Sample Midterm Exam, Fall 2008

Econ 3740

Professor L Grogan

TIME LIMIT 1 hr 15 mins.

1. In words, math, and a diagram, show what it means if an estimator β is an unbiased estimator.

[3 marks]

2. The relationship between house size and price may be described as follows:

SIZEi = 72.2 + 5.77 ∗ PRICEi

SIZEi=size in square feet of ith house.

PRICEi=price in thousands of dollars of ith house.

a. Explain meaning of regression coefficients. [2 marks]

b. Suppose you are told that this equation explains more than 80% of variation in size of the house.

Does this imply that high housing prices cause houses to be large? Explain. [2 marks]

c. Can we say that these results are statistically significant? Economically significant? [2 marks]

3. Graph the decomposition of the variance in Y corresponding to the OLS estimator. ie. show the

elements contributing to TSS=ESS+RSS [5 marks]

4. Explain the Gauss-Markov Theorum. [3 marks]

5. You decide to model sales at the frozen yoghurt store near your university. Owner believes stu-

dents make up the bulk of the business and therefore is very happy to help with data collection. Your

estimated model is:

Yt = 262.5 + 3.9Tt − 46.94Pt + 134.43At − 152.1Ct

(0.7) (20.00) (108.0) (138.3)

N=29, R2=0.78.

a. What are the degrees of freedom in this model? [1 mark]

b. What fraction of the variance in sales is unexplained by the model? [1 mark]

1

c. What might explain the sign and significance of Ct? [1 mark]

2

6. Suppose you estimate a model of house prices to determine the impact of having ocean frontage

on the value of a house.

PRIi = 40 + 35.0 ∗ LOTi − 2.0 ∗AGEi + 10.0 ∗BEDi − 4.0FIREi + 100 ∗BEACHi

(5.0) (1.0) (10.0) (4.0) (10)N=30, R2=.63.where PRIi price of ith house; LOTi size of lot in thousands sq. feet; AGEi age of ith house in years; BEDi

no. bedrooms in ith house; FIREi dummy for fireplace; BEACHi dummy for ocean frontage.a. You expect BED, LOT , and BEACH to have positive coefficients. Write down and test these hypothesesat 5% level using the T-dibn table in the back of the textbook and the appropriate degrees of freedom. [3marks]

b. Does AGE have a negative coefficient? Perform a one-tailed t-test which evaluates this expectationat the 10% level. [1 mark]

c. Perform a two-sided t-test, significant at the 5% level, of the null hypothesis that the coefficient onFIRE is equal to zero. [1 mark]

7. Explain the main differences between these two econometric problems: serial correlation and heteroscedas-ticity. [3 marks].

3

8. Simple regression model (1 explanatory var): Show in a diagram the relationship between actual andpredicted values in the case that TSS=RSS. [2 marks]

9. The correlation coefficient between two variables, X and Y, takes the value r=-1. If a simple regression isrun using these data, assuming Yi = Xi + εi, give the value of R2. Draw the scatterplot and regression lineillustrating this relationship. [4 marks].

10. Name the most important criteria for the inclusion of an explanatory variable in a regression equation.[1 mark].

11. The p-value on my regression coefficient β is 0.02. Would I accept or reject a null hypothesis that β=0using a two-sided t-test and allowing for 5% Type I error?

12. You calculate the 95 % confidence interval (CI) for β using the standard formula CI=β ± tc ∗ s.e.(β).You obtain CI=0.5 ± 0.2. If you were to conduct a two-sided hypothesis test where H0 : β = 0, and allowfor 5% Type I error, would you accept or reject the null hypothesis? [2 marks]

4

13. Illustrate in a diagram the case of a linear regression model in which R2=1, and the case of R2=0.

5