Question

1) The first regression equation in Table II is estimated as quadratic in levels of income (G) with the dependent variable being total automotive lead emissions (in millions of grams). Regression coefficients, standard errors and R 2 are shown below. Lead =25,149−1.58G+0.0003G 2 −11,476D 83 ​ +8.97D 83 ​ G− (9,435) ​ (2.89) ​ (0.0002) ​ (3,623) ​ (2.36) ​ 0.0008D 83 ​ G 2 (0.0002) +38.4 (popdensity) - 308(year) (14.5) (148) ​ where D 83 ​ represents 1 for the post-1983 period and zero for the pre-1983 period R 2 =0.68 N=528 ​ 1 F-Test of hypothesis that all time-interaction coefficients equal to zero =5.14 (a) What does the term "quadratic in levels of income" mean? What is the shape of the polynomial that is estimated in Equation One? What is the value of the leading coefficient of the income terms in the time period from 19831992, when taking into account that the equation has time interaction terms? Do these results seem consistent with the EKC theory? (b) Which of the variables are statistically significant at a 5% significance level in the above Equation? (See the T Distribution Table provided). (c) How do you undertake an F-test of the hypothesis that all time-interaction coefficients with respect to income are equal to zero? What are the results of the F-test of the hypothesis that all time-interaction coefficients with respect to income are equal to zero? How do you interpret these results at a 5% significance level? (See the F Distribution Table provided).

Answer 1

(a) "Quadratic in levels of income" means that the regression equation includes both linear and quadratic terms of income (G). The shape of the polynomial estimated in Equation One is a downward-facing parabola, as evidenced by the negative coefficient (-1.58) for the linear income term and the positive coefficient (0.0003) for the quadratic income term. The leading coefficient of the income terms during the period 1983-1992, accounting for time interaction terms, is 8.97. These results are not consistent with the Environmental Kuznets Curve (EKC) theory, as the quadratic term does not dominate the relationship, indicating a continuous negative impact of income on lead emissions.

(b) Statistically significant variables at a 5% significance level in the equation include the constant term (25,149), linear income term (-1.58G), quadratic income term (0.0003G^2), time indicator variable for the post-1983 period (D83), and the interaction term between the post-1983 period and income (8.97D83G).

(c) To conduct an F-test for the hypothesis that all time-interaction coefficients with respect to income are zero, compare the calculated F-statistic (5.14) to the critical value from the F Distribution Table. The result indicates rejection of the null hypothesis at a 5% significance level, suggesting significant differences in income coefficients between pre-1983 and post-1983 periods.

Step-by-step explanation

The term "quadratic in levels of income" in Equation One signifies the inclusion of both linear and quadratic income terms in the regression. The coefficients (-1.58 and 0.0003) indicate a parabolic relationship between income and total automotive lead emissions, with the parabola facing downward. The leading coefficient of 8.97 for the income terms during 1983-1992, accounting for time interaction terms, emphasizes an amplification effect on lead emissions during that period.

The statistical significance at a 5% level is determined by examining the t-statistics of the coefficients. The constant term, linear and quadratic income terms, time indicator variable (D83), and interaction term (8.97D83G) are all statistically significant, indicating their importance in explaining variations in lead emissions.

The F-test evaluates whether the time-interaction coefficients with respect to income are collectively equal to zero. The calculated F-statistic (5.14) is compared to the critical value from the F Distribution Table. The rejection of the null hypothesis suggests that there are significant differences in income coefficients between the pre-1983 and post-1983 periods. This implies a dynamic relationship between income and lead emissions over time, not aligning with the EKC theory, which predicts a turning point where environmental quality improves with economic development.