Assume There Is A Population Regression Model Β0 Β1x1

Assume That There Is A Population Regression Modely Β0 Β1x1 Β

1. Assume that there is a population regression model y = β0 + β1x1 + β2x2 + β3x3 + u and that the model satisfies assumptions MLR1 through MLR5 in the population. Indicate, without explanation, whether the following statements are true or false (each answer is worth 1 point):

• a. If you take a random sample from the population, and estimate an OLS regression with y as the dependent variable and x1, x2 and x3 as the independent variables, the estimated coefficients of x1, x2 and x3 will be unbiased estimates of β1, β2, and β3.
• b. If you take a random sample from the population, and estimate an OLS regression with y as the dependent variable and x1, x2 and x3 as the independent variables, the estimated coefficients of x1, x2 and x3 will be equal to β1, β2, and β3.
• c. If you take a random sample from the population, and estimate an OLS regression with y as the dependent variable and x1, x2 and x3 as the independent variables, the estimated coefficients of x1, x2 and x3 will be statistically significant.
• d. If you take two random samples from the same population, and use each of the samples to estimate the population model using OLS, you will get the same β estimates from each regression.
• e. If β1 is a positive number, then the OLS estimate of β1 that you see on the output after estimating a regression may be positive or negative.

2. Provide a short answer (one to three sentences) to each of the following questions. (Each part is worth 3 points.)

• a. Suppose you have a sample that tells you the life expectancy of a 60-year-old male in each of the 50 states, along with the average education completed and the average income of people over 60 in each state. You want to estimate the effect of education on life expectancy. Is it a good idea to include the variable measuring average income? Discuss the costs and benefits.
• b. Consider the population regression model: score = β0 + β1classiz + β2faminc + u, where classiz is class size and faminc is family income. The expected value of u is the same across all levels, but the variance of u differs for different classrooms. Does this cause bias in OLS estimators? Explain.
• c. Consider the model with cigs for cigarette consumption, price, and income. Given a sample of 500 people, is it better if price and income are highly correlated or only weakly correlated for estimating β1?
• d. In the model estimating log of fries price with variables prpblck, lincome, prppov, does the high correlation between lincome and prppov violate one of the MLR assumptions?

Paper For Above instruction

Regression analysis is a fundamental statistical tool used extensively across various disciplines to understand the relationships between dependent and independent variables. The classical linear regression model rests on several key assumptions, collectively referred to as the MLR (Multiple Linear Regression) assumptions, which ensure unbiasedness, efficiency, and consistency of the estimators. These assumptions are critical to interpreting the results of any regression analysis accurately and reliably.

Part 1: True/False Statements on Population Regression Model

The first set of questions probes the understanding of the properties of Ordinary Least Squares (OLS) estimators under the assumptions of the classical linear regression model. Assuming the model y = β0 + β1x1 + β2x2 + β3x3 + u satisfies MLR1 through MLR5, some key points emerge:

• a. The statement asserts that if a random sample is taken and an OLS regression is performed, the estimated coefficients of x1, x2, and x3 will be unbiased estimators of the true parameters β1, β2, and β3. This is true under the assumption that the Gauss-Markov assumptions (MLR1-MLR5), particularly that the regressors are uncorrelated with the error term (MLR4), hold in the population, ensuring the unbiasedness of the OLS estimates.
• b. The statement claims the estimates will be exactly equal to the true parameters with a sample, which is false. OLS estimators are random variables subject to sampling variability; they are unbiased but not necessarily equal to the true parameters in any given sample.
• c. The statement suggests the coefficients will be statistically significant, which is false as statistical significance depends on the data, sample size, variability, and true parameters, not just the model assumptions or estimates.
• d. The claim that two separate samples from the same population will produce identical estimates ignores sampling variability. Hence, this statement is false since each sample produces a different estimate due to sampling noise.
• e. If β1 is positive, its OLS estimate may turn out to be negative purely due to sampling variability, especially in small samples. Therefore, the estimated coefficient can be positive or negative, making this statement true.

Part 2: Short Answer Questions

Addressing the second set of questions requires understanding of the properties and implications of regression assumptions.

a. Including Income in the Regression

In the context of estimating the effect of education on life expectancy across states, including the average income variable might control for confounding factors influencing both education and health outcomes. However, it can introduce multicollinearity if income is highly correlated with education, inflating standard errors and complicating interpretation. Moreover, if income is affected by education (endogeneity), including it may bias the estimate of education's effect. Thus, whether to include income depends on the specific research question and the causal structure, but generally, including income can help control for omitted variable bias, provided multicollinearity and endogeneity are properly addressed (Wooldridge, 2010).

b. Heteroscedasticity and Bias in OLS Estimators

The scenario describes heteroscedasticity—variance of the error term u varies across observations—but this violates the assumption of constant variance (homoscedasticity). However, heteroscedasticity does not introduce bias into the OLS estimators; they remain unbiased and consistent. It does, however, lead to inefficient estimates and biased standard errors, which can affect hypothesis testing and confidence intervals (White, 1980).

c. Correlation Between Price and Income in Estimation

When estimating the effect of price on cigarette consumption (β1), the correlation between price and income influences multicollinearity. High correlation (multicollinearity) reduces the precision of the estimate of β1, leading to larger standard errors, which diminishes statistical power. Conversely, low correlation improves estimation precision. So, having only a little correlation between price and income in the sample is preferable for accurately estimating β1 (Gujarati & Porter, 2009).

d. Multicollinearity and MLR Assumptions

The high correlation between lincome and prpov in the fries price regression violates the assumption of no perfect multicollinearity (one of the MLR assumptions). Perfect multicollinearity renders the OLS estimator undefined because it becomes impossible to separately identify the individual effect of multicollinear variables, although just high correlation (not perfect) causes issues in estimation precision.

Conclusion

Understanding the properties of the OLS estimators and the implications of the MLR assumptions is vital for conducting reliable regression analysis. While the assumptions provide the foundation for unbiased and efficient estimation, real-world data often violate these conditions, necessitating robust techniques and careful interpretation of results.

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