Chapter 3: Pooled Regression Model to Estimate Time Effect

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1 Panel Data Structure

In Panel Data Analysis, we observe the behavioral performance of multiple units (e.g., individuals, firms, countries) at mutiple points in time. Unlike cross-sectional data structure $(X_i , Y_i)$ or time-series data structure $(X_t , Y_t)$, panel data has a dual dimensional structure $(X_{i,t} , Y_{i,t})$: a cross-sectional dimension $(i=1,2,...,N)$ and a time dimension $(t=1,2,...,T)$.

Formally, $Y_{i,t}$ denotes the dependent variable for the ith unit at time t and $X_{i,t}$ denote its corresponding explanatory variable (which can be a vector), then the general form of the panel data can be expressed as:

$$(X_{i,t}, Y_{i,t}), i = 1, 2, ... , N; t = 1, 2, ... , T$$

$N$ denotes the number of observation units;

$T$ denotes the number of time periods;

$X_{i,t}$ can include variables that vary over time (e.g., income, employment status) as well as variables that do not vary over time (e.g., gender, place of birth) that are considered in some models;

$Y_{i,t}$ is the outcome variable we are interested in, e.g., wages, output, or health level.

In practice, panel data are stored and expressed in two table structures:

(1) Long format or stacked format

The long format is standard representation of panel data. In this structure, each row corresponds to the observation of a specific unit $i$ t timepoint $t$, and the data are arranged in the form of "periob by period stacking". For example, a 2 units × 3 periods data in long format is as follows:

id	time	X	Y	Sex
1	1	$X_{1,1}$	$Y_{1,1}$	1
1	2	$X_{1,2}$	$Y_{1,2}$	1
1	3	$X_{1,3}$	$Y_{1,3}$	1
2	1	$X_{2,1}$	$Y_{2,1}$	0
2	2	$X_{2,2}$	$Y_{2,2}$	0
2	3	$X_{2,3}$	$Y_{2,3}$	0

(2) Wide format

Wide format is a representation that more closely resembles the cross-sectional data structure. In wide format, each cross-sectional unit corrresponds to a row, and the variable values at each time point are expanded into separate colums. For example, a 2 unitis × 3 periods data in wide format is as follows:

id	Sex	X_t1	X_t2	X_t3	Y_t1	Y_t2	Y_t3
1	1	$X_{1,1}$	$X_{1,2}$	$X_{1,3}$
2	0

2 Pooled Regression Model

In panel data analysis, there are three basic regression models that we can apply to deal with data with both individual and time dimensions, namely the Pooled Regression Model (PRM), the Fixed Effects Model (FE Model), and the random Effects Model (RE Model). The core difference amoung these models lies in their different assumptions about unobservable individual or temporal heterogeneity: while PR Model assumes that all individuals and times share the same regression relationship, FX Moldel and RM Model allow for unobserved heterogeneity in different ways.

In panel data stored in long format, each row corresponds to an observation $(X_{i,t} , Y_{i,t})$ with a unique row identifieer. If we use OLS directly to regress Y on X without accounting for indivdual and time dimension, this procedure effectively treats the data as if it were a flat pooled cross-section. In this case, the model is not really regressing $Y_{i,t}$ on $X_{i,t}$ with recognition of the hierarchical structure, but rather regress $Y_{rowid}$ on $X_{rowid}$, treating each observation as an independent unit.

Let $Y_{i,t}$ denote the outcome variable for unit $i=1,2,...,N$ in time period $t=1,2,...,T$, and let $X_{i,t}$ be a vector of explanatory variables. The pooled regression model is given by:

$$Y_{i,t} = β₀ + β₁X_{i,t} + u_{i,t}$$

This model assumes that all observations are independent and identically distributed, and that error term satisfies the standard Gauss-Markov assumptions.

The pooled regression relies on the following assumption:

(1) No unobserved individual-specific effects: All heterogeneity across units is either captured by observed varibales or is irrrelevant.

(2) Error term properties:

Zero conditional mean: $\mathbb{E}[u_{it} \mid X_{it}] = 0$. This assumption ensures constant variance of the error term aross all observations.

No autocorrelation or cross-sectional dependence: $\mathrm{Cov}(u_{it}, u_{js} \mid X_{it}, X_{js}) = 0 \quad \text{for any } i \ne j \text{ or } t \ne s$

This implies that error terms are uncorrelated both across time (no serial correlation) and across individuals (no cross-sectional correlation).

No serial correlation assumption (Independence within time) and no cross-sectional correlation assumption (Independence within units) can be rewritten though Rubin potentical outcomes framewoork:

$$\left\{ Y_{it}(0), Y_{it}(1) \right\} \perp\!\!\!\perp \left\{ Y_{is}(0), Y_{is}(1) \right\} \quad \text{for all } t \ne s$$

which means that given unit characteristics, the potential outcomes of this unit at different time points are independent.

$$\left\{ Y_{it}(0), Y_{it}(1) \right\} \perp\!\!\!\perp \left\{ Y_{jt}(0), Y_{jt}(1) \right\} \quad \text{for all } i \ne j$$

which indicates that the potential outcomes of different individuals are not correlatesd with each ohter, equivalent to the assumption of no correlation among error terms.

$$\left\{ Y_{it}(0), Y_{it}(1) \right\} \perp\!\!\!\perp D_{it} \mid X_{it}$$

e.g., strong ignorability, corresponding to zero conditional mean $\mathbb{E}[u_{it} \mid X_{it}] = 0$.

3 Life Cycle Effect, Period Effect and Period Effect

3.1 Introduce to Time Effect

In regression analysis of panel data or repeated cross-cross-sectional data, researcher typically wish to control for changes in the life cycle of individuals and shifts in the social and historical context. These two dimensions are modeled using the varibales age and year , respectively. However, there is a fundamental difference between the effects of controlling for age and year in the theory and in empirical practice.

Age primarily reflects an individual's position in the life cycle and controls for the socalled life cycle effect, which refers to the systematic changes that individuals experiecne as they age. In labor market research, this effect typically manifests as wage changes resulting from accumulated experience, improved skills, or cahnges in physical health.

In contrast, the year controls for the period effect, which is the common influence of a specific historical period on all observed units. This effect reflects macro cahnges such as institutioanl changes, technological progress, inflation, and macroeconomic cycles. It affects individual of all ages and birth cohorts and is simultaneous and universal. For example a new minimum wage law impelmented in a given year will affect the income levels of all workers in that year.

Since the cohort (year of birth) can be calculated by subtracting the age from the year, that is:

$$cohort = year-age$$

Age, year and cohort are completely linearly dependent. When age and year are introduced into the regression model at the same time, this identity relationship causes complete muticollinearity in the model, meansing that the explanatory variables have a perfect linear relationship. In this case, regression analysis cannot accurately separate the age effect and the period effect, the standard error may be significantly inflated, the estimation results become unstabl.

Therefore, researchers should select variables based on the analysis objectives: if the focus is on life cycle behavior, age should be retained; if the emphasis is on historical background changes, year should be controlled first; if the aim is to ifentify the independent effects of age, year, and cohort simultaneously, a specific identification strategy should be adopted, such as constraint methods, hierarchical models, or principal component analysis. However, in conventional regression analysis, avoiding the simutaneous inclusion of age and year is an importamt principle for preventing multicollinearity.

3.2

The key to identify cohort effects lies in distinguishing between intergenerational variation (across birth cohorts) and temporal variation (calendar time). In cross-sectional data, this identification almost alway fails because cohorts and time are closely coupled, and it is difficult to control for unobserved heterogeneity. Panel data, however, provide a strategy for identifying these dimensions of variation.

While cohort is ofter included as a regressor in cross-sectional models, a critical identification issue arises in single-period cross-sectional data due to the identity:

$$cohort_i = year_i-age_i$$

This results in a strict linear dependency between age, year, and cohort, which leads to severe multicollinearity, especially when education level (grade) is highly correlated with age. Consequently, it becomes difficult to separately identify the main effects of cohort and grade, as well as their interaction. Moreover, cross-sectional data do not allow us to observe employment decisions, wage structures, or labor supply behavior across cohorts, thus making it impossible to determine whether the cohort effect stems from price changes in education returns or from structural shifts in work participation.

PRM overcomes these limitations in two key ways. First, it allows us to simulate a dynamic tracking structure by using wage information shortly after the completion of education, thus achieving a decoupling of cohort and period effects without relying on true longitudinal data. Second, by incorporating cohort × grade interaction terms, we are able to capture whether the marginal returns to education decline due to macro-structural changes, such as education expansion or increasing rates of short-time employment among women. We can further include short-time work status as a mediating variable, and within the PRM framework, so that we can estimate both direct and indirect effects.

Such an identification strategy would not be feasible in single cross-sectional data. PRM draws its strength from multiple survey years, offering broader variation in cohort composition, while allowing restriction to a common lifecycle phase (early career entry). This setup effectively avoids confounding age and cohort effects and provides more credible identification of cohort-induced structural shifts in education returns. The main trade-off is that PRM does not control for individual fixed effects, leaving estimates potentially vulnerable to unobserved heterogeneity. Nonetheless, we mitigate this concern through sample restriction and robustness tests involving mediation analysis.

4 Demo with Stata

4.1 Introduce to Education Effect on Wage

In consideration of data permission and risk of violations, I can not use SOEP or CFPS to demonstrate how it works in Stata. Fortunately, we have a publicly available dataset that can be practiced and demonstrated.

To demonstrate the PRM in practice, we use classic nlswork dataset, a standard panel dataset included in Stata. It contains longitudinal data from the Nationl Longitudinal Survey of Young Women (NLSW), covering various demographic and labor market characteristics of women in th United States between 1968 and 1988.

We begin by loading the dataset and setting the panel structure:

clear all
set more off
webuse nlswork, clear 
describe
xtset idcode year

I try to make a demonstration using the can't be more classic education sociology example on the effect of years of scholling years grade on wages ln_wage.

tab grade 

Table 1

We can construct a group variable for education:

gen educ_group = .
replace educ_group = 1 if grade <= 10
replace educ_group = 2 if grade > 10 & grade <= 12
replace educ_group = 3 if grade > 12
label define educ_lbl 1 "low" 2 "medium" 3 "high"
label values educ_group educ_lbl
tab educ_group

Table 2

We create the following figure by collapsing the data into group-level year means and using the twoway line command in Stata.

collapse (mean) ln_wage, by(year educ_group)
twoway line ln_wage year if educ_group==1, lcolor(red) ///
    || line ln_wage year if educ_group==2, lcolor(blue) ///
    || line ln_wage year if educ_group==3, lcolor(green) ///
    legend(label(1 "Low") label(2 "Medium") label(3 "High")) ///
    title("Trend of ln_wage over Time by Education Group") ///
    ytitle("Log Wage") xtitle("Year")

This command collapses the dataset to average values of ln_wage for each year–group combination, then overlays three time-series lines representing each education group.

Figure 1: Trend of Wage over time by Education Group

Figure 1 illustrates the evolution of average log wages over time by education group. Obviously, all three goups experience rising wages over time, reflecting general labor market improvements or inflations; The high education group ( > 12 years) consistently earns the most, followed by the medium and low groups; the steeptes growth trajectory is observed for the high education group, suggesting increasing returns to education.

From Figure 1, it is evident that wage levels increase over time across all education groups. Therefore if we want to estimate the effect of education on wages, we must consider time as potential confounding factor.

4.2 Baseline Model (Primary Least Structure)

If we only have access to OLS estimation using cross-sectional or pooled data and we wish to control for general time effects (e.g., trends over calendar years), we can include year as an additional regressor in the model, which may be the most common strategy available in cross-sectional settings.

To estimate the association betweem years of schooling (grade) and wages using pooled OLS, we begin with the following population model:

$$\text{ln\_wage}_i = \beta_0 + \beta_1 \cdot \text{grade}_i + u_i$$

This specification assumes that the variation in wages is soley attributable to differences in education, ignoring potential confounding by time.

To estimate this model we use the following command:

reg ln_wage grade // Model 1

Table 3

Then we get estimated model:

$$\widehat{\ln\ wage}_{i}=0.54+0.09 \cdot grade_i \quad (1)$$

The research question answered by Model 1 is: Whether education is positively correlated with salary for women. However, its explannatory power is extermely limited because confounding variables such as cohort, work experience, and working hours have not been controlled.

4.2 Control for Historical Effect

As we know, wages tend to increase over time due to inflation, economic growth, and labor market changes. Failing to control for these time effects biases the coefficent on education upward.

If we control for time variable year the population model is as follows:

to estimate this model we use the following command:

reg ln_wage grade year

we get

Table 4

Then we get estimated model:

$$\widehat{\ln\ wage}_{i}=-0.46+0.08 \cdot grade_i +0.01\cdot year_i \quad (2)$$

Comparing (1) and (2), we see that the education coefficient decreases from 0.09 to 0.08 when controlling for year. This indicates that part of the original effect of education in (1) was likely due to upward wage trends over time.

4.3 Control for Life Cycle Effect

Wage levels not only change over time due to historical developments, but also vary systematically across individuals depending on their age. This is referred to as the life cycle effect, reflecting how earnings increase as individuals gain more work experience, seniority, or job stability. Failing to control for this effect can also bias the estimated returns to education, especially if education level is correlated with age.

If we control for the variable age, the population model is as follows:

$$\text{ln\_wage}_i = \beta_0 + \beta_1 \cdot \text{grade}_i + \beta_2\cdot \text{age}_i+u_i$$

To estimate this model, we use the following Stata command:

reg ln_wage grade age 

This produces the following result:

Table 5

Then we get the estimated model:

$$\widehat{\ln\ wage}_{i}=0.22+0.08 \cdot grade_i +0.01\cdot age_i \quad (3)$$

Comparing (2) and (3) we find that, the coefficients on grade and the time related variable remain virtually unchanged, while the intercept undergoes a substantial change. The reason is that in the sample, year and age are almost linearly related. Since it is panel data, age an year in each observation usually satisfy the following relationship:

$$age_{it}=year_{it}-cohort_i$$

$$age_{it}=year_{it}-C$$

When the variable age is linearly dependent on year, as is the case in most panel datasets where individuals age one year per time period, substituting one for the other results in equivalent model structures with only a shift in the constant term. This shift reflects a change in the location of the baseline (reference point) rather than a substantive change in the relationship between education, time, and wages.

4.4 Baseline Model of Educational Returns with Cohort Effect

Previous models assumed that the marginal effect of education on wages is constant over time. However, labor markets evolve due to technological change, globalization, policy shifts, and institutional reforms, all of which may alter how education translates into erinings. To assess whether the returns to education have changed historically and illustrate the differences of estimates of education effect, I will try different interaction models, and gradually emphasize heterogeneity.

we extend the regression model by including an interaction term between grade and cohort:

$$\text{ln\_wage}_i = \beta_0 + \beta_1 \cdot \text{grade}_i + \beta_2\cdot \text{cohort}_i+\beta_3\cdot (\text{grade}_i \times \text{cohort}_i )+u_i$$

gen cohort = year-age
reg ln_wage c.grade##c.cohort // Model 4 

We get

Table 6 : Results for Model 4

Model 4 allows for differences in educational returns across birth cohorts, thereby identifying changes in the marginal benefits of education brought about by shifts in social structure (such as educational expansion, technological upgrades, and labor market saturation). The question this model can address is: Do educational returns change systematically across cohorts?

As shown, the estimated coefficient on education is positive and statitically siginificant, suggesting that one more year of schooling are associaltes with 5% higher wages. The coefficient on cohort is negative, indicating a declining wage level across more recent birth cohorts. The interaction term between education and cohort is statistically significant but soc-economically unsignificant, which implies that the wage return to each additional year of schooling has stalled, across successive cohorts. Therefore, there is no systematical change in educational returns across cohorts.

4.5 Age-based Control Mothods

Model 4 provides an initial answer to our question — whether the economic value of education has changed over time for women. In particular, it treats cohort and educational attainment as exogenous and ognoores potential confounders such as labor market experience, sectoral shifts, or regional differences. Furthermore, it does not differentiate between individuals who are still enrolled in school and thoose who habe completed their education, which may bias estimates of educational returns. To address these limitations adn explore heteroogeneity in greater detail, we can extend the analysis in the next section by fefining the sample selction and applying several restrain conditions.

A common strategy is to exclude respondents below a certain age threshold—typically age 25— on the assumption that most individuals will have finished schooling by that point. Accordingly, in the next model we restrict the sample to respondents aged 25 and older.

 reg ln_wage c.grade##c.cohort if age > 24 // Model 5

we can get

Table 7: Results for Model 5

We find that both coeffcients on cohort and interaction term are statistically unsignificant, which implies that there is no time effect on educational return. This approach has the advantage of simplicity. However it also introduces a bias: as age increases, so does the variation in labor markedt experience, which itself is a strong predictor of wages. As a result, the estimated return to education may be confounded with the effects of cumulative work experience, making it difficult to interpret cohort fidderences in educational returns cleanly.

To addresss this concern, we can further refine the age-based sample restriction to further mitigate concerns about heterogeneity in labor market experience. Specifically, we limit the sample to individuals between the ages of 25 and 29, a group that is highly likely to have completed their education but has not yet accumulated substantial experience in the labor market. This narrower age band serves two purposes: it exclues students whos are still enrolled in postsecondary education and at the same time, reduces the variation in work experience that may otherwise confound estimates of educational returns.

While still relying on age as a proxy for labor market entry, the age 15-29 window effectively captures respondents who are plausibly in the transition from school to work.

reg ln_wage c.grade##c.cohort if age > 24 & age < 30 // Model 6

we get

Table 8: Results for Model 6

Table 8 presents the results from the model restricted to individuals aged between 25 adn 29. Consistent with the baseline model, the coeffient on grade is statitically significant and positive but greater (from 5% to 20%), which indicates that educational attainment plays a more pronounced role in wage determination among youn addlts who no longer enrolled in school.

In contrast to baselin model, the coefficient on cohort is positive (0.03) and the coefficient on interaction term is negativ but soc-economically ignorable (-0.003). While the full sample showed a negative trend in wages across birth cohorts, the 25–29 age group reveals a positive and significant cohort effect. This finding implies that, among recent labor market entrants, later-born cohorts may in fact earn more, potentially reflecting improvements in early-career wage conditions or changes in occupational sorting. The interaction term between grade and cohort is negative but small, which indicates that the marginal return to education has remained relativly stable across cohorts within this age group. That is, although wages are higher for more recent cohorts, the incremental wage premium associated with additional schooling does not appear to have changed significantly over time.

4.5 Trajectories-Based Control Methods

To move beyond age-based approximations and more precisely identify the transition from education to work, we can construct a sample based on individual educational trajectories and labor market participation. Specifically, we identify the year in which each respondent completes their formal education by detecting the last year in which their reporteed schooling level increases. We then compute the number of yeras siche educational completion and restrict the sample to the first three yrars follwing this transition.

In order to ensure that individuals have achtually entered the labor market, we further require that respondents report positive work experience ttl_exp >0. This ensures that the sample reflects individuals who are not only done with schooling but also meaningfully engaged in the labor force.

sort idcode year
gen grade_lag = grade[_n-1] if idcode == idcode[_n-1]
gen delta_grade = grade - grade_lag
gen educ_finish_year = .
bysort idcode (year): replace educ_finish_year = year if delta_grade > 0
bysort idcode (year): replace educ_finish_year = educ_finish_year[_n-1] if missing(educ_finish_year)
bysort idcode (year): replace educ_finish_year = educ_finish_year[_n-1] if missing(educ_finish_year)
gen years_after_educ = year - educ_finish_year
gen in_sample = (years_after_educ >= 0 & years_after_educ <= 2 & ttl_exp > 0)	
// Model 7
reg ln_wage c.grade##c.cohort if in_sample == 1 // Model 7
margins, at(cohort=(42(1)54)) dydx(grade)
marginsplot, ytitle("Marginal Return to Education") xtitle("Cohort")

we get

Table 9: Results for Model 7

The question Model 7 can adress is, when educational investments is completed and enters the early stage of the labor market, does the return on education increase during the cohorts. The advantge of Model 7 is to identify the initial wage differential with the cohort interaction. However, it does not account for the cumulative effect of returns to education over time, nor does it capture the role of education in the later wage growth path.

The sample we have restricted by educational trajetories provieds the cleanest identification of early-caeer educational returns while minimizing contamination from ongoing schooling or cumulative experience effects. This suggests that while education remains valuable at career entry, its estimated return does not increase in the more tightly defined sample and may reflect a broader pattern of stability in wage premiums

Unlike in earlier specifications, the coeffiecnt on cohort is now negative (approximately 0.02), suggesting that newer cohorts earn systematically lower wages in their early careers. This result is consistent with broader trends in wage stagnation among younger generations and may reflect structural changes in the labor market.

Importantly, the interaction term between grade and cohort is statistically insignificant. This implies that while wage levels have declined for more recent cohorts, the marginal return to education has remained stable: individuals continue to benefit from additional education at roughly the same rate, even as overall entry-level wages fall.

Finally, to distinguish between part-time and full-time participation, we add an additional condition that the individual must have worked at least 30 weeks in the previous year (wks_work ≥ 30). This ensures that the sample reflects individuals who are not only done with schooling but also meaningfully engaged in the labor force.

// Model 8 
gen in_sample_full = (years_after_educ >= 0 & years_after_educ <= 2 & ttl_exp > 0 & wks_work >= 30)	
reg ln_wage c.grade##c.cohort if in_sample_full == 1

we get

Tbale 10： Results for Model 8

When we further restrict the sample to respondents engaged in standard forms of employment.By excluding respondents working reduced hours, this specification focusesexclusively on individuals with full labor market participation, thereby eliminating wage variation driven by differences in labor suppy intensity.

We find that the coefficient on grade decreased to 0.07. The coefficient on cohort remains negative (-0.02, p = 0.011). The coefficient on the interaction term is statistically unsignificant (p=0.554). This implies that among early-career women, the marginal return to education remains stable across cohorts, yet initial wages have declined significantly. This divergence suggests that education continues to confer similar relative advantages, but that newer cohorts face structural disadvantages in absolute earnings. One plausible explanation is the rising prevalence of short-time work among women, which may depress observed wages even when educational attainment is high. Alternatively, compositional shifts in occupational entry or a general erosion of entry-level job quality could also contribute to this trend. Further analysis is needed to disentangle the roles of labor supply intensity, job sorting, and family-related constraints in shaping these cohort-specific wage patterns.

4.6 Quantity Effect in Cohort Effect on Initical Wage for Women

We have reason to suspect that this systemaic decline in wage levels across cohorts may be due to more women entering short-time work positions, i.e., there is a quantity effect that lowers women's initial wages across cohorts, as we have seen that educational return rate does not decline over time.

// Model 9
reg hours cohort if in_sample == 1
// Model 10
gen short_time = hours <30
logit short_time cohort if in_sample  ==1

we get

Table 11: Results for Model 9

and

Tbale 12: Results for Model 10

We apply Model 9 and Model 10 to anwser this question: Is the younger cohort more inclined to short time work? Clearly, the coefficient on cohort is both statistically and soci-economically significant, which indicates that more and more young women are entering non-standard hours work after completing education.

// Model 11
reg ln_wage short_time grade cohort if in_sample == 1

we get

Table 12: Results for Model 11

Clearly, after controlling for education and cohrt, the initial age penalty associated with short time work as high as 15%.

We use the mediation analysis framework of Baron & Kenny (1986) to estimate the total effect, direct effect and indirect effect under the setting of linear regression. The mediating variable is short_time (whether to engage in short-term work, 0/1 variable), the explanatory variable is cohort, the outcome variable is ln_wage, and control variable is grade.

// Composite Model 12 
reg ln_wage cohort grade if in_sample == 1
reg short_time cohort grade if in_sample == 1
reg ln_wage cohort short_time grade if in_sample == 1

we get

Table 12: Total Cohort Effect TE

and

Table 13

and

Table 14:

To assess whether the cohort effect on wages is mediated by short-time work participaton, we follow the classical three-step approach. All model are estimated using ordinary least squares and are restricted to the subsample f individuals who have completed their education and just entered the labor markt.

In first step, we estimate the total effect TE of cohort and education level on log wages. The estimated coefficient for cohort is negative and statistically significant (β = −0.0168, p < 0.001). In the second step, we regress the proposed mediator short_time on both cohort and grade. the estimated coefficient for coohort is positive and statistically significant (β = 0.00446, p < 0.001), indicating that more recent cohorts are more likely to enter short-time employment. At the same time, the coefficient for grade is negative (β = −0.00740, p < 0.001), suggesting that higher educational attainment reduces the likelihood of short-time work. These results support the hypothsis that short-time work may act as a mediating channel between cohort and wages.

In the third step, we include the mediator short_time alongside cohort and grade in the wage equation. The effect of short-time wokr on log wages is negative and significant (β = −0.1525, p < 0.001), while the direct effect of cohort decreases slightly in magnitude from −0.0168 to −0.0162 but remains significant. This reduction suggests the presence of a partial mediation effect.

To compute the indirect effect, we multiply the cohort → short_time coefficient (a = 0.00446) with the short_time → ln_wage coefficient (b = −0.1525), yielding an estimated indirect effect of:

$ab = 0.00446×(−0.1525)≈−0.00068$

This value quantifies the portion of the cohort effect on wages that operates through increased participation in short-time work. Although statistically siginifican, the size of mediatied effect is relatively small in magnitude (soc-economically not so relevant). It accounts for approximately 4% of the total effect, suggesting htat while changes in work arrangements contribute to wage decline, the do not constitude the primary driver.

4.7 Summary of Section

Through the series of models presented above to examine the dynamics of educational returns over the early career stage, I truely believe that you have gained a clearer understanding of both the analytical power and the limitations inherent in different modelling strategies. Beginning with the simplest baseline speification, which helps to capture the inituitive relationship between education and wages. Building on this, interaction models enabled the indentification of the temporal heterogeneity/consistency in returns to education, uncovering potential stagnation of educational return rate over time, indicates that there is no price effect on educatianl returns for young women. Finally, mediation models were employed to detect quantity effects — specifically, the extent to which increasing participation in non-standard (short-time) employment mediates the observed decline in wages across cohorts, though soc-economically unrelevant.

5 Summary

In this chapter, we adopt the pooled Regression Model PRM as our primary tool to systematically investigate whether the returns to education have changes across cohorts. Unlike tradional panel fixed effects model, our PRM specification does not relay on within-individual variation. Instead, it leverages the structure of pooled cross-sectional data to identify time-based heterogeneity/consistency. Specifically, by restricting the sample to individuals within 0-2 years after completing their education, we focus on the wage outcome at the entry point of labor markt, effectively constructing a quasi-natural experimental setup that mimic cohort-based temporal variation.