fixed parameter for prefecture i; θ _j is the fixed parameter for intercensal period j; and ε _ij is a random variable with a distribution $${\varepsilon}_{{ij}} \!\sim \!N(0,\sigma _{i}^{2} )$$ .