## Question a

Linear model for the natural log of wage as a function of age and education:

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Where:

- W is the wage
- E is education level
- A is age in years
- C is a constant term

### The possible consequence of ignoring age

Age is an important determining factor of wage. It is expected that wage is low among children and teenagers. Wage then starts increasing at young adulthood, peaks at adulthood and then starts declining in old age. This is a reflection of the productivity of people at different times of their lifespan. Excluding the age variable from the model will result in poor fitness of the model. That is, education alone cannot explain wage and therefore without the age variable, the model would be poor. This in practice is normally reflected by the measure of the goodness of fit of the model, which is the R squared. A low R squared reflects a poor fit of the model while a high R squared reflects a good fit of the model (Koop, 2004).

Extend the model in a (i) above to allow for the possibility that wage changes with respect to age, at a different rate depending on the level of age:

The above model shows that the natural log of wage is a function of education level, age in years and age squared. The purpose of including the age squared is to show that wage changes as age increases but in a non-constant manner. The model is in form of a quadratic function, with respect to the age variable.

Extend the model in a (ii) above to allow for the possibility that wage changes with respect to age, at a different rate depending on education levels:

The above model shows that the natural log of wage is not only a function of education level, age in years and age squared, but it is also a function of the interaction between age in years and education level. The interaction term shows that the natural log of wage changes with respect to age, at a different rate depending on education level.

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## Question b

### Model 1

From the Eviews output 1, the coefficient on education is 0.089715, while that on age is 0.014212 and the constant is 0.623976. The standard error associated with the coefficient on education is 0.004948 and the associated t-statistic is 18.13304.

The probability value is 0.0000. The standard error associated with the coefficient on age is 0.001211 and the associated t-statistic is 1173407. The probability value is 0.0000. The standard error associated with the constant is 0.078503 and the associated t-statistic is 7.948386. The probability value is 0.0000. The R-squared is 0.277857 while the F-statistic is 247.4048 with a probability value of 0.000000.

Model 1:

#### Interpretation of the coefficients

The coefficient on education (0.089715) implies that if education level increases by one level, wage will increase by 0.089715 x 100% = 8.9715 percent. The coefficient on age (0.014212) implies that if age increases by one year, wage will increase by 0.014212 x 100% = 1.4212 percent.

### Model 2

From the Eviews output 2, the coefficient on education is 0.085245, while that on age is 0.072534. The coefficient on agesquared is -0.000738 and the constant is -0.370723. The standard error associated with the coefficient on education is 0.004875 and the associated t-statistic is 17.48498 while the probability value is 0.0000. The standard error associated with the coefficient on age is 0.007706 and the associated t-statistic is 9.412955.

The probability value is 0.0000. The standard error associated with the coefficient on agesquared is 9.63E-05 and the associated t-statistic is -7.659714. The probability value is 0.0000. The standard error associated with the constant is 0.150871 and the associated t-statistic is -2.457213. The probability value is 0.0141. The R-squared is 0.309389 while the F-statistic is 191.8903 with a probability value of 0.000000.

Model 2:

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#### Interpretation of the coefficients

The coefficient on education (0.085245) implies that if education level increases by one level, wage will increase by 0.085245 x 100% = 8.5245 percent. The coefficient on age (0.072534) implies that if age increases by one year, wage will increase by 0.072534 x 100% = 7.2534 percent. The coefficient on agesquared (-0.000738) implies that if age increases by one year, wage will reduce by 0.000738 x 100% = 0.0738 percent.

### Model 3

From the Eviews output 3, the coefficient on education is 0.050464, while that on age is 0.063333. The coefficient on agesquared is -0.000760 while that on age_educ is 0.000859. The constant is 0.041101. The standard error associated with the coefficient on education is 0.018958 and the associated t-statistic is 2.661846 while the probability value is 0.0079. The standard error associated with the coefficient on age is 0.009097 and the associated t-statistic is 6.962193 with a probability value of 0.0000.

The standard error associated with the coefficient on agesquared is 9.69E-05 and the associated t-statistic is -7.842321 with a probability value of 0.0000. The standard error associated with the coefficient on age_educ is 0.000452 and the associated t-statistic is 1.898335 with a probability value of 0.0579. The standard error associated with the constant is 0.264157 and the associated t-statistic is -0.155592 with a probability value of 0.8764. The R-squared is 0.311322 while the F-statistic is 145.1102 with a probability value of 0.000000.

Model 3:

#### Interpretation of the coefficients

The coefficient on education (0.050464) implies that if education level increases by one level, wage will increase by 0.050464 x 100% = 5.0464 percent. The coefficient on age (0.063333) implies that if age increases by one year, wage will increase by 0.063333 x 100% = 6.3333 percent. The coefficient on agesquared (-0.000760) implies that if agesquared increases by one unit, wage will reduce by 0.000760 x 100% = 0.076 percent.

The coefficient on age_educ (0.000859) implies that if age_educ increases by one unit, wage will increase by 0.000859 x 100% = 0.0859 percent. Given that age has been interacted with education, this coefficient can also be interpreted as: the return to education increases by 0.0859 percent as age increases.

## Question c

### The t-tests

The t-test is a test for the significance of the estimated coefficients. The null and alternative hypotheses for the t-test are given as:

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αi represents the individual coefficients in each of the three models above. The null hypothesis states that the coefficient is not different from zero hence it is statistically insignificant. The alternative hypothesis on the other hand states that the coefficient is different from zero and is therefore statistically significant. To determine the statistical significance of the t-tests, one compares the p-value with the level of significance (Tinbergen, 2004). If the p-value is less than the level of significance, we reject the null hypothesis and fail to reject the alternative hypothesis (Wooldridge, 2009).

Assuming a significance level of 5% (0.05), the p-values in model 1 are all 0.000 and are therefore less than the p-value. In this case, we reject the null hypothesis that the each of the two coefficients is statistically insignificant and conclude that all the coefficients are statistically significant.

Similarly in model 2, all the p-values are 0.000 and are therefore less than the level of significance. In this case, we reject the null hypothesis and conclude that each of the three coefficients is statistically significant.

In model 3, the p-values associated with the four coefficients are: 0.0079, 0.000, 0.000 and 0.0579, respectively. Comparing the p-values with a 5% level of significance, only the first three coefficients are statistically significant. The last coefficient (on age_educ) is not statistically significant at 5% level of significance. However, it is statistically significant at 10% level of significance.

### The F-test

The F-test measures the joint significance of all the coefficients (Kennedy, 2003). The null and alternative hypotheses are given as:

Like in the t-test, the significance is determined by comparing the p-value (of the F-statistic) with the level of significance (Baltagi, 2011). In model 1, the p-value of the F-statistic is 0.000000 which is less than the level of significance (0.05). The conclusion is that education and age jointly explain changes in wages.

Similarly in model 2, the p-value of the F-statistic is 0.000000, which is less than the level of significance of 0.05 implying that education, age and agesquared jointly explain changes in wages.

Likewise in model 3, the p-value of the F-statistic is 0.000000, which is less than the level of significance of 0.05 implying that education, age, agesquared and age_educ jointly explain changes in wages.

## Question d

Other potentially relevant variables that should have been included in the model include: years of experience, skills level and government policy.

The interaction between age and education (showing the dependence between education and age) is statistically insignificant. This shows that education is indeed independent of age and therefore the models are not mis-specified.

## References

Baltagi, B.H., 2011. *Econometrics.* London: Springer.

Kennedy, P., 2003. *A guide to econometrics*. Cornwall: MPG Books.

Koop, G., 2004. *Analysis of Economic Data. Hoboken.* UK: John Wiley & Sons, Inc.

Tinbergen, J., 2004. *Econometrics*. Oxfordshire: Routledge.

Wooldridge, J.M., 2009. *Introductory econometrics: a modern approach.* Mason, OH: South-Western Cengage Learning.