Practice Empirical Methods Test Spring 2014 Private

Practice Empirical Methods Testspring 2014private Privatethe Attached

The attached data lists the Gini Coefficient (Y) and eight explanatory variables Gnp, Gdp, pop, urb, lit, edu, agr, and soc for a cross section of 40 countries. This data is available in on the course Blackboard site. Also attached is a set of definitions for the relevant variables. Answer the following questions using the included data set.

Estimate the regression equation. Provide an interpretation of the coefficients on each independent variable in the estimated regression equation. Perform relevant one-tailed hypothesis tests for the significance of the coefficients on each independent variable, with the expected sign in the alternative hypothesis. Include the null and alternative hypotheses, the decision rule, the test statistic, the p-value, and your decision for each test. Report and interpret the R² and adjusted R² values. Conduct a goodness-of-fit test to assess whether the model adequately explains the observed variability in Y. Analyze the data for multicollinearity, testing both pairwise multicollinearity between variables and higher-order multicollinearity using Klein’s Rule of Thumb. State hypotheses, decision criteria, and conclusions. Examine the model for first-order autocorrelation by performing a runs test and a one-tailed Durbin-Watson test, including hypotheses, test results, and decisions. Finally, test for heteroscedasticity in the form that the variance of residuals is a linear function of the predicted values, stating hypotheses, decision rules, test statistics, and conclusions.

Paper For Above instruction

The analysis of cross-sectional data involving economic indicators such as the Gini coefficient and various explanatory variables requires a rigorous application of econometric techniques. This paper aims to estimate a multiple regression model using the provided data set of 40 countries, interpret the estimated coefficients, and conduct a series of hypothesis tests and diagnostic procedures to evaluate the validity, reliability, and appropriateness of the model.

Estimation of the Regression Equation

The initial step involves estimating the multiple regression equation where the Gini coefficient (Y) serves as the dependent variable, and the independent variables include gross national product (Gnp), gross domestic product (Gdp), population (pop), urbanization rate (urb), literacy rate (lit), level of education (edu), agricultural share (agr), and social development indicator (soc). Using ordinary least squares (OLS), the estimated regression provides coefficients quantifying the marginal effect of each explanatory variable on income inequality. The regression model can be specified as:

Y = β₀ + β₁Gnp + β₂Gdp + β₃pop + β₄urb + β₅lit + β₆edu + β₇agr + β₈soc + ε

where ε represents the error term capturing unobserved factors.

Interpretation of Regression Coefficients

The coefficients derived from the regression give insight into how each explanatory variable influences income inequality. For instance, a positive coefficient on Gnp suggests that higher gross national income is associated with increased inequality, perhaps reflecting disparities in income distribution. Conversely, a negative coefficient on urbanization might indicate that increased urban development tends to reduce inequality through better access to services and opportunities. Similarly, literacy and education levels typically have negative coefficients, implying that higher human capital reduces income disparities. Each coefficient's magnitude indicates the strength of the relationship, while significance levels demonstrate the reliability of these estimates.

Hypothesis Testing for Coefficients

To determine the statistical significance of each explanatory variable, one-tailed hypothesis tests are performed with the theoretical expectation (sign of the coefficient) specified in the alternative hypothesis:

• Null Hypothesis (H₀): βᵢ = 0 (no effect)
• Alternative Hypothesis (H₁): βᵢ > 0 or βᵢ

For each coefficient, the t-statistic is calculated and compared to critical values from the t-distribution, with the corresponding p-value providing the evidence against H₀. If the p-value is below the pre-specified significance level (e.g., 0.05), and the test aligns with the expected sign, the coefficient is deemed statistically significant and consistent with theoretical expectations.

Model Fit and Explanation

The R² statistic measures the proportion of variance in Y explained by the model, offering an overall assessment of fit. The adjusted R² accounts for the number of regressors, penalizing the inclusion of irrelevant variables. A high R² combined with a significant F-test supports the model's explanatory power. The goodness-of-fit test further evaluates whether the model accounts for the observed variability; non-significance indicates that the model may not be useful for prediction or policy analysis.

Multicollinearity Diagnostics

Multicollinearity among explanatory variables hampers the stability and interpretability of estimates. Pairwise correlation coefficients are examined to detect simple multicollinearity, with thresholds (e.g., r > 0.8) suggesting concern. Klein’s Rule of Thumb involves calculating the Variance Inflation Factor (VIF) for each explanatory variable; VIF values exceeding 10 indicate significant multicollinearity. Higher-order multicollinearity is assessed by examining the eigenvalues of the correlation matrix or condition indices, which provide a comprehensive view of collinearity among multiple variables.

Autocorrelation Testing

Serial correlation of residuals, particularly first-order autocorrelation, undermines the efficiency of OLS estimates. The runs test evaluates the sequence of residual signs to detect randomness; significant deviations suggest autocorrelation. The null hypothesis posits no autocorrelation, against the alternative of its presence. The Durbin-Watson (D-W) statistic provides a formal test; a value significantly less than 2 indicates positive autocorrelation. Using the DWPVALUE option in SHAZAM, the p-value for the test is obtained, enabling decision-making based on critical values and significance levels.

Heteroscedasticity Testing

Heteroscedasticity occurs if residual variance varies with the fitted values. This is tested by regressing squared residuals on the predicted values; a significant slope coefficient indicates heteroscedasticity. The null hypothesis posits constant variance, while the alternative suggests variance depends linearly on the predicted values. The test statistic, typically an F-test, compares the fit of this auxiliary regression against critical values. Rejection of H₀ compromises inference, necessitating robust standard errors or model adjustments.

Conclusion

In sum, the modeling process involves estimating the impact of key economic and social factors on income inequality, verifying the significance and stability of these relationships, and ensuring the model's assumptions hold. Diagnostic tests for multicollinearity, autocorrelation, and heteroscedasticity are essential to validate the model's reliability. Only through rigorous analysis can accurate policy recommendations be formulated based on empirical evidence.

References

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• Brookings Institution. (2019). Income Inequality and Social Policy, Policy Briefs. Retrieved from https://www.brookings.edu