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Properties of R-Square and Adjusted R-Square

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Researcher estimates the following two econometric models
Yt= β1 + β2x2t + β3x3t +ut
Yt= β1 + β2x2t + β3x3t + β4x4t +vt
Ut and Vt are iid disturbances and x3t is an irrelevant variable which does not enter into the data generating process for Yt. Will the value of (a) R^2, (b) Adjusted R^2, be higher for the second model than the first? Explain your answers

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(a) The value of R^2 should be higher for the second model as an additional variable (4t) is added to the regression.
The value of R^2 would be identical for two econometric models only if the coefficient (β4) of the additional variable (4t) equals 0 (i.e. 4t is ...

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Explaining how adding a new independent variable will affect the value of R-Square and Adjusted R-Square

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Regression Analysis of Professor Salaries

See the attached file.

Question 1 - 5
1992-1993 Salary data for a sample of 15 universities was obtained. We are curious about the relationship between mean salaries for assistant professors (junior faculty) and full professors (senior faculty) at a given university. IN particular, do universities pay (relatively) high salaries to both assistant and full professors, or are full professors treated much better than assistant professors? In other words, do senior faculty receive high salaries compared to other universities while junior faculty receive relatively low salaries? Suppose we fit the following simple linear regression model:
Full Prof. Salary = (see attached file) Asist. Prof. Salary + (see attached file). The variables full Prof. Salary and Asst. Prof. Salary are the mean salaries for full and assistant professors at a given university. This model was fit to the data using the method of least squares. The following results were obtained from statistical software. Not that salaries were in thousands of dollars. Mean assistant professors salaries were treated as the explanatory variable and mean full professor salaries as the response variable.

1. The intercept of the least-squares regression line is (approximately)
2. A 90% confidence interval for the slope 1 in the simple linear regression model is
3. Suppose the researchers test the hypotheses H0 : (see attached file).
4. The correlation between mean assistant and full professor salaries is
5. Is there strong evidence (and if so, why ) that the relationship between mean assistant and full professor salaries is adequately described by a straight line?

Questions 6 - 11
A researcher wanted to find out the effect of the number of bedrooms and bathrooms on the annual property taxes of the houses in a county. To do this he tried to create a least squares linear model to predict the property tax from the number of bedrooms and bathrooms. A random sample of 100 residential properties were collected. The annual property taxes were given in dollars. A JMP analysis of the data collected is given below.

6. What are the explanatory variables?
7. What is the equation to predict the sales tax from the number of bedrooms and
bathrooms?
8. What would is the predicted value of the taxes for a house with three bedrooms and
two baths?
9. The F-ratio given in the Analysis of Variance table above is used to test
H0 : (see the attached file) Here we find that the F-ratio is 23.9212 and the
Corresponding p-value is less than 0.0001. Given a 5% significance level would you
Accept or reject the null hypothesis? Explain.
10. What is the value of the test statistic?
11. What is the p-value of the test statistic ?
Consider the following dataset that is a subset of the dataset for the above problem.
12. Find the least squares line or regression line that could be used to predict the number
of bathrooms from the number of bedrooms.
13. Use the regression line to find the predicted number of bathrooms in a house with two
bedrooms.
14. What is the standard error in prediction?
15. Find the least squares line or regression line that could be used to predict the number
of bedrooms from the number of bathrooms.
16. What is the correlation between the number of bedrooms and the number of
bathrooms?
17. Find the least squares linear model used to predict the taxes from the number of
bedrooms and the number of bathrooms.
18. What is the standard error of prediction for the new linear model.
19. Find the least squares quadratic model used to predict the taxes from the
number of bathrooms.
20. What is the Standard Error of prediction for the new linear model?

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