In order to verify our main hypothesis, we consider a main model inspired by Hanushek, Woessmann, and Zhang (2011). The model is built over a difference-in-differences (herein- after, DD) framework, but this is not the archetypical DD design used in applied economet- rics, where the two considered dimensions are a trigger policy which builds up two difference groups, and a time variable which splits the sample to study pre- and post-policy outcomes, as in the classic paper of Card and Krueger (1994). Rather, we use a more general DD set- up, more similar to Angrist and Evans (1999) to study the effect of changes in state abortion laws on teen pregnancy using variation of state and year of birth, or rather to Autor (2003) who implements the Granger test investigating the effect of employment protection policy on firms’ use of temporary help for lags of more years than in a before-after strategy. Without considering a “true” trigger policy to run our DD configuration, we apply the treatment in a continuous and subsequent levels of age, starting from a fixed baseline age to a hypothetical and broad retirement age, which represents our individuals’ life-cycle in the SHIW data- set. The treatment is general education at upper secondary school level, while individuals
with vocational (and “other”, taken as given) education fall into the control group. The main differences with the Hanushek, Woessmann, and Zhang (2011) derive from the availability of information in our dataset built from SHIW waves, as illustrated in Section 3.1, which gives us more information than the source of Hanushek, Woessmann, and Zhang (2011) us- ing IALS-97 dataset (Organization for Economic Co-operation and Development (OECD),
1997) only with cross-section data, exploiting a DD regression for only one wave of analysis. More waves of analysis allows us to partially intercept the impact of selectivity in education through individuals born in different decades, who took the treatment in different periods of time: for this reason, we pool information from different waves of the survey together, taking into account age-invariant wave effects, represented by the coefficients δsin Equation (3.1), where we use the notation of Wooldridge (2010) and Angrist and Pischke (2009).
yi= α0+ α1· agei+ α2· age2i + β0· gi+ β1· gi· agei+ β2· gi· age2i +
+
∑
s
(δs· si) + Xi· γ + εi
(3.1)
In Equation (3.1), yi is the labour-market outcome of interest for the ith individual, age and age-squared capture the normal age-y pattern in economy without treatment, gi is an indic- ator variable equalling 1 if the ith individual has general education type as specified and zero otherwise, identifying treatment and control group, siis an indicator variable equalling 1 if the ith individual belongs to wave s, and εi is simply the the unobserved error term. Xi is a vector of control variables for the ith individual including oi, which is an indicator variable equalling 1 if individual i has “other” education type and zero otherwise, regional and muni- cipality fixed effects to eliminate overall differences between regional and municipal micro- labour markets, and various measures of individual influencing labour-market skills (other than education type), such as education level (including tertiary education), time-invariant birth family background and current household factors. We will look in depth at these con- trols in Section3.2.3. Therefore, in the same fashion of Brunello and Rocco (2015) we use ageing as a synonymous of time, as we do not explicitly distinguish between age and time effects in our empirical study because waves are used only for pooling. Furthermore, in con- trast with Hanushek, Woessmann, and Zhang (2011) and in line with German Microcensus 2006 data of Hanushek et al. (2017), we propose non-linearities in terms of quadratic effects
of age over the education type, in order to better identify the trajectory of the linear ageing effect over the life-cycle discriminating for education type, and the intensity over time of the differential impact between general and vocational education. This is possible because the pooled sample has sufficient power to look at them.
The configuration of the main model in Equation (3.1) varies on the basis of the y labour- market outcome’s choice, as the coefficients’ interpretation.
• Considering employment as labour-market outcome, we obtain a linear probability model (hereafter, LPM) where the dependent variable is a Bernoulli random vari- able empi= 1 indicating the response probability of employment (Wooldridge, 2010, pp. 562-565): this is obtained starting from the variable nonoc in the SHIW (Bank of Italy,2015), which equals to zero if individuals is employed and > 0 if individual is unemployed for the categories shown by Table3.3. The interpretation of coefficients is quite straightforward in terms of employment probability: κ · 100 is the impact of a unit increase of the level variable x at which this coefficient is assigned on the probability of employment in percentage points.
• for wage patterns over the life-cycle, we consider the net wages and salaries extracted from the SHIW yl1 aggregate indicator of the compensation of employees in their jobs (Bank of Italy,2015), and reshaped in natural logarithm form, pushed into a lin- ear regression model. This transformation is applied in empirical labour economics’ researches for several reasons:
– to deal with monetary values;
– to smooth sample outliers without facing other attrition issues;
– to interpret coefficients as the semi–elasticity of wages in a log–level model, for which the parameter on a level variable x has to be approximately inter- preted as the percentage change in y resulting from a one-unit change in x, 100 · κx = 100 · ∂ ln E[y|x,z,...]∂ x , conditional on all covariates (Wooldridge, 2010, pp. 15-19).
In Equation (3.1), the coefficient β0 measures the time-invariant (at age = 0) employment probability, or percentage variation on wages, of those with general education G relative to
those with vocational education V (and “other” education, but hereafter we refer only to
vocational education taking this smaller group as given) at the entrance of labour market.
β0= E[yi| g = 1, age = 0, s, X]
| {z }
time-invariant probability of employment or expected value of ln wages and salaries
for general education
− E[yi| g = 0, age = 0, s, X]
| {z }
α0
floor probability of employment or average value of ln wages and salaries
=
= ¯yG,0|s,X− ¯yV,0|s,X, ∀ age.
(3.2)
As Hanushek, Woessmann, and Zhang (2011) state, we cannot say the overall difference in employment probabilities and wages between general education and vocational education reflected in β0adequately identifies the impact of general education, because this parameter implicitly involves any elements of selectivity in the completion on different types of school- ing which are not captured by Xi, raising an omitted variable bias, and it is difficult to control for any factor of congestion in the labour market at country level, especially for what matters the closest waves of analysis affected by the financial crisis (Crepaldi et al.,2014). Withal, it is doubtful that the quantified effects on employment found from our datasets fully capture the systematic differences across schooling groups, and this is also a common issue on the datasets considered by the literature in Chapter2. Hence, the key parameters of interest are:
• β1, the DD estimator, which captures the differential impact of a general relative to a vocational education on the labour-market outcome for each year of age, drawing the divergence in employment patterns by education type over age cohorts, and identifying a cut-off age of convergence between the two paths.
β1· age + β2· age2= E[(yi| g = 1 − yi|g = 0)
age, s, X ]
| {z }
difference after age-lags in the labour-market outcome of a general relative to a vocational
+
− E[(yi| g = 1 − yi|g = 0)
age= 0, s, X ]
| {z }
time-invariant difference in the labour-market of a general relative to a vocational
=
= ( ¯yG,age|s,X− ¯yV,age|s,X) − ( ¯yG,0|s,X− ¯yV,0|s,X), ∀ age.
(3.3)
• β2, which gives the sensibility of the differential impact among the two groups with respect to ageing. For example, if β2 has a negative direction, it means that the dif-
ferential impact of the labour-market outcome with respect to age for individuals with general education over those with vocational education decreases as age increases.
Therefore, the trajectories of labour-market outcomes over the life-cycle for the two different education groups may have a parabolic shape if the coefficients upon age2i and the interac- tion term gi · age2i are significantly different from zero, as the normal age-y pattern in the economy is the control group ageing effect on the labour-market outcome.
α1· age + α2· age2= E[yi| g = 0, age, s, X]
| {z }
expected value of the labour- market outcome after age-lags
for vocational
− E[yi| g = 0, age = 0, s, X]
| {z }
α0
average floor level of the labour- market outcome in the economy
=
= ¯yV,age|s,X− ¯yV,0|s,X, ∀ age.
(3.4)
However, as in Hanushek, Woessmann, and Zhang (2011) we need a crucial assumption for the identification of the causal impact of education type on changes in labour-market outcome patterns over the life-cycle. The selectivity of people into general and vocational education conditional on all covariates of X should not vary over time, which means we assume that current old people are a good proxy for today’s young people. This is a strong assumption that allows us to estimate the impact of education type by the divergence in age- employment and age-wages patterns across the life-cycle. To validate this assumption, we look at the descriptive statistics in Table3.2: again, we perceive that individuals in youngest cohorts select more in general education than in vocational with respect to older age cohorts over time. As long as the effects of this selectivity are captured by covariates in Xi, the as- sumption holds perfectly and we fall in a quasi-random experiment framework, otherwise changes in labour market may reflect also the varying ability of young and old workers in different education categories.
In addiction to this assumption and the classic assumptions for the consistency of the Or- dinary Least Squares (OLS) estimator of orthogonality and full rank of the expected outer product matrix of the explanatory variables (see Wooldridge,2010, pp. 52-53), we need to state the classic parallel trends assumption (Angrist & Krueger, 1999, pp. 1296-1299) for DD models, for which in absence of treatment the trend of individuals in general education treatment group should be the same of the trend of those in vocational education control group or, in other terms, interaction terms are zero in absence of treatment. This key identi-
fying assumption is usually not testable and often undervalued by researchers, although its assertion is crucial.