## Question

We are interested in the effect on test scores of the student-teadher ratio (STR). The following regresion results have been obtained using the California data set. All the regressions usod average test scores in the district as the dependent variable. Many factors potentially affect the average test scores in a district. Three wariables are cotsidered: (i) the fraction of students who are still learning English/HiEL (a dummy taking 1 if the fraction is larger than 10%. 0 otherwisc), (ii) the percentage of students who are eligible for receiving sabsidised free lunch at school, and (iii) average district income (in logarithm). Different regressions use different combinations of regresoros, and quadratic and cuble terms of STR, and the interaction terms. The entries in the table include estimated coefficients, standard errors (in parenthesis), F-statistios and their p-values (in parenthesis below the values of F-statistic), and other summary statistics (SER, adjusted R2 ). The significance of each variable is indicated by ∗∗(1%) and * (5%). 3 (c) Consider (1) and (2). Note that log (average district income) is omitted in (1), but included in (2) i. Discuss the results in terms of bias of STR. By adding log (average income of district), would you say the omitted variable bias is mitigated ? Why or why not? ii. Discuss whether there exits a concern of multicollinearity problem when adding log (average income of district). Why or why not? iii. Would you conclude regression (1) suffers from an endogeneity problems? Why or why not? iv. Would you conclude regression (2) suffers from an endogeneity problem? Why or why not?

## Answer

Explanation:

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It seems that your question refers to a statistical analysis involving a variable called STR (Student-Teacher Ratio) and another variable log(average district income). The presence or absence of log(average district income) in the analysis can affect the bias in estimating the effect of STR on some outcome. Let’s break down the discussion step by step.

**i. Discuss the results in terms of bias of STR:**

When log(average district income) is omitted in the analysis (equation 1), there is a potential for omitted variable bias. Omitted variable bias occurs when a relevant variable is left out of the analysis, leading to biased estimates of the coefficients of the included variables. In this case, if log(average district income) is a relevant factor associated with the outcome variable, its omission might result in biased estimates for the effect of STR.

**By adding log (average income of the district), would you say the omitted variable bias is mitigated? Why or why not?**

The inclusion of log(average district income) in the analysis (equation 2) has the potential to mitigate omitted variable bias if log(average district income) is indeed a relevant variable. Including this variable allows the model to account for its impact on the outcome, leading to more accurate estimates of the coefficients for other variables, including STR.

The steps to check for omitted variable bias and its mitigation would typically involve running both models (with and without log(average district income)), comparing the results, and conducting statistical tests to assess the impact of including the additional variable.

**ii. Discuss whether there exists a concern of multicollinearity problem when adding log(average income of the district). Why or why not?**

Multicollinearity refers to a situation in which two or more independent variables in a regression model are highly correlated. This can cause issues in estimating the individual coefficients’ impact, leading to imprecise or unstable estimates. To determine multicollinearity, one can examine the variance inflation factor (VIF) for each variable.

The VIF for a variable is calculated as , where

is the coefficient of determination from regressing that variable on all the other independent variables. A high VIF (usually above 10) is often considered an indication of multicollinearity.

**iii. Would you conclude regression (1) suffers from an endogeneity problem? Why or why not?**

Endogeneity occurs when an independent variable is correlated with the error term in a regression model. This can lead to biased and inefficient parameter estimates. To assess endogeneity, researchers often use diagnostic tests, such as the Durbin-Watson test or tests for instrumental variables.

Without specific details on the regression model and the data, it’s challenging to definitively conclude whether regression (1) suffers from endogeneity. However, if there is reason to believe that the omitted variable (log(average income of the district)) is correlated with the error term, there might be an endogeneity problem.

**iv. Would you conclude regression (2) suffers from an endogeneity problem? Why or why not?**

Similarly, without specific details on the model and data, it’s challenging to conclusively determine if regression (2) suffers from endogeneity. The inclusion of log(average income of the district) might address endogeneity if it is a relevant variable and correlated with the error term in the original model. However, the resolution of endogeneity depends on the specific relationships among variables and the model’s overall specification.

**Final Answer:**

The assessment of multicollinearity and endogeneity requires specific details about the data, variables, and regression models. Conducting diagnostic tests, examining correlations, and assessing the model specification are essential steps in determining whether these issues exist in regressions (1) and (2).