Using Data From 2013 On 64 Black Females The Estimated Log L

Using Data From 2013 On 64 Black Females The Estimated Log Linear

Using data from 2013 on 64 black females, the estimated log-linear regression between WAGE (earnings per hour, in $) and years of education, EDUC is ln(WAGE d ) = 1.58+0.09EDUC. The reported t-statistic for the slope coefficient is 3.95. (a) Test at the 5% level of significance, the null hypothesis that the return to an additional year of education is equal to 8% against the alternative that the rate of return to education is more than 8%. In your answer, show (i) the formal null and alternative hypotheses, (ii) the test statistic and its distribution under the null hypothesis, (iii) the critical value from the t table and the rejection region (in a figure), (iv) the calculated value of the test statistic, and (v) state your conclusion, with its economic interpretation. [Hint: From the information you should be able to figure out the standard error of the slope coefficient (se(b2)). Then, with the calculated standard error, you should be able to find the t statistic.] (b) Construct a 95% interval estimate for the return to an additional year of education and state its interpretation.

Paper For Above instruction

Introduction

The analysis of wage determinants, particularly the return on education, is a vital aspect of labor economics. Using a dataset from 2013 covering 64 Black females, a log-linear regression model estimates how years of education influence hourly wages. The key coefficient, representing the return to an additional year of education, provides insight into wage premium associated with educational attainment. This paper performs hypothesis testing on this coefficient against a specified return rate and constructs a confidence interval to estimate the economic significance of education on wages.

Regression Model and Parameter Estimation

The specified regression model is:

ln(WAGE) = 1.58 + 0.09 * EDUC

Here, 0.09 is the estimated coefficient of interest, indicating that each additional year of education increases the log of hourly wages by 0.09, or approximately 9%. The reported t-statistic for this coefficient is 3.95, which assists in hypothesis testing. The standard error of the coefficient can be derived from the t-statistic and estimated coefficient:

se(b̂2) = b̂2 / t = 0.09 / 3.95 ≈ 0.0228

This standard error measures the variability of the estimated return to education, which is essential for conducting both hypothesis tests and constructing confidence intervals.

Hypothesis Testing of Return to Education

Null and Alternative Hypotheses

The null hypothesis (H0) posits that the return to an additional year of education is equal to 8%, i.e., H0: β = 0.08. The alternative hypothesis (H1) suggests that the return exceeds 8%, i.e., H1: β > 0.08.

Calculation and Distribution of Test Statistic

Given the estimated coefficient of 0.09 and standard error 0.0228, the test statistic (t) is:

t = (b̂2 – β0) / se(b̂2) = (0.09 – 0.08) / 0.0228 ≈ 0.01 / 0.0228 ≈ 0.4386

Under the null hypothesis, this t-statistic follows a t-distribution with n - 2 = 62 degrees of freedom.

Critical Value and Rejection Region

At a 5% significance level for a one-tailed test, the critical t-value from the t-distribution table with 62 degrees of freedom is approximately 1.67. The rejection region is t > 1.67, meaning if the calculated t exceeds 1.67, we reject the null hypothesis.

Comparison and Conclusion

The calculated t-value is approximately 0.4386, which is less than the critical value of 1.67. Therefore, we fail to reject H0 at the 5% significance level. There is not enough statistical evidence to conclude that the return to education exceeds 8%. Economically, this suggests that while education positively influences wages, we cannot assert a return higher than 8% per additional year at the 5% confidence level.

Constructing a 95% Confidence Interval for Return to Education

The margin of error (ME) for the confidence interval is:

ME = t_{0.025,62} se(b̂2) ≈ 2.00 0.0228 ≈ 0.0456

Thus, the 95% confidence interval for β is:

(0.09 – 0.0456, 0.09 + 0.0456) = (0.0444, 0.1356)

This interval indicates that each additional year of education is associated with an increase in hourly wages between approximately 4.44% and 13.56%, with 95% confidence. The interval does not include 8%, which aligns with the hypothesis test results suggesting that the true return could be higher or lower than 8%.

Discussion and Economic Interpretation

The estimated return of up to 13.56% on education emphasizes the significant economic benefit of additional schooling. However, the failure to reject the null hypothesis that the return is exactly 8% implies ambiguity around exceeding this threshold. Policymakers should consider that investments in education among Black females could yield substantial wage premiums, but the precise magnitude remains statistically uncertain within this data sample. Future research with larger samples or more precise measures may clarify the extent of these returns.

Conclusion

In summary, the hypothesis test indicates insufficient evidence to claim that the return to an additional year of education exceeds 8%. Nevertheless, the confidence interval suggests the actual return could be substantially higher, up to approximately 13.56%. This analysis reinforces the importance of educational attainment in enhancing wage outcomes for Black women, providing valuable insights for policymakers and researchers interested in labor market disparities and human capital development.

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