Chinese University Major Decision and Its Effect on Wages: Modeling Interaction Between Major Specificity and Education-Job Relevancy Using Machine Learning Approaches

The study of wage determinants is a core topic in Labor Economics both on the supply and demand sides. Analysis of starting wages focuses on human capital theory and family background, given that new employees do not yet have work experience and wage records. However, it is generally understood that one’s first job and starting salary are extremely important, as the former is one’s springboard to a future career, while the latter sets a signal for future earning levels [11].

The field of intergenerational mobility contains many existing studies on the family effect on college graduate wages. Family background is typically measured by parental income, job prestige or education. It has been noted that parental background significantly influences students’ achievement and their future incomes in OECD countries [9, 13, 35]. However, conclusions about causality are mixed due to omitted variable bias, variations in operationalization, and for other reasons. For example, parents’ social class ranking has been found to be significant in determining female (but not male) MBA graduates’ starting salaries [23]. There is also criticism of using annual salaries (including starting salaries) as a measurement, as it leads to a biased calculation of intergenerational elasticity and intergenerational correlation. Permanent earnings or an average for several years have been suggested as proxies [50]. A Chinese study estimates that the wage premium of having a cadre parent (a civil servant in at least the county-level government) is 15% of total earnings [34]. Li claims that 58.1% of “cadre parents’’ are college graduates compared with 19.1% for non-cadre parents, and that parents’ social class is not associated with their children’s major choice.

As starting salary is free from the noise of experience and former salary, some scholars focus on the effect of education on starting salaries, including those who emphasize human capital theories [7, 48], or the alternative screening hypothesis [51]. The former treats education as a good measurement of productivity that is reflected in future salary; while the latter supposes that education merely functions as a signal, given that one’s productivity is innate and unrelated to educational training. A major body of literature indicates a positive correlation and causal relationship between university tiers and earnings. Mark Hoekstra uses SAT scores and GPA to estimate student proximity to flagship university admission rules, and builds a counterfactual by comparing those who are near to but below the cut-off of the estimated admission rule to those who are expected to achieve flagship university enrollment [27]. Those who are predicted to be enrolled according to the model earn around 20% more than those who are below the cut-off. Rodney Andrews compares different types of universities by analyzing the unconditional quantile treatment effect of different university diplomas on earnings [3]. By analyzing the distribution rather than mean regressors, significant wage premium heterogeneity is found among students of the University of Texas at Arlington (UTA), Texas A&M University, and community colleges in Texas across different wage intervals.

Since Becker’s initial work, human capital theory has also continued to explore the relationship between education and occupation [6]. One’s major signals one’s higher education curriculum content. A few seminal studies have shown a remarkable wage gap between different university majors [16, 31]. However, due to data limitations, Chinese scholars have often grouped hundreds of sub-majors into very broad categories such as Science, Engineering, and Social Science [29, 33, 54]. This oversimplified grouping of majors is problematic. For example, according to the official set of discipline genres, social sciences such as Journalism and Languages are grouped as Literature. In addition, most Chinese studies in this field use dummy variables to label major disciplines, and hence focus more on the phenomenon of wage gaps across majors, lacking exploration of major pricing mechanisms within the labor market.

Human capital can be conceptualized as general or specific [6, 7]. Specific human capital is only useful in one firm, while general human capital involves skills that can easily be used in other firms. The difference between the two rests on a measure of the degree of skill transferability [6, 30]. Recent studies indicate that vocational majors have a higher payoff in the earlier period of one’s career while academic majors may pay less at the beginning but overtake vocational majors in later career stages; and that a general major has a similar payoff across a broad occupation choice set, while a more specific major has a good payoff only in a more narrowed occupation choice set [10, 24, 30]. The Hirschman-Herfindahl Index, Gini coefficient and CR (concentration ratio) are used to measure major specificity [1, 10, 30]. According to Leighton, graduates with highly specific majors measured by occupational HHI earn 8% more than the average in the US labor market; while Altonji, using the Concentration Ratio measure, uncovers a positive relationship between major specificity and earnings [1, 30]. Moreover, scholars in this field have been implementing new models and research designs, such as regression discontinuity, instrumental variables, and dynamic models, to strengthen causal inferences [1, 5, 8].

Variable interaction here refers to bivariate interaction. An interaction between two independent variables means that the effect of one variable is dependent on the value of the other. Function f(x) exhibits an interaction between two variables x₁ and x₂ if the difference in the value of f(x) as a result of changing the value of x₁ depends on the value of x₂ [22, 28]. A traditional way of measuring interaction is to add a multiplication term to the model. In terms of machine learning, the partial dependence measure shows the marginal effect that one or two features have on the predicted outcome [21]. Partial dependence measures the change in the average predicted value as specified features vary over their marginal distribution [28]. Based on the partial dependence measure, the Friedman H-index is an interaction measure that computes the marginal effect of two targeted variables. If two variables do not interact, their joint partial dependence function equals the sum of their individual partial dependence functions. Therefore, given two interacted variables, the H-index is calculated as the sum of squares of the difference between their joint and individual partial dependence functions, divided by the sum of squares of their joint partial dependence function. Therefore, the H-index is employed for measuring interaction strength. However, some researchers argue that the original H-index is inefficient and gives a spurious result if the denominator (the sum of squares of the joint partial dependence) is close to zero. As a remedy, the adjusted H-index drops the denominator of the original H-index [4, 28].

Major specificity is measured by the Herfindahl Hirschman Index (HHI). The HHI index is calculated by the following formula:

where m denotes major; o denotes occupation index and S_mo denotes the share of graduates with the major m flowing into the occupation o [10, 30]. This index ranges from zero to one. A bigger HHI means higher major specificity. By this measure, a major that links to a small set of job roles is of high specificity; while a major with wide and even distribution of job roles is a general major. For example, for major x, if 90% of the graduates go to one job, 10% go to another job, then the HHI for major x is 0.9²+0.1²=0.82. Similarly, the industry HHI uses the same method to measure major industry diversification.

Table 3 shows the values for major specificity. The HHI index is a measure of major specificity derived by calculating each major’s job or industry concentration. A major with higher HHI value is a major with a more specific or clearer career decision. As the space is limited, only the twenty most common majors are shown below.

Social science researchers are often confronted with the issues of endogeneity and selection bias [32, 36]. The two-step Heckman selection model can adjust the estimator to deal with potential non-response bias [25, 26], therefore the Heckman models are fitted as the baseline. The dependent variable of interest (step two) will be estimated based on the control of the inverse mills ratio from step one. Around 31,911 out of the 45,661 respondents (69.89%) from Survey One did not answer Survey Two, which collected the data about wages and other job-related variables. The detailed formulations are in Subsection “Heckman model”.

For machine learning approaches, the KNN and Random Forest models are fitted. Then the pair-wise interaction effect is measured by the adjusted H-index based on the prediction of the KNN and Random Forest model fits [22]. The heat map and network plot are displayed using the R package vivid, developed by Inglis et al. [19, 28]. The original H-index for measuring interaction is:

where function f_j and f_k denotes the partial dependence function, and f_jk denotes the two-way partial dependence function [28]. The partial dependence function is estimated as:

where x_c denotes independent variables, x_cⁱ are the values of the independent variables of n observations, x_s denotes the variables of interest, f_s refers to the machine learning model selected, and function g() gives the prediction from the machine learning model. This estimates whether and to what extent two variables in the model interact with each other. The interaction effect is more powerful as the H-index gets larger. The adjusted H-index reduces the identification of spurious interactions by ruling out the denominator of the original H-index [4, 28], which is:

In studies of wages, there are two endogeneity concerns: the family-education relationship and sample selection bias. Both family background and education are significant wage determinants, but family background may affect education. In this study, parents’ education and hukou location are family-related variables, while university tier and major decision are education-related variables. The first potential endogeneity is between family and university tier. However, as shown in Table 4, university tier seems to be unrelated to family background.

The relationship between family and university tier is further analyzed by multinomial logit model in Table 5. The dependent variable university is coded as 0 = private, 1 = ordinary public, 2 = 211 universities. The independent variables are hukou (0 = rural, 1 = county seat, 2 = metropolis), parentedu (0 = high school or below, 1 = above high school), gender (male = 0, female = 1) and year (0 = 2016, 1 = 2017, 2 = 2018). As the VIF values (Variance Inflation Factor) of the independent variables are all smaller than two, multicollinearity is not detected. Ordinary public university (middle-ranking) is set as the base. After the relative risk ratio transformation, cateris paribus, it is found that educated parents are associated with an increased chance (31.4% more likely) of students being enrolled in a private university (low-ranking) vs. an ordinary public university, while educated parents are also associated with an increased chance (60.3% more likely) of being enrolled in a 211 university (high-ranking) vs. an ordinary public university. In other words, with educated parents, graduates are most likely to attend the best universities in the sample, followed by the lowest ranking universities, and least likely to go to a middle-ranking university. Therefore, educated parents do not have a consistent positive impact on university tier. Similarly, the effect of hukou on university tier is not uniform either. The chance of attending private vs. ordinary public universities increases for graduates with more urban hukou; while for 211 universities, hukou switching from rural to county-seat decreases the enrollment rate; while hukou switching from rural to metropolis increases the enrollment rate. Therefore, having urban as opposed to rural hukou does not render a clear positive or negative effect on university tier.

In addition, there is also a natural concern that students’ subjective selection of universities may be affected by tuition expenses. However, the typical public versus private comparison in the U.S. does not apply in China’s case. In 2018, tuition for most majors in China’s private universities was around 12,000 to 30,000 yuan per year without public subsidy. With full government subsidy, tuition for most majors in 211 universities and ordinary public universities was between 5,000 and 6,000 yuan per year. Therefore, in China, the best universities are also the cheapest. Hence the preference for choosing an upper-tier university rather than a lower tier one, especially when they are located in the same city, is extremely straightforward. Therefore, the biases identified by Hoekstra (omitted family bias and university selection bias) are less relevant here [27].

It is also necessary to analyze potential endogeneity in terms of family impact on major choice. Table 6 shows graduates’ discipline choice with different hukou backgrounds. The percentage of major choices from the same hukou cohort is shown in parentheses. With the exception of agriculture (2% of total observations), where graduates from rural hukou are 2.34 times more likely to pick agriculture than their peers from metropolitan backgrounds, there is no discernible pattern of major choice across all hukou origins. To validate this conclusion, the relationship between family and discipline choice is analyzed using a logit model in Table 7. Discipline has nine categories: Economics (Econ.), Law, Education (Edu.), Language (Lang.), Science (Sci.), Engineering (Engr.), Agriculture (Agri.), Medicine (Med.) and Management (Mgmt.). Nine logit models are fitted separately. The dependent variable of each logit model is a dummy variable measuring selection of each discipline. For instance, in the first model, Economics = 1, other disciplines = 0. Explanatory variables are hukou and parentedu, with university, gender and year controlled. Within the nine disciplines, hukou has an uniform effect on choice of Language, Agriculture and Medicine. Agriculture and Medicine are less attractive to graduates with more urban hukou, conversely, Language is preferred as hukou changes from rural to metropolis. According to Table 6, these three disciplines represent 13.52% of the sample or 18.52% of the Chinese bachelor’s degree holding population. Additionally, the Wald Test is used to compare coefficients for different levels of hukou in the models, determining if the coefficient for hukou = 1 equals the coefficient for hukou = 2. The test indicates that only the hukou coefficients for Language and Agriculture differ significantly within the three disciplines, excluding Medicine. We also find that county-seat graduates are significantly more likely to choose Economics and Management compared to metropolis and rural graduates, while they are less interested in Law and Science. For parent education, choices of Economics, Law, Agriculture and Medicine are found to be associated with parentedu. Students with educated parents are more likely to choose Economics and Law, while students whose parents did not attend high school tend to favor Medicine and Agriculture. These disciplines represent 15.38% of the sample or 17.73% of the Chinese bachelor’s degree holding population.

Consequently, in contrast to prior research, we first find hukou and parent education level bring no clear advantage in access to higher university tiers; second, there is no evidence that family variables influence most students’ discipline choice. Admittedly, the models show that family variables may still influence university tier and major choice, though in an inconsistent way. For example, compared to having a rural hukou, having a metropolis hukou has a “polarizing” effect on university tier, as metropolis graduates are least likely to attend middle-ranking universities. Law and Science are the least popular disciplines among county-seat graduates, while their preferred disciplines are Economics and Management.

Another endogeneity concern relates to sample selection bias. Compared with the classic scenario in which a female has a higher threshold for not working than a male, this study of fresh college graduates has somewhat different findings. In this dataset, no single variable overwhelmingly dominates in terms of missing values, but all job-related variables have around 70% missing values. This is because nearly two thirds of the graduates who completed Survey One missed Survey Two. If a respondent missed Survey Two, all the job-related variables are recorded as missing in the data. As illustrated earlier, these job-related missing values can be attributed to multiple sources. To name a few, these include: (1) some graduates did not access their school email; (2) some left for further study hence they were not able to report their occupation-related variables; (3) the unemployed; (4) some may reluctant to disclose personal information; and (5) some of the employed may be too busy to bother with the survey. According to an official report covering all universities in Jiangsu Province, the number of graduates who go on to further study is three times the number of unemployed graduates [52]. However, it is reasonable to assume that the reasons for the missing data are not mutually exclusive. For example, an unemployed graduate could choose further study if she or he did not obtain a satisfying job offer, or a graduate went for further study and so then did not check their former school email. As the sources of each missing value are not mutually exclusive and unobserved, there is no additional information with which to explain the occurrence of the missing data. Given the sample selection bias, the Heckman model is used to control the bias in the second step by including the step-one Inverse Mills Ratio. Please see the model result in Sect. “Heckman model”.

Linear modeling is the traditional and so far the most widely used method family in studies of wages. Hence, we select linear models as a good baseline for comparison with the machine learning models. The first step of the Heckman model is:

where Record_i is the non-missing status of the wage variable (0 = missing, 1 = non-missing). MSpec_i refers to the major specificity measured by job-wise or industry-wise indicator. The vector X_i contains the control variables in both steps collected by Survey One, which are gender, university tier, year and discipline dummies. The vector Z_i includes college education satisfaction (satisfaction, 0 = bad, 1 = neutral, 2 = good and 3 = very good), parent education (parentedu) and hukou, functioned as the exclusion restriction put forward by Lennox et al. [32].

As discussed in Subsection “Family Effect on University Tier and Major Choice”, there are multiple sources that may lead to missing values, therefore the selection of appropriate exclusion restrictions is difficult. The three exclusion restriction variables hukou, parentedu and satisfaction are all significant and have the predicted signs in the Heckman first step (Table 8), while are not significant if added in the Heckman second step. Furthermore, if including the three variables in the second step, the VIF of the inverse Mills ratio rapidly exceeds ten, indicating a high multicollinearity problem. In the survey, to record satisfaction students were asked to evaluate their degree of satisfaction with the college education they received. Students’ satisfaction is highly associated with school status and other social connections [49, 55]. Hence, lower satisfaction with their college education can lead to graduates’ infrequent use of school email after graduation and corresponding tendency to miss Survey 2, corresponding to the missing data source (1) discussed in Subsection “Family Effect on University Tier and Major Choice”: some graduates missed Survey Two through not checking school email. Meanwhile, hukou and parentedu are found to be negatively associated with non-missing values. That is to say, urban graduates or graduates with educated parents were more likely to miss Survey Two. This is explained by the missing data source (2) in subsection “Family Effect on University Tier and Major Choice”, as graduates with a better-off family background are more likely to leave for further study, hence their occupation-related variables are missing. Therefore, these three variables are selected as the potential exclusion restrictions.

Although satisfaction, hukou and parentedu are not significant if added in the Heckman step two, their exogeneity in the wage function is disputed in the academic literature. There are scholars who argue that students’ satisfaction with their college education is affected by their valuation and image of the school and their expectations [2], or that satisfaction may predict students’ productivity in school [47]. Therefore, satisfaction can be a potential wage determinant through its association with students’ valuation of their school or productivity in school. In terms of the family variables, some scholars have argued that wage difference is mainly caused by educational attainment, rather than rural-urban hukou differences [38, 40]. However, some empirical studies also state that hukou is relevant to job and wages, as college education and hukou transfer are found to interact in wage pricing [56], or because hukou can affect graduates’ job seeking intensity [54]. Similarly, the effect of parent education, or socioeconomic status (SES) in a broader sense, may also affect graduates’ job seeking behavior and wages [9, 13, 35]. Therefore, we must admit that the three exclusion restriction variables are ad hoc due to exogeneity concerns.

Table 8 shows the selection equation (first step) of the Heckman model. Model 1 uses job-wise HHI, whereas Model 2 uses industry-wise HHI as the major specificity indicator. The major-related variables in the Heckman first step are major specificity and the discipline dummies. For major specificity, industry-wise major specificity has an insignificant negative sign, but job-wise major specificity is significantly negative, indicating that majors with high specificity may be associated with more missing values. One explanation is that high-specificity majors, such as majors in Medicine and Science, have the highest rate of further study, and thus graduates with these majors are more likely to having missing values for the wage variable [41]. The discipline dummy variables are omitted for brevity, showing that only Agriculture has a significant positive sign (with Economics as the base), indicating that the Agriculture graduates in this dataset were more inclined to answer Survey Two. Besides the major variables, a higher university tier, parent education, and urban hukou all increase the likelihood of missing values for wages, whereas – as illustrated earlier – higher satisfaction with college education decreases it. The AUC stands for the area under the ROC curve. The AUC value of 0.66 indicates that the probit model is with certain discriminatory power.

The second step of the Heckman model is:

where Logwage_i is the logarithm of wage, MSpec_i is the major specificity, and Rel_i refers to the major-occupation relevancy (0 = Not Relevant, 1 = Relevant). The interaction between the MSpec_i and the Rel_i is also included. The vector X_i contains the same control variables as in the first step, collected by Survey One, which are gender, university tier, year and discipline dummies. The vector X’_i contains job search variables (searching time, application number and offer number), job dummies and industry dummies, all collected by Survey Two. IMR_i stands for the inverse Mills ratio.

In Table 9, Models 1 and 3 employ the job-wise HHI index; while Models 2 and 4 use the industry-wise HHI as the indicator of major specificity. The VIF values of MSpec, Rel and IMR are smaller than ten, indicating that the variables of interest do not suffer from multicollinearity. As shown in Models 1 and 2, major-job relevancy and the major specificity are significant wage determinants, as they are both positively correlated with wages. Adding the interaction term, Models 3 and 4 indicate that relevancy or major specificity alone cannot guarantee a high payoff, but their interaction is positive and significant in terms of wage estimation. If major specificity is a measure of major productivity, then the realization of the major premium in the labor market is conditioned by the major job relevancy. For example, according to Model 3, the interaction between major specificity and major job relevancy is 0.254. Since the dependent variable is log-transformed, an increase in the interaction term from zero to one would explain an e^0.254 − 1 = 28.9% increase in wages. Table 12 in the Appendix shows the OLS results for comparison to the Heckman model. The OLS and the Heckman second step give almost the same coefficients for major specificity and its interaction with the major-job relevance. As the dataset does not record more detailed information on the missing values, the missing value analysis cannot go any further. Therefore, the machine learning models below will not control for sample selection bias.

The VIF values of the interaction term MSpec*Rel in Models 3 & 4 are 10.34 are 12.46 due to the existence of MSpec and Rel. The VIF values of MSpec*Rel drop to 3.74 and 2.90 if MSpec and Rel are excluded. As the model results remain almost the same if MSpec and Rel are excluded in Models 3 and 4, they are omitted due to space constraints.

As major specificity measured by industry shows very similar results to that measured by job, we display only the results of the latter. Ten numerical and ordinal independent variables are used to fit the machine learning models. In order to compute the interaction between major specificity and relevance, the KNN and Random Forest regressions are fitted first. As both models are regressions rather than classifications, the tuning metric is MSE (mean square error) or RMSE (root mean square error). As shown by Fig. 3 in the Appendix, the KNN regression uses 5-fold cross validation to tune the k value. The MSE is used as an evaluation metric. The MSE flattens out when k is greater than 25. The optimal k is 63, with the smallest MSE of 0.112 (RMSE = 0.335). As shown by Fig. 4 in the Appendix, the number of trees (n_estimators) and variable splits (max_features) are optimized for Random Forest regression. With 5-fold cross validation, the initial number of trees is set within [1, 1000]. The RMSE flattens out when n_estimators is greater than 300. The optimal n_estimators is 464 with the smallest RMSE, hence the number of trees is set within [1, 500]. Because there are ten independent variables, the range of max_features is tuned within [1, 10]. Using 5-fold cross validation resampling, the optimal value of max_features is 3, with the smallest RMSE of 0.322. Figure 4 also compares the RMSE of max_features = Best (max_features = 3) and the baseline max_features = None (max_features = 10).

Table 10 compares the performance of the KNN regression (k = 63) and the Random Forest regression (n_estimators = 464, max_features = 3) of 5-fold cross validation, using three metrics: RMSE, MAE, and R2 Score. The Random Forest has a higher R2 score, lower RMSE and MAE than the KNN counterpart. As a result, the Random Forest regression outperforms KNN in this case.

After that, the KNN and Random Forest fits are used to calculate variable interaction. For interaction, the partial dependence function is calculated first, to measure the change in the average predicted value as specified variables vary over their marginal distribution. Then for measuring variable interaction, the adjusted H-index is calculated by comparing the joint partial dependence for a pair of variables to their marginal effects respectively [28]. In addition, the Fisher agnostic permutation approach is used to calculate the variable importance [20, 28]. A variable is considered important if permuting its values increases the model prediction error. Table 13 in the Appendix displays the detailed variable importance index. Compared with the education-related variables, family variables such as hukou or parentedu are less important wage determinants.

Table 11 displays part of the adjusted Friedman H-index, which measures major specificity’s interaction with the other variables, calculated using KNN and Random Forest fits. The KNN’s adjusted H-index reveals a stronger interaction between major specificity and major-job relevancy (0.023) than its Random Forest counterpart (0.015). Using the KNN, major specificity mostly interacts with gender, major-job relevancy and university tier. Using Random Forest, major specificity interacts most strongly with university tier, gender and major-job relevancy. According to Table 11, major specificity strongly interacts with major-job relevancy (rel), university tier (university) and gender. If major specificity represents one dimension of major productivity, then university tier is another dimension of the overall quality of graduates, while major-job relevancy measures the match between the major and the job. Major specificity’s interactions with university tier and major job relevancy combine as a measure of a candidate’s productivity in terms of specialized skills, overall abilities and career compatibility. Therefore, graduates’ starting wage determination can be viewed as an integration of major, university tier and education-career relevance.

Figure 1 shows the heatmaps of the KNN and Random Forest fits. The off-diagonal Vint stands for variable interaction; diagonal Vimp stands for variable importance. Comparing the color contrast on the same scale of Vint, the KNN fit identifies more pairwise interactions. Containing the same information as the heatmap, Fig. 2 is better suited to interaction detection. Pairwise interactions are shown by the color depth of the line. Variable importance is indicated by the node size. The interaction between the major specificity (mspec) and the major-job relevancy (rel) is presented in both figures. The figures also show that family factors such as hukou or parentedu do not have an obvious interaction with the educational variables such as university tier or major specificity.