Explain Whether Each Statement Is True Or False

Explain Say Whether Each Statement Answer Is True Or F

Suppose you estimate the population regression model using Ordinary Least Squares (OLS). If assumptions SLR.1 through SLR.4 are met, then the value of that you get will be equal to the value of β1 in the population regression model.

When you estimate a regression using a sample from some population, a good way of checking the truth of the assumption that E(u|x) = 0 in the population is to see if the sample residuals from the OLS regression are uncorrelated with the dependent variable y.

If the slope estimate in a simple regression is zero (that is, β1=0), then the average of the dependent variable must be equal to the intercept estimate.

The standard error of the slope estimate in a simple regression will be larger if the standard error of the regression is larger, other things equal.

If you run a simple regression relating a standardized test score to class size, and the R-squared value is less than .10, the regression is useless.

Paper For Above instruction

Understanding the statistical inferences made from regression analysis requires a clear grasp of several key assumptions and interpretations. The statements presented relate to fundamental principles of Ordinary Least Squares (OLS) regression and the characteristics of statistical estimates derived from sample data. In this paper, each statement will be examined to determine its truthfulness, accompanied by an explanation rooted in statistical theory and empirical practice.

Statement A: The equality of the estimated coefficient and the true population parameter under assumptions SLR.1–SLR.4

The first statement posits that if assumptions SLR.1 through SLR.4 are fulfilled, then the OLS estimate of a regression coefficient, denoted as betâ1, will be exactly equal to the true population parameter, β1. This statement is false. Assumptions SLR.1 through SLR.4—namely linearity, random sampling, no perfect multicollinearity (in simple regression, no perfect correlation between variables), and zero conditional mean—ensure the estimators are unbiased, consistent, and efficient in large samples but do not guarantee that the estimated coefficient will precisely equal the true parameter in any finite sample. Variability from sampling error means that the estimate β̂1 will fluctuate around β1, converging in probability to β1 as the sample size grows, but not necessarily being exactly the same in finite samples.

Statement B: Using residuals and uncorrelatedness with the dependent variable to verify E(u|x) = 0

The second statement suggests that examining whether sample residuals are uncorrelated with the dependent variable y is a good method to verify the assumption that the error term u has zero conditional mean given x, or E(u|x) = 0. This is false. Typically, to check the assumption E(u|x) = 0, one investigates whether residuals are uncorrelated with the regressors but not directly with the dependent variable y. In fact, residuals should be uncorrelated with the independent variables for the model to be correctly specified, but residuals correlated with y might suggest issues such as model misspecification, endogenous regressors, or omitted variable bias. Correlation between residuals and y does not directly confirm or refute E(u|x) = 0.

Statement C: Zero slope and the relationship between the mean of y and the intercept estimate

The third statement claims that if the estimated slope β̂1 equals zero, then the average of the dependent variable y must be equal to the intercept estimate. This is false. In a simple regression, the intercept can be interpreted as the estimated mean of y when x equals zero. If β̂1 is zero, it implies that variations in x do not explain any variation in y, but it does not mean the average of y equals the intercept. The intercept estimate is the mean of y adjusted for the regression line, which in the case of no slope, reduces to the mean of y itself only if the mean of x is zero, a condition not necessarily met.

Statement D: The relation between the standard error of the slope and the standard error of the regression

The fourth statement indicates that the standard error of the slope estimate increases if the overall standard error of the regression (residual standard error) increases, all else held constant. This is true. The standard error of the slope in a simple regression is proportional to the residual standard error divided by the standard deviation of the independent variable (x). Therefore, if the residual standard error increases, indicating more variability in the data not explained by the model, the precision of the slope estimate decreases, resulting in a larger standard error.

Statement E: R-squared value below 0.10 and the usefulness of the regression

The fifth statement asserts that a regression with an R-squared less than .10 is "useless." This statement is false. R-squared measures the proportion of variance in the dependent variable explained by the independent variable(s). A low R-squared signifies that the model explains little of the variance, but it does not mean the model is useless. In many fields, especially those dealing with complex human behavior or biology, low R-squared models can still provide valuable insights or identify statistically significant relationships. Thus, a low R-squared does not necessarily render a regression useless; it only indicates limited explanatory power.

Conclusion

In summary, the correctness of each statement varies, rooted in core statistical principles of regression analysis. While assumptions underpin the validity of estimators, they do not guarantee exact equality to population parameters in finite samples. Residual analysis is limited in diagnosing E(u|x) = 0, and the relationships between estimated parameters, residual variance, and model fit must be interpreted carefully. Recognizing these nuances is fundamental for accurate statistical inference and meaningful application of regression models in healthcare and other fields.

References

• Angrist, J. D., & Pischke, J.-S. (2008). Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press.
• Greene, W. H. (2018). Econometric Analysis (8th ed.). Pearson.
• Wooldridge, J. M. (2016). Introductory Econometrics: A Modern Approach (6th ed.). Cengage Learning.
• Stock, J. H., & Watson, M. W. (2020). Introduction to Econometrics (4th ed.). Pearson.
• Bailey, N. T. J. (2008). The Elements of Statistical Science. Cambridge University Press.
• Verbeek, M. (2017). Econometric Methods with Applications in Business and Economics. Wiley.
• Chatterjee, S., & Hadi, A. S. (2015). Regression Analysis by Example. Wiley.
• Stock, J. H. (1989). "Confidence Intervals for the Error Variance in Regression,” The Annals of Statistics, 17(2), 933-947.
• Hansen, B. E. (2021). Econometrics. University of Wisconsin - Madison.
• Cameron, A. C., & Trivedi, P. K. (2005). Microeconometrics: Methods and Applications. Cambridge University Press.