The following estimates were made from a sample of 100 individuals. Thedependent variable is 1 if the person is self-employed and 0 otherwise.
OLS = - 0.151 + 0.009age + 0.065gender + 0.014Ed1 – 0.006Ed2 (0.016) (0.003)(0.003)(0.008)(0.005)
Logit = -3.780 + 0.097Age + 0.619gender + 0.111Ed1 – 0.043Ed2 (0.168) (0.033)(0.033)(0.062)(0.047)
Where
Age = age in years; gender = 1 if male, 0 if female; Ed1 = 1 if completed a degree, 0 otherwise; Ed2 = 1 if has a non-degree higher educational qualification, 0 otherwise;. The figures in brackets are conventionally calculated standard errors for the OLS estimates, asymptotic standard errors for the probit estimates.
i) Interpret the estimated coefficients of both OLS and Logit model.
ii) What are the estimated probabilities that a man aged 40 with Ed1=1 is self- employed using the above models?
i)
The estimated coefficients in both the OLS and Logit models represent the relationship between the dependent variable (self-employment) and the independent variables (age, gender, Ed1, Ed2).
In the OLS model:
- The coefficient for age (0.009) suggests that for every year increase in age, the probability of being self-employed increases by 0.009, holding all other variables constant.
- The coefficient for gender (0.065) suggests that males are more likely to be self-employed compared to females, holding all other variables constant.
- The coefficient for Ed1 (0.014) suggests that completing a degree increases the probability of being self-employed, holding all other variables constant.
- The coefficient for Ed2 (-0.006) suggests that having a non-degree higher educational qualification decreases the probability of being self-employed, holding all other variables constant.
In the Logit model:
- The coefficient for age (0.097) suggests that for every year increase in age, the odds of being self-employed increase by a factor of exp(0.097), holding all other variables constant.
- The coefficient for gender (0.619) suggests that males have higher odds of being self-employed compared to females, holding all other variables constant.
- The coefficient for Ed1 (0.111) suggests that completing a degree increases the odds of being self-employed, holding all other variables constant.
- The coefficient for Ed2 (-0.043) suggests that having a non-degree higher educational qualification decreases the odds of being self-employed, holding all other variables constant.
ii)
To calculate the estimated probabilities that a man aged 40 with Ed1=1 is self-employed using the above models, we substitute the values into the equations.
For the OLS model:
OLS = - 0.151 + 0.009(40) + 0.065(1) + 0.014(1) – 0.006(0) = -0.151 + 0.36 + 0.065 + 0.014 = 0.288
The estimated probability that a man aged 40 with Ed1=1 is self-employed using the OLS model is 0.288.
For the Logit model:
Logit = -3.780 + 0.097(40) + 0.619(1) + 0.111(1) – 0.043(0) = -3.780 + 3.88 + 0.619 + 0.111 = 1.83
The estimated odds that a man aged 40 with Ed1=1 is self-employed using the Logit model is 1.83.
i) To interpret the estimated coefficients of both the OLS and Logit models, we need to understand their significance and meaning for each independent variable.
In the OLS model:
- The coefficient for "age" is 0.009, meaning that for a one-unit increase in age, the estimated probability of being self-employed increases by 0.009.
- The coefficient for "gender" is 0.065, indicating that being male (gender = 1) is associated with a higher probability of being self-employed.
- The coefficient for "Ed1" (completion of a degree) is 0.014, suggesting that individuals who have completed a degree have a slightly higher probability of being self-employed.
- The coefficient for "Ed2" (non-degree higher educational qualification) is -0.006, indicating that individuals with a non-degree higher educational qualification have a slightly lower probability of being self-employed.
In the Logit model:
- The coefficient for "Age" is 0.097, implying that for a one-unit increase in age, the odds of being self-employed increase by a factor of exp(0.097) (exponential of 0.097).
- The coefficient for "gender" is 0.619, suggesting that being male (gender = 1) is associated with higher odds of being self-employed.
- The coefficient for "Ed1" (completion of a degree) is 0.111, indicating that individuals who have completed a degree have higher odds of being self-employed.
- The coefficient for "Ed2" (non-degree higher educational qualification) is -0.043, implying that individuals with a non-degree higher educational qualification have lower odds of being self-employed.
ii) To calculate the estimated probabilities that a man aged 40 with Ed1=1 is self-employed using the above models, we substitute the given values into the equations.
For the OLS model:
Probability = -0.151 + 0.009 * age + 0.065 * gender + 0.014 * Ed1 – 0.006 * Ed2
Probability = -0.151 + 0.009 * 40 + 0.065 * 1 + 0.014 * 1 – 0.006 * 0
Probability = -0.151 + 0.36 + 0.065 + 0.014 ≈ 0.288 (approximately)
For the Logit model:
Probability = 1 / (1 + exp(-(-3.780 + 0.097 * Age + 0.619 * gender + 0.111 * Ed1 – 0.043 * Ed2)))
Probability = 1 / (1 + exp(-(-3.780 + 0.097 * 40 + 0.619 * 1 + 0.111 * 1 – 0.043 * 0)))
Probability = 1 / (1 + exp(2.26))
Probability = 1 / (1 + 9.564)
Probability ≈ 0.094 (approximately)
Therefore, the estimated probability using the OLS model is approximately 0.288, and the estimated probability using the Logit model is approximately 0.094.