ECON107 ASSIGNMENT 1 Answer Key - mysmu.eduย ยท โด Do not reject H. 0. Question 2 (20 marks) 1....
Click here to load reader
Transcript of ECON107 ASSIGNMENT 1 Answer Key - mysmu.eduย ยท โด Do not reject H. 0. Question 2 (20 marks) 1....
ECON107 ASSIGNMENT 1 Answer Key Question 1 (50 marks)
1. โXt2 = 32+22+12+(-1)2+02 = 15
โXtYt = (5)(3)+(2)(2)+(3)(1)+(2)(-1)+(-2)(0) = 20
โXt = 3+2+1+(-1)+0 = 5
โYt = 5+2+3+2+(-2) = 10
๐ = ฮฃ๐๐ก5
= 1
๐ = ฮฃ๐๐ก5
= 2
2. ๐ฝ1๏ฟฝ = nฮฃ๐๐ก๐๐ก โ ฮฃ๐๐กฮฃ๐๐กnฮฃ๐๐ก2 โ (ฮฃ๐๐ก)2
R = (5)(20)โ(5)(10)(5)(15)โ(52)
= 1
๐ฝ0๏ฟฝ = ๐ โ ๐ฝ1๏ฟฝ.๐ = 1
3.
-2
-1
0
1
2
3
4
5
-2 0 2 4
Yt
Xt
(1,2)
4. ๐ฝ1: An unit increase in X leads to an average of ๐ฝ1 unit increase in Y
๐ฝ0: an average value of Y when X is 0.
5. See above. 6. โ ๐๐ก5
๐ก=1 = โ (๐๐ก5๐ก=1 โ ๐๐ก๏ฟฝ )
= โ (๐๐ก5๐ก=1 โ ๐ฝ0๏ฟฝ โ ๐ฝ1๏ฟฝ.๐๐ก)
= 1+(-1)+1+2+(-3) = 0
Standard error of estimate, ๐๏ฟฝ = ๏ฟฝโ (๐๐ก)25๐ก=1๐โ2
= ๏ฟฝ12+(โ1)2+12+22+(โ3)2
5โ2
= 2.309
Coefficient of determination, r2 = ๐ธ๐๐๐๐๐
= โ (๐๐ก๏ฟฝโ5๐ก=1 ๐)2
โ (๐๐กโ5๐ก=1 ๐)2
= 1026
= 0.385
38.5% of the total variation in Yt is explained by the regression model (or by X).
7. H0: ๐ฝ1=0 H1: ๐ฝ1โ 0
t = ๐ฝ1๏ฟฝโ๐ฝ1๐(๐ฝ1๏ฟฝ)
= (๐ฝ1๏ฟฝโ๐ฝ1)๏ฟฝโ๐๐ก2
๐๏ฟฝ ~ t(n-k-1) ,
where ๐(๐ฝ1๏ฟฝ) = ๐๏ฟฝ
๏ฟฝโ๐๐ก2 = 0.7303. So t=1.3693
Reject H0 if |T| > t0.05 = 2.35336
โด Do not reject H0
Question 2 (20 marks)
1. ๐ฝ1๏ฟฝ = nฮฃ๐๐๐๐ โ ฮฃ๐๐ฮฃ๐๐nฮฃ๐๐2 โ (ฮฃ๐๐)2
R = ฮฃ(๐๐โ๐)(๐๐โ๐)ฮฃ(๐๐โ๐)2
Add 1 to the dependent variable: new ๐ฝ1๏ฟฝ = ฮฃ(๐๐โ๐)(๐๐+1โ(๐+1))ฮฃ(๐๐โ๐)2
= ฮฃ(๐๐โ๐)(๐๐โ๐)ฮฃ(๐๐โ๐)2
โด There is no change in ๐ฝ1๏ฟฝ.
๐ฝ0๏ฟฝ = ๐ โ ๐ฝ1๏ฟฝ.๐
Add 1 to the dependent variable: new ๐ฝ0๏ฟฝ = ๐ + 1 โ ๐ฝ1๏ฟฝ.๐
โด ๐ฝ0๏ฟฝ will increase by 1 unit.
2. ๐ฝ1๏ฟฝ = nฮฃ๐๐๐๐ โ ฮฃ๐๐ฮฃ๐๐nฮฃ๐๐2 โ (ฮฃ๐๐)2
R = ฮฃ(๐๐โ๐)(๐๐โ๐)ฮฃ(๐๐โ๐)2
Add 1 to the independent variable: new ๐ฝ1๏ฟฝ = ฮฃ(๐๐+1โ(๐+1))(๐๐โ๐)ฮฃ(๐๐+1โ(๐+1))2
= ฮฃ(๐๐โ๐)(๐๐โ๐)ฮฃ(๐๐โ๐)2
โด There is no change in ๐ฝ1๏ฟฝ.
๐ฝ0๏ฟฝ = ๐ โ ๐ฝ1๏ฟฝ.๐
Add 1 to the independent variable: new ๐ฝ0๏ฟฝ = ๐ โ ๐ฝ1๏ฟฝ(๐ + 1)
= ๐ โ ๐ฝ1๏ฟฝ.๐ โ ๐ฝ1๏ฟฝ
โด ๐ฝ0๏ฟฝ will increase by ๐ฝ1๏ฟฝ unit.
Question 3 (30 marks)
๐ฝ1๏ฟฝ =ฮฃ(๐๐ โ ๐)(๐๐ โ ๐)
ฮฃ(๐๐ โ ๐)2
Since there is no variation in X, ฮฃ(๐๐ โ ๐)2 = 0. ๐ฝ1๏ฟฝ is then undefined. As a result, residuals are not defined. So are RSS and ESS. Therefore, r2 is naturally undefined as well.
โด The person is not justified as the regression model is not valid.
If observations were lined up at 45ยฐ from origin, ๐๐ = ๐๐
๐ฝ1๏ฟฝ = ฮฃ๏ฟฝ๐๐โ๐๏ฟฝ๏ฟฝ๐๐โ๐๏ฟฝ
ฮฃ๏ฟฝ๐๐โ๐๏ฟฝ2
= ฮฃ๏ฟฝ๐๐โ๐๏ฟฝ๏ฟฝ๐๐โ๐๏ฟฝ
ฮฃ๏ฟฝ๐๐โ๐๏ฟฝ2 = 1
R2 = ๐ธ๐๐๐๐๐
= โ(๐๐ค๏ฟฝโ๐)2
โ(๐๐โ๐)2
= โ(๐ฝ0๏ฟฝ+๐ฝ1๏ฟฝ .๐๐โ๐)2
โ(๐๐โ๐)2 (๐ฝ0๏ฟฝ=0 as intercept is at origin)
= โ(๐๐โ๐)2
โ(๐๐โ๐)2
= โ(๐๐โ๐)2
โ(๐๐โ๐)2
= 1
โด Regression line will be a โperfect fitโ.