Economics2 Multiple Questions

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Surname 1 Firstname Lastname Instructor’s Name Course Number Date Economics: Multiple Questions Question 1 a. The null hypothesis suggests there is no relationship between the dependent and independent variables, attributing any observed relationship in the sample to chance. The alternative hypothesis posits that a real relationship exists between these variables in the population. Null : β 1 = 0 Alternative : β ≠ 0 b. The null hypothesis suggests a negative relationship between the dependent and independent variables, attributing any observed relationship in the sample to chance. The alternative hypothesis indicates that a positive relationship exists between these variables in the population. Null : β 1 ≤ 0 Alternative : β > 0 c. t-stat = standard error = 1.65 d. Degrees of freedom = n-1 = 51-1 = 50 Critical t-values for 1% significance at 50 degrees of freedom = 2.40 Critical t-values for 5% significance at 50 degrees of freedom = 1.68 Critical t-values for 10% significance at 50 degrees of freedom = 1.299

Surname 2 In a two-tailed test, t-stat < t-critical indicates a significant coefficient. Therefore, coefficient is statistically significant at 1% and 5% but insignificant at 10% because 1.65 is less than 1.68 and 2.40. Question 2 a. Intercept The intercept represents the average height of male students (reference category), which is 71.0 inches. The slope indicates the difference in average height between female and male students, with female students being 4.84 inches shorter on average. b. Hypothesis Testing HO: Slope = 0 H1: Slope < Intercept The tabulated t value at 5% significance level is 1.658 while the calculated t value of -8.49. The null hypothesis is rejected, implying that the average height of female students is less than that of male students. c. Due to the small sample size, homoskedasticity, which assumes constant error variance across explanatory variable values (e.g., BFemme), is unlikely in this case, challenging the assumption within classical linear regression models. Question 3 a. A 1% increase in population rate (0.01 to 0.02) reduces per capita income by nearly 20% (0.188), a substantial effect. Countries with the same population growth rate as the United States tend to have about half the per capita income. The t-statistic of 5.93 indicates a statistically significant relationship.

Surname 3 b. With a 1% population growth rate increase, the partial derivative interpretation remains the same. The regression R-squared and t-statistic remain unchanged since only a constant was removed from the explanatory variable, but the intercept changes due to X modification. Question 4 a. The regression equation is given as: RelProd = -0.08 + 2.44SK Each unit of GDP investment share (SK) increases relative per capita income by 2.44 units. Even without investment share (SK = 0), relative per capita income is -0.08. SK explains 46% of relative per capita income variation, according to R-squared = 0.46. Standard error of regression (SER) is 0.21, measuring data point deviation from regression line. b. The t-statistics for the coefficients: t(SK) = (2.44 - 0) / 0.38 = 6.42 t(intercept) = (-0.08 - 0) / 0.04 = -2 Both coefficients are statistically significant at the 5% significance level. A two-sided test is appropriate because the goal is to determine whether the coefficients differ from zero, regardless of their direction (positive or negative). c. Due to the similarity between the homoskedasticity-only standard errors and the heteroskedasticity-robust standard errors, it can be concluded that the errors are approximately homoskedastic, which allowed the coefficients to remain unchanged. However, the results' significance was not affected, as measured by either the t-statistics or the significance levels.

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Surname 4 d. Computing OLS with homoskedasticity-only standard errors is faster and simpler. In cases where error variance depends on development level or political stability, heteroskedasticity-robust standard errors provide more accurate estimates. In this case, heteroskedasticity-robust and homoskedasticity-only standard errors are similar, suggesting it is not a major issue. Question 5 Since economic theory rarely accepts homoskedastic errors, it is usually prudent to assume heteroskedasticity. Using homoskedasticity-only standard errors when heteroskedasticity-robust ones are required can lead to incorrect conclusions, especially because the former are smaller, resulting in exaggerated t-statistics and more frequent null hypothesis rejection. An alternative GLS estimator, weighted least squares, can be optimal but requires knowledge of how error variance varies with X (e.g., X or X2). Earnings functions, cross- country beta-convergence regressions, consumption functions, sports regressions involving teams from markets with varying population sizes, and weight-height relationships for children are all examples of this. Question 6 The average birth weight of all mothers is 3382.934 grams. The average birth weight of smoking mothers isn3178.832 gram Question 7 a. The average value of the birthweight of all the mothers is obtained by using the function y = β 0 + β 1 x 1 + β 2 x 2 + ∈ 1 b. The Standard Error for the estimated difference is obtained by the formula SE = s√ {(1/n1) + (1/n2)} Where:

Surname 5 s = n 1 s 1² ±n 2 s 2² n 1 + n 2

Surname 6

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