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Linear Probability Model

Model A: 𝑦 = Β π‘ƒπ‘Ÿ(π‘šπ‘Žπ‘›π‘Žπ‘”π‘’π‘Ÿ = Β 1|π‘₯!) Β  Β  = Β  𝛽!+ Β  𝛽!π‘™π‘œπ‘!+ 𝛽!π‘ π‘’π‘™π‘“π‘’π‘ π‘‘π‘’π‘’π‘š!+ 𝛽!π‘Ÿπ‘–π‘ π‘˜!+ 𝛽!π‘“π‘’π‘šπ‘Žπ‘™π‘’!+ 𝛽! π‘™π‘œπ‘π‘– Β Γ— Β π‘“π‘’π‘šπ‘Žπ‘™π‘’ ! + Β  Β  𝛽! π‘ π‘’π‘™π‘“π‘’π‘ π‘‘π‘’π‘’π‘šΓ— Β π‘“π‘’π‘šπ‘Žπ‘™π‘’ !+ Β  𝛽! π‘Ÿπ‘–π‘ π‘˜Γ— Β π‘“π‘’π‘šπ‘Žπ‘™π‘’ !+ 𝛽!π‘π‘™π‘Žπ‘π‘˜!+ Β  𝛽!β„Žπ‘–π‘ π‘π‘Žπ‘›π‘–π‘! + Β  𝛽!π‘Žπ‘”π‘’ Β !+ 𝛽!"π‘Žπ‘”π‘’! !+ Β  𝛽!!π‘’π‘Ÿπ‘π‘Žπ‘›!+ Β  𝛽!"π‘β„Žπ‘–π‘™π‘‘π‘Ÿπ‘’π‘›!+ Β  𝛽!"π‘šπ‘Žπ‘Ÿπ‘Ÿπ‘–π‘’π‘‘! + 𝛽!"π‘’π‘‘π‘’π‘π‘Žπ‘‘π‘–π‘œπ‘›! Β  + Β  𝛽!"𝐴𝐹𝑄𝑇!+ 𝑒!

My first model (A), is a linear probability model; the LPM is a linear regression using ordinary least squares that has a binomial response dummy as the outcome variable. The coefficient estimates the effect of a one unit increase in the independent variable on the probability of being a manager, holding the other variables constant. The estimates are irrespective of the base value of x; a unit increase from 0 to 1 has the same effect as, say, an increase from 15 to 16. While the LPM is easy to use and interpret, there are several issues with

the LPM that may cause inefficient, inconsistent, or nonsensical results, and should be addressed.

The first problem with the linear probability model is that it violates the OLS

assumption of homoscedasticity. Because of the dichotomous nature of the dependent variable, which represents the probability that y=1 given x, the error term’s value will be dependent on the value of x. As a result, standard errors will be biased, and hypothesis tests will be incorrect. These issues may be minor, and I correct for them by using weighted standard errors.

The second issue is that the error term of a LPM has a binomial distribution instead of a normal distribution; it can only take on two values, one value for when y = 1, and the other for when y = 0. Consequently, it’s impossible for the error term to have a normal distribution. Additionally, the LPM assumes linearity and thus can give out-of-bounds predictions at extreme values of the independent variables. OLS assumes a continuous y variable, and allows the dependent variable to take on values from -∞ to +∞ . This is nonsensical considering that probabilities must fall within the range [0,1].

The last fault of the LPM is that of functional form. A linear probability model assumes that the effect of a change in x on the value of y is the same for all starting values of x, hence it is linear; however, as often is the case with probabilities, this is not realistic. I expect that within very low (high) values of non-cognitive traits, a one unit decrease (increase) will not have as great of an effect as in the middle of the distribution of test scores. These issues with the LMP motivate the use of my second model.

Probit

Model B: 𝑦 = Β π‘ƒπ‘Ÿ(π‘šπ‘Žπ‘›π‘Žπ‘”π‘’π‘Ÿ = Β 1|π‘₯!) Β  Β  = Β   𝛷(𝛽!+ Β  𝛽!π‘™π‘œπ‘!+ 𝛽!π‘ π‘’π‘™π‘“π‘’π‘ π‘‘π‘’π‘’π‘š!+ 𝛽!π‘Ÿπ‘–π‘ π‘˜!+ 𝛽!π‘“π‘’π‘šπ‘Žπ‘™π‘’!+ 𝛽! π‘™π‘œπ‘π‘– Β Γ— Β π‘“π‘’π‘šπ‘Žπ‘™π‘’ ! + Β  Β  𝛽! π‘ π‘’π‘™π‘“π‘’π‘ π‘‘π‘’π‘’π‘šΓ— Β π‘“π‘’π‘šπ‘Žπ‘™π‘’ !+ Β  𝛽! π‘Ÿπ‘–π‘ π‘˜Γ— Β π‘“π‘’π‘šπ‘Žπ‘™π‘’ !+ 𝛽!π‘π‘™π‘Žπ‘π‘˜!+ Β  𝛽!β„Žπ‘–π‘ π‘π‘Žπ‘›π‘–π‘! + Β  𝛽!π‘Žπ‘”π‘’ Β !+ 𝛽!"π‘Žπ‘”π‘’!!+ Β  𝛽!!π‘’π‘Ÿπ‘π‘Žπ‘›!+ Β  𝛽!"π‘β„Žπ‘–π‘™π‘‘π‘Ÿπ‘’π‘›!+ Β  𝛽!"π‘šπ‘Žπ‘Ÿπ‘Ÿπ‘–π‘’π‘‘! + 𝛽!"π‘’π‘‘π‘’π‘π‘Žπ‘‘π‘–π‘œπ‘›! Β  + Β  𝛽!"𝐴𝐹𝑄𝑇!+ 𝑒!) Β 

My second model, a probit model, uses the cumulative distribution function of a standard normal distribution to transform the linear function into one that more appropriately fits the . The CDF of a normal distribution is s-shaped, and is restricted to the range [0,1], and so it is more appropriate for probabilities. The probit model uses maximum likelihood

estimation instead of ordinary least squares, meaning it does not hold the other independent variables constant to determine the effect of x on y, but finds the combination of x’s that maximize the probability of y equaling 1. Because maximum likelihood estimation is based on the distribution of y given x, the heteroscedasticity is accounted for. The probit regression coefficients give the change in the z-score for a one unit change in the x variable. These on their own are not easily interpretable, so I will use the marginal effects of the independent variables on y, calculated at the means. This should provide results similar to the ones in model 1.

Variables

The motivation of my empirical model is similar to the one used by Cobb-Clark and Tan (2011) in their study on non-cognitive traits and occupational segregation in Australia. The independent variables of interest in my model are locus of control, risk preference, self-esteem. My control variables are the AFQT, gender, marital status, number of children, age, education, location and race.

In addition the above covariates, I include interactions between the non-cognitive traits and the dummy for the variable female. This allows me to test whether or not the returns to these traits differ by sex. This method shows if these traits are valued differently depending on the sex of the person they belong to. If I find they are lower for women, then it may be

appropriate to assume discrimination is affecting women's occupational attainment.

A concern raised in the literature on the effect of personality on economic outcomes is that causality may actually run from wages or occupation to personality, thus causing

endogeneity issues (Borghans et al. 2008). Many researchers have defended the practice of using personality traits measured later in adulthood, as psychological literature has shown personality stabilizes in early adulthood (Costa and McCrae 1997). Although some have argued that self-esteem may be an exception to this finding, as intuitively it seems the most likely personality trait to demonstrate reverse causality. Fortunately, the NLSY79 measures locus of control and self-esteem in 1979 and 1980 respectively when respondents were in their late teens; due to their lack of experience in the labor market, I find it improbable that their labor market experience would affect their locus of control or self-esteem. This is one of the oft- noted benefits of using the NLSY to study the effect of non-cognitive traits on economic

outcomes (Fortin 2008; Osborne 2000). Risk aversion was only measured in 2010, but according to Jung and Treibich (2014), risk aversion is stable enough to be validly used to predict differences in economic outcomes between individuals across time, so this measure should suitably capture differences in the effects of risk preference on labor market decisions.

5.3 Results

(see table 2)

Model 1.a

Model 1.a is a linear probability model with only locus of control, self-esteem, and risk tolerance as independent variables. The coefficient of locus of control is -.0055. This means that a unit change in the Rotter locus of control scale, which ranges from 4 to 16, towards externality (increase in value), will decrease the chance of holding a management position (y=1) by .55 percent. The coefficient is small, but statistically significant (p-value=0.001). Because this model is linear, I can calculate that a move from the lowest value of locus of control to the highest decreases the probability of being a manager by 6.48%.

The coefficient self-esteem is .0089. This means that a unit increase in self-esteem, as measured by the Rosenberg scale, which ranges from 6 to 30 in the data, increases the

probability of holding a management position by .89 of a percentage point. This coefficient is also statistically significant at the 0.000% level. A move from the lowest value of self-esteem to the highest increases the probability of being a manager by 21.36%.

The coefficient of the variable risk is .0066; a unit increase in risk tolerance increases the probability of being a manager by .66 of a percentage point. This coefficient is statistically significant (p-value=0.000). The NLSY risk scale ranges from 0 to 10, and thus a move from extremely risk averse to extremely risk tolerant increase one’s chance of being a manger by 6.6%

I interpret these results cautiously, as they are likely to be biased because of the

excluded variables. Additionally, the R squared is small, at .0228, meaning that only 2.28% of the variance in the manager variable is explained by the variance of non-cognitive traits. However R squared om Thus far, these findings are in line with my hypothesis; the coefficient

on locus of control is negative, while the coefficients on self-esteem and risk tolerance are positive.

Model 1(b)

In Model 1(b), I keep the independent variables of locus of control, risk, and self- esteem from Model 1(a), and add a dummy variable for gender (0=male, 1=female) into the model. The coefficients on locus of control and self-esteem only change ten-thousandths of a percent, and the significance decreases slightly to (p=.003). Risk’s coefficient changes slightly, it decreases from .0066 to .0057, suggesting that the exclusion of gender from Model 1(a) was inflating the effect of risk on the dependent variable, albeit by a small amount. The coefficient for female is -.0281, so being a female lowers the chance of being a manager by 2.81%. The coefficient on the effect of being a woman is statistically significant at the 0.000% level (p- value 0.000)

Model 1.c

In model 1(c) I keep all of the variables used in model 1(b) and incorporate interactions between the non-cognitive traits and gender in order to see if these traits are valued differently between men and women. I interact locus of control, self-esteem, and risk with the female dummy variable. Females are more heavily penalized for a unit shift towards external control than men are, although the difference in slopes is insignificant. Women also have a lower return than men to risk tolerance, but the returns remain positive; however, they are not statistically significantly different either (p-value=0.633). Women and men have significantly different returns to self-esteem (p-value=0.000). Women’s returns are less than half that of men’s. The coefficient for self-esteem when female=0 is 0.012, while for women it is .0051.

The addition of the interaction variables changes the coefficient for female from negative to positive and increases it considerably (.186) and is significant at the 0.005% level; this suggests that differing returns to self-esteem were a driving factor causing women’s lower occupational attainment relative to men.

In Model 1(d-f), I include all of the variables from model 1(c) while adding a series of control variables. These only variables change the coefficients from model slightly, with no overall changes in significance. In model 1(d). These are the demographic variables: race, age, age squared, and urban/rural. Both race variables are negative, and significant. In model 1(e) the added controls are for household factors: number of children and marital status. Marital status has a coefficient of .03 and is significant at the 0.001% level. In model 1(f) I add the variables education and AFQT, which measure cognitive skills. The coefficient of AFQT is significant at the 0.000% level and is .0011. An increase from the mean (40.9) to the top percentile (99) increases the likelihood of being manager by 6.38%. An increase from the lowest to the highest percentile increases the probability of being a manager by 108.9%, illustrating the shortcomings of a linear model. In order to see if cognitive skills were more highly valued, I interacted male and female with AFQT (not shown) Men and women experience small, but statistically significant differing returns to AFQT (p-value=0.000). Women experience almost no benefit from a unit increase (.0002), while men have a modest one of 0.0018.

The structure of the models for my probit regression is the same as that for my linear probability model. I first test the non-cognitive skills alone, then add the dummy for female, then add the interactions, and then add my various controls.

Once I measure the marginal effects of the variables, the overall signs and statistical significance of matched those in the linear probability model. Like the in LMP, once I include interactions in the model the effect of being female becomes positive. Interestingly, the interaction between locus of control and female is significantly different. Suggesting that both locus of control and self-esteem effect women’s likelihood of being managers relative to men. The R-squares are low just as in the LPM, which is perplexing by might be because of the binary variable. Another, possibly more appropriate, measure of goodness of fit for non-linear models is the percent correctly predicted. My model does better by this metric; it predicted whether or not someone would be a manager in 89% of cases.

Conclusion:

My results show that women are less likely to be managers both because of their different levels of non-cognitive traits and their different returns to them. The first finding supports human capital theory. I argue that these differences are formed throughout a lifetime of gendered socialization. The differing returns to these traits, suggests biases that may cause employers to reward women less for traits they possess in equal measure to men. This is in keeping with the out-group homogeneity effect; these traits may be rewarded more highly in females than males, as nuances in personality are taken notice of more in men. Another explanation may be that women perform these traits differently in the marketplace; high self- esteem in a woman may not seem assertive, while in a man it may read that way.

I think that researchers interested in unequal economic outcomes should continue seek non-traditional explanatory variables to incorporate into their analysis, such as non-cognitive skills. I believe that solutions to discriminatory outcomes can be found only through

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