Question

Stata Exercise: Please type your answers into a word processing document (provided separately). Insert ONLY the relevant part of your Stata log file into your document, and clearly explain how you use your Stata output answers the question. Copying and pasting your log file into the word document without an explanation will NOT be considered a complete answer and will not receive full credit. You are encouraged to refer to example Stata answer format file in Canvas. The dataset WAGE2 contains information on individuals' salaries, education levels and IQ test scores in 1980. wage is monthly earnings in dollars, educ is the number of years of education completed, and IQ is the standardized score on an IQ test. Download the dataset from D2L and open it in Stata. In all of the regressions that follow, be sure to include an intercept a. Run a simple regression of IQ on education to obtain the slope coefficient and call it δ ~ . Formally state your result. Are IQ and education positively or negatively related? [2 points] b. Using logarithmic functions, write the simple linear regression model between wages and education (only) that allows increasing returns of education on wages. [2 points] c. Given the correlation between education and IQ, do you expect there to be omitted variable bias in your regression model in part (b)? Explain. [2 points] d. Do you think the OLS estimate of the simple regression model will be an overestimate or underestimate of the true effect of education on wages? [2 points] e. Run the simple regression in part (b) to get an estimate of the slope coefficient, β ~ ​ 1 ​ . Formally state your results. [2 points] f. Run the multiple regression loglog( wage )=β 0 ​ +β 1 ∗ ​ educ +β 2 ∗ ​ IQ to get an estimate of the slope coefficients, β ^ ​ 1 ​ and β ^ ​ 2 ​ . Formally state your results. [2 points] g. Verify the formula we derived in class: β ~ ​ 4 ​ = β ^ ​ 1 ​ + δ ~ 4 ​ ∗ β ^ ​ 2 ​ . [2 points] h. Another way to allow for nonlinear relationship is to include a quadratic terms in education. Run the following regression and report your results. wage =β 0 ​ +β 1 ​ ∗ educ +β 2 ​ ∗ educsq +β 3 ​ ∗ IQ Note that wages are now in level terms. Using your results, explain how predicted wages and education are related at a fixed level of IQ. In doing so, calculate the association between education and wages when educ is at the sample mean and when educ is 12 years and when educ is 18 years. [6 points]

Answer 1

In statistics, determining the relationship between variables involves plotting data, calculating the least-squares line, and assessing the correlation coefficient's significance. Predictions are made using the regression equation, and additional factors must be considered to reduce bias in the model.

Step-by-step explanation:

In the context of the statistics exercise, analyzing the relationship between different variables such as education, IQ, and wages requires first plotting data points on a scatter plot to visually assess the potential relationship. The next step involves calculating the least-squares line, which represents the line of best fit through the data, typically in the form ý = a + bx. After plotting the line on the scatter plot, the significance of the linear relationship is examined by calculating the correlation coefficient, which ranges from -1 to 1, with values closer to the extremes indicating a stronger relationship. Whether positive or negative, the significance of this coefficient can often be verified through statistical tests.

Applying this concept to various scenarios, one needs to determine which variable serves better as the dependent variable, representing the outcome of interest, and the independent variable, acting as the predictor. After establishing the relationship through visual inspection and correlation analysis, predictions can be made using the regression equation, including scenarios where the estimated average height for different ages or the predicted cost of supplies based on distances is calculated.

It's important to note that if other factors are believed to influence the dependent variable, they should be included in the model to reduce omitted variable bias. For example, including both education and IQ in the model when predicting wages accounts for their interrelationship. Finally, for nonlinear relationships, quadratic or logarithmic terms can be included in the regression equation to better capture the real-world dynamics of the data.