Variables From The Heightweight Data File For Regression Ana

Variables from the heightweight.dta file for regression analysis and interpretation

The assignment involves multiple steps of regression analysis using datasets related to height, weight, traffic safety, and speeding tickets. The exercises focus on estimating Ordinary Least Squares (OLS) models, assessing omitted variable bias, understanding the implications of omitted variables, and analyzing how the inclusion of additional control variables affects coefficient estimates. Additionally, it requires evaluating issues of endogeneity, measurement accuracy, and statistical significance, with particular attention to how data limitations influence the results.

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Understanding the relationships between variables such as wages, height, and other demographic factors is crucial in econometric analysis. The initial part of this analysis involves exploring how adult wages relate to physical stature, specifically examining how adult height influences wages and how the inclusion of adolescent height alters the estimated effect. The subsequent interpretation of these models offers insights into potential omitted variable bias and the importance of control variables in regression analysis.

In the first regression, adult wages are modeled as a function of adult height only. The estimated coefficient on adult height indicates the expected change in wages associated with a one-inch increase in height, holding everything else constant. Given this simple bivariate model, the estimate reflects the raw association between height and wages, potentially capturing omitted factors linked to both variables, such as health, nutrition during early childhood, or socio-economic background.

The second model introduces adolescent height as an additional regressor. This step aims to control for factors affecting physical development before adulthood, which could confound the relationship between adult height and wages. When adolescent height is included, the coefficient on adult height typically diminishes or changes because some of the explanatory power initially attributed to adult height was actually attributable to earlier growth patterns or childhood circumstances, which are now controlled for. This highlights the importance of including relevant control variables to obtain a more causally interpretable estimate.

Regarding omitted variable bias, IQ is notably absent from the models. This omission could be problematic because IQ may influence wages and be correlated with height or other included variables. With IQ omitted, the estimated effect of height on wages could be biased if IQ acts as a confounder. The problem's severity depends on the correlation between IQ and height, and the extent to which IQ influences wages independently. If IQ indeed affects wages and correlates with height, then omitting it from the model leads to an omitted variable bias, inflated or deflated estimates of height's true effect.

Similarly, eye color's omission warrants consideration. Eye color is likely uncorrelated with wages directly and is probably not causally related to height or other variables being modeled. As a result, excluding eye color is less likely to bias the coefficient estimates significantly. This classification aligns with the concept of irrelevant variables: their omission does not distort the relationships among the relevant variables in the model, provided they are truly uncorrelated with the independent variables and the dependent variable.

The second dataset concerns the impact of cell phone usage on traffic safety. The initial simple regression model estimates traffic deaths as a function of cell phone subscriptions at a state level. While this approach provides a baseline association, it also raises issues about measurement accuracy and omitted variables that could confound the relationship. For example, population density, safety enforcement, or vehicle miles traveled (VMT) may influence traffic deaths but are not included initially, potentially biasing the estimates.

Adding population to the model serves to control for the size of the state, which is correlated with both the number of cell phones and traffic deaths. The coefficient on cell phone subscriptions may change after this addition because part of the variation previously attributed to cell phones was actually due to differences in population size. Population acts as a confounder, and controlling for it helps isolate the effect of cell phone subscriptions on traffic fatalities.

Furthermore, incorporating total miles driven into the model addresses additional unobserved heterogeneity. Since more miles driven naturally increase the risk of accidents, accounting for this exposure variable helps refine the estimate of the effect of cell phones. Usually, the inclusion of miles driven reduces the coefficient on cell phone subscriptions, indicating that part of the initial association was due to increased driving activity rather than cell phone use alone. This emphasizes the importance of proper controls in causal inference.

The third analysis deals with the determinants of speeding ticket fines, which are only observed when police decide to issue a fine. Modeling ticket amount as a function of age reveals whether age is statistically significant in predicting the fines. Endogeneity may arise if unobserved factors, such as driving behavior or risk tolerance, influence both age and the likelihood or amount of a fine. If such factors exist, the coefficient estimates could be biased, and the apparent significance of age might be spurious.

Introducing additional control variables, such as miles per hour over the speed limit, addresses omitted variable bias related to the severity of the speeding violation. When this variable is included, the estimated effect of age on ticket amount may change—possibly decreasing if some of the original effect was confounded with the extent of speeding over the limit. The inclusion of more controls generally leads to more precise estimates but may also attenuate the significance of age if it was proxying for unaccounted factors.

Finally, reducing the sample size to the first thousand observations affects the statistical properties of the estimation. With fewer data points, standard errors tend to increase, and t-statistics decrease, leading to less statistically significant results. This illustrates the importance of sample size for inference accuracy and the potential for reduced statistical power when data is limited.

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