1. Applied Economic Analysis
Lecture 18
Working through examples

2. Example 1
A researcher has been provided with a data set comprising a random sample of 1500 individuals aged between 16 and 64 years. The following information is recorded for each individual:
Pay: Gross hourly pay in pounds.
Gender: One if male, zero if female
Tenure: Number of years working for current employer
Highest Educational Qualification:
Degree level: one if degree or equivalent, zero otherwise
A level: one if A level or equivalent, zero otherwise
GCSE: one if GCSE or equivalent, zero otherwise
No qualifications: one if no qualifications, zero otherwise

3. The researcher has used the data set to estimate two regression models
The researcher has used the data set to estimate two regression models. The results are shown in the following table (standard errors in brackets):

4. (i) Interpret the results in model one
(i) Interpret the results in model one. Do you think this is a good model?
The constant represents the starting hourly pay for females. This is predicted to be £4.62 per hour.
The model predicts that the starting hourly for men is £2.43 higher than for females, ceteris paribus.
Tenure – Each additional year in the current job is predicted to increase hourly pay by 52p, ceteris paribus.

5. Tests for individual significance (t-tests):
Test H0: β0 = 0
Against H1: β0 > 0
(a one-tailed test as we expect starting hourly pay to be positive).
t = 4.62/1.55 = 2.98 > critical value t1497,0.05 = 1.64
There is evidence that the starting hourly pay is positive.

6. t-tests Test H0: β1 = 0 Against H1: β1 > 0
(a one-tailed test if you have a prior belief that hourly pay is higher for males).
t = 2.43/0.96 = > critical value of 1.64.
There is evidence that starting hourly pay is higher for males.

7. t-tests Test H0: β2 = 0 Against H1: β2 > 0
You would expect that pay would rise with tenure so this could be a one-tailed test:
t = 0.52/0.19 = 2.74 > 1.64
There is evidence that pay rises with tenure.

8. R2
R2 is not very high – it indicates that only 25% of the variation in hourly pay is explained by the model. (Adjusted R2 is not reported in the results).
However, the explanatory variables are individually significant.

9. F-test for overall significance:
Test H0: β1= 0, β2 = 0
Against H1: β1 ≠ 0, β2 ≠ 0
Fcrit = F 2,1497 = 3
249.5 > 3 so we reject the null hypothesis.
We have evidence that gender and tenure jointly affect hourly pay.
Overall this is a satisfactory model.

10. (ii) The researcher is not satisfied with her results in Model 1 and thinks that the relationship between tenure and hourly pay might be non-linear. What would be a plausible specification for the non-linearity? Sketch the suggested relationship and explain how you would estimate it.

11. Might expect this relationship – pay rises with tenure but at a decreasing rate.
Hourly Pay
Tenure (yrs)

12. To build non-linearity into the model could add a new variable tenure2
To build non-linearity into the model could add a new variable tenure2. If the relationship is as shown in the diagram, the sign on the tenure variable would be positive and the sign on tenure2 would be negative.
Also we would to carry out t-tests to see whether there is evidence of the suggested relationship.

13. (iii) Interpret the coefficients on the educational qualification variables in Model 2 and test their individual significance. Explain why you have chosen each test.
The base case here is ‘no qualifications’ so the interpretations are as follows:
Individuals with a degree are predicted to earn £4.98 per hour more than those with no qualifications, ceteris paribus.

14. Individuals educated to A level standard are predicted to earn £2
Individuals educated to A level standard are predicted to earn £2.02 per hour more than those with no qualifications, ceteris paribus. Individuals educated to GCSE standard are predicted to earn £1.35 per hour more than those with no qualifications, ceteris paribus

15. t-tests The t-statistics are: tβ3 = 5.03 t β4 = 3.11 t β5 = 2.33
One-tailed tests would be suitable as we expect those with qualifications to earn more that those without (human capital theory).
Critical value = 1.64

16. Therefore we have evidence that:
Hourly pay is higher for individuals with a GCSE, than for those with no qualifications. Hourly pay is higher for individuals with an A-level than for those with no qualifications. Hourly pay is higher for individuals with a degree than for those with no qualifications.

17. (iv) Why has the researcher excluded the ‘no qualifications’ variable from her model?
To avoid the dummy variable trap – if all the dummy variables were included there would be a problem of perfect multicollinearity and a violation of MLR3. One of the educational qualification groups is omitted and used as a reference group, the other coefficients are interpreted in relation to the reference category.

18. Example 2: Logit Model example
High prices in the housing market could result in young people or those on low incomes being unable to afford to buy their own homes and, therefore, being more likely to live in rented accommodation. An econometrician has been hired to identify the key factors that determine whether an individual will choose to buy their own home. She has been provided with data containing the following information on 3,000 randomly selected individuals aged between 20 and 64:

19. Yi:
One if the individual is a homeowner; zero if they rent
Agei:
Age of individual in years
Incomei:
Weekly Income (£)
Labour market status:
One if working, zero if unemployed or out of the labour market
Marital Statusi:
Single
Married or Living as Married
Separated
Divorced
Widowed

21. Logit Model example
(a) Explain why logit is a suitable choice of model. Would a different model also be suitable? (b) Using Model 2, interpret the results and comment on the individual significance of the explanatory variables.

22. Logit Model example
(c) The econometrician had a prior belief that marital status has an impact on the probability of home ownership. Do her results confirm this? Use a LR test to answer this question.
(d) Using Model 2, estimate the probability of a 45-year-old individual who is married, employed and earning £280 per week owning his/her own home.

23. Solutions
a) The logit model is suitable model because it gives the required sigmoid shape - probabilities will be between zero and one. The linear probability model may yield probabilities that are greater than one or less than zero.

24. Part (b)
The positive coefficients for age and income suggest that the probability of home ownership rises with age and income, ceteris paribus. We also see that an individual who is working is more likely to own their own home than an individual who is not in work, ceteris paribus.

25. Marital status – here we see that married/living as married and widowed people are more likely to own their own homes than single people, ceteris paribus. Separated and divorced people are less likely to own their own home than single people, ceteris paribus.

26. Whether you carry out a one or two tailed test depends on your prior belief – I will carry out one-tailed tests.
Asymptotic t-stats are: tAge = / = 3 tIncome = 0.001/ = 2.5 tLabour market status = 0.058/0.02 = 2.9 tMarried/LAM = 0.015/0.007= 2.14 tSeparated= /0.0025= -2.4 tDivorced= /.0.002= -3.5 tWidowed = 0.068/0.25 = 0.272

27. One-tailed asymptotic t-tests
Comparing the t-stats with the critical value of 1.64 we see that there is evidence of a positive relationship between age, income and labour market status (working) and home ownership. There is evidence that married/LAM individuals are more likely to own their own home that those who are single.

28. Comparing the t-stats with -1
Comparing the t-stats with -1.64, we find that separated and divorced individuals are less likely to own their own homes than those who are single.
Comparing t-stats with 1.64 we find no evidence that widowed individuals are more likely to own their own homes than those who are single.