Open Data Ch8 CPS On RStudio: How Many Variables?

Open The Data Ch8 Cps On Rstudiohow Many Variable Number Of Observa

Open the data “ch8_cps” on RStudio. How many variables, observations, and dummy variables are in the dataset? Run a regression of ahe on age; what is the model and the predicted equation? What is the effect of age on ahe? Provide the scatter plot and the fitted regression line. Run a non-linear cubic regression of ahe on age (including age, age squared, and age cubed). Write down the model and the predicted equation for this cubic regression. Based on the significance of the cubic term in this regression, do you prefer this cubic model, a quadratic model, or a linear model? Which model is easier to interpret? What is the effect of age on ahe in the chosen model? Is this effect easy to interpret? Add the fitted curve from the cubic regression to the scatter plot. Run a multiple regression of ahe on age, age squared, age cubed, female, and yrseduc all together. Which coefficients are statistically significant? Why? What is the R-squared of this regression, and what does it indicate?

Paper For Above instruction

The analysis begins with inspecting the dataset “ch8_cps” loaded into RStudio. This dataset comprises several variables, observations, and dummy variables, which are essential for comprehensive regression analysis. The first step involves determining the number of variables, observations, and dummy variables present in the dataset. Typically, functions like dim() and str() in R provide these insights efficiently. Assuming the dataset contains a mixture of continuous variables and categorical dummy variables, identifying the dummy variables is crucial for understanding their influence during regression modeling.

Next, the focus shifts to examining the relationship between the average hourly earnings (ahe) and age. Running a simple linear regression of ahe on age provides a foundational understanding of their relationship. The regression model can be expressed as:

ahe = β₀ + β₁ * age + ε

where β₀ is the intercept, β₁ is the slope coefficient indicating the average change in ahe for each additional year of age, and ε is the error term.

The regression results will include the estimated coefficients, their statistical significance, and overall model fit. The predicted equation from the regression might look like:

ahê = intercept + (β₁ * age)

This model helps in understanding the linear effect of age on hourly earnings. The sign and magnitude of β₁ indicate whether age positively or negatively impacts earnings. Typically, the regression output provides the estimate of β₁, its standard error, t-statistic, and p-value.

A scatter plot of ahe against age, along with the fitted regression line, visually illustrates the relationship. The scatter plot reveals the spread and any apparent linear trend, while the regression line summarizes this trend numerically.

To explore potential non-linear relationships, a cubic regression model is estimated by including age, age squared, and age cubed. The model takes the form:

ahe = γ₀ + γ₁ age + γ₂ age² + γ₃ * age³ + ε

This model captures more complex, non-linear patterns in the data. The predicted equation based on regression estimates might look like:

ahê = γ̂₀ + γ̂₁ age + γ̂₂ age² + γ̂₃ * age³

Examining the significance of the cubic term (γ̂₃) allows us to determine whether this non-linear term adds explanatory power. If γ̂₃ is statistically significant, it suggests that the cubic model better captures the relationship between age and ahe than linear or quadratic models. Conversely, if it is not significant, a simpler quadratic or linear model may suffice.

Comparing models, the cubic model may offer a better fit if the cubic term is significant, indicating a more complex relationship. If the cubic term is not significant, the quadratic or linear models are preferable due to simplicity and interpretability. The effect of age on ahe in the cubic or quadratic models can be summarized as the derivative of the predicted equation with respect to age, which accounts for the non-linear effects.

The fitted cubic curve can be visualized by overlaying the predicted curve onto the scatter plot of ahe versus age, providing a clear visual comparison of how well the cubic model captures the data pattern.

Further, a comprehensive multiple regression model includes not only age, age squared, and age cubed, but also other covariates such as gender (female) and years of education (yrseduc). The model is specified as:

ahe = δ₀ + δ₁ age + δ₂ age² + δ₃ age³ + δ₄ female + δ₅ * yrseduc + ε

Estimation results reveal which predictors are statistically significant. If certain variables, such as female or yrseduc, have p-values below a significance threshold (e.g., 0.05), they are considered significant contributors to explaining variations in ahe. The significance might reflect differences in earnings based on gender or educational attainment, which are pertinent in labor economics.

The R-squared statistic indicates the proportion of variance in ahe explained by the model. A higher R-squared signifies a better fit, implying the included predictors collectively account for a substantial part of the variation in hourly earnings. This measure helps evaluate the model's explanatory power, although it must be interpreted alongside other diagnostics for a comprehensive assessment.

In conclusion, this analysis employs a series of regression models—linear, quadratic, cubic, and multiple—to explore the relationship between age and hourly earnings. Visualizations such as scatter plots and fitted curves are crucial for interpreting these models. The significance tests for polynomial terms guide the choice of the most appropriate model, balancing complexity and interpretability. Ultimately, the regression results shed light on how age, education, and gender influence earnings, informing economic theory and policy considerations.

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