1
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 y i = β 1 x 1i + β 2 x 2i + β 3 x 3i +e i with the
following nine observations.
y i x 1i x 2i x 3i
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 y i
*
= y i − y , x 2i
*
= x 2i − x 2 and x 3i
*
= x 3i − x 3 .
2
b. (2 marks) Calculate y i
* x
2i
*
i=1
N
∑
, x 2i
*2
i=1
N
∑
, y i
* x
3i
*
i=1
N
∑
, x 3i
*2
i=1
N
∑
and x 2i
*
x 3i
*
i=1
N
∑
.
3
c. (2 marks) Find least square estimates b 1 , b 2 and b 3 . (Hint: See Appendix 5A.)
4
d. (2 marks) Find the lease squares residuals
ê
1 , ê 2 ,…, ê 9 .
e. (2 marks) Find the variance estimate ˆ σ
2
.
f. (2 marks) Find the sample correlation between x 2 and x 3 .
5
g. (2 marks) Find the standard error for b 2 .
h. (2 marks) Find SSE, SST, SSR and R 2 .
6
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 = β 1 + β 2 AGE+ β 3 AGE 2 + β 4 AGE 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?
7
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.)
8
4. (6 marks) Output for the two models y = β 1 + β 2 x+ β 3 w+e and y = β 1 + β 2 x+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
ˆ y
2
) and 8.72 (for
ˆ y
2
and
ˆ y
3 ). The correlation between x and w is r
xw = 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?
9
5. (16 marks) Using data in the file beer.dat, apply RESET to the following two alternative models.
a. (2 marks) lnQ = β 1 + β 2 lnPB+ β 3 lnPL+ β 4 lnPR+ β 5 lnI +e .
b. (2 marks) Q = β 1 + β 2 PB+ β 3 PL+ β 4 PR+ β 5 I +e .
c. (2 marks) Which model seems to better reflect the demand for beer?