Advanced Applied Econometrics Econ 6511 Winter 2015 Midterm

Advanced Applied Econometrics Econ 6511winter 2015 Midterm Examinst

Suppose that the true relationship between high school graduation rates, school funding, population density, and construction wages for all counties in the United States is given by hsgrad = β0 + β1sfunding + β2popdensity + β3cwage + u, and suppose that E(u|sfunding, popdensity, cwage) = 0. (a) (5 points) Suppose the sample only includes urban counties (counties with popdensity > 10). Would OLS produce an unbiased estimate of β1? Why or why not? (b) (5 points) Suppose the sample only includes urban counties (counties with popdensity > 10). Would OLS produce an unbiased estimate of β2? Why or why not? (c) (5 points) Suppose the sample only includes counties where hsgrad

Paper For Above instruction

The given assignment revolves around the principles of Ordinary Least Squares (OLS) estimation within the context of selective sampling and potential biases in econometric modeling. It emphasizes understanding the conditions under which OLS yields unbiased estimates and how sample restrictions can influence the validity of inference.

In part (a), the core question is whether restricting the sample to urban counties, defined as those with population density greater than 10, affects the unbiasedness of the OLS estimator of β1, the coefficient on school funding. Since the assumption is that the conditional expectation of the error term u, given the regressors, is zero (E(u|sfunding, popdensity, cwage) = 0), the unbiasedness of OLS hinges on the regressors being uncorrelated with the error term in the selected sample. If the sample excludes rural counties where the relationship between funding and high school graduation might differ, this could induce sample selection bias, especially if the omitted rural counties have systematically different funding effects. However, assuming that the true model holds across the population and that the sample restriction does not induce correlation between the regressors and the omitted factors within urban counties, OLS would remain unbiased for β1 in this restricted sample.

Similarly, part (b) explores whether the OLS estimate of β2, the effect of population density, remains unbiased under the same sample restriction. The answer depends on whether population density within urban counties remains uncorrelated with the error term after sample restriction. If urban counties with higher population densities have uncorrelated unobserved factors affecting high school graduation beyond density itself, then the estimate remains unbiased. Conversely, if the restriction introduces correlation—perhaps because unobserved factors influencing graduation are related to population density—then the estimator may be biased.

Moving to parts (c) and (d), the analysis shifts to sample restriction based on the high school graduation rate (hsgrad). When limiting the sample to counties where hsgrad

Overall, the main issue centers on whether sample restrictions induce correlation between the regressors and the error term, violating the exogeneity condition necessary for unbiased OLS estimates. Ensuring that the sample selection process does not correlate with the regressors or the unobserved factors is critical for validity. When sample restrictions are purely descriptive and do not induce bias, the estimators remain unbiased; otherwise, the estimates are biased and inconsistent, limiting the causal interpretation of the estimated coefficients.

In practical applications, researchers should carefully consider sample selection issues, potentially employ methods such as sample weighting, instrumental variables, or selection models to address bias caused by sample restrictions or to verify the robustness of their inferences across different subsamples.

References

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