LEGISLACIÓ VIGENT

The theoretical model presented in section 3 shows how family income influences the participation decision through its impact on factors such as time preferences, access to credit or family resources, and debt

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aversion. Having seen in the LSYPE data that there is a strong correlation between family income and debt attitudes, this section accounts for that relationship explicitly and seeks to measure the impact on the university participation decision of family income working through debt aversion. Estimating this effect for different pairs of family income groups allows us to see if low family income influences participation through its effect on debt attitudes more strongly than high family income does. Using this framework makes it possible to incorporate the relationship between debt aversion and family income directly rather than modelling interaction effects between the two variables.

Figure 3-9: Direct and Indirect Impact of Family Income of University Participation

Source: own representation based on Preacher and Hayes (2008)

This framework can be depicted as follows: The total effect of family income on university participation (c above) is assumed to consist of a direct effect (c’ in the diagram above) and an indirect effect working through debt aversion (ab above).

In the initial stage, several logit models were estimated, all of which used “atuni” as the dependent variable, which is equal to 1 if the young person was at university when interviewed in wave 6 and zero otherwise. The first model was run with family income but without any debt aversion variable, as follows:

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= + ∑ + ∑ + (3.15) The second included both family income and debt aversion as explanatory variables and has the following functional form:

= + ∑ + ∑ + ∑ + (3.16)

where the variables on family income groups and debt aversion are sets of dummy variables and the Xs are controls.

In linear regression, it would be possible to break down the effect of family income into a direct component and the indirect component coming through debt aversion by simply examining the change in the family income coefficients between models such as the ones described above, where one model includes debt aversion as an explanatory variable and the other excludes it. However, this is not possible in a logit context, due to a structural bias which can be explained as follows. A logistic regression is a comparison of proportions that have first been transformed into log-odds ratios. Probabilities at each possible value of the mediating variable (debt aversion in this case) are transformed into log-odds ratios. When this transformation is performed for probabilities close to 0 or 1, they become less tightly clustered together than they were as probabilities (i.e. values at the extremity are more extreme in the log odds metric than in the probability metric). When the mediating variable (debt aversion) is left out of the regression, the model in effect takes an average of the proportions before transforming this average into a log odds ratio. Computing the average proportion before transforming the proportions into log odds means that the extreme values are less influential than they would have been if the means were computed in the log odds metric, so the average is pulled towards the less extreme categories. The consequence of this is that the effect in terms of log odds will be less when the mediating variable is left out of the model, even if there is no indirect effect (Buis, 2008).

Erikson et al (2005) develop a solution to this problem which uses counterfactuals. In their study of student choices to progress to A-levels based on performance at key stage 3, they assume that the choice characteristics of students of one class can be combined with the

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performance distribution of students of another class to produce a counterfactual or potential outcome. They implement this using numerical integration to produce a hypothetical intervention in which the choice characteristics change but the performance distribution is unchanged (and vice versa). This makes it possible to investigate the relative contributions of choice and performance.

Based on Erikson et al (2005), Buis (2008) presents a generalisation that allows the variable through which the indirect effect occurs to follow any distribution (not just a normal distribution as per Erikson et al). This is useful for this chapter as the debt aversion variable from the LSYPE is a categorical variable that only loosely follows a normal distribution. Buis (2008) also suggests bootstrapping as a method for obtaining standard errors and shows how to control for other variables. Using an explanatory variable producing a direct effect that is a categorical variable, it is possible to produce a decomposition for all pairwise combinations of categories. The explanatory variable is family income group – performing a decomposition using this variable makes it possible to see the relative contribution of debt aversion for different pairs of family income groups. Factors described above suggest the possibility that debt aversion is a greater hindrance for poorer families. This would be confirmed by a falling indirect effect for groups of progressively higher family income compared to a base group with the lowest income.

For clarity, the following points describe the components required for the comparison of income groups 1 and 4:

 The total effect is given by log odds of success for family income group 4 minus the log odds of success for family income group 1.  The indirect effect is given by the log odds of success of family

income group 1 with the debt aversion profile of family income group 4 minus the log odds of success of family income group 1 (using their own debt aversion profile)

 The direct effect is given by the log odds of success for family income group 4 minus the log odds of success of family income group 1, given the debt aversion profile of family income group 4.

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These calculations should also be carried out for the complimentary counterfactual (i.e. using family income groups 1 and 4 the other way around). As these two methods produce similar but not identical results, an average of the two can be taken as the final result.

The equations below provide more detail:

(3.17) Using the rule that ln(a) – ln(b) = ln(a/b),

(3.18) By exponentiating both sides of this equation, the decomposition can also be presented in terms of odds ratios. Since exp(a + b) = exp(a) x exp(b), the total effect is given by the product of the two effects:

(3.19)

Using this technique will make it possible to examine and compare the direct effect of family income on university participation with the indirect effect working through debt aversion, whilst avoiding any bias arising from the functional form of the model.