QUESTION FIVE Using data on wage and IQ, the following simple regression model is estimated explaining monthly salary (wage) in terms of IQ score (IQ). \[ \text { lwage }=\beta_{0}+\beta_{1} I Q+u \] Estimated model: \[ \text { lwage }=5.8869+0.00881 I Q \] \[ S E(0.0890)(0.0008694 \text { ) } \] \[ n=935, R^{2}=0.0991 \] a) Calculate the t-statistics for \( I Q \) and the constant parameters/coefficient. (5 marks) b) Write down the hypothesis testing rule for testing \( I Q \) has no impact on wage. (5 marks) c) Interpret the ceteris paribus coefficient for education \( \beta_{1} \), is the interpretation statistically significant? (7.5 marks) d) If \( I Q \) increases by 15 points, what is approximate percentage increase in predicted wage? (7.5 marks)

To begin with, let’s talk about t-statistics. To calculate the t-statistics for IQ, you would divide the estimated coefficient by its standard error. For IQ, it’s \( t_{IQ} = \frac{0.00881}{0.0008694} \approx 10.13 \). For the constant coefficient, \( t_{0} = \frac{5.8869}{0.0890} \approx 66.14 \). Both t-statistics suggest that the parameters are statistically significant. Now, regarding the hypothesis testing rule, you would generally set \( H_0: \beta_1 = 0 \) (no impact of IQ on wage) and \( H_a: \beta_1 \neq 0 \) (impact exists). You compare the t-statistic with the critical value from the t-distribution based on your significance level (commonly 0.05) and degrees of freedom. For interpreting the ceteris paribus coefficient \( \beta_1 = 0.00881 \), it implies that for every one-point increase in IQ, the log of wage increases by about 0.00881, holding other factors constant. In terms of statistical significance, with a high t-statistic (10.13), we would usually reject the null hypothesis, suggesting the result is statistically significant. Finally, if IQ increases by 15 points, the predicted log-wage increases by \( 0.00881 \times 15 = 0.13215 \). To find the percentage increase in wage, we take \( e^{0.13215} - 1 \approx 0.1416 \) or roughly a 14.16% increase in predicted wage.