Department of Economics

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CONCORDIA UNIVERSITY Department of Economics

ECON 222/2 SECTIONS A and B

STATISTICAL METHODS II FALL 2015 – ASSIGNMENT 3

Due: Monday, November 23, before 3:00 pm 1. (14 marks) Consider the multiple regression model yi = β1x1i +β2x2i +β3x3i +ei with the

following nine observations.

yi x1i x2i x3i 1 1 0 1 2 1 1 -2 3 1 2 1

-1 1 -2 0 0 1 1 -1

-1 1 -2 -1 2 1 0 1 1 1 -1 1 2 1 1 0

Use a hand calculator to answer the following questions.

a. (2 marks) Calculate the observations in terms of deviations from their means. That is, find yi

* = yi − y , x2i * = x2i − x2 and x3i

* = x3i − x3 .

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b. (2 marks) Calculate yi *x2i *

i=1

N

∑ , x2i*2 i=1

N

∑ , yi*x3i* i=1

N

∑ , x3i*2 i=1

N

∑ and x2i* x3i* i=1

N

∑ .

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c. (2 marks) Find least square estimates b1 , b2 and b3 . (Hint: See Appendix 5A.)

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d. (2 marks) Find the lease squares residuals ê1, ê2,..., ê9 .

e. (2 marks) Find the variance estimate σ̂ 2 .

f. (2 marks) Find the sample correlation between x2 and x3 .

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g. (2 marks) Find the standard error for b2 .

h. (2 marks) Find SSE, SST, SSR and R2 .

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2. (10 marks) Use a computer to verify your answers to problem 1, parts (c) and (e) – (h). Attach your results.

3. (14 marks) Lion Forest has been a very successful golf professional. However, at age 45 his

game is not quite what it used to be. He started the pro tour when he was only 20 and he has been looking back examining how his scores have changed, as he got older. In the file golf.dat, the first column contains his final score (relative to par) for 150 tournaments. The second column contains his age (in units of 10 years). There are scores for 6 major tournaments in each year for the last 25 years. Denoting his score by SCORE and his age by AGE, estimate the following model and obtain the within-sample predictions:

SCORE = β1 +β2AGE +β3AGE

2 +β4AGE 3 +e

a. (2 marks) Test the null hypothesis that a quadratic function is adequate against the cubic

function as an alternative. What are the characteristics of the cubic equation that might make it appropriate?

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b. (10 marks) Use the within-sample predictions to answer the following questions:

(i) (2 marks) At what age was Lion at the peak of his career?

(ii) (2 marks) When was Lion’s game improving at an increasing rate?

(iii) (2 marks) When was Lion’s game improving at a decreasing rate?

(iv) (2 marks) At what age did Lion start to play worse than he had played when he was 20 years old?

(v) (2 marks) When could he no longer score less than par (on average)?

c. (2 marks) When he is aged 70, will he be able to break 100? (Assume par is 72.)

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4. (6 marks) Output for the two models y = β1 +β2x+β3w+e and y = β1 +β2x+e , based on 35 observations, are given below.

Variable Coefficient Std Error t-value Coefficient Std Error t-value C X W

3.6356 -0.99845 0.49785

2.763 1.235 0.1174

1.316 -0.8085 4.240

-5.8382 4.1072

2.000 0.3383

-2.919 12.14

RESET applied to the second model yields F-values of 17.98 (for ŷ2 ) and 8.72 (for ŷ2 and ŷ3 ). The correlation between x and w is rxw = 0.975 . Discuss the following questions.

a. (2 marks) Should w be included in the model?

b. (2 marks) What can you say about omitted-variable bias?

c. (2 marks) What can you say about the existence of collinearity and its possible effect?

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5. (16 marks) Using data in the file beer.dat, apply RESET to the following two alternative models.

a. (2 marks) lnQ = β1 +β2 lnPB+β3 lnPL+β4 lnPR+β5 lnI +e .

b. (2 marks) Q = β1 +β2PB+β3PL+β4PR+β5I +e .

c. (2 marks) Which model seems to better reflect the demand for beer?